18 May midnight study (Options)

OPTIONS STRATEGIES

by Adam Schwartz, PhD, CFA and Barbara Valbuzzi, CFA

Adam Schwartz, PhD, CFA, is at Bucknell University (USA). Barbara Valbuzzi, CFA (Italy).

LEARNING OUTCOMES

The candidate should be able to:

• demonstrate how an asset’s returns may be replicated by using options

• discuss the investment objective(s), structure, payoff, risk(s), value at expiration, profit, maximum profit, maximum loss, and breakeven underlying price at expiration of a covered call position

• discuss the investment objective(s), structure, payoff, risk(s), value at expiration, profit, maximum profit, maximum loss, and breakeven underlying price at expiration of a protective put position

• compare the delta of covered call and protective put positions with the position of being long an asset and short a forward on the underlying asset

• compare the effect of buying a call on a short underlying position with the effect of selling a put on a short underlying position

• discuss the investment objective(s), structure, payoffs, risk(s), value at expiration, profit, maximum profit, maximum loss, and breakeven underlying price at expiration of the following option strategies: bull spread, bear spread, straddle, and collar

• describe uses of calendar spreads

• discuss volatility skew and smile

• identify and evaluate appropriate option strategies consistent with given investment objectives

• demonstrate the use of options to achieve targeted equity risk exposures

INTRODUCTION

Derivatives are financial instruments through which counterparties agree to exchange economic cash flows based on the movement of underlying securities, indexes, currencies, or other instruments or factors.

A derivative’s value is thus derived from the economic performance of the underlying.

Derivatives may be created directly by counterparties or may be facilitated through established, regulated market exchanges.

Direct creation between counterparties has the benefit of tailoring to the counterparties’ specific needs but also the disadvantage of potentially low liquidity.

Exchange-traded derivatives often do not match counterparties’ specific needs but do facilitate early termination of the position, and, importantly, mitigate counterparty risk.

Derivatives facilitate the exchange of economic risks and benefits where trades in the underlying securities might be less advantageous because of poor liquidity, transaction costs, regulatory impediments, tax or accounting considerations, or other factors.

Options are an important type of contingent-claim derivative that provide their owner with the right but not an obligation to a payoff determined by the future price of the underlying asset.

Unlike other types of derivatives (i.e., swaps, forwards, and futures), options have nonlinear payoffs that enable their owners to benefit from movements in the underlying in one direction without being hurt by movements in the opposite direction.

The cost of this opportunity, however, is the upfront cash payment required to enter the options position.

Options can be combined with the underlying and with other options in a variety of different ways to modify investment positions, to implement investment strategies, or even to infer market expectations.

Therefore, investment managers routinely use option strategies for hedging risk exposures, for seeking to profit from anticipated market moves, and for implementing desired risk exposures in a cost-effective manner.

The main purpose of this reading is to illustrate how options strategies are used in typical investment situations and to show the risk–return trade-offs associated with their use.

Importantly, an informed investment professional should have such a basic understanding of options strategies to competently serve his investment clients.

Section 2 of this reading shows how certain combinations of securities (i.e., options, underlying) are equivalent to others. Sections 3–6 discuss two of the most widely used options strategies, covered calls and protective puts. In Sections 7 and 8, we look at popular spread and combination option strategies used by investors. The focus of Section 9 is implied volatility embedded in option prices and related volatility skew and surface. Section 10 discusses option strategy selection. Sections 11 and 12 demonstrate a series of applications showing ways in which an investment manager might solve an investment problem with options. The reading concludes with a summary.

POSITION EQUIVALENCIES

Learning Outcome

• demonstrate how an asset’s returns may be replicated by using options

ChatGPT (On PC parity and PC Forward Parity) - https://chatgpt.com/c/43a3fb6f-d32b-4662-b8a2-6b2c8bd6d00c

It is useful to think of derivatives as building blocks that can be combined to create a specific payoff with the desired risk exposure. A synthetic position can be created for any option or stock strategy. Most of the time, market participants use synthetic positions to transform the payoff profile of their positions when their market views change. We cover a few of these relationships in the following pages. First, a brief recap of put–call parity and put–call–forward parity will help readers to understand such synthetic positions.

As you may remember, put–call parity shows the equivalence (or parity) of a portfolio of a call and a risk-free bond with a portfolio of a put and the underlying, which leads to the relationship between put and call prices.

Put–call parity can be expressed in the following formula, where S0 is the price of the underlying; p0 and c0 are the prices (i.e., premiums) of the put and call options, respectively; and X/(1 + r)T is the present value of the risk-free bond: S0 + p0 = c0 + X/(1 + r)T.

A closely related concept is put–call–forward parity, which identifies the equivalence between buying a fiduciary call, given by the purchase of a call and the risk-free bond, and a synthetic protective put. The latter involves the purchase of a put option and a forward contract on the underlying that expires at the same time as the put option. In the put–call–forward parity formula, S0 is replaced with a forward contract to buy the underlying, where the forward price is given by F0(T) = S0(1 + r)T. Therefore, put–call–forward parity is: F0(T)/(1 + r)T + p0 = c0 + X/(1 + r)T.

Synthetic Forward Position

The combination of a long call and a short put with identical strike price and expiration, traded at the same time on the same underlying, is equivalent to a synthetic long forward position. In fact, the long call creates the upside and the short put creates the downside on the underlying.

Consider an investor who buys an at-the-money (ATM) call and simultaneously sells a put with the same strike and the same expiration date. Whatever the stock price at expiration, one of the two options will be in the money. If the contract has a physical settlement, the investor will buy the underlying stock by paying the strike price. In fact, on the expiration date, the investor will exercise the call she owns if the stock price is above the strike price. Otherwise, if the underlying price is below the strike price, the put owner will exercise his right to deliver the stock and the investor (who sold the put) must buy it for the strike price. Exhibit 1 shows the values of the two options and the combined position at expiration, compared with the value of the stock purchase at that same time. The stock in this case does not pay dividends.

Exhibit 1:

Synthetic Long Forward Position at Expiration

Stock price at expiration:	40	50	60
Alternative 1:
Long 50-strike call payoff	0	0	10
Short 50-strike put payoff	−10	0	0
Total value	−10	0	10
Alternative 2:
Long stock at 50	−10	0	10
Total value	−10	0	10

We now compare the same option strategy with the payoff of a forward or futures contract in Exhibit 2. The motivation to create a synthetic long forward position could be to exploit an arbitrage opportunity presented by the actual forward price or the need for an alternative to the outright purchase of a long forward position. Frequently, a forward contract is used instead of futures to acquire a stock position because it allows for contract customization.

Exhibit 2:

Synthetic Long Forward Position vs. Long Forward/Futures

Stock price at expiration:	40	50	60
Alternative 1:
Long 50-strike call payoff	0	0	10
Short 50-strike put payoff	−10	0	0
Total value	−10	0	10
Alternative 3: Long forward/futures at 50
Value	−10	0	10

EXAMPLE 1

Synthetic Long Forward Position vs. Long Forward/Futures

A market maker has sold a three-month forward contract on Vodafone that allows the client (counterparty) to buy 10,000 shares at 200.35 pence (100p = £1) at expiration. The current stock price (S0) is 200p, and the stock does not pay dividends until after the contract matures. The annualized interest rate is 0.70%. The cost (i.e., premium) of puts and calls on Vodafone is identical.

1. Discuss (a) how the market maker can hedge her short forward position upon the sale of the forward contract and (b) the market maker’s position upon expiration of the forward contract.

   Solution 1:

   1. To offset the short forward contract position, the market maker can borrow ₤20,000 (= 10,000 × S0/100) and buy 10,000 Vodafone shares at 200p. There is no upfront cost because the stock purchase is 100% financed.

   2. At the expiry of the forward contract, the market maker delivers the 10,000 Vodafone shares she owns to the client that is long the forward, and then the market maker repays her loan. The net outflow for the market maker is zero because the following two transactions offset each other:Amount received for the delivery of shares: 10,000 × 200.35p = £20,035Repayment of loan: 10,000 × 200p [1 + 0.700% × (90/360)] = £20,035

2. Discuss how the market maker can hedge her short forward contract position using a synthetic long forward position, and explain what happens at expiry if the Vodafone share price is above or below 200.35p.

   Solution 2:

   To hedge her short forward position, the market maker creates a synthetic long forward position. She purchases a call and sells a put, both with a strike price of 200.35p and expiring in three months.

   At the expiry of the forward contract, if the stock price is above 200.35p, the market maker exercises her call, pays £20,035 (=10,000 × 200.35p), and receives 10,000 Vodafone shares. She then delivers these shares to the client and receives £20,035.

   At the expiry of the forward contract, if the stock price is below 200.35p, the owner of the long put will exercise his option, and the market maker receives the 10,000 Vodafone shares for £20,035. She then delivers these shares to the client and receives £20,035.

Consider now a trader who wants to short a stock over a specified period. He needs to borrow the stock from the market and then sell the borrowed shares. Instead, the trader can create a synthetic short forward position by selling a call and buying a put at the same strike price and maturity. When using options to replicate a short stock position, it is important to be aware of early assignment risk that could arise with American-style options. As Exhibit 3 shows, the payoff is the exact opposite of the synthetic long forward position.

The same outcome can be achieved be selling forwards or futures contracts (as seen in Exhibit 3). These instruments are also commonly used to eliminate future price risk. Consider an investor who owns a stock and wants to lock in a future sales price. The investor might enter into a forward or futures contract (as seller) requiring her to deliver the shares at a future date in exchange for a cash amount determined today. Because the initial and final stock prices are known, this investment should pay the risk-free rate. For a dividend-paying stock, the dividends expected to be paid on the stock during the term of the contract will decrease the price of the forward or futures.

Exhibit 3:

Synthetic Short Forward Position

Stock price at expiration:	40	50	60
Alternative 1:
Short 50-strike call payoff	0	0	−10
Long 50-strike put payoff	10	0	0
Total value	10	0	−10
Alternative 2:
Short stock at 50	10	0	−10
Value	10	0	−10
Alternative 3:
Short forward/futures at 50	10	0	−10
Value	10	0	−10

Synthetic forwards on stocks and equity indexes are often used by market makers that have sold a forward contract to customers—to hedge the risk, the market-maker would implement a synthetic long forward position—or by investment banks wishing to hedge forward exposure arising from structured products.

Synthetic Put and Call

As already described, market participants can use synthetic positions to transform the payoff and risk profile of their positions. The symmetrical payoffs of long and short stock, forward, and futures positions can be altered by implementing synthetic options positions. For example, the symmetric payoff of a short stock position can become asymmetrical if the investor transforms it into a synthetic long put position by buying a call.

Exhibit 4 shows the payoffs of a synthetic long put position that consists of short stock at 50 and a long call with an exercise price of 50. It can be seen that the payoffs from this synthetic put position at various stock prices at option expiration are identical to those of a long put with a 50-strike price. Of course, all positions are assumed to expire at the same time. Note that the same transformation of payoff and risk profile for a position of short forwards or futures can also be accomplished using long call options.

Exhibit 4:

Synthetic Long Put

Stock price at expiration:	40	50	60
Alternative 1:
Short stock at 50	10	0	−10
Long 50-strike call payoff	0	0	+10
Total value	10	0	0
Alternative 2
Long 50-strike put payoff	10	0	0
Value	10	0	0

EXAMPLE 2

Synthetic Long Put

Three months ago, Wing Tan, a hedge fund manager, entered into a short forward contract that requires him to deliver 50,000 Generali shares, which the fund does not currently own, at €18/share in one month from now. The stock price is currently €16/share. The hedge fund’s research analyst, Gisele Rossi, has a non-consensus expectation that the company will report an earnings “beat” next month. The stock does not pay dividends.

1. Under the assumption that Tan maintains the payoff profile of his current short forward position, discuss the conditions for profit or loss at contract expiration.

   Solution 1:

   If Tan decides to keep the current payoff profile of his position, at the expiry date, given a stock price of ST, the profit or loss on the short forward will be 50,000 × (€18 − ST). The position will be profitable only if ST is below €18; otherwise the manager will incur in a loss.

