Final Project for Empirical Finance course - MaFin - UdeSA

1. EmpiricalFinanceMaestría en Finanzas Universidad de San Andrés Efficient Monte Carlo Pricer March 2009 Pablo Siber Prof: M. Azmy

2. Objective Developa modular Monte Carlo (MC) pricer. Designappropiatebuilding blocks: Random Number Generator (RGN) Stochastic Process (SP) Payoff Pricer

3. Usage Examples Variance Reduction (VR) Techniques Antithetic Approach Control Variate Importance Sampling Payoff Structures European American Asian Underlying dynamics Geometric Brownian Motion (GBM) Heston Process Correlated Processes Misc Implied Volatility Greeks Estimation

4. Naive Estimation No VR Technique Efficiency of Estimation does not improve with N (num. of samples)!!

5. Control Variate Idea Use payoff of “known-how-to price” security in order to get a proxy for option prices Efficency improves ITM for obvious reasons (greater correlation)

6. Importance Sampling Idea Shift probability distribution taking prices more ITM. Then, bigger proportion pdf mass takes significant values for option pricing purposes

7. VR Techniques Comparison IS, CV & Antithetic Approach (AC) Relationship with moneyness

8. American Payoff Implement Longstaff-Schwartz (LS) algorithm Idea Simulate process step-wise Check for worth to exercise realizations Backwards Induction

9. American Payoff Premium relationship with moneyness Consider Put Prices, not Calls

10. Asian Payoff Implement Discrete Averaging Need to simulate whole path Comparison of two different CV proxies (analytic formulae) Vanilla Call Geometric Averaging (achieve better results because of greater correlation)

11. Underlying Dynamics Heston Process Simulate two correlated processes One path example

12. Underlying Dynamics Heston Process Effects of dynamics according to r, s Effect on Skew Effect on Kurtosis

13. Underlying Dynamics Correlated paths Implemented Cholesky Decomposition Precaution: check Correlation Matrix is definitive-positive (historical estimates can’t guarantee this feature) Application: Basket of options. Margabe model to check results in 2-D

14. Misc Greeks Estimation Pathwise Differentiation Method No need to re-sample

15. Misc Implied Volatilities According to Heston model Generation of smiles Calibration to option prices

16. Conclusions Possible extensions are countless Always check for robusteness with known examples Modular design is crucial Fully implemented in Matlab (2008a), under the OO paradigm. “Best of two worlds”