Delta-Weighted Stock Control Variate

Article Quant Q&A · Author: Lin Lex

Summary

The document discusses reducing Monte Carlo pricing variance for a European put under a stochastic volatility simulation. The questioner uses the terminal stock price as a control variate and reports a modest reduction in the estimated standard error after many simulated paths. They also say that using a plain European put payoff as the control performed poorly because it was weakly correlated with the target payoff.

The proposed alternative is a discretized delta-weighted stock increment. In a continuous-time idealization, the delta hedge’s gains process is expected to leave a deterministic risk-free drift contribution, making it a potentially more effective control. The suggestion is conceptual rather than worked through: the document supplies no implementation details, derivation for the stated simulation, or comparative results. Its practical benefit depends on the model, discretization, and how the control coefficient is estimated.

Key ideas

A terminal stock price can serve as a control variate, but its effectiveness depends on its correlation with the option payoff.
The example reports only a modest standard error reduction with the chosen stock-price control.
A delta-weighted stock increment is suggested as a potentially stronger control variate.
In the continuous-time limit, the hedged gains contribution is associated with risk-free drift.
The proposed alternative is not demonstrated with implementation details or new results.

Tags

Options Statistics Backtesting Derivatives pricing

Full text

# Improving control variate for variance reduction


# Improving control variate for variance reduction












I have tried stock price as control variate for my monte carlo simulation, and I am trying to reduce the variance of my estimated price for European Put option. And the code look like this:

```
path <- function(rho,T, S0, r, kappa, theta,xi,v0, Maturity ) { 
  
  dt = Maturity / T
  COV <- matrix(c(1,rho,rho,1), nrow = 2,ncol = 2)
  W <- rmvnorm(T, , COV)
  W_v <- W[,1]
  W_s <- W[,2]
  
  Vt <- rep(0,T)
  Vt[1] <- v0
  St <- rep(0,T)
  St[1] <- S0
  
  for (i in 2:T){
    Vt[i] <- Vt[i-1] + kappa* (theta - Vt[i-1])*dt + xi* sqrt(Vt[i-1]) * W_v[i-1] * sqrt(dt)
    St[i] <- St[i-1]*exp((r-0.5*Vt[i-1]) *dt + sqrt(dt*Vt[i-1]) * W_s[i-1])
    
  }
  St
}
set.seed(214)
N<-2500
Stm <- replicate(100000, path(0.5,N,45,-0.02,10,0.09,0.2,0.09,2))
payoff <- pmax (40 - Stm[N,], 0 ) *exp(-0.02 * 2)
Hest_MC1 <- mean(payoff)
S_aver <- mean(Stm[N,])

ST <- 45* exp(-0.02 *2 )

hest_mc <- Hest_MC1 + -reg$coefficients[[2]]*(S_aver - ST)
hest_mc
reg <- lm(payoff~Stm[N,])

reg$coefficients

sd_old <- sqrt(var(payoff)/length(payoff))
sd_reduced <- sqrt(var(payoff)/length(payoff) + reg$coefficients[[2]]^2 *var(Stm[N,])/length(Stm[N,])
                         + 2*( -reg$coefficients[[2]] )*cov(Stm[N,],payoff)/length(payoff))
# result
        sd_old sd_reduced
[1,] 0.06926151 0.04898853
```

As you can see, the result only improved marginally after 100,000 Simulation. Do anyone has any Idea how to continue improve on this? I did try the European Put (plain) as control variate, however, the payoff is very weakly correlated with the payoff of the option I tried to reduce variance on.

Any comment would be appreciated.

## Answer by Arshdeep (score 0)

https://quant.stackexchange.com/a/60132

Try a discretized form of the $Put Delta*Stock increment$, since the injection of this term should produce ideally (in the limit case) a 0 variance variable, namely the risk free drift.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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

Using a Delta-Weighted Stock Increment as an Option Control Variate

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