2. After discussing with Rossi her earnings outlook, Tan remains bearish on Generali. He decides to hedge his risk, however, in case the stock does report a positive earnings surprise. Discuss how Tan can modify his existing position to produce an asymmetrical, risk-reducing payoff.

   Solution 2:

   Tan decides to modify the payoff profile on his short forward position so that, at expiration, it will benefit from any stock price decrease below €16 while avoiding losses if the stock rises above that price. He purchases a call option with a strike price €16 and one month to maturity at a cost (premium) of €0.50. At expiration, the payoffs are as follows:

   • On the short forward contract: 50,000 × (€18 − ST)

   • On the long call: 50,000 × {Max[0,(ST − €16)] − €0.50}

   • On the combined position: 50,000 × {(€18 − ST) + [Max[0,(ST − €16)] − €0.50]}

   If ST ≤ €16, the call will expire worthless and the profit will amount to 50,000 × (€18 − ST + 0 − €0.50).

   If ST > €16, the call is exercised and the Generali shares delivered for a maximum profit of 50,000 × (€18 − €16 − €0.50) = €75,000.

In similar fashion, an investor with a long stock position can change his payoff and risk profile into that of a long call by purchasing a put (“protective put” strategy). The long put eliminates the downside risk, whereas the long stock leaves the profit potential unlimited. As shown in Exhibit 5, the strategy has a payoff profile resembling that of a long call. Again, all positions are assumed to expire at the same time. We will have much more to say about the protective put strategy later in this reading. Finally, the payoff profile of a long call can also be achieved by adding a long put to a long forward or futures position, all with the same expiration dates and the same strike and forward (or futures) prices.

Exhibit 5:

Synthetic Long Call

Stock price at expiration:	40	50	60
Alternative 1:
Long stock at 50	−10	0	10
Long 50-strike put payoff	10	0	0
Total value	0	0	10
Alternative 2
Long 50-strike call payoff	0	0	10
Value	0	0	10

COVERED CALLS AND PROTECTIVE PUTS

Learning Outcome

• discuss the investment objective(s), structure, payoff, risk(s), value at expiration, profit, maximum profit, maximum loss, and breakeven underlying price at expiration of a covered call position

Writing a covered call is a very common option strategy used by both individual and institutional investors.

In this strategy, a party that already owns shares sells a call option, giving another party the right to buy their shares at the exercise price. The investor owns the shares and has taken on the potential obligation to deliver the shares to the call option buyer and accept the exercise price as the price at which she sells the shares. For her willingness to do this, the investor receives the premium on the option.

When someone simultaneously holds a long position in an asset and a long position in a put option on that asset, the put is often called a protective put. The name comes from the fact that the put protects against losses in the value of the underlying asset.

The examples that follow use the convention of identifying an option by the underlying asset, expiration, exercise price, and option type. For example, in Exhibit 6, the PBR October 16 call option sells for 1.42. The underlying asset is Petróleo Brasileiro (PBR) common stock, the expiration is October, the exercise price is 16, the option is a call, and the call premium is 1.42. It is important to note that even though we will refer to this as the October 16 option, it does not expire on 16 October. Rather, 16 reflects the price at which the call owner has the right to buy, otherwise known as the exercise price or strike.

Petróleo Brasileiro (PBR)	October	16	Call
Underlying asset	Expiration	Exercise price	Option type

On some exchanges, certain options may have weekly expirations in addition to a monthly expiration, which means investors need to be careful in specifying the option of interest. For a given underlying asset and exercise price, there may be several weekly and one monthly option expiring in October. The examples that follow all assume a single monthly expiration.

Investment Objectives of Covered Calls

Consider the option data in Exhibit 6. Suppose there is one month until the September expiration. By convention, option listings show data for a single call or put, but in practice, the most common trading unit for an exchange-traded option is one contract covering 100 shares. Besides call and put premiums for various strike (i.e., exercise) prices and monthly expirations, the option data also shows implied volatilities as well as the “Greeks” (variables so named because most of the common ones are denoted by Greek letters). Implied volatility is the value of the unobservable volatility variable that equates the result of an option pricing model—such as the Black–Scholes–Merton (BSM) model—to the market price of an option, using all other required (and observable) input variables, including the option’s strike price, the price of the underlying, the time to option expiration, and the risk-free interest. Before proceeding further, we provide a brief review of the Greeks because they will be an integral part of the discussion of the various option strategies to be presented.

• Delta (Δ) is the change in an option’s price in response to a change in price of the underlying, all else equal. Delta provides a good approximation of how an option’s price will change for a small change in the underlying’s price. Delta for long calls is always positive; delta for long puts is always negative. Delta (Δ) ≈ Change in value of option/Change in value of underlying.

• Gamma (Γ) is the change in an option’s delta for a change in price of the underlying, all else equal. Gamma is a measure of the curvature in the option price in relationship to the underlying price. Gamma for long calls and long puts is always positive. Gamma (Γ) ≈ Change in delta/Change in value of underlying.

• Vega (ν) is the change in an option’s price for a change in volatility of the underlying, all else equal. Vega measures the sensitivity of the underlying to volatility. Vega for long calls and long puts is always positive. Vega (ν) ≈ Change in value of option/Change in volatility of underlying.

• Theta (Θ) is the daily change in an option’s price, all else equal. Theta measures the sensitivity of the option’s price to the passage of time, known as time decay. Theta for long calls and long puts is generally negative.

Assume the current PBR share price is 15.84 and the risk-free rate is 4%. Now let us consider three different market participants who might logically use covered calls.

Exhibit 6:

PBR Option Prices, Implied Volatilities, and Greeks

Call Prices	Exercise Price	Put Prices
SEP	OCT	NOV		SEP	OCT	NOV
1.64	1.95	2.44	15	0.65	0.99	1.46
0.97	1.42	1.90	16	1.14	1.48	1.96
0.51	1.02	1.44	17	1.76	2.09	2.59

Call Implied Volatility		Put Implied Volatility
SEP	OCT	NOV		SEP	OCT	NOV
64.42%	57.33%	62.50%	15	58.44%	56.48%	62.81%
55.92%	56.11%	60.37%	16	59.40%	56.35%	62.27%
51.07%	55.87%	58.36%	17	59.59%	56.77%	63.40%

Delta: change in option price per change of +1 in stock price, all else equal
Call Deltas		Put Deltas
SEP	OCT	NOV		SEP	OCT	NOV
0.657	0.647	0.642	15	−0.335	−0.352	−0.359
0.516	0.540	0.560	16	−0.481	−0.460	−0.438
0.351	0.434	0.475	17	−0.620	−0.564	−0.513

Gamma: change in delta per change of +1 in stock price, all else equal
Call Gammas		Put Gammas
SEP	OCT	NOV		SEP	OCT	NOV
0.125	0.100	0.075	15	0.136	0.102	0.075
0.156	0.109	0.082	16	0.147	0.109	0.080
0.159	0.109	0.086	17	0.140	0.107	0.079

Theta: daily change in option price, all else equal
Call Thetas (daily)		Put Thetas (daily)
SEP	OCT	NOV		SEP	OCT	NOV
−0.019	−0.012	−0.011	15	−0.015	−0.010	−0.009
−0.018	−0.013	−0.011	16	−0.017	−0.011	−0.010
−0.015	−0.012	−0.011	17	−0.016	−0.011	−0.010

Vega: change in option price per 1% increase in volatility, all else equal
Call Vegas (per %)		Put Vegas (per %)
SEP	OCT	NOV		SEP	OCT	NOV
0.017	0.024	0.030	15	0.017	0.024	0.030
0.018	0.026	0.031	16	0.018	0.026	0.031
0.017	0.025	0.032	17	0.017	0.025	0.032

Market Participant #1: Yield Enhancement

The most common motivation for writing covered calls is cash generation in anticipation of limited upside moves in the underlying. The call option writer keeps the premium regardless of what happens in the future. Some covered call writers view the premium they receive as an additional source of income in the same way they view cash dividends. For a covered call, a long position in 100 shares of the underlying is required for each short call contract. No additional cash margin is needed if the long position in the underlying is maintained. If the stock price exceeds the strike price at expiry, the underlying shares will be “called away” from the covered call writer and then delivered to satisfy the option holder’s right to buy shares at the strike price. It is important to recognize, however, that when someone writes a call option, he is essentially giving up the returns above the strike price to the call holder.

Consider an individual investor who owns PBR and believes the stock price is likely to remain relatively flat over the next few months. With the stock currently trading at just under 16, the investor might think it unlikely that the stock will rise above 17. Exhibit 6 shows that the premium for a call option expiring in September with an exercise price of 17, referred to as the SEP 17 call, is 0.51. She could write that call and receive this premium. Alternatively, she could write a different call, say the NOV 17 call, and receive 1.44. There is a clear trade-off between the size of the option premium and the likelihood of option exercise. The option writer would get more cash from writing the longer-term option (because of a larger time premium), but there is a greater chance that the option would move in the money, resulting in the option being exercised by the buyer and, therefore, the stock being called away from the writer. The view of the covered call writer can be understood in terms of the call option’s implied volatility. Essentially, writing the call expresses the view that the volatility of the underlying asset will be lower than the pricing of the option suggests. As shown in Exhibit 6, the implied volatility of the NOV 17 options is 58.36%. By writing the NOV 17 call for 1.44, the covered call investor believes that the volatility of the underlying asset will be less than the option’s implied volatility of 58.36%. The call buyer believes the stock will move far enough above the strike price of 17 to provide a payoff greater than the 1.44 cost of the call.

Although it may be acceptable to think of the option premium as income, it is important to remember that the call writer has given up an important benefit of stock ownership: capital gains above the strike price. This dynamic can be seen in Exhibit 7. Consider an investor with a long position in PBR stock (with delta of +1) and a short position in a PBR NOV 17 call. The investor enjoys the benefit of the call premium of 1.44. This cushions the value of the position (Stock − Call, or S − C) as the PBR share price drops. If the PBR stock price drops to 5, the call option will drop to essentially 0. The portfolio will be worth about 6.44, as shown in Exhibit 7. As the stock price increases, however, the short call position begins to limit portfolio gains. If the price of PBR shares rises to 30, the call option delta approaches 1, so the delta of the portfolio (S − C) approaches 0. The portfolio gains from the long PBR stock position will be reduced by losses on the short call position. As the in-the-money option expires, the maximum value of the portfolio will approach 18.44, the exercise price of 17 plus the 1.44 premium, as in Exhibit 7.

Exhibit 7:

Covered Call Portfolio Value: Long PBR Stock—NOV 17 Call

Market Participant #2: Reducing a Position at a Favorable Price

Next, consider Sofia Porto, a retail portfolio manager with a portfolio that has become overweighted in energy companies. She wants to reduce this imbalance. Porto holds 5,000 shares of PBR, an energy company, and she expects the price of this stock to remain relatively stable over the next month. She may decide to sell 1,000 shares for 15.84 each. As an alternative, Porto might decide to write 10 exchange-traded PBR SEP 15 call contracts. This means she is creating 10 option contracts, each of which covers 100 shares. In exchange for this contingent claim, she receives the option premium of 1.64/call × 100 calls/contract × 10 contracts = 1,640. Because the current PBR stock price (15.84) is above the exercise price of 15, the options she writes are in the money. Given her expectation that the stock price will be stable over the next month, it is likely that the option will be exercised. Because Porto wants to reduce the overweighting in energy stocks, this outcome is desirable. If the option is exercised, she has effectively sold the stock at 16.64. She receives 1.64 when she writes the option, and she receives 15 when the option is exercised. Porto could have simply sold the shares at their original price of 15.84, but in this specific situation, the option strategy resulted in a price improvement of 0.80 ([15 + 1.64] − 15.84) per share, or 5.05% (0.80/15.84), in a month’s time.2 By maintaining the stock position and selling a 15 call, she still risks the possibility of a stock price decline during the coming month resulting in a realized price lower than the current market price of 15.84. For example, if the PBR share price declined to 10 over the next month, Porto would realize only 10 + 1.64 =11.64 on her covered call position.

An American option premium can be viewed as having two parts: exercise value (also called intrinsic value) and time value.3

Call Premium = Time Value + Intrinsic Value = Time Value + Max(0,S − X)

In this case, the right to buy at 15 when the stock price is 15.84 has an exercise (or intrinsic) value of 0.84. The option premium is 1.64, which is 0.80 more than the exercise value. This difference of 0.80 is called time value.

