1 / 12

Monte Carlo: Option Pricing

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.

jamar
Download Presentation

Monte Carlo: Option Pricing

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Monte Carlo: Option Pricing Reference: Option Pricing by Simulation, Bernt Arne Ødegaard (http://finance.bi.no/~bernt/gcc_prog/recipes/recipes/node12.html)

  2. 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.

  3. … 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))]

  4. 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.

  5. 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.

  6. 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 • St+1=Ste(r-½2)+x

  7. …..Lognormally distributed randoms • Or more generally,

  8. 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.

  9. …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

  10. …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.

  11. Price of the Call ( and r constant) • 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

  12. Results: • S = 100; X = 110; r = 0.1; • sigma = 0.4; t = 6 • Exact ct = 53.4636 • Monte Carlo: • # sims: 10 ct = 15.4533 • # sims: 1,000 ct = 54.9804 • # sims: 1,000,000 ct = 53.5126 • # sims: 100,000,000 ct = 53.4593 • # sims: 1,000,000,000 ct = 53.4722

More Related