19 May noon study - volatility
Last updated
Last updated
Learning Outcome
discuss volatility skew and smile
An important factor in the current price of an option is the outlook for the future volatility of the underlying asset’s returns, the implied volatility.
Implied volatility is not observable per se, but it is derived from an option pricing model—such as the Black–Scholes–Merton (BSM) model—and it is value that equates the model price of an option to its market price.
Note that all other input variables to the BSM model, including the option’s strike price, the price of the underlying, the time to option expiration, and the risk-free interest rate, are observable.
Implied volatilities incorporate investors’ expectations about the future course of financial asset returns and the level of market uncertainty associated with them.
Implied volatilities for options on a specific asset may differ with strike price (i.e., moneyness), side (i.e., put or call), and time to expiration.
In particular, out-of-the-money (OTM) puts typically command higher implied volatilities than ATM or OTM calls.
This phenomenon is attributed to investors’ reassessments of the probabilities of negative “fat-tailed” market events such as the 2007–2009 global financial crisis.
Implied volatility is often compared with realized volatility (i.e., historical volatility), which is the square root of the realized variance of returns and measures the range of past returns for the underlying asset.
To calculate the historical volatility for the given option, stock, or equity index, a series of past prices is needed. For example, to calculate the volatility of the S&P 500 Index over the past month (i.e., 21 trading days), it is necessary to first calculate the daily percentage change for each day’s index closing price.
This is done using the following formula, where Pt is the closing price and Pt−1 is the prior day’s closing price: (Pt − Pt−1)/Pt−1.
The next step is to apply the standard deviation formula you learned in Level I Quantitative Methods to the daily percentage change data to calculate the standard deviation (i.e., volatility) of the S&P 500 for the selected period, which in this case is the past month.
The standard deviation over the past month is then annualized by multiplying by the square root of the number of periods in a year. Because we assume the average number of trading days in a year and in a month are 252 and 21 (excluding weekends and holidays), respectively, the formula is:
𝜎𝐴𝑛𝑛𝑢𝑎𝑙(%)=𝜎𝑀𝑜𝑛𝑡ℎ𝑙𝑦(%)2522113
Note that this example uses one month of daily return data, but the process is equally applicable to any other period.
Obviously, we cannot use the previous formula for realized volatility, which is based on past prices, to obtain implied volatility, which is the expected volatility of future returns of the underlying asset.
Instead, the one-month annualized implied volatility can be derived from the current price of an option maturing in one month by using the BSM model.
Once the one-month annualized implied volatility is obtained, it can then be converted into an estimate of the volatility expected on the underlying asset over the 21-day life of the option.
This expected monthly volatility is given by the one-month annualized implied volatility divided by the square root of the number of 21-day periods in a 252-day trading year, as follows:
𝜎𝑀𝑜𝑛𝑡ℎ𝑙𝑦(%)=𝜎𝐴𝑛𝑛𝑢𝑎𝑙(%)/2522114
Exhibit 27:
One-Month Annualized Implied Volatility
Underlying Index
Implied Volatility
Three-Year Low
Three-Year High
Euro Stoxx 50
9.56
9.0
17.9
FTSE MIB
14.22
11.0
21.7
DAX
9.29
7.3
13.9
Using the one-month annualized implied volatility for DAX index options of 9.29%, the volatility expected to materialize in the DAX index over the next month (21 trading days) can be calculated as follows:
𝜎𝑀𝑜𝑛𝑡ℎ𝑙𝑦(%)=9.29%/25221=2.68%
So, for example, if an investor buys an ATM one-month (21-day) straddle using puts and calls on the DAX, in order for the strategy to be profitable at expiration, the index must move up or down by at least 2.68%. The investor can compare this price movement needed to reach breakeven with the DAX’s realized volatility over similar time horizons in the past. If such a price change is considered reasonable, then the investor can elect to implement the strategy.
The implied volatility of ATM options, calculated from the options’ market prices using the BSM model, remains the simplest way to measure the prevailing volatility level. The BSM model assumes that volatility is constant, and notably, before the 1987 stock market crash, there was little volatility skew in equity index markets. Today, however, options prices on several asset classes display persistent volatility skew and, in some circumstances, volatility smile. The implied volatilities of options of a given expiration are thus dependent on their strike prices.
The more common shape of the implied volatility curve, however, is a volatility skew, where
the implied volatility increases for OTM puts and decreases for OTM calls, as the strike price moves away from the current price. This shape persists across asset classes and over time because
investors have generally less interest in OTM calls whereas OTM put options have found universal demand as portfolio insurance against a market sell-off.
Exhibit 28:
Implied Volatility Curves for Three-Month Options on FTSE MIB
The extent of the skew depends on several factors, including investor sentiment and the relative supply/demand for puts and calls, among others.
Several theoretical models try to use these factors to forecast skew variation. However, these models do not lead to unique predictions.
In general, we can say that when the implied volatility is significantly higher (relative to historical levels) for puts with strike prices below the underlying asset's price, it means that there is an imbalance in the supply and demand for options.
In fact, when investors are looking to hedge the underlying asset, the demand for put options exceeds that for call options.
Option traders, who meet this excess demand by selling puts, increase the relative price of these options, thereby raising the implied volatility.
A sharp increase in the level of the skew, accompanied with a surge in the absolute level of implied volatility, is an indicator that market sentiment is turning bearish.
In contrast, higher implied volatilities (relative to historical levels) for calls with strike prices above the underlying asset's price indicate that investors are bullish and the demand for OTM calls to take on upside exposure is strong.
The 90% moneyness option is a put with strike (X) equal to 90% of the current underlying price (S); thus X/S = 90%.
The 110% moneyness of the call is calculated similarly. Also shown is the skew, calculated as the difference between the implied volatilities of the 90% put and the 110% call.
Exhibit 29:
Implied Volatilities and Skew for Three-Month Options
Implied Volatility by Moneyness
90%–110%
Index
ATM
Put: 90%
Call: 110%
Volatility Skew
Nikkei 225
12.9
18.9
12.4
6.5
S&P 500
10.3
17.7
9.4
8.3
Euro Stoxx 50
12.3
17.8
9.3
8.5
DAX
14.5
20.0
11.0
9.0
Exhibit 30:
90% Put–110% Call Implied Volatility Skew for Three-Month Options on S&P 500
There are trading strategies that attempt to profit from the existence of an implied volatility skew and from changes in its shape over time.
A common strategy is to take a long or short position in a risk reversal, which is then delta hedged.
Using OTM options, a combination of long (short) calls and short (long) puts on the same underlying with the same expiration is a long (short) risk reversal.
In particular, when a trader thinks that the put implied volatility is too high relative to the call implied volatility, she creates a long risk reversal, by selling the OTM put and buying the same expiration OTM call.
The options position is then delta-hedged by selling the underlying asset.
The trader is not aiming to profit from the movement in the overall level in implied volatility.
In fact, depending on the strikes of the put and the call, the trade could be vega-neutral.
For the trade to be profitable, the trader expects that the call will rise more (or decrease less) in implied volatility terms relative to the put.
Typically, implied volatility is not constant across different maturities, which means that options with the same strike price but with different maturities display different implied volatilities.
This determines the term structure of volatility, which is often in contango, meaning that the implied volatilities for longer-term options are higher than for near-term ones.