1.64 = Time Value + (15.84 − 15)

Someone who writes covered calls to improve on the market is capturing the time value, which augments the stock selling price. Remember, though, that giving up part of the return distribution would result in an opportunity loss if the underlying goes up.

Market Participant #3: Target Price Realization

A third popular use of options is really a hybrid of the first two objectives. This strategy involves writing calls with an exercise price near the target price for the stock. Suppose a bank trust department holds PBR in many of its accounts and that its research team believes the stock would be properly priced at 16 per share, which is only slightly higher than its current price. In those accounts for which the investment policy statement permits option activity, the manager might choose to write near-term calls with an exercise price near the target price, 16 in this case. Suppose an account holds 500 shares of PBR. Writing 5 SEP 16 call contracts at 0.97 brings in 485 in cash. If the stock is above 16 in a month, the stock will be called away at the strike price (target price), with the option premium adding an additional 6% positive return to the account.4 If PBR fails to rise to 16, the manager might write a new OCT expiration call with the same objective in mind.

Although this strategy is popular, the investor should not view it as a source of free money. The stock is currently very close to the target price, and the manager could simply sell it and be satisfied. Although the covered call writing program potentially adds to the return, there is also the chance that the stock could experience bad news or the overall market might pull back, resulting in an opportunity loss relative to the outright sale of the stock. The investor also would have an opportunity loss if the stock rose sharply above the exercise price and it was called away at a lower-than-market price.

The exposure from the short position in the PBR SEP 16 call can be understood in terms of the Greeks in Exhibit 6. Delta measures how the option price changes as the underlying asset price changes, and gamma measures the rate of change in delta.5 A PBR SEP 16 call has a delta = 0.516 and a gamma of 0.156. A short call will reduce the delta of the portfolio (S − C) from +1 to +0.484 (= +1[Share] − 0.516[Short Call]). The lower portfolio delta will reduce the upside opportunity. A share price increase of 1 will result in a portfolio gain of approximately 0.484.6 The delta of the portfolio is not constant. By selling the PBR 16 call, the portfolio is now “short gamma”. Remember, gamma is the rate of change of delta. Although the underlying PBR share has a gamma of 0, the short call will make the gamma of the portfolio −0.156. As the price of PBR shares increases above 16, the delta of the PBR call position will change, at a rate of gamma. Gamma is greatest for a near-the-money option and becomes progressively smaller as the option moves either into or out of the money (as seen in Exhibit 8).

Gamma of an ATM option can increase dramatically as the time to expiration approaches or volatility increases. Traders with large gamma exposure (especially large negative gamma) should be aware of the speed with which the position values can change. The change in portfolio delta and gamma for a PBR SEP 16 covered call as a function of share price can be seen in Exhibit 8. As the price of PBR shares increase, the portfolio delta changes at a rate of gamma. As the share price moves above the exercise price of 16, the portfolio (S − C) delta drops at a rate gamma towards its eventual limit of 0, effectively eliminating any remaining upside in the position.

Exhibit 8:

Delta vs. Gamma for PBR 16 Covered Call Portfolio

Profit and Loss at Expiration

In the process of learning option strategies, it is always helpful to look at a graphical display of the profit and loss possibilities at the option expiration. Suppose an investor owns PBR, currently trading at 15.84. The investor believes gains may be limited above a price of 17 and decides to write a call against the long share position. The 17 strike calls will have no intrinsic value because the share price is currently 15.84. The investor must now consider the available option maturities (SEP, OCT, and NOV) as shown in Exhibit 6. In deciding which option to write, the investor may consider the option premiums and implied volatilities. Based on the investor’s view that volatility will remain low over the next three months, the investor chooses to write the NOV call. At 58.36%, the NOV 17 call has highest implied volatility of the available 17 strike options, so it would be the most overvalued assuming low volatility. The option premium of 1.44 is completely explained by the time value of the NOV option, because the NOV 17 option has no exercise value (Option premium = Time value + Intrinsic value; 1.44 = Time value + Max[0,15.84 − 17]). If the stock is above 17 at expiration, the option holder will exercise the call option and the investor will deliver the shares in exchange for the exercise price of 17. The maximum gain with a covered call is the appreciation to the exercise price plus the option premium.7

Some symbols will be helpful in learning these relationships:

S0 = Stock price when option position opened

ST = Stock price at option expiration

X = Option exercise price

c0 = Call premium received or paid

The maximum gain is (X − S0) + c0. With a starting price of 15.84, a sale price of 17 results in 1.16 of price appreciation. The option writer would keep the option premium of 1.44 for a total gain of 1.16 + 1.44 = 2.60. This is the maximum gain from this strategy because all price appreciation above 17 belongs to the call holder. The call writer keeps the option premium regardless of what the stock does, so if it were to drop, the overall loss is reduced by the option premium received. Exhibit 9 shows the situation. The breakeven price for a covered call is the stock price minus the premium, or S0 − c0. In other words, the breakeven point occurs when the stock falls by the premium received—in this example, 15.84 − 1.44 = 14.40. The maximum loss would occur if the stock became worthless; it equals the original stock price minus the option premium received, or S0 − c0.8 In this single unlikely scenario, the investor would lose 15.84 on the stock position but still keep the premium of 1.44, for a total loss of 14.40.

At option expiration, the value of the covered call position is the stock price minus the exercise value of the call. Any appreciation beyond the exercise price belongs to the option buyer, so the covered call writer does not earn any gains beyond that point. Symbolically,

Covered Call Expiration Value = ST − Max[(ST − X),0].1

The profit at option expiration is the covered call value plus the option premium received minus the original price of the stock:

Covered Call Profit at Expiration = ST − Max[(ST − X),0] + c0 − S0.2

In summary:

Maximum gain = (X − S0) + c0

Maximum loss = S0 − c0

Breakeven price = S0 − c0

Expiration value = ST − Max[(ST − X),0]

Profit at expiration = ST − Max[(ST − X),0] + c0 − S0

Exhibit 9:

Covered Call P&L Diagram: Stock at 15.84, Write 17 Call at 1.44

It is important to remember that these profit and loss diagrams depict the situation only at the end of the option’s life.9 Most equity covered call writing occurs with exchange-traded options, so the call writer always has the ability to buy back the option before expiration. If, for instance, the PBR stock price were to decline by 1 shortly after writing the covered call, the call value would most likely also decline. If this investor correctly believed the decline was temporary, he might buy the call back at the new lower option premium, making a profit on that trade, and then write the option again after the share price recovered.

EXAMPLE 3

Characteristics of Covered Calls

S0 = Stock price when option position opened = 25.00

X = Option exercise price = 30.00

ST = Stock price at option expiration = 31.33

c0 = Call premium received = 1.55

1. Which of the following correctly calculates the maximum gain from writing a covered call?

   1. (ST − X) + c0 = 31.33 − 30.00 + 1.55 = 2.88

   2. (ST − S0) − c0 = 31.33 − 25.00 −1.55 = 4.78

   3. (X − S0) + c0 = 30.00 − 25.00 + 1.55 = 6.55

   Solution to 1:

   C is correct. The covered call writer participates in gains up to the exercise price, after which further appreciation is lost to the call buyer. That is, X − S0 = 30.00 − 25.00 = 5.00. The call writer also keeps c0, the option premium, which is 1.55. So, the total maximum gain is 5.00 + 1.55 = 6.55.

2. Which of the following correctly calculates the breakeven stock price from writing a covered call?

   1. S0 − c0 = 25.00 − 1.55 = 23.45

   2. ST − c0 = 31.33 − 1.55 = 29.78

   3. X + c0 = 30.00 + 1.55 = 31.55

   Solution to 2:

   A is correct. The call premium of 1.55 offsets a decline in the stock price by the amount of the premium received: 25.00 − 1.55 = 23.45.

3. Which of the following correctly calculates the maximum loss from writing a covered call?

   1. S0 − c0 = 25.00 − 1.55 = 23.45

   2. ST − c0 = 31.33 − 1.55 = 29.78

   3. ST − X + c0 = 31.33 − 30.00 + 1.55 = 2.88

   Solution to 3:

   A is correct. The stock price can fall to zero, causing a loss of the entire investment, but the option writer still keeps the option premium received: 25.00 − 1.55 = 23.45

INVESTMENT OBJECTIVES OF PROTECTIVE PUTS

https://study.cfainstitute.org/app/cfa-institute-program-level-iii-for-august-2024#read/study_task/2562663/investment-objectives-of-protective-puts-1

Learning Outcome

• discuss the investment objective(s), structure, payoff, risk(s), value at expiration, profit, maximum profit, maximum loss, and breakeven underlying price at expiration of a protective put position

The protective put is often viewed as a classic example of buying insurance. The investor holds a risky asset and wants protection against a loss in value. He then buys insurance in the form of the put, paying a premium to the seller of the insurance, the put writer. The exercise price of the put is similar to the coverage amount for an insurance policy. The insurance policy deductible is similar to the difference between the current asset price and the strike price of the put. A protective put with a low exercise price is like an insurance policy with a high deductible. Although less expensive, a low strike put involves greater price exposure before the payoff function goes into the money. For an insurance policy, a higher deductible is less expensive and reflects the increased risk borne by the insured party. For a protective put, a lower exercise price is less costly and has a greater risk of loss in the position.

Like traditional term insurance, this form of insurance provides coverage for a period of time. At the end of the period, the insurance expires and either pays off or not. The buyer of the insurance may or may not choose to renew the insurance by buying another put. A protective put can appear to be a great transaction with no drawbacks, because it provides downside protection with upside potential, but let us take a closer look.

Loss Protection/Upside Preservation

Suppose a portfolio manager has a client with a 50,000 share position in PBR. Her research suggests there may be a negative shock to the stock price in the next four to six weeks, and he wants to guard against a price decline. Consider the put prices shown in Exhibit 6; the purchase of a protective put presents the manager with some choices. Puts represent a right to sell at the strike price, so higher-strike puts will be more expensive. For this reason, the put buyer may select the 15-strike PBR put. Longer-term American puts are more expensive than their equivalent (same strike price) shorter-maturity puts. The put buyer must be sure the put will not expire before the expected price shock has occurred. The portfolio manager could buy a one-month (SEP) 15-strike put for 0.65. This put insures against the portion of the underlying return distribution that is below 15, but it will not protect against a price shock occurring after the SEP expiration.

Alternatively, the portfolio manager could buy a two-month option, paying 0.99 for an OCT 15 put, or she could buy a three-month option, paying 1.46 for a NOV 15 put. Note that there is not a linear relationship between the put value and its time until expiration. A two-month option does not sell for twice the price of a one-month option, nor does a three-month option sell for three times the price of a one-month option. The portfolio manager can also reduce the cost of insurance by increasing the size of the deductible (i.e., the current stock price minus the put exercise price), perhaps by using a put option with a 14 exercise price. A put option with an exercise price of 14 would have a lower premium but would not protect against losses in the stock until it falls to 14.00 per share. The option price is cheaper, but on a 50,000 share position, the deductible would be 50,000 more than if the exercise price of 15 were selected.10

Because of the uncertainty about the timing of the “shock event” she anticipates, the manager might consider the characteristics of the available option maturities. Given our assumptions, three of the BSM model inputs for the available 15 strike options are the same (PBR stock price 15.84, the strike price 15 and the risk-free rate of interest 4%). The difference in the cost of the SEP, OCT, and NOV options will be explained by the differences in time and the term structure of volatility. The BSM model assumes option volatility does not change over time or with strike price. In practice, volatility can vary across time and strike prices. For the 15 puts, the implied volatility is slightly greater for the NOV option, perhaps reflecting other traders’ concerns about a shock event before expiration. Because the PBR stock price is 15.84 and the put options are all 15 strike, all three maturities have no intrinsic value.

