Efficient Monte Carlo Pricer

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Final Project for Empirical Finance course - MaFin - UdeSA

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Efficient Monte Carlo Pricer

  1. 1. EmpiricalFinanceMaestría en Finanzas Universidad de San Andrés<br />Efficient Monte Carlo Pricer<br />March 2009<br />Pablo Siber<br />Prof: M. Azmy<br />
  2. 2. Objective<br />Developa modular Monte Carlo (MC) pricer.<br />Designappropiatebuilding blocks:<br />Random Number Generator (RGN)<br />Stochastic Process (SP)<br />Payoff<br />Pricer<br />
  3. 3. Usage Examples<br />Variance Reduction (VR) Techniques<br />Antithetic Approach<br />Control Variate<br />Importance Sampling<br />Payoff Structures <br />European<br /> American<br />Asian<br />Underlying dynamics<br />Geometric Brownian Motion (GBM)<br />Heston Process<br />Correlated Processes<br />Misc<br />Implied Volatility<br />Greeks Estimation<br />
  4. 4. Naive Estimation<br />No VR Technique<br /> Efficiency of Estimation does not improve with N (num. of samples)!!<br />
  5. 5. Control Variate<br />Idea<br />Use payoff of “known-how-to price” security in order to get a proxy for option prices<br />Efficency improves ITM for obvious reasons (greater correlation)<br />
  6. 6. Importance Sampling<br />Idea<br />Shift probability distribution taking prices more ITM.<br />Then, bigger proportion pdf mass takes significant values for option pricing purposes<br />
  7. 7. VR Techniques Comparison<br /> IS, CV & Antithetic Approach (AC)<br />Relationship with moneyness<br />
  8. 8. American Payoff<br />Implement Longstaff-Schwartz (LS) algorithm<br />Idea<br />Simulate process step-wise<br />Check for worth to exercise realizations<br />Backwards Induction<br />
  9. 9. American Payoff<br />Premium relationship with moneyness<br />Consider Put Prices, not Calls<br />
  10. 10. Asian Payoff<br />Implement Discrete Averaging<br />Need to simulate whole path<br />Comparison of two different CV proxies (analytic formulae)<br />Vanilla Call<br />Geometric Averaging (achieve better results because of greater correlation)<br />
  11. 11. Underlying Dynamics<br />Heston Process<br />Simulate two correlated processes<br />One path example<br />
  12. 12. Underlying Dynamics<br />Heston Process<br />Effects of dynamics according to r, s<br />Effect on Skew<br />Effect on Kurtosis<br />
  13. 13. Underlying Dynamics<br />Correlated paths<br />Implemented Cholesky Decomposition<br />Precaution: check Correlation Matrix is definitive-positive (historical estimates can’t guarantee this feature)<br />Application: Basket of options. Margabe model to check results in 2-D<br />
  14. 14. Misc <br />Greeks Estimation<br />Pathwise Differentiation Method<br />No need to re-sample<br />
  15. 15. Misc<br />Implied Volatilities<br />According to Heston model<br />Generation of smiles<br />Calibration to option prices<br />
  16. 16. Conclusions <br />Possible extensions are countless<br />Always check for robusteness with known examples<br />Modular design is crucial<br />Fully implemented in Matlab (2008a), under the OO paradigm. “Best of two worlds”<br />

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