When markets are in stress and de-risking sentiment prevails, however, market participants demand short-term options, pushing up their prices and causing the term structure of volatility to invert.
Exhibit 31:
12M–3M Implied Volatility Term Structure of S&P 500 Options
The implied volatility surface can be thought of as a three-dimensional plot, for put and call options on the same underlying asset, of days to expiration (x-axis), option strike prices (y-axis), and implied volatilities (z-axis).
It simultaneously shows the volatility skew (smile) and the term structure of implied volatility.
Considering that implied volatility varies across different option maturities and displays skew, the implied volatility surface is typically not flat as the BSM model may suggest.
By observing the implied volatility surface, one can infer changes in market expectations.
Several studies have focused on ways to extract information embedded in option market prices.
The skew can provide insight into market participants’ perceptions about the price movement of the underlying asset over a specified horizon.
The general interpretation is that the shape of the volatility skew reflects varying degrees of market participants’ fear about future market stress.
Learning Outcome
identify and evaluate appropriate option strategies consistent with given investment objectives
The Necessity of Setting an Objective
Every trade is based on an outlook on the market. With stocks and most assets, one thinks about the direction of the market: Is it going up or down, or is it stable?
When dealing with options, it is not enough to think about the market direction; it is also important to think about the Greeks beyond delta: gamma, theta, and vega.
Where option valuation is concerned, what matters is not only the direction in which the asset underlying the derivative contract is headed but also the volatility of the underlying and even other investors’ perception of that volatility.
The investor’s investment objective may not be achieved if one of these factors moves in an undesired direction.
Gamma could lead to a faster loss or gain, depending on whether the investor is short or long the option,
whereas theta could lead to a loss despite the underlying asset moving in the right direction.
Vega could rise or fall if market expectations of implied volatility change, leading to a loss or profit for the options position.
Furthermore, the option premium paid (if long) or received (if short) must be considered when calculating a position’s total profit and loss at maturity.
For example, in a simple call option purchase, the underlying asset must go up enough so that the call option reaches breakeven by overcoming the premium paid for the call.
Moreover, with the introduction of volatility-based derivatives, investors increasingly view these investments as a way to protect their portfolios against downside risk and also as a method to improve their portfolios’ efficiency.
When considering hedging strategies, it is important to differentiate between situations in which the investor’s goal is to benefit from rising volatility (long volatility)—for example, in hedging a long stock position—and situations in which the investor wants to benefit from falling volatility (short volatility)—for example, by writing a short straddle position.
Derivatives are used by portfolio managers, traders, and corporations to adjust their risk exposures, to achieve a specific investment objective, or even to infer market expectations in the short term (for example, inferring market expectations for central banks’ interest rate decisions).
The main advantage of derivatives is that they allow two parties with differing needs and market views to adjust quickly without having to enter into potentially costly and difficult trades in the underlying.
Another advantage of using derivatives instead of investing in the physical underlying securities is leverage. The fact that investors can take on a large exposure to the underlying asset by putting up only a fraction of the amount of risk capital is an important feature of derivatives.
Liquidity is another key aspect favoring derivatives usage in many markets—for example, the ability to buy or sell credit protection using index credit derivatives (CDS) instead of trading the actual underlying, and likely less liquid, bonds provides a huge liquidity advantage to traders and investors.
In sum, given an actual or anticipated portfolio of equities, bonds, rates, currencies, or other assets; a market outlook and a timeframe; and an understanding of the benefits and limitations of derivatives, it is important to set realistic investment objectives, be they for hedging, for taking direction bets, or for capturing arbitrage opportunities.
Criteria for Identifying Appropriate Option Strategies
Exhibit 32:
Choosing Options Strategies Based on Direction and Volatility of the Underlying Asset
Outlook on the Trend of Underlying Asset
Bearish
Trading Range/Neutral View
Bullish
Expected Move in Implied Volatility
Decrease
Write calls
Write straddle
Write puts
Remain Unchanged
Write calls and buy puts
Calendar spread
Buy calls and write puts
Increase
Buy puts
Buy straddle
Buy calls
Consider an investor who is bearish about a market.
If he expects that implied volatility will increase as the market sells off, then he will buy a put to protect his investments. If instead he believes that volatilities are expected to fall, writing a call would likely be a profitable strategy.
Investors need to keep in mind, however, that the two strategies have two different payoffs and risk management implications.
As we have seen previously, a position in which the investor writes the call and buys the put is a collar, and it is often associated with holding a long position in the underlying asset.
Investors use collars against a long stock position to hedge risk and smooth volatility.
In fact, by selling a covered call while purchasing a protective put, the investor establishes a combined position with fixed downside protection while still providing some opportunity for profits.
Now consider an investor who has a bullish view.
Buying calls is an option strategy that will allow the investor to benefit if her outlook is correct and implied volatility increases.
When implied volatilities are expected to fall, writing a put represents an alternative strategy to take advantage of a bullish market view with declining volatility.
If this happens, the position will likely be profitable, because as the stock rises and implied volatility decreases, the short put moves farther out of the money and its price will decrease.
The combined position in which the investor is long calls and short puts (i.e., a long risk reversal) is used to implement a bullish view (the investor buys the calls) while lowering the cost (the investor sells the puts) of the long position in the underlying asset that will be established upon exercising the options.
Investors can also decide to sell puts when they want to buy a stock only if the price declines below a determined target price (the strike price of the puts will be the same as this target price).
In this case, by selling the put options when implied volatilities are elevated (and the puts are expensive), the investor can realize an effective stock purchase price that is less than the target (strike) price by the size of the premium received.
The purchase of call or put spreads tends to be most appropriate when the investor has a bullish view (call bull spread) or a bearish view (put bear spread) but the underlying market is not clearly trending upward or downward. Furthermore, such spreads are a way to reduce the total cost, given that the spread is normally constructed by buying one option and writing another. Importantly, if the implied volatility curve is skewed, with the implied volatility of OTM puts relatively higher than for nearer-to-the-money options, the cost of a bearish spread is even lower. This is because in the put bear spread, the lower exercise price, more OTM (but relatively more expensive) put is sold and the higher exercise price, nearer-to-the-money (but relatively less expensive) put is purchased.
Now suppose that the market is expected to trade in range. Again, the investor should consider volatility. Suppose he believes the current consensus estimate of volatility implied in option prices is too low and the rate of the change in underlying prices will increase—that is, vega will increase. If the investor is neutral on market direction, the appropriate strategy will be a long straddle, because this strategy takes advantage of an increase in the long call or long put option prices in response to realization of the expected surge in volatility, vega, and gamma. At expiry, the long straddle will be profitable if the price of the call (put) is greater (less) than the upper (lower) breakeven price. In contrast, an investor who expects the market to trade in range and volatility to fall may want to write the straddle instead.
We now consider a strategy that combines a longer-term bullish/bearish outlook on the underlying asset with a near-term neutral outlook. This approach is the calendar spread (or time spread), a strategy using two options of the same type (puts or calls) and same strike price but with different maturity dates. Typically, a long calendar spread—wherein the shorter-maturity option is sold and the longer-maturity option is purchased—is a long volatility trading strategy because the longer-term option has a higher vega than the shorter-term option. The maximum profit is obtained when the short-term option expires worthless, then implied volatility surges, increasing the price of the remaining long-term option. Although the delta for a calendar spread is approximately zero, gamma is not, so the main risk for the calendar strategy is that the underlying stock price moves too fast and too far from the strike prices. For this reason, the calendar spread is typically implemented in option markets characterized by low implied volatility when the underlying stock is expected to remain in a trading range, but only until the maturity of the short-term option.