The cost of each PBR 15 strike option is entirely explained by the remaining time value. If the stock price does not fall below 15, the SEP, OCT, and NOV put option values will erode to 0 as they approach their expiration dates. The erosion of the options value with time is approximated by the theta. The daily thetas (Theta/365) for the PBR puts and calls are given in Exhibit 6. Notice, all the theta values in the table are negative. These values approximate the daily losses on the option positions as time passes, all else equal. The NOV 15 put (90 days) has a theta of −0.009 and the SEP 15 put (30 days) has a theta of −0.015. If the NOV 15 option is held for one day, and the price and volatility of the underlying do not change, the put value will decline by approximately 0.009 to approximately 1.45 (= 1.46 − 0.009).

The graph of the BSM theta function for the PBR NOV 15 option as it approaches maturity is shown in Exhibit 10. Notice how the rate of decline changes as maturity approaches. If the PBR price does not drop below 15, the NOV 15 put will expire out-of-the-money and the option price will gradually fall to 0. All else equal, the sum of the daily losses approximated by theta will explain the entire loss of 1.46 in option value over that time. The complex shape of the theta graph in Exhibit 10 results from the nature of the BSM theta formula, which includes terms to reflect the probability that the stock price will fall below the strike price during the remaining time. Note that if the price of PBR remains at 15.84 for the last 10 days to maturity, the BSM put option value will erode to 0 at varying rates averaging about −0.03/day. Assumptions of the BSM model explain the negative peak in theta around three days prior to maturity as the remaining time value rapidly decays to 0. Theta values might help the investor decide which maturity to choose. If he were to buy the cheaper SEP put, the daily erosion of value (−0.015) would be greater than for the more expensive NOV put (−0.009).

Exhibit 10:

PBR 15 Put Theta over Time

Given the four- to six-week time horizon for the shock event anticipated by the portfolio manager, the OCT put seems appropriate, but there is still the potential to lose the premium without realizing any benefit. With a 0.99 premium for the OCT 15 put and 50,000 shares to protect, the cost to the account would be almost 50,000. One advantage of the NOV option is that although it is more expensive, it has the smallest daily loss of value, as captured by theta. This option also has a greater likelihood of not having expired before the news hits. Also, although the portfolio manager could hold onto the put position until its expiration, she might find it preferable to close out the option prior to maturity and recover some of the premium paid.11

Profit and Loss at Expiration

Exhibit 11 shows the profit and loss diagram for the protective put.12 The stock can rise to any level, and the position would benefit fully from the appreciation; the maximum gain is unlimited. On the downside, losses are “cut off” once the stock price falls to the exercise price. With a protective put, the maximum loss is the depreciation to the exercise price plus the premium paid, or S0 − X + p0. At the option expiration, the value of the protective put is the greater of the stock price or the exercise price. The reason is because the stock can rise to any level but has a floor value of the put exercise price. In symbols,

Value of Protective Put at Expiration = ST + Max[(X − ST),0].3

The profit or loss at expiration is the ending value minus the beginning value. The initial value of the protective put is the starting stock price minus the put premium. In symbols,

Profit of Protective Put at Expiration = ST + Max[(X − ST),0] − S0 − p0.4

Exhibit 11:

Protective Put P&L Diagram: Stock at 15.84, Buy 15 Put at 1.46

To break even, the underlying asset must rise by enough to offset the price of the put that was purchased. The breakeven point is the initial stock price plus the option premium. In symbols, Breakeven Price = S0 + p0.

In summary:

Maximum gain = ST − S0 − p0 = Unlimited

Maximum loss = S0 − X + p0

Breakeven price = S0 + p0

Expiration value = ST + Max[(X − ST),0]

Profit at expiration = ST + Max[(X − ST),0] − S0 − p0

EXAMPLE 4

Characteristics of Protective Puts

S0 = Stock price when option position opened = 25.00

X = Option exercise price = 20.00

ST= Stock price at option expiration = 31.33

p0 = Put premium paid = 1.15

1. Which of the following correctly calculates the gain with the protective put?

   1. ST − S0 − p0 = 31.33 − 25.00 − 1.15 = 5.18

   2. ST − S0 + p0 = 31.33 − 25.00 + 1.15 = 7.48

   3. ST − X − p0 = 31.33 − 20.00 − 1.15 = 10.18

   Solution to 1:

   A is correct. If the stock price is above the put exercise price at expiration, the put will expire worthless. The profit is the gain on the stock (ST − S0) minus the cost of the put. Note that the maximum profit with a protective put is theoretically unlimited, because the stock can rise to any level and the entire profit is earned by the stockholder.

2. Which of the following correctly calculates the breakeven stock price with the protective put?

   1. S0 − p0 = 25.00 − 1.15 = 23.85

   2. S0 + p0 = 25.00 + 1.15 = 26.15

   3. ST + p0 = 31.33 + 1.15 = 32.48

   Solution to 2:

   B is correct. Because the option buyer pays the put premium, she does not begin to make money until the stock rises by enough to recover the premium paid.

3. Which of the following correctly calculates the maximum loss with the protective put?

   1. S0 − X + p0 = 25.00 − 20.00 + 1.15 = 6.15

   2. ST − X − p0 = 31.33 − 20.00 − 1.15 = 10.18

   3. S0 − p0 = 25.00 − 1.15 = 23.85

   Solution to 3:

   A is correct. Once the stock falls to the put exercise price, further losses are eliminated. The investor paid the option premium, so the total loss is the “deductible” plus the cost of the insurance.

EQUIVALENCE TO LONG ASSET/SHORT FORWARD POSITION

https://study.cfainstitute.org/app/cfa-institute-program-level-iii-for-august-2024#read/study_task/2562673/equivalence-to-long-assetshort-forward-position-1

Learning Outcome

• compare the delta of covered call and protective put positions with the position of being long an asset and short a forward on the underlying asset

All investors who consider option strategies should understand that some options are more sensitive to changes in the underlying asset than others. As we have seen, this relationship is measured by delta, an indispensable tool to an options user. Because a long call increases in value and a long put decreases in value as the underlying asset increases in price, call deltas range from 0 to 1 and put deltas range from 0 to −1. (Naturally, the signs are reversed for short positions in these options.) A long position in the underlying asset has a delta of 1.0, whereas a short position has a delta of −1.0. When the share price is close to the strike price, a rough approximation is that a long ATM option will have a delta that is approximately 0.5 (for a call) or −0.5 (for a put). Exhibit 12 shows the delta for the PBR SEP 16 put and call versus share price. As the stock price moves toward 16 (the strike price), the call option delta is approximately 0.52 and the put delta is −0.48. In general, Call Delta − Put Delta = 1 for options on the same underlying with the same BSM model inputs.

Exhibit 12:

Delta of PBR Options vs. Stock Price

Delta can be applied to a portfolio as well. Suppose on the Tokyo Stock Exchange, Honda Motor Company stock sells for ¥3,500. A portfolio contains 100 shares, and the manager writes one exchange-traded covered call contract with a ¥3,500 strike. The delta of the 100-share position will be 100 × +1 = +100. Because the call is at the money, meaning that the stock price and exercise price are equal, it will have a delta of approximately 0.5. The portfolio, however, is short one call contract. From the perspective of the portfolio, the delta of the short call contract is −0.5 × 100 = −50. A short call loses money as the underlying price rises. So, this covered call has a position delta (which is an overall or portfolio delta) of 50, consisting of +100 points for the stock and −50 points for the short call. Compare this call with a protective put, in which someone buys 100 shares of stock and one contract of an ATM put. Its position delta would also be 50: +100 points for the stock and −50 points for the long put.

Finally, consider a long stock position of 100 shares and a short forward position of 50 shares. Because futures and forwards on non-dividend-paying stocks are essentially proxies for the stock, their deltas are also 1.0 for a long position and −1.0 for a short position. In this example, the short forward position “cancels” half the long stock position, so the position delta is also 50. These examples show three different positions: an ATM covered call, an ATM protective put, and a long stock/short forward position that all have the same delta. For small movements in the price of the underlying asset, these positions will show very similar gains and losses.

Writing Puts

If someone writes a put option and simultaneously deposits an amount of money equal to the exercise price into a designated account, it is called writing a cash-secured put.13 This strategy is appropriate for someone who is bullish on a stock or who wants to acquire shares at a particular price. The fact that the option exercise price is escrowed provides assurance that the put writer will be able to purchase the stock if the option holder chooses to exercise. Think of the cash in a cash-secured put as being similar to the stock part of a covered call. When an investor sells a covered call, she takes on the obligation to sell a stock, and this obligation is covered by ownership in the shares. When a put option is sold to create a new position, the obligation that accompanies this position is to purchase shares. In order to cover the obligation to purchase shares, the portfolio should have enough cash in the account to make good on this obligation. The short put position is covered or secured by cash in the account.

Now consider two slightly different scenarios using the price data from Exhibit 6. In the first scenario, one investor might be bullish on PBR and is interested in buying the stock at a cheaper price. With the stock at 15.84, she writes the SEP 15 put for 0.65, which is purchased by another investor who is bearish on PBR stock. The option writer will keep the option premium regardless of what the stock price does. If the stock is below 15 at expiration, however, the put would be exercised and the option writer would be obliged to purchase shares from the option holder at the exercise price of 15.

Possible small (and independent) changes to the variables from Exhibit 6 are simulated in Exhibit 13 for the long PBR SEP 15 put position. The long put is illustrated here for simplicity—these statistics for the long put position should also help the put writer to understand the risks and returns for her position, because a short position is simply the mirror image of the long position. The initial values are 15.84 for the stock and 0.65 for the put, and the put buyer has acquired a delta of −0.335 and a gamma of 0.136.14

As demonstrated in change #1, if the stock price rises by 0.10 from 15.84 to 15.94, the long (short) put will lose (gain) approximately −0.335 × 0.10 = −0.0335 (+0.0335), as the put value drops from 0.65 to approximately 0.617 (≈ 0.6165 = 0.65 − 0.0335). Remember, this approximation is good for only a small change in the underlying share price. As the stock price rises, the long put’s initial delta, −0.335, will change at a rate of gamma, 0.136, so the delta then becomes −0.321.

Exhibit 13:

Long PBR SEP 15 PUT, Greeks and Put Price Changes for Small, Independent Changes in Inputs

	Stock Price (S)	Delta (Δ)	Gamma (Γ)	Option Price (p)
Initial Values	15.84	−0.335	0.136	0.65
Change #1: Stock Price Increases by 0.10, from 15.84 to 15.94
ΔS =+0.10 Δt = 0 ΔVol = 0	15.94	Δ changes at rate of Γ, so: Δ1 ≈ Δ0 + (Γ × ΔS) −0.321 ≈ −0.335 + (0.136 × 0.10)	Γ changes slightly to 0.133	p1 ≈ p0 + (Δ0 × ΔS) 0.617 ≈ 0.65 + (−0.335 × 0.10)
Change #2: Time to Expiration Changes by 1 Day, from 30 to 29 Days
ΔS = 0 Δt = 1 day (to 29 Days) ΔVol = 0	15.84	−0.335	0.136	p1 ≈ p0 + (Θ × Δt) 0.635 ≈ 0.65 + (−0.015 × 1)
Change #3: Implied Volatility Increases by 1 Percentage Point, from 58.44% to 59.44%
ΔS = 0 Δt = 0 ΔVol = +1% (to 59.44%)	15.84	−0.335	0.136	p1 ≈ p0 + (ν × ΔVol) 0.667 ≈ 0.65 + (0.017 × 1)

The long SEP 15 put position also has a theta of −0.015 and a vega of +0.017.15 As time decays, the long (short) put option will lose (gain) value at a rate of theta, so the value of long (short) position will decrease (increase) by approximately 0.015/per day. As demonstrated in change #2 (which is separate and independent from change #1), all else equal, the long put value would drop from about 0.65 to 0.635 as the put moves one day closer to expiration (from 30 to 29 days). If the implied volatility of the SEP 15 put were to increase by 1% (from 58.44% to 59.44%), all else equal, the option price would increase by 0.017 to approximately 0.667, as demonstrated in change #3. The increase in volatility would benefit the put holder at the expense of the writer, because the short put position would lose 0.017.

If the stock is above 15 at expiration, the put option will expire unexercised. At the expiration date, the put writer will either keep the premium or have PBR shares put to her at 15. Because the put writer was bullish on PBR and wanted to purchase it at a cheaper price, she may be happy with this result. Netting out the option premium received by the put writer would make her effective purchase price 15.00 − 0.65 = 14.35.

