Introduction to Binomial Trees

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1 11 C H A P T E R Introduction to Binomial Trees A useful and very popular technique for pricing an option involves constructing a binomial tree. This is a diagram that represents di erent possible paths that might be followed by the stock price over the life of the option. In this chapter we will take a rst look at binomial trees and their relationship to an important principle known as riskneutral valuation. The general approach adopted here is similar to that in an important paper published by Cox, Ross, and Rubinstein in The material in this chapter is intended to be introductory. Numerical procedures involving binomial trees are discussed in more detail in Chapter A ONE-STEP BINOMIAL MODEL We start by considering a very simple situation. A stock price is currently \$20, and it is known that at the end of three months it will be either \$22 or \$18. We are interested in valuing a European call option to buy the stock for \$21 in three months. This option will have one of two values at the end of the three months. If the stock price turns out to be \$22, the value of the option will be \$1; if the stock price turns out to be \$18, the value of the option will be zero. The situation is illustrated in Figure It turns out that a relatively simple argument can be used to price the option in this example. The only assumption needed is that arbitrage opportunities do not exist. We set up a portfolio of the stock and the option in such a way that there is no uncertainty about the value of the portfolio at the end of the three months. We then argue that, because the portfolio has no risk, the return it earns must equal the risk-free interest rate. This enables us to work out the cost of setting up the portfolio and therefore the option's price. Because there are two securities (the stock and the stock option) and only two possible outcomes, it is always possible to set up the riskless portfolio. Consider a portfolio consisting of a long position in shares of the stock and a short position in one call option. We calculate the value of that makes the portfolio riskless. If the stock price moves up from \$20 to \$22, the value of the shares is 22 and the value of the option is 1, so that the total value of the portfolio is If the stock price moves down from \$20 to \$18, the value of the shares is 18 and the value of the option is zero, so that the total value of the portfolio is 18. The portfolio is 247

2 248 CHAPTER 11 Figure 11.1 Stock price movements in numerical example Stock price = \$22 Option price = \$1 Stock price = \$20 Stock price = \$18 Option price = \$0 riskless if the value of is chosen so that the nal value of the portfolio is the same for both alternatives. This means 22 1 ˆ 18 or ˆ 0:25 A riskless portfolio is therefore Long: 0.25 shares Short: 1 option If the stock price moves up to \$22, the value of the portfolio is 22 0:25 1 ˆ 4:5 If the stock price moves down to \$18, the value of the portfolio is 18 0:25 ˆ 4:5 Regardless of whether the stock price moves up or down, the value of the portfolio is always 4.5 at the end of the life of the option. Riskless portfolios must, in the absence of arbitrage opportunities, earn the risk-free rate of interest. Suppose that in this case the risk-free rate is 12% per annum. It follows that the value of the portfolio today must be the present value of 4.5, or 4:5e 0:123=12 ˆ 4:367 The value of the stock price today is known to be \$20. Suppose the option price is denoted by f. The value of the portfolio today is It follows that or 20 0:25 f ˆ 5 f 5 f ˆ 4:367 f ˆ 0:633 This shows that, in the absence of arbitrage opportunities, the current value of the option must be If the value of the option were more than 0.633, the portfolio would cost less than to set up and would earn more than the risk-free rate. If the

3 Introduction to Binomial Trees 249 value of the option were less than 0.633, shorting the portfolio would provide a way of borrowing money at less than the risk-free rate. A Generalization We can generalize the argument just presented by considering a stock whose price is S 0 and an option on the stock whose current price is f. We suppose that the option lasts for time T and that during the life of the option the stock price can either move up from S 0 to a new level, S 0 u, or down from S 0 to a new level, S 0 d (u > 1; d < 1). The proportional increase in the stock price when there is an up movement is u 1; the proportional decrease when there is a down movement is 1 d. If the stock price moves up to S 0 u, we suppose that the payo from the option is f u ; if the stock price moves down to S 0 d, we suppose the payo from the option is f d. The situation is illustrated in Figure As before, we imagine a portfolio consisting of a long position in shares and a short position in one option. We calculate the value of that makes the portfolio riskless. If there is an up movement in the stock price, the value of the portfolio at the end of the life of the option is S 0 u f u If there is a down movement in the stock price, the value becomes The two are equal when or S 0 d f d S 0 u f u ˆ S 0 d f d ˆ fu f d S 0 u S 0 d 11:1 In this case, the portfolio is riskless and must earn the risk-free interest rate. Equation (11.1) shows that is the ratio of the change in the option price to the change in the stock price as we move between the nodes at time T. If we denote the risk-free interest rate by r, the present value of the portfolio is S 0 u f u e rt Figure 11.2 Stock and option prices in a general one-step tree S 0 u f u S 0 f S 0 d f d