Learning Outcome
demonstrate the use of options to achieve targeted equity risk exposures
This section uses “mini cases” to illustrate some of the ways in which different market participants use derivative products to solve a problem or to alter a risk exposure. Note that with the wide variety of derivatives available, there are almost always multiple ways in which derivatives might logically be used in a particular situation. These mini cases cover only a few of them.
Covered Call Writing
Carlos Rivera is a portfolio manager in a small asset management firm focusing on high-net-worth clients. In mid-April, he is preparing for an upcoming meeting with Parker, a client whose daughter is about to marry. Parker and her husband have just decided to pay for their daughter’s honeymoon and need to raise $30,000 relatively quickly. The client’s portfolio is 70% invested in equities and 30% in fixed income and is by policy slightly aggressive. Currently the Parkers are “asset rich and cash poor,” having largely depleted their cash reserves prior to the wedding expenses. The recently revised investment policy statement permits most option activity except the writing of naked calls.
Exhibit 33:
Manzana Inc. May Options With 44 Days to Expiration, MNZA Stock = $169
Call Premium
Call Delta
Exercise Price
Put Premium
Put Delta
Put or Call Vega
12.55
0.721
160
3.75
−0.289
0.199
9.10
0.620
165
5.30
−0.384
0.224
6.45
0.504
170
7.69
−0.494
0.234
4.03
0.381
175
10.58
−0.604
0.225
2.50
0.271
180
14.10
−0.702
0.199
Discuss the factors that Rivera should consider and the strategy he should recommend to Parker.
Solution:
To generate cash, Rivera will want to write options. The account permits the writing of covered calls. Manzana options trade on an organized exchange with a standard contract size of 100 shares per contract. With 5,000 shares in the account, 50 call contracts would be covered.
Exhibit 34:
Profit and Loss for MNZA May 170 Covered Call (S0 = 169, c0 = 6.45)
Put Writing
Oscar Quintera is the chief financial officer for Tres Jotas, a private investment firm in Buenos Aires. He wants to purchase 50,000 MNZA shares for the firm, but at the current price he considers MNZA shares to be a bit expensive. The current share price is $169, and Quintera is willing to buy the stock at a price not higher than $165. Quintera decides to write out-of-the-money puts on MNZA shares.
Discuss the outcome of the transaction, a short position in MNZA May 165 puts, assuming two scenarios:
Scenario A: MNZA is $163 per share on the option expiration day.
Scenario B: MNZA is $177 per share on the option expiration day.
Solution:
Exhibit 35:
Short Position Profit for 500 MNZA May 165 Put Contracts
Scenario A:
Scenario B:
Long Straddle
Hamlet expects that the stock will move at least 10% either way once the product announcement is made, making the straddle strategy potentially appropriate. The vega of her position would be 0.234 + 0.234= +0.468, meaning a 1% move in the options’ volatility would result in a gain of about $0.468 in the value of the straddle. The straddle’s delta would be approximately zero, at +0.01 (Call Delta + Put Delta = (0.504 + [−0.494]). This strategy is long volatility. After the market close, Hamlet hears a news story indicating that the product will be unveiled at a trade show in two weeks. The following morning after the market opens, she goes to place her trade and finds that although the stock price remains at $169.00, the option prices have adjusted upward to $10.20 for the call and $10.89 for the put.
Discuss whether the new option premiums have any implications for Hamlet’s intended straddle strategy.
Exhibit 36:
Long Manzana Straddle
Solution:
The news report about the imminent product unveiling, however, has increased the implied volatility in the options, from about 30% to about 45%, raising their prices and making it more difficult to achieve the new breakeven points. After the news report, Hamlet finds the MAY 170 call now costs $10.20 and the MAY 170 put is trading for $10.89, so the MAY 170 straddle costs $21.09 ($10.20 + $10.89) to implement. Relating back to the vega she calculated the day before, Hamlet computed an initial vega (pre-announcement) of +0.468 for the straddle. Now she sees the approximate 15 percentage point rise in implied volatility (to 45%) in both the put and call. According to her initial vega calculation, she would expect an increase of 15 × 0.468 = $7.02 in her straddle value after the announcement. The announcement increased the price of the straddle by $6.95 (= 21.09 − 14.14), which is very close to the $7.02 increase predicted by the vega calculation. To reach the new breakeven points (170 ± 21.09), she now needs the stock to move by more than 12%, a larger move than 10% from the current level of 169. Given that Hamlet expects only a 10% price movement, she decides against executing this straddle trade.
EXAMPLE 7
Straddle Analytics
Use the following information to answer Questions 1 to 3 on straddles.
XYZ stock price = 100.00100-strike call premium = 8.00100-strike put premium = 7.50Options expire in three months
If Yelena Strelnikov, a portfolio manager, buys a straddle on XYZ stock, she is best described as expecting a:
higher volatility market.
lower volatility market.
stable volatility market.
Solution to 1:
A is correct. A straddle is directionally neutral in terms of price; it is neither bullish nor bearish. The straddle buyer wants higher volatility and wants it quickly but does not care in which direction the price of the underlying moves. The worst outcome is for the underlying asset to remain stable.
This strategy will break even at expiration stock prices of:
92.50 and 108.50.
92.00 and 108.00.
84.50 and 115.50.
Solution to 2:
C is correct. To break even, the stock price must move enough to recover the cost of both the put and the call. These premiums total to $15.50, so the stock must move up at least to $115.50 or down to $84.50.
Reaching an upside breakeven point implies an annualized rate of return on XYZ stock closest to:
16%.
31%.
62%.
Solution to 3:
C is correct. The price change to a breakeven point is 15.50 points, or 15.5% on a 100 stock. This is for three months. Ignoring compounding, this outcome is equivalent to an annualized rate of 62% on XYZ stock, found by multiplying by 12/3 (15.5% × 4 = 62%).
Collar
Bernhard Steinbacher has a client with a holding of 100,000 shares in Tundra Corporation, currently trading for €14 per share. The client has owned the shares for many years and thus has a very low tax basis on this stock. Steinbacher wants to safeguard the position’s value because the client does not want to sell the shares. He does not find exchange-traded options on the stock. Steinbacher wants to present a way in which the client could protect the investment portfolio from a decline in Tundra’s stock price.
Discuss an option strategy that Steinbacher might recommend to his client.
Solution:
Calendar Spread
Exhibit 37:
3,500 Strike Put Options on Euro Stoxx 50
Option A
Option B
Current Price
€119
€173
Time to Maturity
3 months
6 months
Discuss how can Dubois take advantage of her out-of-consensus view.
Analyze four scenarios that Dubois might likely face for the Stoxx 50 index at the expiry of the three-month option (these scenarios are provided at the beginning of the Solution to 2).
Solution to 1:
Dubois’s view is best implemented with a long position in a calendar spread that combines a longer-term bearish outlook on the underlying asset with a near-term neutral outlook. She is bearish long-term and so would buy a calendar put spread. A long calendar spread is a long volatility trading strategy whereby the maximum profit is obtained when the short-term at the money option expires worthless with the underlying almost unchanged.