In another scenario, an institutional investor might be interested in purchasing PBR. Suppose the investor wrote the SEP 17 put for 1.76. This strategy will have slightly different values for the Greeks compared with the previous strategy. The delta of the SEP 17 short put position will be +0.62, gamma will be −0.140, and theta will equal +0.016. This position will be more sensitive to changes in the stock price than the SEP 15 put. If the PBR share price increases 0.10 from 15.84 to 15.94, the put writer will now profit by approximately +0.62 × 0.10 = 0.062. The higher strike price makes the short SEP 17 put a more bullish position than the SEP 15 put. This dynamic is reflected in the larger delta for the short SEP 17 put at +0.620 (versus +0.335 for the short SEP 15 put).

If the stock is below 17 at expiration, the SEP 17 puts will be exercised and the investor (i.e., put writer) will pay 17 for the shares, resulting in a net price of 17.00 − 1.76 = 15.24. Anytime someone writes an option, the maximum gain is the option premium received, so in this case, the maximum gain is 1.76. The maximum loss when writing a put occurs when the stock falls to zero. The option writer pays the exercise price for worthless stock but still keeps the premium. In this example, the maximum loss would be 17.00 − 1.76 = 15.24. Exhibit 14 shows the corresponding profit and loss diagram.

Exhibit 14:

Short Put P&L Diagram: Write SEP 17 Put at 1.76

Note the similar shape of the covered call position in Exhibit 9 and the short put in Exhibit 14. Writing a covered call and writing a put are very similar with regard to their risk and reward characteristics.16

RISK REDUCTION USING COVERED CALLS AND PROTECTIVE PUTS

https://study.cfainstitute.org/app/cfa-institute-program-level-iii-for-august-2024#read/study_task/2562683/risk-reduction-using-covered-calls-and-protective-puts-1

Learning Outcome

• compare the effect of buying a call on a short underlying position with the effect of selling a put on a short underlying position

Covered calls and protective puts may both be viewed as risk-reducing or hedging strategies. In the case of a covered call, some price uncertainty is eliminated for price increases. For a protective put, the price uncertainty is eliminated for price decreases. The risk reduction can be understood by considering hedge statistics.

Covered Calls

Consider the individual who owns 100 shares of a PBR stock at 15.84. The long position has a delta of +100. Suppose the investor now writes a NOV 17 call contract against this entire position. These options have a delta of 0.475. This covered call position has a position delta of (100 × +1.0) − (100 × 0.475) = +52.5. A position delta of 52.5 is equivalent (for small changes) to owning 52.5 shares of the underlying asset. An investor can lose more money on a 100-share position than on a 52.5-share position. Even if the stock declines to nearly zero, the loss is reduced only by the amount of the option premium received. Viewed this way, the covered call position is less risky than the underlying asset held alone. The lower position delta will work against the investor if the share price increases. A PBR share price above 17 would result in the shares getting called away, and portfolio gains per share are limited to 2.60 = (X − S0) + c0 = (17 − 15.84) + 1.44.

Protective Puts

Similar logic applies to the use of protective puts. An investor who buys a put is essentially buying insurance on the stock. An investor owning PBR stock could purchase a NOV 15 put with an option delta of −0.359. The position delta from 100 shares of PBR stock and one NOV 15 put contract would be +100 + (−0.359 × 100) = +64.1 For small changes in price, the protective put portfolio reduces the risk of the 100-share PBR position to the equivalent of a 64.1 share position. This insurance lasts only until NOV. One buys insurance to protect against a risk, and the policyholder should not feel bad if the risk event does not materialize and he does not get to use the insurance. Stated another way, a homeowner should be happy if the fire insurance on his house goes unused. Still, we do not want to buy insurance we do not need, especially if it is expensive. Continually purchasing puts to protect against a possible stock price decline will result in lower volatility in the overall portfolio, but the trade-off between premium cost and risk reduction must be carefully considered. Such continuous purchasing of puts to protect against a possible stock price decline is an expensive strategy that would wipe out most of the long-term gain on an otherwise good investment. The occasional purchase of a protective put to manage a temporary situation, however, can be a sensible risk-reducing activity.

Buying Calls and Writing Puts on a Short Position

The discussion on protective puts (Stock + Put) and covered calls (Stock − Call) describes risk-reduction strategies for investors with long positions in the underlying asset. How can investors reduce risk when they are short the underlying asset? The short investor is worried the underlying stock will go up and profits if the underlying stock goes down. To offset the risks of a short position, an investor may purchase a call. The new portfolio will be (Call − Stock). The long call will offset portfolio losses when the share price increases.

To generate income from option premiums, the investor may also sell a put. As the stock drops in value, the investor profits from the short stock position, but the portfolio (− Put − Stock) gains will be reduced by the short put. When the share price increases, the short position loses money. The put expires worthless, meaning the investor will keep the put premium. The loss on the short position can still be substantial but is somewhat reduced by the put option premium.

Let us consider these two scenarios using the price data from Exhibit 6. In both cases, the investor is bearish on PBR and shorts the stock at 15.84. In the first case, she purchases the SEP 16 call for 0.97. As the share price increases above 16, the payoff from the call will act to offset losses in the short position. Exhibit 15 illustrates this dynamic. As the share price increases, portfolio losses never exceed 1.13. The profit on the short stock position plus the profit from the in-the-money call equals (15.84 − S) + [(S − 16) − 0.97] = −1.13. If the share price decreases, the investor profits from the short but loses the call premium of 0.97. The delta from the short PBR shares is −1. The SEP 16 call delta is 0.516. The overall portfolio delta is still negative at −0.484, making this a bearish strategy. The investor is also long vega from purchasing the call, 0.018, and the position is exposed to time decay, because theta is −0.018 per day. So, she is hoping to profit from increased downside volatility from the short PBR shares while the long call cushions losses from increased upside volatility.

Exhibit 15:

P&L of Long PBR SEP 16 Call and Short PBR Stock

In the second scenario, the investor writes the SEP 15 put for 0.65 and collects the put premium. The upside protection from the long call in the first scenario is not provided by writing a put. The short stock position can have potentially unlimited losses. As shown in Exhibit 16, the potential gain from a falling PBR price now belongs to the put owner. The maximum gain from this strategy is given by the profit on the short stock position plus the profit from the out-of-the-money short put, which equals (15.84 − S) − [(15 − S) − 0.65] = 1.49. Losses from the short stock position will be cushioned only by the 0.65 premium collected from writing the put. The delta of the short PBR shares is −1, and the delta of the short put is − (−0.335), so the position delta is −1 + 0.335 = −0.665. The investor is bearish and hoping to profit from a downward price move. She is also short vega from writing the put, (−0.017), and benefits from time decay, as theta of the short put is +0.015 (= − [−0.015]). So, she is hoping for reduced volatility to give her an opportunity to collect the put premium without losing from the short on PBR shares.

Exhibit 16:

 P&L of Short PBR SEP 15 Put and Short PBR Stock

EXAMPLE 5

Risk-Reduction Strategies

Janet Reiter is a US-based investor who holds a limited partnership investment in a French private equity firm. She has received notice from the firm’s general partner of an upcoming capital call. Reiter plans to purchase €1,000,000 in three months to meet the capital call due at that time. The current exchange rate is US$1.20/€1, but Reiter is concerned the euro will strengthen against the US dollar. She considers the following instruments to reduce the risk of the planned purchase:

• A three-month USD/EUR call option (to buy euros) with a strike rate X = US$1.25/€1 and costing US$0.02/€1

• A three-month EUR/USD put option (to sell dollars) with a strike rate X = €0.8080/US$1 priced at €0.0134/US$1

• A three-month USD/EUR futures contract (to buy euros) with f0 = US$1.2052/€1

1. Discuss the position required in each instrument to reduce the risk of the planned purchase.

   Solution to 1:

   Reiter could purchase a €1,000,000 call option struck at US$1.25/€1 for US$20,000. If the EUR price were to increase above US$1.25, she would exercise her right to buy EUR for US$1.25. She would also benefit from being able to purchase EUR at a cheaper price should the exchange rate weaken. A call on the euro is like a put on the US dollar. So, a put to sell dollars struck at an exchange rate of X = €0.8000/US$1 can be viewed as a call to buy Euro at an exchange rate of US$1/€0.8000 = US$1.25/€1. Reiter could also buy a put option on USD struck at X = €0.8080/US$1 which would allow her to sell US$1,237,624 (= €1,000,000/[€0.8080/$1]) to receive the €1,000,000 should the dollar weaken below that level. This would cost her €0.0134/US$1 × US$1,237,624 = €16,584 or US$19,901 upfront. If USD appreciated against the EUR, Reiter would still be able to benefit from the lower cost to purchase the EUR. She could instead enter a long position in a three-month futures contract at US$1.2052. Reiter would have the obligation to purchase €1,000,000 at US$1.2052 regardless of the exchange rate in three months. The futures position requires a margin deposit, but no premium is paid.

2. Reiter purchases call options for US$20,000, and the exchange rate increases to US$1.29/€1 (EUR currency strengthens) over the next three months. The effective price Reiter pays for her 1,000,000 EUR purchase is closest to:

   1. US$1,270,000.

   2. US$1,290,000.

   3. US$1,310,000.

   Solution to 2:

   A is correct. At an exchange rate of US$1.29/€1, the call with strike of X = US$1.25/€1 will be exercised. Including the call premium (US$0.02/€1), the price effectively paid for the euros is US$1.27/€1 × €1,000,000 = US$1,270,000.

3. Calculate the price Reiter will pay for the EUR using the three instruments if the exchange rate in three months falls to US$1.10/€1 (EUR currency weakens).

   Solution to 3:

   Both the call and the put options will expire unexercised and Reiter benefits from the lower rate by purchasing €1,000,000 for US$1,100,000. However, she will lose the premiums she paid for the options. For the futures contract, she pays US$1.2052/€1 or US$1,205,200 for €1,000,000 regardless of the more favorable rate.

SPREADS AND COMBINATIONS

https://study.cfainstitute.org/app/cfa-institute-program-level-iii-for-august-2024#read/study_task/2562693/spreads-and-combinations-1

Learning Outcome

• discuss the investment objective(s), structure, payoffs, risk(s), value at expiration, profit, maximum profit, maximum loss, and breakeven underlying price at expiration of the following option strategies: bull spread, bear spread, straddle, and collar

Option spreads and combinations can be useful option strategies. We first consider money spreads, in which the two options differ only by exercise price. The investor buys an option with a given expiration and exercise price and sells an option with the same expiration but a different exercise price. Of course, the options are on the same underlying asset. The term spread is used here because the payoff is based on the difference, or spread, between option exercise prices. For a bull or bear spread, the investor buys one call and writes another call option with a different exercise price, or the investor buys one put and writes another put with a different exercise price.17 Someone might, for instance, buy a NOV 16 call and simultaneously write a NOV 17 call, or one might buy a SEP 17 put and write a SEP 15 put. An option combination typically uses both puts and calls. The most important option combination is the straddle, on which we focus in this reading. We will investigate spreads first.

Bull Spreads and Bear Spreads

Spreads are classified in two ways: by market sentiment and by the direction of the initial cash flows. A spread that becomes more valuable when the price of the underlying asset rises is a bull spread; a spread that becomes more valuable when the price of the underlying asset declines is a bear spread. Because the investor buys one option and sells another, there is typically an initial net cash outflow or inflow. If establishing the spread requires a cash payment by the investor, it is referred to as a debit spread. Debit spreads are effectively long because the long option value exceeds the short option value. If the spread initially results in a cash inflow to the investor, it is referred to as a credit spread. Credit spreads18 are effectively short because the short option value exceeds the long option value. Any of these strategies can be created with puts or calls. The motivation for a spread is usually to place a directional bet, giving up part of the profit potential in exchange for a lower cost of the position. Some examples will help make this clear.

Bull Spread

Regardless of whether someone constructs a bull spread with puts or with calls, the strategy requires buying one option and writing another with a higher exercise price. Because the higher exercise price call is less expensive than the lower strike, a call bull spread involves an initial cash outflow (debit spread). A bull spread created from puts also requires the investor to write the higher-strike option and buy the lower-strike one. Because the higher-strike put is more expensive, a put bull spread involves an initial cash inflow (credit spread).

