# Using stock prices as control variate

In this paper , the author suggested using terminal stock price as control variates. However, I do not understand as we only observe stock price distribution at the terminal, and we do not have any analytical solutions for the stock price, therefore E(F(a*) = F(a) + (S(T) - E(S(T))) and the last bit will always be the same (S(T) = E(S(T))), thus it does not provide correction for our estimate. Thank for any comments.

• In a risk-neutral world with constant interest rates and dividend yield, we have $\mathbb{E}_t^\mathbb{Q}[S_T]=S_te^{(r-q)(T-t)}$, irrespective of the distribution of the stock price. Using arithmetic Asian options as an example, the paper also highlights that European call options or geometrically averaged options would be better suited. The idea is that $\mathbb{C}\text{ov}(S_T,\xi_T)>0$, where $\xi_T$ is the payoff of your derivative. Thus, using the stock price itself may help to reduce the bias a bit – Kevin Dec 23 '20 at 12:07
• Hi, Kevin, I tried the stock price as control variate, however, I found the stock prices itself carries very large variance, which is higher than the covariance can off-set, thus the variance actually increases. Do you have any suggestions for variance reduction for the Heston model where it depends on two stochastic processes. And I have tried BS European and low /no correlation was found. @ Kevin – Lin Lex Dec 23 '20 at 12:37
• You can't use the Black-Scholes option price in a Heston model simulation. Suppose you value an exotic option. Find another asset with similar payoff which has a closed-form solution. Typical examples include the stock price and a vanilla option. These payoffs will positively correlate with the payoffs of your exotic option. Value the auxiliary derivative using your MC simulations and compute the bias. Subtract this bias from your MC estimate for the exotic option (you can scale the bias by a $\beta$-style coefficient to further reduce variance, akin to a OLS regression or minimum hedge ratio) – Kevin Dec 23 '20 at 13:15