Dubois can implement a put calendar spread trade by selling the three-month put option (A) for €119 and buying the six-month same strike put option (B) at the price for €173. Therefore, the cost of establishing this strategy is a net debit of €54 per contract (given by €173 − €119). Remember, Dubois has a bearish long-term outlook. If the put calendar spread is not profitable at the expiry of the three-month put, the short option expires worthless and then she owns the longer-term option free and clear. Thus, Dubois has managed to lower the cost of purchasing a longer-term put option, which could be kept for hedging her portfolio’s downside risk.
Solution to 2:
If the put calendar spread position is held until the expiry of the three-month put, then Dubois might likely face one of the four following scenarios for the Euro Stoxx 50. In the first three scenarios, the implied volatility is assumed to remain constant:
Scenario 1
The index is still trading at 3500 as expected.
Scenario 2
The index has increased and is trading at 4200.
Scenario 3
The index has decreased and is trading at 3000.
Scenario 4
The index has decreased and is trading at 3000, but the implied volatility has significantly increased.
Scenario 1:
Exhibit 38:
P&L for 3,500 Strike Calendar Spread at Expiration of Three-Month Put
Scenario 2:
The Euro Stoxx 50 has increased and is trading at 4200. The three-month put option expires worthless. Also, the value of the six-month put is near zero, and if Dubois unwinds her (long) put option position she will lose all €54 (given by €173 − €119), the cost of the put calendar spread.
Scenario 3:
The Euro Stoxx 50 index has decreased and is trading at 3000. Dubois must pay €500 (€3,500 − €3,000) to settle the (short) three-month put option at expiration. The (long) put option with three months remaining to expiration is deep in the money and, assuming volatility is still unchanged and €15 of time value, it is worth €515 (given by Intrinsic Value of €500 + €15 of Time Value). If Dubois sells this put to a dealer, she will lose €39 (= €515 − €500 − €54) on the put calendar spread.
Scenario 4:
Exhibit 39:
P&L for 3,500 Strike Calendar Spread at Expiration of Three-Month Put Assuming an Increase in Implied Volatility
Learning Outcome
demonstrate the use of options to achieve targeted equity risk exposures
Jack Wu is a fund manager who oversees a stock portfolio valued at US$50 million that is benchmarked to the S&P 500. He expects an imminent significant correction in the US stock market and wants to profit from an anticipated jump in short-term volatility to hedge his portfolio’s tail risk.
At maturity, the options’ payoffs will depend on the settlement price of the relevant VIX futures contracts. The options will expire one month from now, and the contract size is 100.
Exhibit 40:
Options on VIX Index
Call Option
Put Option
Option Strike
15.60
14.75
Option Price
2.00
1.55
Discuss the following:
A strategy Wu can implement to hedge tail risk in his equity portfolio, by taking advantage of his expected increase in volatility while lowering his hedging cost
Profit and loss on the strategy at options expiration
Relevant issues and advantages of this strategy
Solution to 1:
Wu decides to purchase the 15.60 call on the VIX and, to partially finance the purchase, he sells an equal number of the 14.75 VIX puts. The total cost of the options strategy is 0.45 (= 2.00 − 1.55) per contract.
Solution to 2:
Exhibit 41:
Profit and Loss of the VIX Options Strategy
In this case, at the expiry the strategy will be profitable if volatility spikes up (as anticipated) and the VIX futures increase above 16.05. This is calculated as the call strike of 15.60 plus the net cost of the options (15.60 + 0.45). Above this level, the strategy will gain proportionally. In contrast, Wu’s option strategy will lose proportionally to its exposure to the short puts if the VIX futures’ settlement price is below 14.75 (put strike).
Solution to 3:
The hedge ratio that determines the number of calls to buy could be determined based on regression and scenario analysis on the portfolio’s profit and loss versus rates of increase in implied volatility during significant stock market sell-offs in the past. Of course, a risk is that past correlations may not be indicative of future correlations. Importantly, the relative advantage of implementing this long volatility hedging strategy by purchasing calls on the VIX over buying VIX futures depends on the difference in leverage available, the difference in payoff profiles (asymmetrical for options and symmetrical for futures), and the shape of the volatility futures term structure, as well as the cost of the options compared with the cost of the index futures.
Establishing or Modifying Equity Risk Exposure
In this section, we examine some examples in which investors use derivatives for establishing an equity risk exposure, for risk management, or for implementing tactical asset allocation decisions. The choice of derivative that will satisfy the investment goal depends on the outlook for the underlying asset, the investment horizon, and expectations for implied volatility over that horizon.
Long Call
Exhibit 42:
Three-Month Call Options on Markle Co. Ltd.
Option A
Option B
Option C
Strike
£58.00
£60.00
£70.00
Price
£4.00
£3.00
£0.40
Delta
0.6295
0.5227
0.1184
Gamma
0.0304
0.0322
0.0160
Discuss the option strategy that Sanchez should recommend to his clients.
Solution:
Sanchez has a bullish view because he anticipates a nearly 17% price increase in Markle shares over the next three months, from the current price of £60. He expects that implied volatilities of the options on Markle shares will stay unchanged, making the purchase of options a profitable strategy if his outlook materializes within the given timeframe.
The best strategy would be a long position in the £60 strike calls (option B). The breakeven price of the position is £63 (£60 + £3), so at option expiry, the overall position would profitable at any stock price above £63. In contrast, the breakeven price of the 58-strike call (option A) is £62 (£58 + £4) and for the 70-strike call (option C) the breakeven is £70.40 (£70 + £0.40). Therefore, Sanchez would not use option C to implement his strategy because the breakeven price is above his target price of £70/share.
Given the £60 strike call has a lower price (premium) than the £58 strike call, Sanchez’s clients can purchase more of these lower-priced options for a given investment size. Moreover, the 60-strike call offers the largest profit potential per unit of premium paid if the stock price increases to £70 (as expected).
Call strike £58: (£70 − £62)/£4 = 2.0
Call strike £60: (£70 − £63)/£3 = 2.3
Purchasing the £60 strike call (option B) is the most profitable strategy given that Sanchez’s expectations are realized. A position in the £60 strike call has a lower delta (= 0.5227) compared with the £58 strike call (delta = 0.6295), so at current prices of the underlying, the change in value of the £60 strike call is lower. If Markle’s stock reaches £70 per share during the life of the option, however, the £60 strike call will benefit from having a larger gamma (= 0.0322) compared with the £58 strike call (gamma = 0.0304).
Risk Management: Protective Put Position
Investors use protective puts, collars, and equity swaps against a long stock position to hedge market risk. Here we turn to a practical application of protective puts.
Eliot McLaire manages a Glasgow-based hedge fund that holds 100,000 shares of Relais Corporation, currently trading at €42.00.
Situation A: Before Relais Corporation’s quarterly earnings release:
Exhibit 43:
One-Month Put Options on Relais Corporation
Option A
Option B
Option C
Strike
€40.00
€42.50
€45.00
Price
€1.45
€1.72
€3.46
Delta
−0.4838
−0.5385
−0.7762
Gamma
0.0462
0.0460
0.0346
Discuss an options strategy that McLaire can implement to hedge his fund’s portfolio against a short-term decline in the share price of Relais Corp.