Let’s consider a call bull spread. Suppose, for instance, an investor thought it likely that by the September option expiration, PBR would rise to around 17 from its current level of 15.84. Based on the price data in Exhibit 6, what option strategy would capitalize on this anticipated price movement? If he were to buy the SEP 15 call for 1.64 and the stock rose to 17 at expiration, the call would be worth ST − X = 17 − 15 = 2. If the price of the option was 1.64, the profit is 0.36. The maximum loss is the price paid for the option, or 1.64. If, instead, an investor bought the SEP 16 call for 0.97, at an expiration stock price of 17, the call would be worth 1.00 for a gain of 0.03. A spread could make more sense with the following option values. If he believes the stock will not rise above 17 by September expiration, it may make sense to “sell off” the part of the return distribution above that price. The investor would receive 0.51 for each SEP 17 call sold.

The value of the spread at expiration (VT) depends on the stock price at expiration ST. For a bull spread, the investor buys the low strike option (struck at XL) and sells the high strike option (struck at XH), so that:VT = Max(0,ST − XL) − Max(0,ST − XH).5Therefore, the value depends on the terminal stock price ST:

VT = 0 − 0 = 0 if ST ≤ XL

VT = ST − XL − 0 = ST − XL if XL < ST < XH

VT = ST − XL− (ST − XH) = XH − XL if ST ≥ XH

The profit is obtained by subtracting the initial outlay for the spread from the foregoing value of the spread at expiration. To determine the initial outlay, recall that a call option with a lower exercise price will be more expensive than a call option with a higher exercise price. Because we are buying the call with the lower exercise price (for cL) and selling the call with the higher exercise price (for cH), the call we buy will cost more than the call we sell (cL > cH). Hence, the spread will require a net outlay of funds. This net outlay is the initial value of the position, V0 = cL − cH, which we call the net premium. The profit is:Π = Max(0,ST − XL) − Max(0,ST − XH) − (cL − cH).6In this manner, we see that the profit is the profit from the long call, Max(0,ST − XL) − cL, plus the profit from the short call, −Max(0,ST − XH) + cH. Broken down into ranges, the profit is as follows:

Π = −cL + cH if ST ≤ XL

Π = ST − XL − cL + cH if XL < ST < XH

Π = XH − XL − cL + cH if ST ≥ XH

If ST is below XL, the strategy will lose a limited amount of money. When both options expire out of the money, the investor loses the net premium, cL − cH. The profit on the upside, if ST is at least XH, is also limited to the difference in strike prices minus the net premium.

Consider two alternatives for the call purchase leg of the bull spread: 1) buy the SEP 15 call or 2) buy the SEP 16 call instead. Which is preferred? With Alternative 1, the SEP 15 call costs 1.64. Writing the SEP 17 call brings in 0.51, so the net cost is 1.64 − 0.51 = 1.13. Traders would refer to this position as a PBR SEP 15/17 bull call spread. The maximum profit would occur at or above the exercise price of 17 because all gains above this level belong to the owner of the PBR SEP 17 call. At an underlying price of 17 or higher, from the trader’s perspective, the position is worth 2, which represents the price appreciation from 15 to 17 (i.e., the difference in strikes). The maximum profit is

Π = XH − XL − cL + cH = 17 − 15 − 1.64 + 0.51 = 0.87.

Another way to look at it is that at a price above 17, the trader exercises the long call, buying the stock at 15, and is forced to sell the stock at 17 to the holder of his short call.

With Alternative 2, the investor buys the SEP 16 call and pays 0.97 for it. Writing the SEP 17 call brings in 0.51, so the net cost would be 0.97 − 0.51 = 0.46. At an underlying price of 17 or higher, the spread would be worth 1.00, so the maximum profit is

Π = XH − XL − cL + cH = 17 − 16 − 0.97 + 0.51 = 0.54.

Exhibit 17 compares the profit and loss diagrams for these two alternatives.

To determine the breakeven price with a spread, find the underlying asset price that will cause the exercise value of the two options combined to equal the initial cost of the spread. A spread has two exercise prices. There are also two option premiums. Mathematically, the breakeven price for a call bull spread can be derived from Π = ST* − XL − cL + cH = 0 and isST* = XL + cL − cH,which represents the lower exercise price plus the cost of the spread. In the examples here, Alternative 1 costs 1.13 (= 1.64 − 0.51). The breakeven ST* = XL + cL − cH = 15 + 1.64 − 0.51 = 16.13. If at option expiration the stock is 16.13, the 15-strike option would be worth 1.13 and the 17-strike call would be worthless. The breakeven price ST* is 15.00 + 1.13 = 16.13, as Exhibit 17 shows.

Exhibit 17:

Bull Spreads: Current PBR Stock Price = 15.84

Which of the alternatives is preferable? There is no clear-cut answer. As Exhibit 17 shows, the maximum loss for alternative 1 is 1.64 − 0.51= 1.13, compared with a maximum loss of 0.97 − 0.51 = 0.46 for Alternative 2. However, Alternative 1 is potentially more profitable for a move above 17 and has a lower breakeven price.

With Alternative 1, the breakeven point of 16.13 is less than 2% above the current level of 15.84, whereas with Alternative 2, reaching the breakeven point requires almost a 4% rise in the stock price. There is some additional information in Exhibit 6 the investor may wish to consider. The SEP 15/17 spread involves buying the SEP15 call with implied volatility of 64.42% and selling the SEP 17 call option with implied volatility of 51.07%. The investor may believe the SEP 15 call being purchased is relatively expensive compared with the SEP 17 call being sold. The PBR SEP 16/17 involves buying a SEP 16 call at a cost of 0.97 with an implied volatility of 55.92%. The investor may believe the SEP 16 call represents a better value than the SEP 15 call and so may choose the PBR SEP 16/17 spread.

We can calculate the Greek values for the spread. For example, using Exhibit 6, we see the theta of the PBR SEP 15/17 spread is −0.004 = −0.019 − (−0.015), and the theta of the PBR SEP 16/17 is −0.003 = −0.018 − (−0.015). Therefore, the SEP 16/17 should experience slightly less erosion of value resulting from time decay. The investor may also consider the delta and gamma that each spread would add to her PBR position. The delta of the PBR SEP 15/17 spread is +0.306 = 0.657 − 0.351, and the delta of the PBR SEP 16/17 spread is +0.165 = 0.516 − 0.351. From the current PBR price of 15.84, the long position in the PBR 15 call will make the SEP 15/17 PBR spread slightly more sensitive to an increase in share price than the SEP 16/17 spread. For the SEP 15/17, we have gamma = −0.034 = 0.125 − 0.159 and for the SEP 16/17 gamma = −0.003 = 0.156 − 0.159. The more negative gamma value for the SEP 15/17 spread means that the position delta will decrease at a faster rate than the SEP 16/17 spread as the price of PBR shares increase. By carefully selecting the expiration and exercise prices for the options for the spread, an investor can choose the risk–return mix that most closely matches her investment outlook.

Bear Spread

With a bull spread, the investor buys the lower exercise price and writes the higher exercise price. It is the opposite with a bear spread: buy the higher exercise price and sell the lower. Because puts with higher exercise prices are (all else equal) more expensive, a put bear spread will result in an initial cash outflow (be a debit spread). For a call bear spread, the investor buys a higher exercise price call and sells the lower exercise price call. Because the higher exercise price call being purchased is less expensive than the lower strike being sold, a call bear spread will result in an initial cash inflow (credit spread).

If a trader believed PBR stock would be below 15 by the November expiration, one strategy would be to buy the PBR NOV 16 put at 1.96 and write the NOV 15 put at 1.46. This spread has a net cost of 0.50; this amount is the maximum loss, and it occurs at a PBR stock price of 16 or higher. The maximum gain is also 0.50, which occurs at a stock price of 15 or lower. (A useful way to see this result is to realize that reversing the signs of the trades leaves the horizontal axes in a diagram like Exhibit 17 intact, but it flips the profit/loss and cost lines vertically! A debit from buying a spread must be consistent with the seller of the same spread receiving a credit.) Finding the breakeven price uses the same logic as with a bull spread: find the underlying asset price at which the exercise value equals the initial cost. Let pL represent the lower-strike put premium and pH the higher-strike put premium. Mathematically, the value of this bear spread position at expiration is:

VT = Max(0,XH − ST) − Max(0,XL − ST).7

Broken down into ranges, we have the following relations:

VT = XH − ST − (XL − ST) = XH − XL if ST ≤ XL

VT = XH − ST − 0 = XH − ST if XL < ST < XH

VT = 0 − 0 = 0 if ST ≥ XH

To obtain the profit, we subtract the initial outlay. Because we are buying the put with the higher exercise price and selling the put with the lower exercise price, the put we are buying is more expensive than the put we are selling. The initial value of the bear spread is V0 = pH − pL. The profit is, therefore, VT − V0, which is:Π = Max(0,XH − ST) − Max(0,XL − ST) − (pH − pL).8We see that the profit is that on the long put, Max(0,XH − ST) − pH, plus the profit from the short put, −Max(0,XL − ST) + pL. Broken down into ranges, the profit is as follows:

Π = XH − XL − pH + pL if ST ≤ XL

Π = XH − ST − pH + pL if XL < ST < XH

Π = −pH + pL if ST ≥ XH

The breakeven point, ST* = XH − pH + pL, sets the profit equal to zero between the strike prices. In this example, 16 − 1.96 + 1.46 = 15.50. That is, at a stock price of 15.50 on the expiration day, the 16-strike put would be worth 0.50 and the 15-strike put would be worthless. Exhibit 18 shows the profit and loss for a NOV 15/16 bear spread.19

Exhibit 18:

Bear Spread: Current PBR Stock Price = 15.84

Refining Spreads

It is not necessary that both legs of a spread be established at the same time or maintained for the same period. Options are very versatile, and positions can typically be quickly adjusted as market conditions change. Here are a few examples of different tactical adjustments an option trader might consider.

Adding a Short Leg to a Long Position

Consider Carlos Aguila, a trader who in September paid a premium of 1.50 for a NOV 40 call when the underlying stock was selling for 37. A month later, in October, the stock has risen to 48. He observes the following premiums for one-month call options.

Strike	Premium
40	8.30
45	4.42
50	1.91

This position has become very profitable. The call he bought is now worth 8.30. He paid 1.50, so his profit at this point is 8.30 − 1.50 = 6.80. He thinks the stock is likely to stabilize around its new level and doubts that it will go much higher. Aguila is considering writing a call option with an exercise price of either 45 or 50, thereby converting his long call position into a bull spread. Looking first at the NOV 50 call, he notes that the 1.91 premium would more than cover the initial cost of the NOV 40 call. If he were to write this call, the new profit and loss diagram would look like Exhibit 19. To review, consider the following points:

• At stock prices of 50 or higher, the exercise value of the spread is 10.00. The reason is because both options would be in the money, and a call with an exercise price of 40 would always be worth 10 more at exercise than a call with an exercise price of 50. The initial cost of the call with an exercise price of 40 was 1.50, and there would be a 1.91 cash inflow after writing the call with an exercise price of 50. The profit is 10.00 − 1.50 + 1.91 = 10.41.

• At stock prices of 40 or lower, the exercise value of the spread is zero; both options would be out of the money. The initial cost of the call with an exercise price of 40 was 1.50, and there would be a 1.91 cash inflow after writing the call with an exercise price of 50. The profit is 0 − 1.50 + 1.91 = 0.41.

• Between the two strike prices (40 and 50), the exercise value of the spread rises steadily as the stock price increases. For every unit increase up to the higher strike price, the exercise value of this spread increases by 1.0.

For instance, if the stock price remains unchanged at 48, the exercise value of the spread is 8.00. The reason is because the call with an exercise price of 40 would be worth 8.00 and the call with an exercise price of 50 would be worthless. The initial cost of the 40-strike call was 1.50, and there would be a 1.91 cash inflow when the 50-strike call was written. The profit is 8.00 − 1.50 + 1.91 = 8.41.

