Risk-Neutral Drift in Monte Carlo Option Pricing

# option pricing: monte carlo simulations that include expected return


# option pricing: monte carlo simulations that include expected return












In the book Derivatives Markets (McDonald, 3rd edition), there's a chapter on Monte Carlo valuation of option prices. It starts with simulating stock prices (p578) with the following equation:

```
St = price after time T
So = starting price
alpha = risk-free rate
delta = dividend yield
sigma = volatility
Z = random variable drawn from standard normal distribution
```

To simplify things lets say that T = 1, alpha and delta = 0. But I also add in an expected underlying return variable, u. That would give the simulated stock price at end of period T to be:

> St = S0^(u - 0.5 * sigma^2) + sigma * Z)

If I do 10000 simulations with the above equation, using S0 = 100, u = 0.03, sigma = 0.1, I get a mean St of 103.10.

If strike price (K) for a call was 100, I get a mean max[0, St - K] of 5.79. I interpret this as the fair price of the option.

If I instead set u = 0, I get mean St of 100.13 and mean max[0, K - 100] of 4.12.

According to this, the price of the option is depending on the expected return, u, of the underlying. But I've read 1000 times that the option price does not depend on expected return. Can someone clarify what I'm doing wrong?

## Answer by emot (score 1)

https://quant.stackexchange.com/a/66324

Option price does not depend on $\mu$, but it depends on the risk free rate $r$. This means that when you do simulation, you should simulate: $$S_T=S_0 e^{(r-0.5\sigma^2)T+\sigma W_t}$$ rather than: $$S_T=S_0 e^{(\mu-0.5\sigma^2)T+\sigma W_t}$$.

When we want to price options we start with assumption that the stock prices follows the process with mean $\mu$. But then we change the drift parameter $\mu$ to risk-free rate $r$ because this allows us to calculate replicating portfolio value. We call this step - changing real world measure to risk neutral measure. This comes from the fact that we assume that the market is arbitrage-free and we can replicate the option by proper trading strategy (replicating portfolio). If it is possible, then the option price is just a price of replicating portfolio that replicates the option payoff and it turns out that to calculate the price of replicating portfolio (and the option itself) we have to change the real drift parameter to risk free rate and then evaluate expectation of the discounted payoff. Therefore the price of replicating portfolio is: $$e^{-rT}E[max(S_T-K,0)]=e^{-rT}E[max(S_0 e^{(r-0.5\sigma^2)T+\sigma W_t}-K,0)]$$ And by the arbitrage-free argument this is equal to option price.

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