Binomial Tree Model for Financial Derivatives Valuation

Introduction to Binomial Trees Chapter 12 12.1

Goals of Chapter 12 Introduce the binomial tree model ( ) in the one-period case Discuss the risk-neutral valuation relationship ( ) Introduce the binomial tree model in the two- period case and the CRR binomial tree model Consider the continuously compounding dividend yield in the binomial tree model 12.2

12.1 One-Period Binomial Tree Model 12.3

One-Period Binomial Tree Model The binomial tree model represents possible stock price at any time point based on a discrete-time and discrete-price framework For a stock price at a time point, a binomial distribution ( ) models the stock price movement at the subsequent time point That is, there are two possible stock prices with assigned probabilities at the next time point The binomial tree model is a general numerical method for pricing derivatives with various payoffs The binomial tree model is particularly useful for valuing American options, which do not have analytic option pricing formulas 12.4

One-Period Binomial Tree Model One-period case for the binomial tree model The stock price ? is currently $20 After three months, it will be either $22 or $18 for the upper and lower branches ?? = 22 ? = 20 ?? = 18 12.5

One-Period Binomial Tree Model Consider a 3-month call option on the stock with a strike price of 21 Corresponding to the upper and lower movements in the stock price, the payoffs of this call option are ?? = $1 and ?? = $1 ?? = 22 ??= 1 ? = 20 ? = ? ?? = 18 ??= 0 What is the theoretical value of this call today? 12.6

One-Period Binomial Tree Model Consider a portfolio P: long shares short 1 call option 18 22 1 Portfolio P is riskless when 22 1 = 18 , which implies = 0.25 The value of Portfolio P after 3 months is 22 x 0.25 1 = 18 x 0.25 = 4.5 12.7

One-Period Binomial Tree Model Since Portfolio P is riskless, it should earn the risk- free interest rate according to the no-arbitrage argument If the return of Portfolio P is higher (lower) than the risk- free interest rate Portfolio P is more (less) attractive than other riskless assets Buying (Shorting) Portfolio P and shorting (buying) the riskless asset can arbitrage Buying (selling) pressure bid up (drive down) the price of Portfolio P, which causes the decline (rise) of the return of Portfolio P The value of the portfolio today is 4.5? 12% 0.25= 4.367, where 12% is the risk-free interest rate The amount of 4.367 should be the fair cost (or the initial investment) to construct Portfolio P 12.8

One-Period Binomial Tree Model The riskless Portfolio P consists of long 0.25 shares and short 1 call option The cost to construct Portfolio P equals 0.25 x 20 ? Solve for the theoretical value of this call today to be ? = 0.633 by equalizing 0.25 x 20 ? with 4.367 12.9

Generalization of One-Period Binomial Tree Model Consider any derivative ? lasting for time T and its payoff is dependent on a stock ??= ? ? ?? ? ? ??= ? ? ?? Assume that the possible stock price at ? are ??= ?? and ??= ??, where ? and ? are constant multiplication factors for the upper and lower branches ??and ??are payoffs of the derivative ? corresponding to the upper and lower branches 12.10

Generalization of One-Period Binomial Tree Model Construct Portfolio P that longs shares and shorts 1 derivative. The payoffs of Portfolio P are ?? ?? ?? ?? Portfolio P is riskless if ?? ??= ?? ?? and thus ?? ?? ?? ?? = Recall that in the prior numerical example, ? = 20, ? = 1.1, ? = 0.9, ??= 1, and ??= 0, so the solution of for generating a riskless portfolio is 0.25 12.11

Generalization of One-Period Binomial Tree Model Value of Portfolio P at time Tis ?? ??(or equivalently ?? ??) Value of Portfolio P today is thus (?? ??)? ?? The cost for Portfolio P is ? ? If Portfolio P is fairly priced: ? ? = (?? ??)? ?? ? = ? (?? ??)? ?? Substituting for ?? ?? we obtain ? = ? ??[? ??+ 1 ? ??], where ? =??? ? ? ? ?? ??in the above equation, 12.12

12.2 Risk-Neutral Valuation Relationship ( ) 12.13

Risk-Neutral Valuation Relationship ( ) Risk-averse ( ), risk-neutral ( ), and risk-loving ( ) behaviors A flipping-coin game vs. a riskless payoff For risk-averse investors, they accept a risky game if its expected payoff is higher than the riskless payoff by a required amount which can compensate them for the risk they bear For different investors, they have different tolerance for risk, i.e., they require different expected payoff to accept the same risky game 12.14