4 250 CHAPTER 11 The cost of setting up the portfolio is It follows that or S 0 f S 0 f ˆ S 0 u f u e rt f ˆ S 0 1 ue rt f u e rt Substituting from equation (11.1) for and simplifying, we can reduce this equation to where f ˆ e rt pf u 1 p f d Š 11:2 p ˆ ert d 11:3 u d Equations (11.2) and (11.3) enable an option to be priced when stock price movements are given by a one-step binomial model. In the numerical example considered previously (see Figure 11.1), u ˆ 1:1, d ˆ 0:9, r ˆ 0:12, T ˆ 0:25, f u ˆ 1, and f d ˆ 0. From equation (11.3), and, from equation (11.2), p ˆ e0:123=12 0:9 1:1 0:9 ˆ 0:6523 f ˆ e 0:120:25 0: : ˆ0:633 The result agrees with the answer obtained earlier in this section. Irrelevance ofthe Stock's Expected Return The option pricing formula in equation (11.2) does not involve the probabilities of the stock price moving up or down. For example, we get the same option price when the probability of an upward movement is 0.5 as we do when it is 0.9. This is surprising and seems counterintuitive. It is natural to assume that as the probability of an upward movement in the stock price increases, the value of a call option on the stock increases and the value of a put option on the stock decreases. This is not the case. The key reason is that we are not valuing the option in absolute terms. We are calculating its value in terms of the price of the underlying stock. The probabilities of future up or down movements are already incorporated into the price of the stock. It turns out that we do not need to take them into account again when valuing the option in terms of the stock price RISK-NEUTRAL VALUATION Although we do not need to make any assumptions about the probabilities of up and down movements in order to derive equation (11.2), it is natural to interpret the variable p in equation (11.2) as the probability of an up movement in the stock price. The variable 1 p is then the probability of a down movement, and the expression pf u 1 p f d

5 Introduction to Binomial Trees 251 is the expected payo from the option. With this interpretation of p, equation (11.2) then states that the value of the option today is its expected future payo discounted at the risk-free rate. We now investigate the expected return from the stock when the probability of an up movement is assumed to be p. The expected stock price at time T, E S T, is given by or E S T ˆpS 0 u 1 p S 0 d E S T ˆpS 0 u d S 0 d Substituting from equation (11.3) for p, we obtain E S T ˆS 0 e rt 11:4 showing that the stock price grows on average at the risk-free rate. Setting the probability of the up movement equal to p is therefore equivalent to assuming that the return on the stock equals the risk-free rate. In a risk-neutral world all individuals are indi erent to risk. In such a world investors require no compensation for risk, and the expected return on all securities is the riskfree interest rate. Equation (11.4) shows that we are assuming a risk-neutral world when we set the probability of an up movement to p. Equation (11.2) shows that the value of the option is its expected payo in a risk-neutral world discounted at the risk-free rate. This result is an example of an important general principle in option pricing known as risk-neutral valuation. The principle states that we can with complete impunity assume the world is risk neutral when pricing options. The resulting prices are correct not just in a risk-neutral world, but in other worlds as well. The One-Step Binomial Example Revisited We now return to the example in Figure 11.1 and show that risk-neutral valuation gives the same answer as no-arbitrage arguments. In Figure 11.1, the stock price is currently \$20 and will move either up to \$22 or down to \$18 at the end of three months. The option considered is a European call option with a strike price of \$21 and an expiration date in three months. The risk-free interest rate is 12% per annum. We de ne p as the probability of an upward movement in the stock price in a riskneutral world. We can calculate p from equation (11.3). Alternatively, we can argue that the expected return on the stock in a risk-neutral world must be the risk-free rate of 12%. This means that p must satisfy 22p 18 1 p ˆ20e 0:123=12 or 4p ˆ 20e 0:123=12 18 That is, p must be At the end of the three months, the call option has a probability of being worth 1 and a probability of being worth zero. Its expected value is therefore 0: : ˆ 0:6523 In a risk-neutral world this should be discounted at the risk-free rate. The value of the option today is therefore 0:6523e 0:123=12