Solution to 1:
McLaire can purchase a protective put with the intent of selling it soon after the earnings announcement. He expects that the maximum drawdown from the current price of €42.00 will be 10%, to €37.80. This expectation narrows the choice of put options based on the following breakeven prices:
Put strike 40.0 (option A): €40.00 − €1.45 = €38.55
Put strike 42.5 (option B): €42.50 − €1.72 = €40.78
Put strike 45.0 (option C): €45.00 − €3.46 = €41.54
The put with strike price of 42.50 (option B) best fits the objective of keeping the cost of adequate protection to a minimum. This is because the 40-strike put (option A) offers limited protection because it is profitable only below €38.55, offering a profit of just €0.75 if the stock falls to €37.80. Furthermore, the 42.50-strike put offers a larger profit per unit of premium paid than the 45.00-strike put if the stock price decreases to €37.80.
Put strike 42.5: (€40.78 − €37.80)/€1.72 = 1.73
Put strike 45.0: (€41.54 − €37.80)/€3.46 = 1.08
The 42.50 strike put has a lower delta (= −0.5385) in absolute value terms than the 45.0 strike put (delta = −0.7762), but it has more gamma. If Relais Corporation’s stock falls to €37.80 per share during the life of the option, a position in the 42.50 strike put will benefit from having a larger gamma (= 0.0460) compared with the 45.0 strike put (gamma = 0.0346).
Therefore, McLaire purchases 100,000 of the 42.50-strike puts at €1.72 for a total cost of €172,000.
If Relais Corporation’s soon-to-be announced earnings miss the market’s expectations, the stock is likely to fall, thereby increasing the long put value and partially offsetting the loss on the stock. If the earnings meet market expectations, then the put may be sold at a price near its purchase price. If the earnings are better than expected and the stock price rises, then the put will decline in value. McLaire would no longer need the “insurance,” and he would sell the put position, thereby recovering part of the purchase price.
Situation B: One week later, just after Relais Corporation’s earnings release:
Exhibit 44:
23-Day Put Options on Relais
Option A
Option B
Option C
Strike
€40.00
€42.50
€45.00
Price
€0.15
€0.66
€1.85
Delta
−0.0923
−0.3000
−0.5916
Gamma
0.0218
0.0460
0.0514
Price Change
−90%
−62%
−47%
Loss on 100,000 Puts from Price Change
€130,000
€106,000
€161,000
Discuss how the strategy fared and how McLaire should proceed, assuming earnings beat the consensus estimate and Relais’s stock price rises by 5% to €44.10.
Solution to 2:
The 42.50-strike put held by McLaire has 23 days to expiration. The price has declined from €1.72 to €0.66 (−62%), for a total loss of €106,000 (= [€1.72 − €0.66] × 100,000). This is less than the loss McLaire would have incurred if he had purchased the other options. At the same time the value of the 100,000 Relais shares held by the fund has increased by €210,000 (= ($44.10 − $42.00) × 100,000). Now that the earnings announcement has been made, McLaire no longer needs the protection from the put options, so he should sell them and recover €66,000 from the original €172,000 put purchase price.
Learning Outcome
demonstrate the use of options to achieve targeted equity risk exposures
Jack Wu is a fund manager who oversees a stock portfolio valued at US$50 million that is benchmarked to the S&P 500. He expects an imminent significant correction in the US stock market and wants to profit from an anticipated jump in short-term volatility to hedge his portfolio’s tail risk.
At maturity, the options’ payoffs will depend on the settlement price of the relevant VIX futures contracts. The options will expire one month from now, and the contract size is 100.
Exhibit 40:
Options on VIX Index
Call Option
Put Option
Option Strike
15.60
14.75
Option Price
2.00
1.55
Discuss the following:
A strategy Wu can implement to hedge tail risk in his equity portfolio, by taking advantage of his expected increase in volatility while lowering his hedging cost
Profit and loss on the strategy at options expiration
Relevant issues and advantages of this strategy
Solution to 1:
Wu decides to purchase the 15.60 call on the VIX and, to partially finance the purchase, he sells an equal number of the 14.75 VIX puts. The total cost of the options strategy is 0.45 (= 2.00 − 1.55) per contract.
Solution to 2:
Exhibit 41:
Profit and Loss of the VIX Options Strategy
In this case, at the expiry the strategy will be profitable if volatility spikes up (as anticipated) and the VIX futures increase above 16.05. This is calculated as the call strike of 15.60 plus the net cost of the options (15.60 + 0.45). Above this level, the strategy will gain proportionally. In contrast, Wu’s option strategy will lose proportionally to its exposure to the short puts if the VIX futures’ settlement price is below 14.75 (put strike).
Solution to 3:
The hedge ratio that determines the number of calls to buy could be determined based on regression and scenario analysis on the portfolio’s profit and loss versus rates of increase in implied volatility during significant stock market sell-offs in the past. Of course, a risk is that past correlations may not be indicative of future correlations. Importantly, the relative advantage of implementing this long volatility hedging strategy by purchasing calls on the VIX over buying VIX futures depends on the difference in leverage available, the difference in payoff profiles (asymmetrical for options and symmetrical for futures), and the shape of the volatility futures term structure, as well as the cost of the options compared with the cost of the index futures.
Establishing or Modifying Equity Risk Exposure
In this section, we examine some examples in which investors use derivatives for establishing an equity risk exposure, for risk management, or for implementing tactical asset allocation decisions. The choice of derivative that will satisfy the investment goal depends on the outlook for the underlying asset, the investment horizon, and expectations for implied volatility over that horizon.
Long Call
Exhibit 42:
Three-Month Call Options on Markle Co. Ltd.
Option A
Option B
Option C
Strike
£58.00
£60.00
£70.00
Price
£4.00
£3.00
£0.40
Delta
0.6295
0.5227
0.1184
Gamma
0.0304
0.0322
0.0160
Discuss the option strategy that Sanchez should recommend to his clients.
Solution:
Sanchez has a bullish view because he anticipates a nearly 17% price increase in Markle shares over the next three months, from the current price of £60. He expects that implied volatilities of the options on Markle shares will stay unchanged, making the purchase of options a profitable strategy if his outlook materializes within the given timeframe.
The best strategy would be a long position in the £60 strike calls (option B). The breakeven price of the position is £63 (£60 + £3), so at option expiry, the overall position would profitable at any stock price above £63. In contrast, the breakeven price of the 58-strike call (option A) is £62 (£58 + £4) and for the 70-strike call (option C) the breakeven is £70.40 (£70 + £0.40). Therefore, Sanchez would not use option C to implement his strategy because the breakeven price is above his target price of £70/share.
Given the £60 strike call has a lower price (premium) than the £58 strike call, Sanchez’s clients can purchase more of these lower-priced options for a given investment size. Moreover, the 60-strike call offers the largest profit potential per unit of premium paid if the stock price increases to £70 (as expected).
Call strike £58: (£70 − £62)/£4 = 2.0
Call strike £60: (£70 − £63)/£3 = 2.3
Purchasing the £60 strike call (option B) is the most profitable strategy given that Sanchez’s expectations are realized. A position in the £60 strike call has a lower delta (= 0.5227) compared with the £58 strike call (delta = 0.6295), so at current prices of the underlying, the change in value of the £60 strike call is lower. If Markle’s stock reaches £70 per share during the life of the option, however, the £60 strike call will benefit from having a larger gamma (= 0.0322) compared with the £58 strike call (gamma = 0.0304).