Risk-Neutral Valuation Relationship ( ) For risk-neutral investors, they accept a risky game even if its expected payoff equals the riskless payoff That is, they care about only the levels of the (expected) payoffs In other words, they require no compensation for bearing risk For risk-loving investors, they accept a risky game (enjoy the feeling of gamble) even if its expected payoff is lower than the riskless payoff That is, they would like to sacrifice some benefit for entering a risky game We do not discuss risk-loving behavior in this chapter 12.15

Risk-Neutral Valuation Relationship ( ) In a risk-averse financial market, securities with higher degree of risk need to offer higher expected returns So, our real world is a risk averse world, i.e., most investors are risk averse and require compensation for the risk they tolerate In a risk-neutral financial market (where every trader is risk neutral), the expected returns of all securities equal the risk free rate regardless of their degrees of risk That is, even for high-risk-level derivatives, their expected returns equal the risk free rate in the risk-neutral world The risk-neutral market only exists in our imagination 12.16

Risk-Neutral Valuation Relationship ( ) Interpret ?in ? = ? ??[???+ 1 ? ??] as a probability in the risk-neutral world If the expected return of the stock price is ? in the real world (a risk averse world), the expected stock price at the end of the period is ? ?? = ???? ?? ? ?? ? ?? + 1 ? ?? = ???? ? =??? ? ? ? 12.17

Risk-Neutral Valuation Relationship ( ) Comparing with ? =??? ? and 1 ? as probabilities of upward and downward movements in the risk-neutral world This is because that the expected return of any security in the risk-neutral world is the risk free rate The formula ? = ? ??[???+ 1 ? ??] is consistent with the general rule to price derivatives Note that in the risk-neutral world, [???+ 1 ? ??] is the expected payoff of a derivative and ? ?? is the correct discount factor to derive its PV today The general derivative pricing method given a constant interest rate ? is that any derivative can be priced as the PV of its expected payoff in the risk-neutral world ? ?, it is natural to interpret ? 12.18

Risk-Neutral Valuation Relationship ( ) Risk-neutral valuation relationship (RNVR) It states that any derivative can be priced with the general derivative pricing method as if it and its underlying asset were in the risk-neutral world Since the expected returns of both the derivative and its underlying asset are the risk free rate The probability of the upward movement in the prices of the underlying asset is ? =??? ? ? ? The discount rate for the expected payoff of the derivative is also ? When we are evaluating an option, the expected return on the underlying asset, ?, is irrelevant 12.19

Risk-Neutral Valuation Relationship ( ) Revisit the original numerical example in the risk-neutral world ?? = 22 ??= 1 ? = 20 ? = ? ?? = 18 ??= 0 Calculate ? =??? ? ? ?=?12% 0.25 0.9 = 0.6523 1.1 0.9 Calculate the option value according to the RNVR ? 12% 0.250.6523 1 + 1 0.6523 0 = 0.633 12.20

12.3 Multi-Period Binomial Tree Model 12.21

Two-Period Binomial Tree Model Illustration of the two-period binomial tree ? = 20, ? = 12%, ? = 1.1, ? = 0.9, ? = 0.5, the number of time steps is ? = 2, and thus the length of each time step is ? = ?/? = 0.25 24.2 22 Note the recombined feature can limit the growth of the number of nodes on the binomial tree in a acceptable manner 19.8 20 18 16.2 By decomposing it into three one-period binomial trees, one can calculate the risk-neutral probability as ? =?? ? ? ? ? 1.1 0.9 =?12% 0.25 0.9 = 0.6523 12.22

Two-Period Binomial Tree Model node D node B 24.2 3.2 22 2.0257 node E node A 19.8 20 0 1.2823 node C node F 18 0 16.2 0 For a European call option with ? = 21, determine its payoffs at terminal nodes first and then perform backward induction method ( ) to obtain option value of each node Option value at B: ? 12% 0.250.6523 3.2 + 0.3477 0 = 2.0257 Option value at C: ? 12% 0.250.6523 0 + 0.3477 0 = 0 Option value at A (the initial or root node): ? 12% 0.25( ) 2.0257 + 0.3477 0 = 1.2823 0.6523 12.23

Two-Period Binomial Tree Model node B node D 72 0 60 1.4147 node A node E 50 48 4 4.1923 node C node F 40 32 20 9.4636 For a European put with ? = 52 and ? = 2 ? = 50, ? = 5%, ? = 1.2, ? = 0.8, ? = 2, ? = 1, and ? = 0.6282 Option value at B: ? 5% 10.6282 0 + 0.3718 4 = 1.4147 Option value at C: ? 5% 10.6282 4 + 0.3718 20 = 9.4636 Option value at A: ? 5% 10.6282 1.4147 + 0.3718 9.4636 = 4.1923 12.24

