1. Laurence Booth Sean Cleary

2. LEARNING OBJECTIVES Options 12 12.1 Describe the basic nature of call options and the factors that influence their value. 12.2 Describe the basic nature of put options and the payoffs associated with long and short positions in put options. 12.3 Explain how to use put-call parity to estimate call and put prices, and explain how it can be used to synthetically create call, put, and underlying positions. 12.4 Explain how to use the Black-Scholes option pricing model to price call options. 12.5 Explain how options are traded and what is meant by implied volatility. 12.6 Understand by means of the simplest option pricing model what factors affect the value of a call option and provide a guide to their order of magnitude.

3. 12.1 CALL OPTIONS Call options give the holder the right, but not the obligation, to buy an underlying asset at a fixed price for a specified time The price at which an investor can buy the underlying asset is called the exercise or strike price and the 3© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition last date the option can be converted or exercised is the expiration date Example: A call option on an underlying asset has a $50 exercise price; the payoff for the buyer of the call is depicted in Figure 12-1

4. Example: A call option on an underlying asset has a $50 exercise price. The payoff for the buyer of the call is depicted in Figure 12-1. This option is in the money, or would generate a 4© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition positive payoff if exercised today, for any underlying asset price above $50 This option is out of the money for any underlying asset price below $50 When the underlying asset price equals the strike price an option is at the money 12.1 CALL OPTIONS

5. An investor who buys a call option takes the long position and has the right but not the obligation to exercise the option if it is in the money The counterparty is the option writer who takes the short position; if the 5© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition option owner exercised the option, it would be exercised against the writer who would have to sell the underlying asset to the holder of the call option for the strike price (even though the strike would be less than the market value of the underlying asset if the option were in the money) The payoff diagram for an option writer is given in Figure 12-2 12.1 CALL OPTIONS

6. Example: A call option on an underlying asset has a $50 exercise price. Consider the following table showing the payoffs to both the holder and the writer of the option and different asset prices 6© John Wiley & Sons Canada, Ltd. Asset Price ($)304050556070 Call holder’s payoff ($)00051020 Call writer’s payoff ($)000-5-10-20 Notice that when the underlying asset price < the strike price, the option is not exercised Notice also that the net payoff between the holder and the writer is zero in aggregate because options are an example of a zero sum game Booth Cleary – 3rd Edition 12.1 CALL OPTIONS

7. Call Option Values The intrinsic value (IV) is the value of the option at expiration and is positive when the option is in the money and zero when it is out of the money At expiration, when the call is in the money, the value of the option is the asset price minus the exercise price, S – X At expiration, when the call is out of the money, the value is zero because it will expire unexercised 7© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition 12.1 CALL OPTIONS

8. Call Option Values Equation 12-1: IV(Call) = Max (S – X, 0) The option premium is the market value of the option, and the time value (TV) is the difference between the option premium and intrinsic value Equation 12-2: Option premium = IV + TV Equation 12-3: TV = Option premium – IV Note: options that are deep are so far in (or out of) the money that they are almost certain to be (or not to be) exercised 8© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition 12.1 CALL OPTIONS

9. Figure 12-3 shows that the call option’s value is non- linearly related to the underlying asset price Table 12-1 shows how various factors affect call and put prices 9© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition 12.1 CALL OPTIONS

10. 12.2 PUT OPTIONS Put options give the holder the right, but not the obligation, to sell an underlying asset at a fixed price for a specified time The price at which an investor can buy the underlying asset is called the exercise or strike price and the last date the option can be converted or exercised is the expiration date Example: A put option on an underlying asset has a $50 exercise price; the payoff for the buyer of the put is depicted in Figure 12-4 and the payoff for the writer of the put is depicted in Figure 12-5 10© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition

11. 11© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition 12.2 PUT OPTIONS

12. Example: A put option on an underlying asset has a $50 exercise price. Consider the following table showing the payoffs to both the holder and the writer of the option and different asset prices 12© John Wiley & Sons Canada, Ltd. Asset Price ($)304050556070 Put holder’s payoff ($)20100000 Put writer’s payoff ($)-20-100000 Notice that when the underlying asset price > the strike price, the option is not exercised Notice also that the net payoff between the holder and the writer is zero because in aggregate options are an example of a zero-sum game Booth Cleary – 3rd Edition 12.2 PUT OPTIONS

13. The intrinsic value of a put option is the strike price minus the underlying asset price X – S when it is in the money, and zero when it is out of the money Equation 12-4: IV(Put) = Max(X – S, 0) The time value and option premium for put options is calculated in the same manner as for call options (see above) 13© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition 12.2 PUT OPTIONS

