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Monte Carlo: Option Pricing Reference: Option Pricing by Simulation, Bernt Arne Ødegaard (http://finance.bi.no/~bernt/gcc_prog/recipes/recipes/node12.html) Introduction Use Monte Carlo to estimate the price of a Vanilla European option priced by Black Scholes equation. Already a closed form solution, therefore, no need to simulate, But, for an illustrative process. … Introduction At maturity, a call option is worth: CT = max (0, ST – X) At an earlier date t, the option will be the expected present value of this: Ct = E[PV(max (0, ST – X))] Risk Neutral Result: Simplify Decision made by a risk neutral investor Also modify the expected return of the underlying asset such that it earns the risk free rate. ct = e-r(T – t)E*[(max (0, ST – X))] Where E*[.] is a transformation of the original expectation. Monte Carlo One way to estimate the value of the call is to simulate a large number of sample values of ST according to the assumed price process, and find the estimated call price as the average of the simulated values. According the “law of large numbers”, this average will converge to the actual call value, depending on the number of simulations that are performed. Lognormally distributed randoms Let x be normally distributed with mean zero and variance one. If St follows a lognormal distribution, the one-period-later price St+1 is simulated as 2)+x (r-½ St+1=Ste …..Lognormally distributed randoms Or more generally, ST St e 1 2 ( r )(T t ) T t x 2 Pricing of European Call Options ct = e-r(T – t)E*[(max (0, ST – X))] Note that here one merely needs to simulate the terminal price of the underlying, ST, the price of ST at time between t and T is not relevant for pricing. …Pricing of European Call Options Proceed by simulating lognormally distributed random variables. Let ST,1, ST,2, …. ST,n denote the n simulated ST values …Pricing of European Call Options We estimate E*[max (0, ST – X)] as the average of option payoffs at maturity, discounted at the risk free rate. n r (T t ) ct e (( max( 0, ST ,i X ) / n) i 1 Price of the Call ( and r constant) Ct SN (d1 ) Xe S 2 log( ) (r )(T t ) X 2 d1 T t d 2 d1 T t 1 N (d ) 2 d e x2 2 dx r (T t ) N (d 2 ) C=Price of the Call S=Current Stock Price T=Time of Expiration X=Strike Price r=Risk-free Interest Rate N()=Cumulative normal distribution function e=Exponential term (2.7183) =Volatility Results: S = 100; X = 110; r = 0.1; sigma = 0.4; t = 6 Exact ct = 53.4636 Monte Carlo: # sims: 10 # sims: 1,000 # sims: 1,000,000 ct = 15.4533 ct = 54.9804 ct = 53.5126 # sims: 100,000,000 ct = 53.4593 # sims: 1,000,000,000 ct = 53.4722