6 252 CHAPTER 11 or \$ This is the same as the value obtained earlier, demonstrating that noarbitrage arguments and risk-neutral valuation give the same answer. Real World vs. Risk-Neutral World It should be emphasized that p is the probability of an up movement in a risk-neutral world. In general this is not the same as the probability of an up movement in the real world. In our example p ˆ 0:6523. When the probability of an up movement is , the expected return on both the stock and the option is the risk-free rate of 12%. Suppose that in the real world the expected return on the stock is 16%, and q is the probability of an up movement in the real world. It follows that 22q 18 1 q ˆ20e 0:163=12 so that q ˆ 0:7041. The expected payo from the option in the real world is then q 1 1 q 0 This is Unfortunately it is not easy to know the correct discount rate to apply to the expected payo in the real world. A position in a call option is riskier than a position in the stock. As a result the discount rate to be applied to the payo from a call option is greater than 16%. Without knowing the option's value, we do not know how much greater than 16% it should be. 1 Using risk-neutral valuation is convenient because we know that in a risk-neutral world the expected return on all assets (and therefore the discount rate to use for all expected payo s) is the risk-free rate TWO-STEP BINOMIAL TREES We can extend the analysis to a two-step binomial tree such as that shown in Figure Here the stock price starts at \$20 and in each of two time steps may go up by 10% or down by 10%. We suppose that each time step is three months long and the risk-free interest rate is 12% per annum. As before, we consider an option with a strike price of \$21. The objective of the analysis is to calculate the option price at the initial node of the tree. This can be done by repeatedly applying the principles established earlier in the chapter. Figure 11.4 shows the same tree as Figure 11.3, but with both the stock price and the option price at each node. (The stock price is the upper number and the option price the lower.) The option prices at the nal nodes of the tree are easily calculated. They are the payo s from the option. At node D the stock price is 24.2 and the option price is 24:2 21 ˆ 3:2; at nodes E and F the option is out of the money and its value is zero. At node C the option price is zero, because node C leads to either node E or node F and at both nodes the option price is zero. We calculate the option price at node B by focusing our attention on the part of the tree shown in Figure Using the notation introduced earlier in the chapter, u ˆ 1:1, d ˆ 0:9, r ˆ 0:12, and T ˆ 0:25, so that 1 Because the correct value of the option is 0.633, we can deduce that the correct discount rate is 42.58%. This is because 0:633 ˆ 0:7041e 0:42583=12.

7 Introduction to Binomial Trees 253 Figure 11.3 Stock prices in a two-step tree p ˆ 0:6523 and equation (11.2) gives the value of the option at node B as e 0:123=12 0:6523 3:2 0: ˆ2:0257 It remains for us to calculate the option price at the initial node A. We do so by focusing on the rst step of the tree. We know that the value of the option at node B is and Figure 11.4 Stock and option prices in a two-step tree. The upper number at each node is the stock price; the lower number is the option price D B A E C F

8 254 CHAPTER 11 Figure 11.5 Evaluation of option price at node B D B E that at node C it is zero. Equation (11.2) therefore gives the value at node A as e 0:123=12 0:6523 2:0257 0: ˆ1:2823 The value of the option is \$ Note that this example was constructed so that u and d (the proportional up and down movements) were the same at each node of the tree and so that the time steps were of the same length. As a result, the risk-neutral probability, p, as calculated by equation (11.3) is the same at each node. A Generalization We can generalize the case of two time steps by considering the situation in Figure The stock price is initially S 0. During each time step, it either moves up to u times its Figure 11.6 Stock and option prices in general two-step tree S 0 u 2 f uu S 0 u f u S 0 f S 0 ud f ud S 0 d f d S 0 d 2 f dd