Now assume that he has written the NOV 50 call. Aguila needs to be careful how he views this new situation. No matter what happens to the stock price between now and expiration, the position is profitable, relative to his purchase price of the calls with an exercise price of 40. If the stock were to fall by any amount from its current level, however, he would have an opportunity loss. His profit would decrease progressively if the price trended back to 40. Aguila would be correct in saying that the bull spread will make a profit of at least 0.41. But, writing the NOV 50 call only partially hedges against a decline in the value of his new strategy. The position can still lose about 96% of its maximum profit, because only about 4% (0.41/10.41) has been hedged.

Exhibit 19:

Spread Creation: Buy a Call with Exercise Price of 40 at 1.50; Write a Call Later with Exercise Price of 50 at 1.91

Spreads and Delta

A spread strategy may be adapted to a changing market view. Suppose the market has been rising, and Lars Clive, an options trader, expects this trend to continue. Hypothetical company ZKQ currently sells for $44. Suppose Clive buys a NOV 45 call for 5.25. He computes the delta of this call as +0.55 and gamma as +0.028. Initially, Clive will profit at a rate of 0.55 for an increase of $1 in the price of ZKQ stock. For small changes, the delta of his position will increase at a rate of 0.028 for an increase of $1 in the price of ZKQ shares.

Three days later, the stock price has risen to $49, the value of the NOV 45 call has increased to 8.18, and the call delta has increased to +0.68. For the NOV 45 call, the option price increased by 2.93 (= 8.18 − 5.25) instead of by 2.75 (= 5 × 0.55, the stock price change multiplied by the initial delta value). Because delta is changing at a rate gamma, the approximation works best for small changes in share prices. With the stock at $49, a higher-strike NOV 50 call sells for 5.74 and has a delta of +0.55. Now Clive establishes a 45/50 bull call spread by writing the NOV 50 call. Clive is less bullish at the price of 49, and his 45/50 spread portfolio now has a delta of +0.13 = +0.68 − 0.55.

Now suppose another five days pass and the stock price falls to 45. The new option values would be 5.41 for the NOV 45 call and 3.55 for the NOV 50 call. Clive closes out the NOV 50 short call by buying it back. He sold the call for 5.74 and bought it back for 3.55, so he makes 5.74 − 3.55 = 2.19, or 2.19 per contract. He still holds the long position in the NOV 45 call, and his portfolio delta increases to +0.57.

Another four days pass, and ZKQ has risen to 48. The new price for the NOV 50 call is 4.71 with a delta of 0.51. Clive owns the NOV 45 with a price of 7.10 and a delta of 0.66. He then decides to write a NOV 50 call and lower his position delta to 0.15 (= 0.66 − 0.51).

At this point, Clive has had two cash outflows totaling 8.80: the initial 5.25 plus the 3.55 to buy back the NOV 50 call. He has two cash inflows totaling 10.45: the premium income of 5.74 and then 4.71 from the two instances of writing the NOV 50 calls. Exhibit 20 provides a summary of the results of Clive’s trades. Because the inflows of 10.45 exceed the outflows of 8.80, he has a resulting profit and loss diagram similar to the plot in Exhibit 19 that we saw in the previous example. Clive’s timing was excellent—in each case, he increased his portfolio delta prior to an increase in ZKQ stock and decreased delta before the share price decreased. The important point is that increasing portfolio delta will result in greater profits (losses) when the underlying asset value increases (decreases).

Exhibit 20:

Spreads and Deltas: A Summary of Results of Clive’s Trades

Day	ZKQ Price	Activity	Portfolio Delta	Cash Out	Cash In
1	44	Buy NOV 45 call	0.55	5.25	—
4	49	Sell NOV 50 call	0.13 (= 0.68 − 0.55)	—	5.74
9	45	Buy NOV 50 call	0.57	3.55	—
13	48	Sell NOV 50 call	0.15 (= 0.66 − 0.51)	—	4.71
Total	8.80	10.45
Net Inflow	1.65

Spreads are primarily a directional play on the underlying asset’s spot price (and also potentially on its volatility); still, spread traders can attempt to take advantage of changes in price, and it is easy to create a hypothetical example like this one. There obviously is no guarantee that any assumed price trend will continue. In fact, in actual practice, the excellent results shown in Exhibit 20 are exceedingly difficult to achieve. Still, the experienced option user knows to look for opportunistic plays that arise from price swings.

EXAMPLE 6

Spreads

Use the following information to answer questions 1 to 3 on spreads.

S0 = 44.50OCT 45 call = 2.55, OCT 45 put = 2.92OCT 50 call = 1.45, OCT 50 put = 6.80

1. What is the maximum gain with an OCT 45/50 bull call spread?

   1. 1.10

   2. 3.05

   3. 3.90

   Solution to 1:

   C is correct. With a bull spread, the maximum gain occurs at the high exercise price. At an underlying price of 50 or higher, the spread is worth the difference in the strike prices, or 50 − 45 = 5. The cost of establishing the spread is the price paid for the lower-strike option minus the price received for the higher-strike option: 2.55 − 1.45 = 1.10. The maximum gain is 5.00 − 1.10 = 3.90.

2. What is the maximum loss with an OCT 45/50 bear put spread?

   1. 1.12

   2. 3.88

   3. 4.38

   Solution to 2:

   B is correct. With a bear spread, an investor buys the higher exercise price and writes the lower exercise price. When this strategy is done with puts, the higher exercise price option costs more than the lower exercise price option. Thus, the investor has a debit spread with an initial cash outlay, which is the most he can lose. The initial cash outlay is the cost of the OCT 50 put minus the premium received from writing the OCT 45 put: 6.80 − 2.92 = 3.88.

3. What is the breakeven price with an OCT 45/50 bull call spread?

   1. 46.10

   2. 47.50

   3. 48.88

   Solution to 3:

   A is correct. An investor buys the OCT 45 call for 2.55 and sells the OCT 50 call for 1.45, for a net cost of 1.10. She breaks even when the position is worth the price she paid. The long call is worth 1.10 at a stock price of 46.10, and the OCT 50 call will expire out of the money and thus be worthless. The breakeven price is the lower exercise price of 45 plus the 1.10 cost of the spread, or 46.10.

STRADDLE

https://study.cfainstitute.org/app/cfa-institute-program-level-iii-for-august-2024#read/study_task/2562703/straddle-1

Learning Outcomes

• discuss the investment objective(s), structure, payoffs, risk(s), value at expiration, profit, maximum profit, maximum loss, and breakeven underlying price at expiration of the following option strategies: bull spread, bear spread, straddle, and collar

• describe uses of calendar spreads

A long straddle is an option combination in which one buys both puts and calls, with the same exercise price and same expiration date, on the same underlying asset.20 If someone writes both options, it is a short straddle. Because a long call is bullish and a long put is bearish, this strategy may seem illogical. When the Greeks are considered, the trader’s position becomes clearer. The classic example is in anticipation of some event in which the outcome is uncertain but likely to significantly affect the price of the underlying asset, regardless of how the event is resolved.

A straddle is an example of a directional play on the underlying volatility, expressing the view that volatility will either increase, for a long straddle, or decrease, for a short straddle, from its current level. A profitable outcome from a long straddle, however, usually requires a significant price movement in the underlying asset. The straddle buyer pays the premium for two options, so to make a profit, the underlying asset must move either above or below the option exercise price by at least the total amount spent on the straddle. As an example, suppose in the next few days there is a verdict expected in a liability lawsuit against an automobile manufacturer. An investor expects the stock to move sharply one way or the other once the verdict is revealed. When the exercise price is chosen close to the current stock price, the straddle is neither a bullish nor a bearish strategy—the delta of the straddle is close to zero. With any other exercise price, there may be a directional bias (non-zero delta) because one of the options will be in the money and one will be out of the money. If the price increases (decreases) significantly, the delta of the call will approach +1 (0), and the put delta will approach 0 (−1), making the delta of the position approximately +1 (−1).

Experienced option traders know that it is difficult to make money with a straddle. In the example, other people will also be watching the court proceedings. The market consensus will predict higher volatility once the verdict is announced, and option prices rise when volatility expectations rise. This increased volatility means that both the puts and the calls become expensive well before the verdict is revealed, and the long straddle requires the purchase of both options. To make money, the straddle buyer must be correct in his view that the “true” underlying volatility is higher than the market consensus. Essentially, the bet is that the straddle buyer is right and the other market participants, on average, are wrong by underestimating volatility.

Suppose the underlying stock sells for 50, and an investor selects 30-day options with an exercise price of 50. The call sells for 2.29 and the put for 2.28, for a total investment of 4.57. At prices above 50, the call is in the money. At prices below 50, the put is in the money. For the straddle to be profitable, one of these two options must be profitable enough to pay for the costs of both the put and call. To recover this cost, the underlying asset must either rise or fall by at least 4.57, as shown by the breakeven points in Exhibit 21.

Exhibit 21:

Long Straddle: Current Stock Price = 50; Buy 50-Strike Call at 2.29, Buy 50-Strike Put at 2.28

The straddle portfolio and Greeks are shown in Exhibit 22. The long straddle initially has a very low delta (+0.069 for this example) with a high gamma (0.139). The trader does not initially favor an increase or a decrease in the share price but knows the delta may quickly change. Once a direction (increase or decrease in the underlying price) asserts itself, the trader’s position will take on a non-zero delta value. The trader’s view can be better understood from a vega perspective. The long straddle will be profitable only if the stock price moves enough to recover both premiums. The short straddle writer collects the put and call premiums but will lose if the stock price moves more than 4.57 away from the strike price. The long (short) straddle trade is said to be long (short) volatility. The long straddle is a bet that increased volatility will move the stock price strongly above or below the strike price. The sensitivity of the straddle to changes in volatility is measured by vega. As shown in Exhibit 22, vega for our long straddle is +0.114, meaning the portfolio will profit by approximately 0.114 from increased volatility of 1% in the underlying. A stock price change large enough to cause the price of one option to exceed the cost of the combined premiums is needed to make the straddle trade profitable. A large increase in the underlying price will cause the delta of the call option to approach +1 and the delta of the put to approach 0. A large decrease in the underlying price will cause the delta of the call to drop to 0 and the put delta to approach −1.

Exhibit 22:

Long and Short Straddle Greeks

	Call	Put	Long Straddle = Call + Put	Short Straddle = −Call + −Put
Cost	2.29	2.28	4.570	−4.570
Delta	0.534	−0.465	0.069	−0.069
Gamma	0.072	0.067	0.139	−0.139
Vega	0.057	0.057	0.114	−0.114
Theta	−0.039	−0.036	−0.075	0.075
Implied Volatility	38%	41%	—	—

Theoretically, the stock can rise to any level, so the maximum profit with the long call is unlimited. If the stock declines, it can fall to no lower than zero. If that happens, the long put would be worth 50. Subtracting the 4.57 cost of the straddle gives a maximum profit of 45.43 from a stock drop. The value of a straddle at expiration is the combined value of the call and the put:VT = Max(0,ST − X) + Max(0,X − ST).9Broken down into ranges,

VT = X − ST if ST < X, and

VT = ST − X if ST > X.

The profit is VT − V0, or Π = Max(0,ST − X) + Max(0,X − ST) − c0 − p0.10

Broken down into ranges,

Π = X − ST − c0 − p0 if ST < X, and

Π = ST − X − c0 − p0 if ST > X.

As can be seen in Exhibit 21, the straddle has two breakeven points. The lower breakeven for the straddle is STL* = X − c0 − p0, and the upper breakeven is STH* = X + c0 + p0.

For the straddle buyer, the worst outcome is if the stock closes exactly at 50, meaning both the put and the call would expire worthless. At any other price, one of the options will have a positive exercise value. Note that at expiration, the straddle is not profitable if the stock price is in the range 45.43 to 54.57. The long straddle shown in Exhibit 21 requires more than a 9% price move in one month to be profitable. A trader who believed such a move was unlikely might be inclined to write the straddle, in which case the profit and loss diagram in Exhibit 21 is reversed, with a maximum gain of 4.57 and a theoretically unlimited loss if prices rise. The risk of a long straddle is limited to the amount paid for the two option positions. The straddle can also be understood in terms of theta. As shown in Exhibit 22, theta of the long straddle is −0.075. All else equal, the long put and call positions will lose their time value as expiration approaches. The long straddle buyer is betting on a large price move in the underlying prior to expiration. If the stock price does not change significantly, the short straddle, which has a positive theta, will benefit from the erosion of time value of the short put and call positions.

