1. EmpiricalFinanceMaestra en Finanzas Universidad de San Andrs
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 cant 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