Binomial Tree Model for American Options node B node D 72 0 60 1.4147 node A node E 50 48 4 5.0894 node C node F 40 12 32 20 For an American put with ? = 52 and ? = 2 Option value at B: ? 5% 10.6282 0 + 0.3718 4 = 1.4147 Option value at C: ? 5% 10.6282 4 + 0.3718 20 = 9.4636 (the put value if one chooses to hold it), which is smaller than the exercise value max ? ??,0 = 12 it is optimal to early exercise Option value at A: ? 5% 10.6282 1.4147 + 0.3718 12 = 5.0894 12.25

Delta Delta ( ) The formula to calculate in the binomial tree model is ?? ?? ?? ?? on Slide 12.11 In the binomial tree model, is the number of shares of the stock we should hold for each option shorted in order to create a riskless portfolio For the one-period example on Slide 12.6, the delta of the call option is 22 18= 0.25 Theoretically speaking, is defined as the ratio of the change in the price of a stock option with respect to the change in the price of the underlying stock, i.e., =?? ?? 1 0 12.26

Delta The delta hedging strategy (delta ) eliminates the price risk by constructing a riskless portfolio for a period of time (discussed in Ch. 17) The method to decide the value of the delta in the binomial tree model is in effect to perform the delta hedging strategy Short 1 derivative and long shares form a portfolio to be ? + ? Since we determine such that the portfolio ( ? + ?)is riskless, it implies that the value of this portfolio is immunized to the change in stock prices, i.e., ?? ??+ ?? ?? since ?? ??= 0 Thus, we can solve =?? ??= 1 by definition 12.27

Delta The value of varies from node to node (revisit the call option on Slide 12.23) node B node D 24.2 3.2 22 2.0257 node E node A 19.8 20 0 1.2823 node C node F 18 0 16.2 0 at A: 2.0257 0 = 0.5064 22 18 3.2 0 24.2 19.8= 0.7273 0 0 19.8 16.2= 0 at B: at C: 12.28

Delta Since the value of changes over time, the delta hedging strategy needs rebalances over time For node A, is decided to be 0.5024 such that the portfolio is riskless during the first period of time If the stock price rises (falls) to reach node B (C), changes to 0.7273 (0), which means that we need to increase (reduce) the number of shares held to make the portfolio risk free in the second period 12.29

CRR Binomial Tree Model How to determine ? and ? In practice, given any stock price at the time point ?, ? and ? are determined to match the variance of the stock price at the next time point ? + ? ??? ?? ??? ? ? + ? 2 ?[??+ ?]2 var ??+ ? = ? ??+ ? ??2?2 ? (approximately correct given a short ?) = ???2?2+ 1 ? ??2?2 (???? ?)2 ?2 ? = ??2+ 1 ? ?2 ?2? ?12.30

CRR Binomial Tree Model With ? =?? ? ? ? ? and the assumption of ?? = 1 ? = ?? ? and ? = ? ? ? This method is first proposed by Cox, Ross, and Rubinstein (1979), so this method is also known as the CRR binomial tree model The validity of the CRR binomial tree model depends on the risk-neutral probability ? being in [0,1] In practice, the life of the option is typically partitioned into hundreds time steps First, ensure the validity of the risk-neutral probability, ?, which approaches 0.5 if ? approaches 0 Second, ensure the convergence to the Black Scholes model (introduced in Ch. 13) 12.31

12.4 Dividend Yield in the Binomial Tree Model 12.32

Effect of Dividend Yield on Risk-Neutral Probabilities ??? ?? ? ??+ ? = ???? ? ? ??? ? ? + ? In the risk-neutral world, the total return including dividend and capital gains is ? If the continuously compounding dividend yield is ?, the ex- dividend ( ) return of in the stock price should be ? ? Hence, ???? + 1 ? ??? = ???? ? ? ? =?? ? ? ? The dividend yield does NOT affect the volatility of the stock price and thus does NOT affect the multiplying factors ? = ?? ?and ? = ? ? ? Note that the discount rate for the expected payoff is still ? ? ? 12.33

Applications of Binomial Tree Model with Dividend Yield For options on a stock paying a continuously compounding dividend yield (?) or options on a stock index ? =?? ? ? ? ? ? For options on a currency (regarded as an asset paying continuously compounding dividend yield ?? (the foreign risk free rate)) ? =?? ?? ? ? ? ? 12.33