Risk Management: Protective Put Position
Investors use protective puts, collars, and equity swaps against a long stock position to hedge market risk. Here we turn to a practical application of protective puts.
Eliot McLaire manages a Glasgow-based hedge fund that holds 100,000 shares of Relais Corporation, currently trading at €42.00.
Situation A: Before Relais Corporation’s quarterly earnings release:
Exhibit 43:
One-Month Put Options on Relais Corporation
Option A
Option B
Option C
Strike
€40.00
€42.50
€45.00
Price
€1.45
€1.72
€3.46
Delta
−0.4838
−0.5385
−0.7762
Gamma
0.0462
0.0460
0.0346
Discuss an options strategy that McLaire can implement to hedge his fund’s portfolio against a short-term decline in the share price of Relais Corp.
Solution to 1:
McLaire can purchase a protective put with the intent of selling it soon after the earnings announcement. He expects that the maximum drawdown from the current price of €42.00 will be 10%, to €37.80. This expectation narrows the choice of put options based on the following breakeven prices:
Put strike 40.0 (option A): €40.00 − €1.45 = €38.55
Put strike 42.5 (option B): €42.50 − €1.72 = €40.78
Put strike 45.0 (option C): €45.00 − €3.46 = €41.54
The put with strike price of 42.50 (option B) best fits the objective of keeping the cost of adequate protection to a minimum. This is because the 40-strike put (option A) offers limited protection because it is profitable only below €38.55, offering a profit of just €0.75 if the stock falls to €37.80. Furthermore, the 42.50-strike put offers a larger profit per unit of premium paid than the 45.00-strike put if the stock price decreases to €37.80.
Put strike 42.5: (€40.78 − €37.80)/€1.72 = 1.73
Put strike 45.0: (€41.54 − €37.80)/€3.46 = 1.08
The 42.50 strike put has a lower delta (= −0.5385) in absolute value terms than the 45.0 strike put (delta = −0.7762), but it has more gamma. If Relais Corporation’s stock falls to €37.80 per share during the life of the option, a position in the 42.50 strike put will benefit from having a larger gamma (= 0.0460) compared with the 45.0 strike put (gamma = 0.0346).
Therefore, McLaire purchases 100,000 of the 42.50-strike puts at €1.72 for a total cost of €172,000.
If Relais Corporation’s soon-to-be announced earnings miss the market’s expectations, the stock is likely to fall, thereby increasing the long put value and partially offsetting the loss on the stock. If the earnings meet market expectations, then the put may be sold at a price near its purchase price. If the earnings are better than expected and the stock price rises, then the put will decline in value. McLaire would no longer need the “insurance,” and he would sell the put position, thereby recovering part of the purchase price.
Situation B: One week later, just after Relais Corporation’s earnings release:
Exhibit 44:
23-Day Put Options on Relais
Option A
Option B
Option C
Strike
€40.00
€42.50
€45.00
Price
€0.15
€0.66
€1.85
Delta
−0.0923
−0.3000
−0.5916
Gamma
0.0218
0.0460
0.0514
Price Change
−90%
−62%
−47%
Loss on 100,000 Puts from Price Change
€130,000
€106,000
€161,000
Discuss how the strategy fared and how McLaire should proceed, assuming earnings beat the consensus estimate and Relais’s stock price rises by 5% to €44.10.
Solution to 2:
The 42.50-strike put held by McLaire has 23 days to expiration. The price has declined from €1.72 to €0.66 (−62%), for a total loss of €106,000 (= [€1.72 − €0.66] × 100,000). This is less than the loss McLaire would have incurred if he had purchased the other options. At the same time the value of the 100,000 Relais shares held by the fund has increased by €210,000 (= ($44.10 − $42.00) × 100,000). Now that the earnings announcement has been made, McLaire no longer needs the protection from the put options, so he should sell them and recover €66,000 from the original €172,000 put purchase price.
This reading on options strategies shows a number of ways in which market participants might use options to enhance returns or to reduce risk to better meet portfolio objectives. The following are the key points.
Buying a call and writing a put on the same underlying with the same strike price and expiration creates a synthetic long position (i.e., a synthetic long forward position).
Writing a call and buying a put on the same underlying with the same strike price and expiration creates a synthetic short position (i.e., a synthetic short forward position).
A synthetic long put position consists of a short stock and long call position in which the call strike price equals the price at which the stock is shorted.
A synthetic long call position consists of a long stock and long put position in which the put strike price equals the price at which the stock is purchased.
Delta is the change in an option’s price for a change in price of the underlying, all else equal.
Gamma is the change in an option’s delta for a change in price of the underlying, all else equal.
Vega is the change in an option’s price for a change in volatility of the underlying, all else equal.
Theta is the daily change in an option’s price, all else equal.
A covered call, in which the holder of a stock writes a call giving someone the right to buy the shares, is one of the most common uses of options by individual investors.
Covered calls can be used to change an investment’s risk–reward profile by effectively enhancing yield or reducing/exiting a position when the shares hit a target price.
A covered call position has a limited maximum return because of the transfer of the right tail of the return distribution to the option buyer.
The maximum loss of a covered call position is less than the maximum loss of the underlying shares alone, but the covered call carries the potential for an opportunity loss if the underlying shares rise sharply.
A protective put is the simultaneous holding of a long stock position and a long put on the same asset. The put provides protection or insurance against a price decline.
The continuous purchase of protective puts maintains the upside potential of the portfolio, while limiting downside volatility. The cost of the puts must be carefully considered, however, because this activity may be expensive. Conversely, the occasional purchase of a protective put to deal with a bearish short-term outlook can be a reasonable risk-reducing strategy.
The maximum loss with a protective put is limited because the downside risk is transferred to the option writer in exchange for the payment of the option premium.
With an option spread, an investor buys one option and writes another of the same type. This approach reduces the position cost but caps the maximum payoff.
A bull spread expresses a bullish view on the underlying and is normally constructed by buying a call option and writing another call option with a higher exercise price (both options have same underlying and same expiry).
A bear spread expresses a bearish view on the underlying and is normally constructed by buying a put option and writing another put option with a lower exercise price (both options have same underlying and same expiry).
With either a bull spread or a bear spread, both the maximum gain and the maximum loss are known and limited.
A long (short) straddle is an option combination in which the investor buys (sells) puts and calls with the same exercise price and expiration date. The long (short) straddle investor expects increased (stable/decreased) volatility and typically requires a large (small/no) price movement in the underlying asset in order to make a profit.
A collar is an option position in which the investor is long shares of stock and simultaneously writes a call with an exercise price above the current stock price and buys a put with an exercise price below the current stock price. A collar limits the range of investment outcomes by sacrificing upside gain in exchange for providing downside protection.
A long (short) calendar spread involves buying (selling) a long-dated option and writing (buying) a shorter-dated option of the same type with the same exercise price. A long (short) calendar spread is used when the investment outlook is flat (volatile) in the near term but greater (lesser) return movements are expected in the future.
Implied volatility is the expected volatility an underlying asset’s return and is derived from an option pricing model (i.e., the Black–Scholes–Merton model) as the value that equates the model price of an option to its market price.