14. There is a major difference between put and call options that can only be exercised on the expiration date and those that can be exercised at any time before the expiration date European options can only be exercised at maturity American options can be exercised at any time up to and including the expiration date The distinction was originally geographic, but no longer describes anything but the presence of absence of the ability to exercise the option prior to the expiration date 14© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition 12.2 PUT OPTIONS

15. 12.3 PUT-CALL PARITY There are four basic option positions: long call, short call, long put, short put These can be combined as shown in Figure 12-6: 15© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition

16. Put-call parity is the relationship between the price of a call option and a put option that have the same strike price and expiry dates; it assumes that the options are not exercised before expiration Consider two portfolios: –Portfolio A has a long put (P) with X = $50 and the underlying asset (S) –Portfolio B has a long call (C) with X = $50 and has invested the present value of the exercise price, PV(X) in the risk free asset paying interest at RF –Assume all options are European (cannot be exercised prior to maturity), and that the underlying asset does not pay dividends or other income 16© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition 12.3 PUT-CALL PARITY

17. Table 12-2 shows the payoffs from these two portfolios if the underlying asset price is either $55 or $45. Do you notice anything? 17© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition 12.3 PUT-CALL PARITY

18. Table 12-3 shows that the payoff from either strategy will always be the same. 18© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition 12.3 PUT-CALL PARITY

19. If the put premium is P, the call premium is C, the strike price is X and the underlying asset price is S, then equation 12-5 gives put-call parity: P + S = C + PV(X) Rearranging equation 12-5 gives equation 12-6, the basic put-call relationship: C – P = S – PV(X) We can also use equations 12-7,12-8 and 12-9 to show how put-call parity can be used to solve for the call and put premiums, and the underlying asset price, respectively: C = P + S – PV(X) P = C – S + PV(X) S = C – P + PV(X) 19© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition 12.3 PUT-CALL PARITY

20. Equation 12-9 shows that a long position in the underlying asset gives the same payoffs as a portfolio with a long call, a short put and an investment of PV(X) in the risk-free asset S = C – P + PV(X) Equivalently, a short position in the underlying asset gives the same payoffs as a portfolio with a short call, a long put and borrowing PV(X) at the risk-free rate – S = P – C – PV(X) We will consider the construction of three types of synthetic positions: –Protective Puts –Covered Calls –Collars 20© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition 12.3 PUT-CALL PARITY

21. A protective put is a long position in both a put option and the underlying asset The investor benefits from the underlying asset if its price appreciates, and 21© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition the put option establishes a price floor for the underlying asset limiting potential loss The net payoff position of a protective put resembles a long call position, as shown in Figure 12-7 12.3 PUT-CALL PARITY

22. A covered call is a long position in the underlying asset and a short position (writing) in a call option The investor benefits from the underlying asset if its price appreciates, and also 22© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition gets to pocket the call premium The investor loses if the underlying asset falls in price. The net payoff position of a covered call resembles a short put position, as shown in Figure 12-8 12.3 PUT-CALL PARITY

23. A collar is a position between a floor and ceiling price established by buying both the underlying asset and a put option and selling a call option to finance the purchase of the put option Collars are a type of self-financing portfolio insurance The net payoff position of a collar is shown in Figure 12-9: 23© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition 12.3 PUT-CALL PARITY

24. 12.4 OPTION PRICING Equation 12-10 is the Black-Scholes model for valuing European call options on non-dividend paying stocks: Based on put-call parity, the price of a corresponding put option is: 24© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition

25. Use software of a table of cumulative normal probabilities, such as Table 12-4 to calculate N[d 1 ], which is the cumulative probability of the option being in the money at expiration Notice that the more risky the underlying asset (the larger the standard deviation), the larger N[d 1 ] and the moneyness of the call 25© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition 12.4 OPTION PRICING

26. Example: Find the value of a three-month call option with a $20 strike price if the price of the non-dividend paying underlying asset is $20.50 and its standard deviation is 25%. Assume the risk-free rate is 5%. S = $20.50, X = $20, r = 5%, t = 3/12 = 0.25, σ = 25% 26© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition 12.4 OPTION PRICING

27. The Greeks indicate the sensitivity of the option price to changes in the underlying parameters Delta (Δ) is the change in the price of an option for a given change in the price of the underlying asset Theta (θ) is the change in the option value with time Gamma (γ) is the change in delta with respect to a change in the underlying asset; i.e., the rate of change in the rate of change in the price of an option for a given change in the price of the underlying asset Rho (ρ) is the change in the option value with respect to a change in the interest rate Vega is the change in the option value with respect to a change in the volatility of the underlying asset 27© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition 12.4 OPTION PRICING

