3. Monte Carlo Simulation

Transcription

1 3. Monte Carlo Simulation 3.7 Variance Reduction Techniques Math443 W08, HM Zhu Variance Reduction Procedures (Chap 4.5., 4.5.3, Brandimarte) Usually, a very large value of M is needed to estimate V with reasonable accuracy. Variance reduction techniques lead to dramatic savings in computational time Antithetic variate technique Control variate technique Importance sampling Stratified sampling Moment matching Using quasi-random sequences Antithetic Variate Technique Consider to estimate V = E( f ( U) ) where U N ( 0, ). The standard Monte Carlo estimate: M VMC = f ( Ui ) with i.i.d. Ui N ( 0, ). M i= The antithetic variate technique: M f ( Ui) + f ( Ui) VAV = with i.i.d. Ui N ( 0, ) M i= We can prove that f ( Ui) + f ( Ui) var = var ( f ( Ui) ) + cov ( f ( U i), f ( U i) ) var ( f ( Ui )) if f is monotonic.

2 Antithetic Variate Technique It involves calculating two values of the derivative. V : calculated in the usual way V : calculated by the changing the sign of all the random standard normal samples used for V V is average of the V and V, the final estimate of the value of the derivative Antithetic Variate Technique Note:. Monte-Carlo works when the simulated variables "spread out" as closely as possible to the true distribution.. Antithetic variates relies upon finding samplings that are anticorrelated with the original random variable. 3. Works well when the payoff is monotonic w.r.t. S 4. Further reading: --P.P. Boyle, Option: a Monte Carlo approach, J. of Finanical Economics, 4: (977) --Boyle, Broadie and Glasserman, Monte Carlo methods for security pricing J. of Economic Dynamics and Control, : 67-3 (997) --N. Madras, Lectures on Monte Carlo Methods, 00 Example: European call payment. S(0) = $50, K = 5, T = 5 months, and annual risk-free interest rate r = 0% and a volatility σ= 40% per Black-Schole Price: $5.9 MC Price with 00,000 simulations: $5.780 Confidence Interval [5.393, 5.67] MCAV Price with 00,000 simulations: $5.837 Confidence Interval [5.65, 5.058] (ratio:.75) 6

3 Example: Butterfly Spread Consider a butterfly spread with S(0) = $50, K = $55, K = $60, K 3 = 65, T = 5 months, and annual risk-free interest rate r = 0% and a volatility σ= 40% per Exact price: $0.64 MC price with 00,000 simulations: $ Confidence Interval [0.607, 0.673] MCAV price with 50,000 simulations: $ Confidence Interval [0.598, 0.698] 7 ( A) ( ) Instead of estimate EV, we can consider another random variable ( ) = EV ( ) var ( ) where V is "close" to V with known mean E V. In this case, and Z= V + E V V A B B B A B EZ A ( Z) = var ( VA VB) = var ( VA) cov ( VA, VB) + var ( VB) ( Z) < ( V ) We want that var var. A 8 The control variate technique is applicable when there are two derivatives, say A,B, which have some general known positive correlation B has an analytic solution available Simulations for A and B use the same random number streams and t and are carried out in parallel Control variate estimate gives a better estimate to the value of derivative A using: V + V V A B_ true B 9 3

4 Example: European call payment. S(0) = $50, K = 5, T = 5 months, and annual risk-free interest rate r = 0% and a volatility σ= 40% per Black-Schole Price: $5.9 MC Price with 00,000 simulations: $5.780 Confidence Interval [5.393, 5.67] MCCV Price with 00,000 simulations: $5.9 Confidence Interval [5.880, 5.963] (ratio: 9.4) 0 We can generalize the previous technique to for any θ. In this case, var We can prove θ ( ( ) ) Z = V + E V V θ A B B ( Zθ ) = var ( VA θvb) = var ( VA) θcov ( VA, VB) + θ var ( VB) that var ( Zθ ) < var ( VA ) if and only if cov ( VA, VB) θ. ( V ) 0< < var B Example: European call payment. S(0) = $50, K = 5, T = 5 months, and annual risk-free interest rate r = 0% and a volatility σ= 40% per Black-Schole Price: $5.9 MC Price with 00,000 simulations: $5.780 Confidence Interval [5.393, 5.67] MCCV Price with 00,000 simulations: $5.883 Confidence Interval [5.7, 5.054] (ratio:.645) 4

5 Example: Arithmetic Average Asian Option Consider the option on a stock with no-dividend payment. S(0) = $50, K = 50, T = 5 months, and annual risk-free interest rate r = 0% and a volatility σ= 40% per We could use the sum of the asset prices as a control variate as we know its expected value and Y is correlated to the option itself N N N r( N+ ) t ri t e E S( ti ) = E S( i t) = S0 e = S0 r t i= 0 i= 0 i= 0 e Another choice of the control variate is the payoff of a geometric average option as this is known analytically N N max S( ti ) K, 0 i= 3 5