Sep
2

## Pathwise Derivative vs Finite Difference For Greeks Computation

I was asked how to improve the convergence speed of Greeks calculation with Monte Carlo simulation. Besides those variance reduction techniques such as antithetic, or low discrepancy random numbers, one efficient way is to use pathwise derivative instead of finite difference.

This is the most widely used & straightforward method, as its name suggests, basically, to estimate dy/dx, we increase x by a very small quantity to x1, re-calculate the option value y1, and then estimate the sensitivity as (y-y1)/(x1-x). Thus this method requires us to calculate the option value at least twice (three times for central difference method), and obviously is a big challenge when we have to simulate lots of times.

contrary to finite difference approximation, pathwise derivative estimate derivative directly, without simulating multiple times. It takes advantage of additional information about the dynamics and parameter dependence of a simulated process. Simply put, by the chain rule, if we could find another variable z such that , and there are solutions to the two derivatives at the right hand side, the pathwise derivative estimator can be applied, and for most cases, stock price S(T) for European option or S(tau) for American option is an excellent choice of z, tau is the optimal timing for exercise. Please read the chapter 7 of Monte Carlo Methods in Financial Engineering (Stochastic Modelling and Applied Probability) (v. 53) for detail.

Below are the sample results for the Greeks calculation for an American option without dividend, time to maturity 1 year, 20% volatility. Pathwise derivative estimator saves 2/3 ~ 3/4 computation time.

Delta, Gamma and Vega converge to their true values much quicker, here old and new code refer to

Not bad.

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**1, Finite Difference approximation**This is the most widely used & straightforward method, as its name suggests, basically, to estimate dy/dx, we increase x by a very small quantity to x1, re-calculate the option value y1, and then estimate the sensitivity as (y-y1)/(x1-x). Thus this method requires us to calculate the option value at least twice (three times for central difference method), and obviously is a big challenge when we have to simulate lots of times.

**2, pathwise derivative estimate**contrary to finite difference approximation, pathwise derivative estimate derivative directly, without simulating multiple times. It takes advantage of additional information about the dynamics and parameter dependence of a simulated process. Simply put, by the chain rule, if we could find another variable z such that , and there are solutions to the two derivatives at the right hand side, the pathwise derivative estimator can be applied, and for most cases, stock price S(T) for European option or S(tau) for American option is an excellent choice of z, tau is the optimal timing for exercise. Please read the chapter 7 of Monte Carlo Methods in Financial Engineering (Stochastic Modelling and Applied Probability) (v. 53) for detail.

Below are the sample results for the Greeks calculation for an American option without dividend, time to maturity 1 year, 20% volatility. Pathwise derivative estimator saves 2/3 ~ 3/4 computation time.

Delta, Gamma and Vega converge to their true values much quicker, here old and new code refer to

**Finite Difference approximation**and**pathwise derivative estimate**, respectively, and the yellow line is the true value.Not bad.

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