9 Introduction to Binomial Trees 255 initial value or moves down to d times its initial value. The notation for the value of the option is shown on the tree. (For example, after two up movements the value of the option is f uu.) We suppose that the risk-free interest rate is r and the length of the time step is t years. Because the length of a time step is now t rather than T, equations (11.2) and (11.3) become f ˆ e rt pf u 1 p f d Š 11:5 p ˆ ert d u d 11:6 Repeated application of equation (11.5) gives f u ˆ e rt pf uu 1 p f ud Š f d ˆ e rt pf ud 1 p f dd Š f ˆ e rt pf u 1 p f d Š 11:7 11:8 11:9 Substituting from equations (11.7) and (11.8) into (11.9), we get f ˆ e 2rt p 2 f uu 2p 1 p f ud 1 p 2 f dd Š 11:10 This is consistent with the principle of risk-neutral valuation mentioned earlier. The variables p 2,2p 1 p, and 1 p 2 are the probabilities that the upper, middle, and lower nal nodes will be reached. The option price is equal to its expected payo in a risk-neutral world discounted at the risk-free interest rate. As we add more steps to the binomial tree, the risk-neutral valuation principle continues to hold. The option price is always equal to its expected payo in a riskneutral world, discounted at the risk-free interest rate A PUT EXAMPLE The procedures described in this chapter can be used to price puts as well as calls. Consider a two-year European put with a strike price of \$52 on a stock whose current price is \$50. We suppose that there are two time steps of one year, and in each time step the stock price either moves up by a proportional amount of 20% or moves down by a proportional amount of 20%. We also suppose that the risk-free interest rate is 5%. The tree is shown in Figure In this case, u ˆ 1:2, d ˆ 0:8,t ˆ 1, and r ˆ 0:05. From equation (11.6), the value of the risk-neutral probability, p, is given by p ˆ e0:051 0:8 1:2 0:8 ˆ 0:6282 The possible nal stock prices are: \$72, \$48, and \$32. In this case, f uu ˆ 0, f ud ˆ 4, and f dd ˆ 20. From equation (11.10), f ˆ e 20:051 0: :6282 0: : ˆ4:1923 The value of the put is \$ This result can also be obtained using equation (11.5)

10 256 CHAPTER 11 Figure 11.7 Use of two-step tree to value European put option. At each node the upper number is the stock price; the lower number is the option price and working back through the tree one step at a time. Figure 11.7 shows the intermediate option prices that are calculated AMERICAN OPTIONS Up to now all the options we have considered have been European. We now move on to consider how American options can be valued using binomial trees such as those in Figures 11.4 and The procedure is to work back through the tree from the end to the beginning, testing at each node to see whether early exercise is optimal. The value of the option at the nal nodes is the same as for the European option. At earlier nodes the value of the option is the greater of: 1. The value given by equation (11.5) 2. The payo from early exercise Figure 11.8 shows how Figure 11.7 is a ected if the option considered is American rather than European. The stock prices and their probabilities are unchanged. The values for the option at the nal nodes are also unchanged. At node B, equation (11.2) gives the value of the option as , whereas the payo from early exercise is negative (ˆ 8). Clearly early exercise is not optimal at node B, and the value of the option at this node is At node C, equation (11.5) gives the value of the option as , whereas the payo from early exercise is 12. In this case, early exercise is optimal and the value of the option at the node is 12. At the initial node A, the value given by equation (11.5) is e 0:051 0:6282 1:4147 0: :0 ˆ5:0894

11 Introduction to Binomial Trees 257 Figure 11.8 Use of two-step tree to value American put option. At each node the upper number is the stock price; the lower number is the option price B A C and the payo from early exercise is 2. In this case early exercise is not optimal. The value of the option is therefore \$ DELTA At this stage it is appropriate to introduce delta, an important parameter in the pricing and hedging of options. The delta of a stock option is the ratio of the change in the price of the stock option to the change in the price of the underlying stock. It is the number of units of the stock we should hold for each option shorted in order to create a riskless hedge. It is the same as the introduced earlier in this chapter. The construction of a riskless hedge is sometimes referred to as delta hedging. The delta of a call option is positive, whereas the delta of a put option is negative. From Figure 11.1, we can calculate the value of the delta of the call option being considered as ˆ 0:25 This is because when the stock price changes from \$18 to \$22, the option price changes from \$0 to \$1. In Figure 11.4, the delta corresponding to stock price movements over the rst time step is 2: ˆ 0:5064