28. Options are traded over the counter, mainly by major financial institutions, and on organized exchanges like the Montreal Exchange (MX) in Canada. Options on all stocks comprising the S&P/TSX 60 Index and the S&P/TX MidCap Index trade on the MX. Table 12-5 shows quotations for options (c = call, p = put) on Teck common shares. The ask is the price a seller wants, while the bid is what a buyer is prepared to pay. The last price is the value of the last transaction. The volume shows the number of contracts traded and open (the open interest) is the total number of contracts outstanding. 28© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition 12.5 OPTIONS MARKET

29. Implied volatility is an estimate of the price volatility of the underlying asset based on observed option prices. The other determinants of option prices are observable, but forecasts of price volatility are not and so must be implied by option prices. Figure 12-11 shows the implied volatilities of the S&P/TSX 60 Index over time. Implied volatility tends to increase when investors are less certain about the future of the underlying asset’s price. 29© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition 12.5 OPTIONS MARKET

30. Appendix 12A BINOMIAL OPTION PRICING AND RISK-NEUTRAL PROBABILITIES While the Black-Scholes model is a continuous time model, the binomial model is an option pricing model that uses discrete time and only two future states of the world (up and down) Example: Figure 12A-1 shows a call option with a $50 strike and $50 current underlying asset value. There is a 60% probability the underlying asset’s price increases to $55 next period, and a 40% chance it falls to $45. If the price increases to $55, the call option will have a $5 payoff. If the price falls, the option will expire worthless. 30© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition

31. The hedge ratio (h) is the number of calls an investor must sell to hedge a long position in the underlying asset, and is given by equation 12A-1: where PU = price up or the price in the up-state PD = price down or the price in the down-state X = strike price Example: In the previous example, X = $50, PU = $55, PD = $45: Therefore an investor must sell two calls to hedge every long position in the underlying asset 31© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition Appendix 12A BINOMIAL OPTION PRICING AND RISK-NEUTRAL PROBABILITIES

32. Equation 12A-2 gives the value of the call option, where h is the hedge ratio, S is the underlying asset price, PD is the price in the down-state and r is the risk-free rate: Example: In our example, X = $50, PU = $55, PD = $45, h = 2 and suppose the risk-free rate is 1%. 32© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition Appendix 12A BINOMIAL OPTION PRICING AND RISK-NEUTRAL PROBABILITIES

33. Unlike direct approaches to valuation, where we estimate expected cash flow on the call and then use a risk-adjusted discount rate, here we use a hedge to generate a risk-free payoff discounted at the risk-free rate This is an indirect approach to valuation. We can also observe that the call option will be worth more if: –The price of the underlying asset increases –The risk-free rate increases –The risk of the asset increases –The strike price increases –The up-price increases These observations are consistent with the impact of these variables discussed earlier, but time to expiration cannot be included because the binomial model used here is a single period model 33© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition Appendix 12A BINOMIAL OPTION PRICING AND RISK-NEUTRAL PROBABILITIES

34. Risk neutral is the state of ignoring the risk involved in determining expected rates of return. Risk-neutral probabilities are derived and ensure that the asset price goes up with the risk-free rate. These are not the true or actual probabilities of the asset price increasing or decreasing, but rather the probabilities that would exist if the investor were risk neutral instead of risk averse. Since the call option is valued as if the investor were risk-neutral, we can use risk-neutral probabilities to determine the expected payoff on the call and value it directly. 34© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition Appendix 12A BINOMIAL OPTION PRICING AND RISK-NEUTRAL PROBABILITIES

35. Example: As before, there is a 60% the price increases from $50 to $55 and a 40% chance it falls to $45. The risk-free rate is 1%. We can calculate the probabilities that will generate a 1% return on the asset given values of either $55 or $45 next period. 50(1.01) = 55P + 45(1 – P), or P = (50.5 – 45) / 10 = 0.55 So, there is a 55% chance of increasing to $55 and a 45% chance of getting zero payoff from the option The expected payoff is: 0.45 ($0) + 0.55 ($5) = $2.75 We can now value the option directly by discounting the expected payoff next period of $2.75 by 1%: –Value of call option = $2.75 / 1.01 = $2.72 35© John Wiley & Sons Canada, Ltd.Booth Cleary – 3rd Edition Appendix 12A BINOMIAL OPTION PRICING AND RISK-NEUTRAL PROBABILITIES

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