When implied volatilities of OTM options exceed those of ATM options, the implied volatility curve is a volatility smile. The more common shape is a volatility skew, in which implied volatility increases for OTM puts and decreases for OTM calls, as the strike price moves away from the current price.
The implied volatility surface is a 3-D plot, for put and call options on the same underlying, showing expiration time (x-axis), strike prices (y-axis), and implied volatilities (z-axis). It simultaneously displays volatility skew and the term structure of implied volatility.
Options, like all derivatives, should always be used in connection with a well-defined investment objective. When using options strategies, it is important to have a view on the expected change in implied volatility and the direction of movement of the underlying asset.
When option prices are compared within or across asset classes or relative to their historical values, they are assessed by their implied volatility. shows a comparison of the one-month annualized (ATM) implied volatility at a given point in time for options across three European equity indexes (Euro Stoxx 50, FTSE MIB, and DAX). The DAX shows the lowest implied volatility among the three indexes.
plots implied volatility (y-axis) against strike price (x-axis) for options on the same underlying, the FTSE MIB (trading at 19,000), with the same expiration. When the implied volatilities priced into both OTM puts and calls trade at a premium to implied volatilities of ATM options (those with strike price at 19,000), the curve is U-shaped and is called a volatility smile, because it resembles the shape of a smile.
To better understand how to measure the volatility skew, consider , which shows the levels of implied volatility at different degrees of moneyness for options expiring in three months on equity indexes where liquid derivatives markets exist.
For most asset classes, the level of option skew varies over time. presents the skew on the S&P 500, measured as the difference between the implied volatilities of options with 90% (puts) and 110% (calls) moneyness, and with three months to expiration.
shows, for options on the S&P 500, a common indicator watched by market participants: the spread between the implied volatilities of 12-month and 3-month ATM options. Values below zero indicate that the term structure is inverted with 3-month options having a higher implied volatility than the 12-month options. In periods when equity markets experience large sell-offs, such as during the 2007–2009 global financial crisis, the term structure of implied volatility typically shows significant inversion. In such periods, the general level of equity volatility and skew also remain high.
Investors use derivatives to achieve a target exposure based on their outlook for the underlying asset. shows one way of looking at the interplay of direction and volatility. The strategies identified are most profitable given the expected change in implied volatility and the outlook for the direction of movement of the underlying asset.
Parker’s account contains 5,000 shares of Manzana (MNZA) stock, a stock that she is considering selling in the near future. Rivera’s firm has a bearish market outlook for MNZA shares over the next six months. Rivera reviews information on the 44-day exchange-listed options, which expire in May (shown in ). He is considering writing MNZA calls, which will accomplish two objectives. First, the sale of calls will generate the required cash for his client. Secondly, the sale will reduce the delta of Parker’s account in line with his firm’s bearish short-term outlook for MNZA shares. The current delta of Parker’s MNZA position is 5,000(+1) or +5,000. contains call and put price information for May MNZA options with strike prices close to the current market price of MNZA shares (S0 = $169).
If Rivera were to write the MAY 180 calls, doing so would not generate the required cash, $2.50 × 100 × 50 = $12,500. Writing the MAY 165 calls would generate more than enough income: $9.10 × 100 × 50 = $45,500. However, the May 165 call is in the money. Although the firm’s outlook is bearish for the shares, Rivera feels there is a high likelihood of Parker’s MNZA shares being called away at expiration. Writing the May 175 call would generate only $4.03 × 100 × 50 = $20,150. Although Rivera believes the 175 call would not be exercised, the cash generated would be only about two-thirds of the client’s projected need. The May 170 call would generate $6.45 × 100 × 50 = $32,250, which looks good from a cash-generation perspective. The current price of MNZA shares is 169, so there is a considerable risk that MNZA shares will sell above that level at expiration. If MNZA stock trades above the strike price at the May expiration, then Parker would be exposed to having her shares called at 170. Given the firm’s bearish outlook for MNZA shares and, as stated previously, “Parker is considering selling the stock in the near future,” this risk might be acceptable. Using the May 170 call option data from , the delta of Parker’s MNZA position would be reduced from +5,000 to 5,000 × (+1 − 0.504) = +2,480.23 The profit graph for the recommended sale of 50 May 170 call contracts against Parker’s 5,000 share long position at 169 is shown in .
The $32,250 generated by the call option sales is Parker’s to keep. There are two risks, however, that Rivera should point out to Parker using . The first risk is that his outlook for MNZA shares might be incorrect. The reduced delta may cost his client a potential gain on the long position. If MNZA shares increase above 170, the profits will go to the call owner. Parker’s MNZA shares may be called away at 170, limiting her profit to the option premium of $32,250 plus the $5,000 from selling her MNZA shares at a profit of $1 (= $170 − $169) as shown in .24 The second risk is that to write the covered call, Parker must continue to hold 5,000 MNZA shares. If the firm’s bearish outlook is correct, the shares may drop in value during the next month, resulting in a loss on the long stock position. The loss would be cushioned somewhat by the 6.45 call premium, but a drop of more than 3.8% (below 162.55 = 169 − 6.45; 6.45/169 = 3.8%) results in an overall loss on the position. After Rivera explains the risks to Parker, she elects to write the MNZA 170 calls.
Quintera can write OTM puts to effectively “get paid” to buy the stock. He sells puts and the firm keeps the cash regardless of what happens in the future. If the stock is above the exercise price at expiration, the put options will not be exercised. Otherwise, the option is exercised, Quintera purchases the stock and, as desired, Tres Jotas becomes an owner of the stock. shows the options information for 44-day MAY put options. Because his target price is 165, Quintera writes 500 May MNZA 165 put contracts and receives premium income of 500 × 100 × $5.30 = $265,000. The company keeps these funds regardless of future stock price movements. But, the firm is obligated to buy stock at $165 if the put holder chooses to exercise. By writing the puts, Quintera has established a bullish position in MNZA stock. The delta of this MNZA position is −500 × 100 × −0.384 = +19,200, the equivalent of a long position in 19,200 MNZA shares. The portfolio profit on the short put is shown in .
The stock is $163 per share on the option expiration day. With an exercise price of 165, the put is in the money and will be exercised. Quintera will be assigned to buy 50,000 shares at the exercise price of 165. The cost is 50,000 × $165 = $8,250,000. Quintera is satisfied with the outcome, because the firm keeps the premium income of $265,000, so the net cost of purchase is $8,250,000 − $265,000 = $7,985,000. On 50,000 shares, this means the effective purchase price is $7,985,000/50,000 = $159.70, which is below the maximum $165 price Quintera was willing to pay. If the price of MNZA shares drops below 165, the effective purchase price will always be X − p = 165 − 5.30 = $159.70. This is the breakeven point for the short put position. At prices below the breakeven amount, Quintera would have been better off not writing the put and just buying MNZA shares outright. For example, if the MNZA price fell to $150, Quintera would have been obligated to buy shares at 165, $15 more than the market price. When the $5.30 premium is considered, the $15 difference would amount to a loss of $9.70 per share or $485,000 which can be seen in .
The stock price is $177 on the option expiration day. With an exercise price of 165.00, the MNZA puts are out of the money and would not be exercised. Tres Jotas keeps the $265,000 premium received from writing the option. This approach adds to the company’s profitability, but Tres Jotas did not acquire the MNZA shares and experienced an opportunity cost relative to an outright purchase of the stock at $169. Any price above 165, will result in earning the premium of $265,000, as can be seen in .