12 258 CHAPTER 11 The delta for stock price movements over the second time step is 3:2 0 24:2 19:8 ˆ 0:7273 if there is an upward movement over the rst time step, and :8 16:2 ˆ 0 if there is a downward movement over the rst time step. From Figure 11.7, delta is 1:4147 9: at the end of the rst time step and either or ˆ 0: ˆ 0: ˆ 1:0000 at the end of the second time step. The two-step examples show that delta changes over time. (In Figure 11.4, delta changes from to either or 0; in Figure 11.7, it changes from 0:4024 to either 0:1667 or 1:0000.) Thus, in order to maintain a riskless hedge using an option and the underlying stock, we need to adjust our holdings in the stock periodically. This is a feature of options that we will return to in Chapters 12 and DETERMINING u AND d Up to now we have assumed values for u and d. For example, in Figure 11.8, we assumed that u ˆ 1:2 andd ˆ 0:8. In practice, u and d are determined from the stock price volatility,. The formulas are u ˆ e p t 11:11 d ˆ 1 11:12 u where t is the length of one time step on the tree. These formulas will be explained further in Chapter 16. The complete set of equations de ning the tree are equation (11.11) and equation (11.12), together with equation (11.6), which can be written p ˆ a d 11:13 u d where a ˆ e rt 11:14

13 Introduction to Binomial Trees 259 Figure 11.9 Two-step tree to value an two-year American put option when stock price is 50, strike price is 52, risk-free rate is 5%, and volatility is 30% Consider again the American put option in Figure 11.8, where the stock price is \$50, the strike price is \$52, the risk-free rate is 5%, the life of the option is two years, and there are two time steps. In this case, t ˆ 1. Suppose that the volatility is 30%. Then, from equations (11.11) to (11.14), we have u ˆ e 0:31 ˆ 1:3499 d ˆ 1 1:3499 ˆ 0:7408 a ˆ e 0:051 ˆ 1:0513 and 1:053 0:7408 p ˆ 1:3499 0:7408 ˆ 0:5097 The tree is shown in Figure The value of the option is This is di erent from the value obtained in Figure 11.8 by assuming u ˆ 1:2 and d ˆ 0: INCREASING THE NUMBER OF TIME STEPS The binomial models presented so far have been unrealistically simple. Clearly an analyst can expect to obtain only a very rough approximation to an option price by assuming that stock price movements during the life of the option consist of one or two binomial steps.

14 260 CHAPTER 11 When binomial trees are used in practice, the life of the option is typically divided into 30 or more time steps. In each time step there is a binomial stock price movement. With 30 time steps, there are 31 terminal stock prices and 2 30, or about 1 billion, possible stock price paths are considered. The equations de ning the tree are equations (11.11) to (11.14), regardless of the number of time steps. Suppose, for example, that there are ve steps instead of two in the example we considered in Figure The parameters would be t ˆ 2=5 ˆ 0:4, r ˆ 0:05, and ˆ 0:3. These give u ˆ e 0:3 p 0:4 ˆ 1:2089, d ˆ 1=1:2089 ˆ 0:8272, a ˆ e 0:050:4 ˆ 1:0202, and p ˆ 1:0202 0:8272 = 1:2089 0:8272 ˆ0:5056. Use ofderivagem The software accompanying this book, DerivaGem, is a useful tool for becoming comfortable with binomial trees. After loading the software in the way described at the end of this book, go to the Equity_FX_Index_Futures_Options worksheet. Choose Equity as the Underlying Type and select Binomial American as the Option Type. Enter the stock price, volatility, risk-free rate, time to expiration, exercise price, and tree steps as 50, 30%, 5%, 2, 52, and 2, respectively. Click on the Put button and then on Calculate. The price of the option is shown as in the boxlabeled Price. Now click on Display Tree and you will see the equivalent of Figure (The red numbers indicate the nodes where the option is exercised.) Return to the Equity_FX_Index_Futures_Options worksheet and change the number of time steps to 5. Hit Enter and click on Calculate. You will nd that the value of the option changes to By clicking on Display Tree the ve-step tree is displayed together with the values of u, d, a, and p we have just calculated. DerivaGem can display trees that have up to 10 steps, but the calculations can be done for up to 500 steps. In our example, 500 steps gives the option price (to two decimal places) as This is an accurate answer. By changing the Option Type to Binomial European, we can use the tree to value a European option. Using 500 time steps, the value of a European option with the same parameters as the American option is (By changing the option type to Analytic European, we can display the value of the option using the Black±Scholes formula that will be presented in the next chapter. This is also 6.76.) By changing the Underlying Type, we can consider options on assets other than stocks. We will now discuss these types of option OPTIONS ON OTHER ASSETS We introduced options on indices, currencies, and futures contracts in Chapter 8 and will cover them in more detail in Chapters 13 and 14. It turns out that we can construct and use binomial trees for these options in exactly the same way as for options on stocks except that the equations for p change. As in the case of options on stocks, equation (11.2) applies so that the value at a node (before the possibility of early exercise is considered) is p times the value if there is an up movement plus 1 p times the value if there is a down movement, discounted at the risk-free rate.