Collars

A collar is an option position in which the investor is long shares of stock and then buys a put with an exercise price below the current stock price and writes a call with an exercise price above the current stock price.21 Collars allow a shareholder to acquire downside protection through a protective put but reduce the cash outlay by writing a covered call. By carefully selecting the options, an investor can often offset most of the put premium paid by the call premium received. Using a collar, the profit and loss on the equity position is limited by the option positions.

For equity investors, the collar typically entails ownership of the underlying asset and the purchase of a put, which is financed with the sale of a call. In a typical investment or corporate finance setting, an interest rate collar may be used to hedge interest rate risk on floating-rate assets or liabilities. For example, a philanthropic foundation funds the grants it makes from income generated by its investment portfolio of floating-rate securities. The foundation’s chief investment officer (CIO) wants to hedge interest rate risk (the risk of rates falling on its floating securities) by buying an interest rate floor (a portfolio of interest rate puts) and paying for it by writing a cap (a portfolio of interest rate calls). Should the rates on the portfolio fall, the long floor will provide a lower limit for the income generated by the portfolio. To finance the floor purchase, the foundation sells a cap. The cap will limit the income generated from the floating rate portfolio in the event the floating rate rises. The CIO is still holding a floating-rate securities portfolio but has restricted the returns using the collar. By setting both a minimum and a maximum portfolio return, the CIO may be better able to plan funding requests.

Collars on an Existing Holding

A zero-cost collar involves the purchase of a put and sale of a call with the same premium. In Exhibit 6, for instance, the NOV 15 put costs 1.46 and the NOV 17 call is 1.44, very nearly the same. A collar written in the over-the-counter market can be easily structured to provide a precise offset of the put premium with the call premium.22

The value of the collar at expiration is the sum of the value of the underlying asset, the value of the long put (struck at X1), and the value of the short call (struck at X2):VT = ST + Max(0,X1 − ST) − Max(0,ST − X2), where X2 > X1.11The profit is the profit on the underlying share plus the profit on the long put and the short call so that:Π = ST + Max(0,X1 − ST) − Max(0,ST − X2) − S0 − p0 + c0.12Broken down into ranges, the total profit on the portfolio is as follows:

Π = X1 − S0 − p0 + c0 if ST ≤ X1

Π = ST − S0 − p0 + c0 if X1 < ST < X2

Π = X2 − S0 − p0 + c0 if ST ≥ X2

Consider the risk–return trade-off for a shareholder who previously bought PBR stock at 12 and now buys the NOV 15 put for 1.46 and simultaneously writes the NOV 17 covered call for 1.44. Exhibit 23 shows a profit and loss worksheet for the three positions. Exhibit 24 shows the profit and loss diagram.

Exhibit 23:

Collar P&L: Stock Purchased at 12, NOV 15 Put Purchased at 1.46, NOV 17 Call Written at 1.44

Stock price at expiration	5	10	15	16	17	20
Profit/loss from long stock	−7.00	−2.00	3.00	4.00	5.00	8.00
Profit/loss from long 15 put	8.54	3.54	−1.46	−1.46	−1.46	−1.46
Profit/loss from short 17 call	1.44	1.44	1.44	1.44	1.44	−1.56
Total	2.98	2.98	2.98	3.98	4.98	4.98

At or below the put exercise price of 15, the collar realizes a profit of X1 − S0 − p0 + c0 = 15 − 12 − 1.46 + 1.44 = 2.98. At or above the call exercise price of 17, the profit is constant at X2 − S0 − p0 + c0 = 17 − 12 − 1.46 + 1.44 = 4.98.

In this example, because the stock price had appreciated before establishing the collar, the position has a minimum gain of at least 2.98 as shown in Exhibit 24. Investors typically establish a collar on a position that is already outstanding.

Exhibit 24:

Collar P&L Diagram: Stock Purchased at 12, NOV 15 Put Purchased at 1.46, NOV 17 Call Written at 1.44

The Risk of a Collar

We have already discussed the risks of covered calls and protective puts. The collar is essentially the simultaneous holding of both of these positions. See Exhibit 25 for the return distribution of a collar. A collar sacrifices the positive part of the return distribution in exchange for the removal of the adverse portion. With the short call option, the option writer sold the right side of the return distribution, which includes the most desirable outcomes. With the long put, the investor is protected against the left side of the distribution and the associated losses. The option premium paid for the put is largely and, often precisely, offset by the option premium received from writing the call. The collar dramatically narrows the distribution of possible investment outcomes, which is risk reducing. In exchange for the risk reduction, the return potential is limited.

The risks of the collar can be understood in terms of the Greeks from Exhibit 6. For example, with a long share the delta of the portfolio = +1. With a collar, the portfolio delta is equal to the delta of the share plus the delta of the long NOV 15 put (−0.359) and short NOV 17 call (−0.475), so the portfolio delta = +1 + (−0.359) + (−0.475) = +0.166. Portfolio gamma, which equals put gamma minus call gamma, will be close to zero, −0.011 (= 0.075 − 0.086). By writing a call and buying a put, the investor reduces the portfolio’s delta at a price of 15.84 from +1 to +0.166 and the gamma is very close to zero. If the stock price moves outside the range depicted in Exhibit 25, the delta of the position will approach zero over time. If the share price moves above 17, the NOV 17 call approaches a delta of +1, and the put will approach a delta of 0. The collar is short the call, so above 17 the portfolio delta (long stock with delta = +1 plus a short call with delta = −1) will approach zero. At prices below 15, the NOV 17 call will have a delta approaching 0, but the long put approaches a delta of −1, so the portfolio delta will again approach zero (long stock with delta = +1 plus the long put with delta = −1). The portfolio’s sensitivity to changes in the stock price will be limited, as shown in Exhibit 25.

Exhibit 25:

Collars and Return Distribution: Stock at 15.84, Write NOV 17 Call and Buy NOV 15 Put

As the chosen put and call exercise prices move successively farther in opposite directions from the current price, the combined collar position begins to replicate the underlying gain/loss pattern of a long position in the underlying security. Conversely, as the chosen strike prices approach and meet each other, the expected returns and volatility become less and less equity-like and eventually converge on those of a risk-free, fixed income return to the time horizon. Thus, a collar position is, economically, intermediate between pure equity and fixed-income exposure.

The Risk of Spreads

Note that the shape of the profit and loss diagram for the bull spread in Exhibit 17 is similar to that of the collar in Exhibit 24. The upside return potential is limited, but so is the maximum loss. As with the risk–return tradeoff with the collar, an option spread takes the tails of the distribution out of play and leaves only price uncertainty between the option exercise prices. Looking at this scenario another way, if someone were to simply buy a long call, the maximum gain would be unlimited and the maximum loss would be the option premium paid. If someone decides to convert this to a spread, doing so limits the maximum gain while simultaneously reducing the total cost.

Calendar Spread

A strategy in which one sells an option and buys the same type of option but with different expiration dates, on the same underlying asset and with the same strike, is commonly referred to as a calendar spread. When the investor buys the more distant call and sells the near-term call, it is a long calendar spread. The investor could also buy a near-term call and sell a longer-dated one, known as a short calendar spread. Calendar spreads can also be done with puts; the investor would still buy a long-maturity put and sell a near-term put with the same strike and underlying to create a long calendar spread. As discussed previously, a portion of the option premium is time value. Time value decays over time and approaches zero as the option expiration date nears. Taking advantage of this time decay is a primary motivation behind a calendar spread. Time decay is more pronounced for a short-term option than for one with a long time until expiration. A calendar spread trade seeks to exploit this characteristic by purchasing a longer-term option and writing a shorter-term option.

Here is an example of how someone might use a calendar spread. Suppose XYZ stock is trading at 45 a share in August. XYZ has a new product to be introduced to the public early the following year. A trader believes this new product introduction will have a positive effect on the share price. Until the excitement associated with this announcement starts to affect the stock price, the trader believes that the stock will languish around the current level. See the option prices, deltas and thetas in Exhibit 26. Based on the bullish outlook for the stock going into January, the trader purchases the XYZ JAN 45 call with a theta (indicator of daily price erosion) of −0.014 for a price of 3.81. Noting that the near-term price forecast is neutral, the trader also decides to sell the XYZ SEP 45 call for 1.55. The theta for the XYZ SEP 45 Call is −0.029. The position costs 2.26 (= 3.81 − 1.55) to create and has an initial theta of +0.015 (= −0.014 − (−0.029)). If the stock price of XYZ remains constant over the next 30 days, the XYZ SEP 45 call will lose time value more rapidly than the JAN 45 call. The delta of calendar spread equals the delta of the JAN 45 call less the delta of the SEP 45 call and will be very low (Delta = +0.041 = 0.572 − 0.531).

Now move forward to the September expiration and assume that XYZ is trading at 45. The September option will now expire with no value, which is a good outcome for the calendar spread trader. The value of the position (now just the XYZ JAN 45 call) is 3.48, a gain of 1.22 over the position cost of 2.26. If the trader still believes that XYZ will stay around 45 into October before starting to move higher, the trader may continue to execute this strategy. An XYZ OCT 45 call might be sold for 1.55 with the hope that it also expires with no value.

Exhibit 26:

Calendar Spread Call Option Prices, Deltas, and Thetas

150 days until January option expiration. Underlying stock price = 45

Exercise Price	SEP	OCT	JAN
40	5.15	5.47	6.63
45	1.55	2.19	3.81
50	0.22	0.62	1.99

	Delta
40	0.975	0.902	0.800
45	0.531	0.545	0.572
50	0.121	0.217	0.363

	Theta (daily)
40	−0.007	−0.011	−0.011
45	−0.029	−0.020	−0.014
50	−0.014	−0.014	−0.013

Just before September option expiration. Underlying stock price = 45

Exercise Price	SEP	OCT	JAN
40	5.00	5.15	6.39
45	0	1.55	3.48
50	0	0.22	1.69

In this example, the calendar spread trader has a directional opinion on the stock but does not believe that the price movement is imminent. Rather, the trader sees an opportunity to capture time value in one or more shorter-lived options that are expected to expire worthless.

A short calendar spread is created by purchasing the near-term option and selling a longer-dated option. Thetas for in-the-money calls may provide motivation for a short calendar spread. Assume a trader purchases the XYZ SEP 40 call with a theta of −0.007 for a price of 5.15. The trader sells the OCT 40 call with a theta of −0.011 for 5.47 to offset the cost of the SEP 40 call. The position nets the trader a cash inflow of 0.32 (= 5.47 − 5.15), and the initial position theta is slightly positive −0.007 − (−0.011) = +0.004.

If the stock price of XYZ remains at 45 (above the strike of 40) at the SEP expiration, the XYZ OCT 40 call will lose time value more rapidly than the SEP 40 call. The trader may close the position at the SEP expiration and make a profit of 0.17 = 0.32 + (5 − 5.15). Note that the profit consists of the 0.32 initial inflow plus the net cost of selling the SEP 40 call (at 5.00) and buying the OCT 40 call (at 5.15). In the event of a larger move, the position values will vary. For a large down move, for example an extreme case in which XYZ loses all of its value (so S = 0 at expiration), the long and the short call positions will be approximately worthless and the profit on the spread will be around 0.32 (= 0.32 + [0 − 0])). For a smaller down move to the strike price (S = 40 at expiration), the short calendar spread may result in a loss. If the XYZ stock price were to fall to 40 at the SEP expiration, the long position in the SEP 40 call would expire worthless but the OCT 40 call would still have a BSM model value of about 1.00 (not shown in Exhibit 26). This scenario would result in a loss of 0.68 = 0.32 + (0 − 1) to close the position. The writer of a calendar spread would typically be looking for a large move away from the strike price in either direction.

In sum, a big move in the underlying market or a decrease in implied volatility will help a short calendar spread, whereas a stable market or an increase in implied volatility will help a long calendar spread. Thus, calendar spreads are sensitive to movement of the underlying but also sensitive to changes in implied volatility.