Katrina Hamlet has been following Manzana stock for the past year. She anticipates the announcement of a major new product soon, but she is not sure how the critics will react to it. If the new product is praised, she believes the stock price will increase dramatically. If the product does not impress, she believes the share price will fall substantially. Hamlet has been considering trading around the event with a straddle. The stock is currently priced at $169.00, and she is focused on close-to-the-money (170) calls and puts selling for 6.45 and 7.69, respectively. Her initial strategy is presented as .
Hamlet is betting on a substantial price movement in the underlying MNZA shares to make money with this trade. That price movement, up or down, must be large enough to recover the two premiums paid. In her earlier planning, that total was $6.45 + $7.69 = $14.14. She expects at least a 10% price movement, which on a stock selling for $169.00 would be an increase of $16.90. This price movement would be sufficient to recover the $14.14 cost of the straddle and make her strategy profitable. The breakeven points were $155.86 and $184.14, as shown in .
In the over-the-counter market, Steinbacher might buy a put and then write an out-of-the money call. This strategy is a collar. The put provides downside protection below the put exercise price, and the call generates income to help offset the cost of the put. The investor decides the strike prices of the put and the call to achieve a specific level of downside protection, while still keeping some benefit from an increase in the stock price. When the strike of the call is set so that the call premium (to be received) exactly offsets the put premium (to be paid), then the position is called a “zero-cost collar.” Recalling and the underlying return distribution, this strategy effectively sells the right tail of the distribution, which represents potential large gains, in exchange for eliminating the left tail, which represents potential large losses.
Ivanka Dubois is a professional advisor to high-net-worth investors. She expects little price movement in the Euro Stoxx 50 in the next three months but has a bearish long-term outlook. The consensus sentiment favoring a flat market shows no signs of changing over the next few months, and the Euro Stoxx 50 is currently trading at 3500. shows prices for two put options with strike price of 3500 that are available on the index. Both options have the same implied volatility.
The Euro Stoxx 50 is still trading at 3500 as expected. The three-month put option expires worthless, and the original longer-term six-month option, which now has three months remaining to expiration, is worth €119 (because the implied volatility has remained constant). The total cost of the calendar spread was €54, and Dubois can sell the remaining put to a dealer for a profit of €65 (given by €119 − €54). As can be seen in , which shows the profit and loss diagram for the calendar spread at the time of the expiration of the three-month put, this corresponds to the level of maximum payoff for this strategy.
The Euro Stoxx 50 has decreased and is trading at 3000, and the implied volatility has significantly increased. Dubois must pay €500 (€3,500 − €3,000) to settle the (short) three-month put option at expiration. The (long) put option with three months remaining to expiration is deep in the money and, assuming volatility has increased and €30 of time value, it is worth €530 (given by Intrinsic Value of €500 + €30 of Time Value). If Dubois sells this put to a dealer, she will realize a loss of €24 (= €530 − €500 − €54) on the put calendar spread. adds the profit and loss diagram for the calendar spread at the time of the expiration of the three-month put, assuming that implied volatility of the six-month put has significantly increased.
The VIX Index is currently at 14.87, and the front-month VIX futures trades at 15.60. Wu observes the quotes shown in for options on the VIX (these options have same implied volatility). It is important to note that VIX option prices reflect the VIX futures prices. Given that the VIX futures trade at 15.60 while the spot VIX is 14.87, the call is at the money while the put is out of the money by 5.45% (= [14.75 − 15.60]/15.60).
At maturity, the options’ payoffs will depend on the settlement price of the relevant VIX futures. shows the profit and loss diagram for the option strategy at the time of the options’ expiration. In particular, the horizontal axis shows the values corresponding to the relevant VIX futures contracts.
Armando Sanchez is a private wealth advisor working in London. He expects the shares of Markle Co. Ltd. will move from the current price of £60 a share to £70 a share over the next three months, thanks to an increase of positive news flows regarding the company’s new fintech services. He also expects that the implied volatilities of options on Markle’s stock will stay almost unchanged over the same period. Prices for three-month call options on the stock are shown in (note that each call contract represents one share). For his high-net-worth clients whose investment policy statements allow the use of derivatives, Sanchez plans to recommend that they purchase the call option that, based on the budget they intend to spend for implementing the strategy, would maximize profits if the stock price increases to £70 a share or more over the next three months.
Relais has a quarterly earnings announcement scheduled in one week. Although McLaire expects an earnings increase, he believes the company will miss the consensus earnings estimate, in which case he expects that the maximum drawdown from the current price of €42.00 would be 10%. He would like to protect the fund’s position in the company for several days around the earnings announcement while keeping the cost of the protection to a minimum. provides information on options prices for Relais Corporation. Note that each put contract represents one share.
McLaire holds the 100,000 puts with the exercise price of €42.50. Seven days have passed since the options’ purchase; Relais has just released its earnings, and they turn out to be surprisingly good. Earnings beat the consensus estimate, and immediately after the announcement the stock price rises by 5% to €44.10. shows the options’ prices on the day of the announcement, after the stock price has increased to €44.10 (the implied volatility remains unchanged).
The VIX Index is currently at 14.87, and the front-month VIX futures trades at 15.60. Wu observes the quotes shown in for options on the VIX (these options have same implied volatility). It is important to note that VIX option prices reflect the VIX futures prices. Given that the VIX futures trade at 15.60 while the spot VIX is 14.87, the call is at the money while the put is out of the money by 5.45% (= [14.75 − 15.60]/15.60).
At maturity, the options’ payoffs will depend on the settlement price of the relevant VIX futures. shows the profit and loss diagram for the option strategy at the time of the options’ expiration. In particular, the horizontal axis shows the values corresponding to the relevant VIX futures contracts.
Armando Sanchez is a private wealth advisor working in London. He expects the shares of Markle Co. Ltd. will move from the current price of £60 a share to £70 a share over the next three months, thanks to an increase of positive news flows regarding the company’s new fintech services. He also expects that the implied volatilities of options on Markle’s stock will stay almost unchanged over the same period. Prices for three-month call options on the stock are shown in (note that each call contract represents one share). For his high-net-worth clients whose investment policy statements allow the use of derivatives, Sanchez plans to recommend that they purchase the call option that, based on the budget they intend to spend for implementing the strategy, would maximize profits if the stock price increases to £70 a share or more over the next three months.
Relais has a quarterly earnings announcement scheduled in one week. Although McLaire expects an earnings increase, he believes the company will miss the consensus earnings estimate, in which case he expects that the maximum drawdown from the current price of €42.00 would be 10%. He would like to protect the fund’s position in the company for several days around the earnings announcement while keeping the cost of the protection to a minimum. provides information on options prices for Relais Corporation. Note that each put contract represents one share.
McLaire holds the 100,000 puts with the exercise price of €42.50. Seven days have passed since the options’ purchase; Relais has just released its earnings, and they turn out to be surprisingly good. Earnings beat the consensus estimate, and immediately after the announcement the stock price rises by 5% to €44.10. shows the options’ prices on the day of the announcement, after the stock price has increased to €44.10 (the implied volatility remains unchanged).