15 Introduction to Binomial Trees 261 Options on Stocks Paying a Continuous Dividend Yield Consider rst a stock paying a known dividend yield at rate q. The total return from dividends and capital gains in a risk-neutral world is r. The dividends provide a return of q. Capital gains must therefore provide a return of r q. If the stock starts at S 0, its expected value after one time step of length t must be S 0 e r q t. This means that so that ps 0 u 1 p S 0 d ˆ S 0 e r q t p ˆ e r q t d u d As in the case of options on non-dividend-paying stocks, we match volatility by setting u ˆ e p t and d ˆ 1=u. This means that we can use equations (11.11) to (11.14) except that we set a ˆ e r q t. Options on Stock Indices When calculating a futures price for a stock indexin Chapter 5, we assumed that the stocks underlying the indexprovided a dividend yield at rate q. We make a similar assumption here. The valuation of an option on a stock indexis therefore very similar to the valuation of an option on a stock paying a known dividend yield. Example 11.1 illustrates this. Options on Currencies As pointed out in Section 5.10, a foreign currency can be regarded as an asset providing a yield at the foreign risk-free rate of interest, r f. By analogy with the stock indexcase, we can construct a tree for options on a currency by using equations (11.11) to (11.14) and setting a ˆ e r r f t. Example 11.2 illustrates this. Options on Futures It costs nothing to take a long or a short position in a futures contract. It follows that in a risk-neutral world a futures price should have an expected growth rate of zero. (We discuss this point in more detail in Chapter 14.) As above, we de ne p as the probability of an up movement in the futures price, u as the amount by which the price is multiplied in an up movement, and d as the amount by which it is multiplied in a down movement. If F 0 is the initial futures price, the expected futures price at the end of one time step of length t should also be F 0. This means that so that pf 0 u 1 p F 0 d ˆ F 0 p ˆ 1 d u d and we can use equations (11.11) to (11.14) with a ˆ 1. Example 11.3 illustrates this.

16 262 CHAPTER 11 Example 11.1 Option on a stock index A stock indexis currently 810 and has a volatility of 20% and a dividend yield of 2%. The risk-free rate is 5%. The gure below shows the output from DerivaGem for valuing a European six-month call option with a strike price of 800 using a two-step tree. In this case, p 0:20 0:25 t ˆ 0:25; u ˆ e ˆ 1:1052; d ˆ 1=u ˆ 0:9048; a ˆ e 0:05 0:02 0:25 ˆ 1:0075; p ˆ The value of the option is DerivaGem Output: At each node: Upper value = Underlying Asset Price Lower value = Option Price Shading indicates where option is exercised Strike price = 800 Discount factor per step = Time step, dt = years, days Growth factor per step, a = Probability of up move, p = Up step size, u = Down step size, d = :0075 0:9048 1:1052 0:9048 ˆ 0: Node Time:

17 Introduction to Binomial Trees 263 Example 11.2 Option on a foreign currency The Australian dollar is currently worth U.S. dollars and this exchange rate has a volatility of 12%. The Australian risk-free rate is 7% and the U.S. risk-free rate is 5%. The gure below shows the output from DerivaGem for valuing a threemonth American call option with a strike price of using a three-step tree. In this case, p 0:12 0:08333 t ˆ 0:08333; u ˆ e ˆ 1:0352; d ˆ 1=u ˆ 0:9660; a ˆ e 0:05 0:07 0:08333 ˆ 0:9983; p ˆ The value of the option is DerivaGem Output: At each node: Upper value = Underlying Asset Price Lower value = Option Price Shading indicates where option is exercised Strike price = 0.6 Discount factor per step = Time step, dt = years, days Growth factor per step, a = Probability of up move, p = Up step size, u = Down step size, d = Node Time: :9983 0:9660 1:0352 0:9660 ˆ 0:4673

18 264 CHAPTER 11 Example 11.3 Option on futures A futures price is currently 31 and it has a volatility of 30%. The risk-free rate is 5%. The gure below shows the output from DerivaGem for valuing a ninemonth American put option with a strike price of 30 using a three-step tree. In this case, p 0:3 0:25 t ˆ 0:25; u ˆ e ˆ 1:1618; d ˆ 1=u ˆ 1=1:1618 ˆ 0:8607; a ˆ 1; p ˆ 1 0:8607 1:1618 0:8607 ˆ 0:4626 The value of the option is DerivaGem Output: At each node: Upper value = Underlying Asset Price Lower value = Option Price Shading indicates where option is exercised Strike price = 30 Discount factor per step = Time step, dt = years, days Growth factor per step, a = Probability of up move, p = Up step size, u = Down step size, d = Node Time:

19 Introduction to Binomial Trees 265 SUMMARY This chapter has provided a rst look at the valuation of options on stocks and other assets. In the simple situation where movements in the price of a stock during the life of an option are governed by a one-step binomial tree, it is possible to set up a riskless portfolio consisting of a stock option and the stock. In a world with no arbitrage opportunities, riskless portfolios must earn the risk-free rate of interest. This enables the stock option to be priced in terms of the stock. It is interesting to note that no assumptions are required about the probabilities of up and down movements in the stock price at each node of the tree. When stock price movements are governed by a multistep binomial tree, we can treat each binomial step separately and work back from the end of the life of the option to the beginning to obtain the current value of the option. Again only no-arbitrage arguments are used, and no assumptions are required about the probabilities of up and down movements in the stock price at each node. An equivalent approach to valuing stock options involves risk-neutral valuation. This very important principle states that it is permissible to assume the world is risk neutral when valuing an option in terms of the underlying stock. This chapter has shown, through both numerical examples and algebra, that no-arbitrage arguments and riskneutral valuation are equivalent and lead to the same option prices. The delta of a stock option,, considers the e ect of a small change in the underlying stock price on the change in the option price. It is the ratio of the change in the option price to the change in the stock price. For a riskless position an investor should buy shares for each option sold. An inspection of a typical binomial tree shows that delta changes during the life of an option. This means that to hedge a particular option position, we must change our holding in the underlying stock periodically. Constructing binomial trees for valuing options on stock indices, currencies, futures contracts is very similar to doing so for valuing options on stocks. In Chapter 16, we will return to binomial trees and give a further details on how they can be used in practice. FURTHER READING Coval, J. E., and T. Shumway. ``Expected Option Returns,'' Journal of Finance, 56, 3 (2001), 983±1009. Cox, J., S. Ross, and M. Rubinstein. ``Option Pricing: A Simpli ed Approach.'' Journal of Financial Economics, 7 (October 1979): 229±64. Rendleman, R., and B. Bartter. ``Two State Option Pricing.'' Journal of Finance, 34 (1979): 1092±1110. Quiz (Answers at End ofbook) A stock price is currently \$40. It is known that at the end of one month it will be either \$42 or \$38. The risk-free interest rate is 8% per annum with continuous compounding. What is the value of a one-month European call option with a strike price of \$39? Explain the no-arbitrage and risk-neutral valuation approaches to valuing a European option using a one-step binomial tree.

