Transcript S - My LIUC
4.1 Option Prices: numerical approach Lecture 4 4.2 Pricing: 1.Binomial Trees Binomial Trees • Binomial trees are frequently used to approximate the movements in the price of a stock or other asset • In each small interval of time the stock price is assumed to move up by a proportional amount u or to move down by a proportional amount d 4.4 A Simple Binomial Model • A stock price is currently $20 • In three months it will be either $22 or $18 Stock Price = $22 Stock price = $20 Stock Price = $18 probabilities can’t be 50%50%, unless you are riskneutral 4.5 A Call Option A 3-month call option on the stock has a strike price of 21. Stock Price = $22 Option Price = $1 Stock price = $20 Option Price=? if you were risk-neutral (and r=0), you could say that the option is worth: 0.5=50%*1$+50%*$0 Stock Price = $18 Option Price = $0 • prob(22) has to be > prob (18), because otherwise Utility function is linear 4.6 U(22) U(20) Expcted U (50% prob) U(18) 22 18 20 • Then, in order to know prob we need to know the Utility function. But this is an impossible task, and we have to find a shortcut .... i.e. we have to find a way of “linearizing” the world Setting Up a Riskless Portfolio • Consider the Portfolio: long D shares short 1 call option 22D – 1 18D • Portfolio is riskless when 22D – 1 = 18D or D = 0.25 4.7 4.8 Valuing the Portfolio • risk-fre rate=12% p.a. ---> 3% quarterly ---> disc. factor=exp(-0.12*0.25)=0.970446 • The riskless portfolio is: long 0.25 shares short 1 call option • The value of the portfolio in 3 months is 220.25 – 1 = 4.50 • Note that this pay-off is deterministic, so its PV is obtained by simple discounting 4.9 Valuing the Option • The value of the portfolio today is 4.5e – 0.120.25 = 4.3670 • The portfolio that is long 0.25 shares short 1 option is worth 4.367 • The value of the shares is 5.000 (= 0.2520 ) • The value of the option is therefore 0.633 (= 5.000 – 4.367 ) 4.10 Valuing the Option • note that the value of the option has been obtained without knowing the shape of the utility function • but if the solution is independent of preferences functional form, then it is valid also for all utility function • Then, it is valid also for risk-neutral preferences ..... • ... eureka !!! let’s imagine a risk-neutral world ---> derive risk-neutral probabilities Summing up ... Movements in Time Dt Su S Sd Risk-neutral Evaluation hyp: risk-free rate=12% p.a.; t = 3m Su = 22 ƒu = 1 S ƒ Sd = 18 ƒd = 0 • Since p is a risk-neutral probability 20e0.12 0.25 = 22p + 18(1 – p ); p = 0.6523 • p is called the risk-neutral probability • show simple_example.xls 4.12 Tree Parameters for a Nondividend Paying Stock • We choose the tree parameters p, u, and d so that the tree gives correct values for the mean & standard deviation of the stock price changes in a risk-neutral world er Dt = pu + (1– p )d s2Dt = pu 2 + (1– p )d 2 – [pu + (1– p )d ]2 • A further condition often imposed is u = 1/ d Tree Parameters for a Nondividend Paying Stock • When Dt is small a solution to the equations is ue s Dt s Dt d e ad p ud a e r Dt The Complete Tree S0u 3 S0u 4 S0u S0u 2 S0u 2 S0u S0 S0d S0 S0d S0d 2 S0d 3 S0 S 0d 2 S 0d 4 Backwards Induction • We know the value of the option at the final nodes • We work back through the tree using risk-neutral valuation to calculate the value of the option at each node, testing for early exercise when appropriate 4.17 Valuing the Option Su = 22 ƒu = 1 S ƒ Sd = 18 ƒd = 0 The value of the option is 0.120.25 [0.65231 + 0.34770] = 0.633 e– A Two-Step Example 24.2 22 19.8 20 18 16.2 • Each time step is 3 months 4.18 Valuing a Call Option D 22 20 1.2823 A B 2.0257 18 24.2 3.2 E 19.8 0.0 C 0.0 F 16.2 0.0 • Value at node B = e–0.120.25(0.65233.2 + 0.34770) = 2.0257 • Value at node C =0 • Value at node A = e–0.120.25(0.65232.0257 + 0.34770) = 1.2823 4.19 4.20 Pricing: 2.Monte Carlo An Ito Process for Stock Prices (See pages 225-6) dS mSdt sSdz where m is the expected return s is the volatility. The discrete time equivalent is DS mSDt sS Dt Monte Carlo Simulation • We can sample random paths for the stock price by sampling values for • Suppose m= 0.14, s= 0.20, and Dt = 0.01, then DS 0.0014 S 0.02 S see simple_example.xls Monte Carlo Simulation – One Path (continued. See Table 10.1) Period Stock Price at Random Start of Period Sample for Change in Stock Price, DS 0 20.000 0.52 0.236 1 20.236 1.44 0.611 2 20.847 -0.86 -0.329 3 20.518 1.46 0.628 4 21.146 -0.69 -0.262 Monte Carlo Simulation When used to value European stock options, this involves the following steps: 1. Simulate 1 path for the stock price in a risk neutral world 2. Calculate the payoff from the stock option 3. Repeat steps 1 and 2 many times to get many sample payoff 4. Calculate mean payoff 5. Discount mean payoff at risk free rate to get an estimate of the value of the option A More Accurate Approach (Equation 16.15, page 407) Use d ln S m s 2 / 2 dt s dz The discrete version of this is ln S DS ln( S ) m s 2 / 2 Dt s or m s / 2 Dt s S DS S e 2 Dt Dt Extensions When a derivative depends on several underlying variables we can simulate paths for each of them in a risk-neutral world to calculate the values for the derivative To Obtain 2 Correlated Normal Samples Obtain independent normal samples 1 and 2 and set 1 x1 2 x1 x2 1 2 A procedure known a Cholesky's decomposition can be used when samples are required from more than two normal variables Standard Errors The standard error of the estimate of the option price is the standard deviation of the discounted payoffs given by the simulation trials divided by the square root of the number of observations. Application of Monte Carlo Simulation • Monte Carlo simulation can deal with path dependent options, options dependent on several underlying state variables, & options with complex payoffs • It cannot easily deal with Americanstyle options