Using a Similar European Call as a Control Variate

Summary

The document asks how to reduce Monte Carlo pricing variance for a European call by using a second call with a nearby strike as a control variate. It specifies a stock model with normally distributed log returns and a call payoff, and notes that the second option’s price is known. The central question is how to combine simulated payoffs and apply the covariance-based adjustment coefficient.

The source contains no worked algorithm or answer, so it does not establish how to estimate the coefficient, form the adjusted price estimator, or assess the variance reduction. It identifies high correlation as a condition for effectiveness, but leaves implementation details unresolved. In practice, the payoff simulations must be paired under the same underlying paths, and the known control price and estimated regression coefficient must be used consistently; those steps are not supplied in the document.

Key ideas

• A second call option with a similar strike is proposed as a Monte Carlo control variate.
• The target contract is a European call with a maximum-of-intrinsic-value payoff.
• The stock model assumes normally distributed log returns with risk-free-rate drift and volatility.
• The question notes that high correlation between the option payoffs is needed for useful variance reduction.
• The document poses the application problem but does not provide a pricing algorithm or worked solution.

Tags

# Use of second similar European Option as control variate to simulate a European option


# Use of second similar European Option as control variate to simulate a European option












I understand the idea and math behind the concept of control variate for the sake of variance reduction, but I struggle to apply it to option pricing.

I need to simulate an European option of a stock which has the traditional charateristics:

- $B(S_t) = max(S_t -K,0)$

- The log-returns are normally distributed with parameters $(R-(\sigma^2/2))t$ and $\sigma_t$, with $R$ being the risk-free rate

I need to use a second European call option as control variate with strike $K_2$ and price $C_0$ which is assumed ot be close enough to the first option.

I need to give a algorithhm to simulate the price of the first option using the strike price and price.

I know that I need an high correlation to have an effective variance reduction and that the variance reduction can be computed as $Cov(X,Y)/Var(Y)$ but i don't see how to apply it in that context

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.