20 266 CHAPTER What is meant by the delta of a stock option? A stock price is currently \$50. It is known that at the end of sixmonths it will be either \$45 or \$55. The risk-free interest rate is 10% per annum with continuous compounding. What is the value of a six-month European put option with a strike price of \$50? A stock price is currently \$100. Over each of the next two six-month periods it is expected to go up by 10% or down by 10%. The risk-free interest rate is 8% per annum with continuous compounding. What is the value of a one-year European call option with a strike price of \$100? For the situation considered in Problem 11.5, what is the value of a one-year European put option with a strike price of \$100? Verify that the European call and European put prices satisfy put±call parity What are the formulas for u and d in terms of volatility? Questions and Problems (Answers in Solutions Manual/Study Guide) Consider the situation in which stock price movements during the life of a European option are governed by a two-step binomial tree. Explain why it is not possible to set up a position in the stock and the option that remains riskless for the whole of the life of the option A stock price is currently \$50. It is known that at the end of two months it will be either \$53 or \$48. The risk-free interest rate is 10% per annum with continuous compounding. What is the value of a two-month European call option with a strike price of \$49? Use no-arbitrage arguments A stock price is currently \$80. It is known that at the end of four months it will be either \$75 or \$85. The risk-free interest rate is 5% per annum with continuous compounding. What is the value of a four-month European put option with a strike price of \$80? Use no-arbitrage arguments A stock price is currently \$40. It is known that at the end of three months it will be either \$45 or \$35. The risk-free rate of interest with quarterly compounding is 8% per annum. Calculate the value of a three-month European put option on the stock with an exercise price of \$40. Verify that no-arbitrage arguments and risk-neutral valuation arguments give the same answers A stock price is currently \$50. Over each of the next two three-month periods it is expected to go up by 6% or down by 5%. The risk-free interest rate is 5% per annum with continuous compounding. What is the value of a six-month European call option with a strike price of \$51? For the situation considered in Problem 11.12, what is the value of a six-month European put option with a strike price of \$51? Verify that the European call and European put prices satisfy put±call parity. If the put option were American, would it ever be optimal to exercise it early at any of the nodes on the tree? A stock price is currently \$25. It is known that at the end of two months it will be either \$23 or \$27. The risk-free interest rate is 10% per annum with continuous compounding. Suppose S T is the stock price at the end of two months. What is the value of a derivative that pays o ST 2 at this time?

21 Introduction to Binomial Trees Calculate u, d, and p when a binomial tree is constructed to value an option on a foreign currency. The tree step size is one month, the domestic interest rate is 5% per annum, the foreign interest rate is 8% per annum, and the volatility is 12% per annum. Assignment Questions A stock price is currently \$50. It is known that at the end of sixmonths it will be either \$60 or \$42. The risk-free rate of interest with continuous compounding is 12% per annum. Calculate the value of a six-month European call option on the stock with an exercise price of \$48. Verify that no-arbitrage arguments and risk-neutral valuation arguments give the same answer A stock price is currently \$40. Over each of the next two three-month periods it is expected to go up by 10% or down by 10%. The risk-free interest rate is 12% per annum with continuous compounding. a. What is the value of a six-month European put option with a strike price of \$42? b. What is the value of a six-month American put option with a strike price of \$42? Using a ``trial-and-error'' approach, estimate how high the strike price has to be in Problem for it to be optimal to exercise the put option immediately A stock price is currently \$30. During each two-month period for the next four months it will increase by 8% or decrease by 10%. The risk-free interest rate is 5%. Use a two-step tree to calculate the value of a derivative that pays o max 30 S T ; 0 Š 2, where S T is the stock price in four months. If the derivative is American style, should it be exercised early? Consider a European call option on a non-dividend-paying stock where the stock price is \$40, the strike price is \$40, the risk-free rate is 4% per annum, the volatility is 30% per annum, and the time to maturity is sixmonths. a. Calculate u, d, and p for a two-step tree. b. Value the option using a two-step tree. c. Verify that DerivaGem gives the same answer. d. Use DerivaGem to value the option with 5, 50, 100, and 500 time steps Repeat Problem for an American put option on a futures contract. The strike price and the futures price are \$50, the risk-free rate is 10%, the time to maturity is six months, and the volatility is 40% per annum.

22

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