Monte Carlo Flashcards
MC price convergence
At the beginning we have big variations in estimated prices from MC and the true price
When we increase the numbers of simulations our Monte Carlo estimated price gets closer to the true price.
The variance of muh decreases linearly with N. Central Limit Theorem.
The confidence bands also get narrower as N increases.
So in sum in order to achieve a precise estimate we need a large N! Then MC provides us with an unbiased and consistent estimate of the price of any given european option.
When to use MC simulation?
Monte-Carlo is useful for pricing path-dependent derivatives
Monte-Carlo is easy to implement and it can also be easily extended to simulate multiple correlated processes:
→ Stochastic volatility models, where volatility follows its own process (like the Heston model)
→ Options on multiple underlyings
It can be easily used to simulate processes other than Geometric Brownian Motions
It can also be used to compute the Greeks by changing the parameter under consideration by a small amount
Why is MC not used all the time?
→ American options are difficult to price using Monte-Carlo
→ Monte-Carlo can take a long time
However, there are many ways to make Monte-Carlo more efficient, e.g., control variates: correct sample bias using a related product with a known price, antithetic variates, etc
What are two methods for reducing variance without increasing computational time to much?
A. Antithetic variables: recycle the draws X1, … Xn to reduce the variance of the estimator. We have to suppose that the distribution X is symmetric that X = -X. Then we can use the same set of draws and calculate two estimators.
B. Control variates: use the same draws to compute the unknown option price and the known option price where the two options are similar.
Variance reduction technqiue: Antithetic Variables
A. Antithetic Variables:
This method involves using pairs of negatively correlated variables to calculate complementary estimates. For instance, if one random variable in the simulation leads to a high estimate, its antithetic counterpart (often calculated by subtracting the random number from one) would produce a low estimate. By averaging these complementary estimates, the variance of the simulation outcome can be reduced. This is particularly effective because the errors in the complementary estimates often cancel each other out, leading to a more stable and accurate mean estimate
Control Variates:
This technique involves using a variable (the control variate) that is highly correlated with the output of interest, but whose expected value is known. By building a linear model that predicts the simulation results from the control variate, and adjusting the simulation outcomes based on the differences observed in the known values of the control variates, the variance of the estimator can be reduced. Essentially, the known behavior of the control variate is used to reduce uncertainty in the estimates of the target variable, thus improving the precision of the simulation.
