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Given the following code (S0 = Initial Share Price, r= (risk-free) interest rate, K=Strike, Sigma= Standard Deviation, T=years, nExp=Number of Experiments)

def MonteCarlo_OptionPricing(S0, K, r, sigma, T, nExp=100000):

    rMC = rd.randn(n_exp) * sigma * np.sqrt(T) + (r - sigma**2 / 2) * T
    ST = S0 * np.exp(rMC) 
    
    cT = np.maximum(ST-K, 0)

    c0 = np.mean(cT) * np.exp(-r*T)

    return c0

For large nExp it will basically return almost the same value for European options as the (standard) Black-Scholes-Model.

My question concerns (r - sigma**2 / 2) * T: What exactly is this part accounting for? Is that taking care of the drift?

Any input is welcome!

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The term $r - \frac{\sigma^2}{2}$ is used to account for the risk-neutral drift in the spot price evolution.

We do not use drift $\mu$ because we are simulating the stochastic process in the risk-neutral world, which states that evolution occurs at the risk-free rate.

You penalize this riskless rate using the volatility term $\frac{\sigma^2}{2}$ because of Ito's calculus, which suggests that we need this second term, full explanation here - Geometric Brownian motion - Volatility Interpretation (in the drift term).

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  • $\begingroup$ So Iet's denote $\mu = a + \frac{\sigma^2}{2}$ and $a = \mu - \frac{\sigma^2}{2}$. Then we would consider $\mu$ to be some kind of real world drift. Whereas $a$ denotes the risk-neutral drift? $\endgroup$
    – Marlon Brando
    Commented May 2 at 21:30
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    $\begingroup$ And the intuition behind this, is that in the risk-neutral world you mentioned, where also the Black-Scholes model takes place, we are prohibited from benefiting from the real world drift, if we set up a replication portfolio by selling the call and buying the underlying asset (which would yield a riskless profit)? $\endgroup$
    – Marlon Brando
    Commented May 2 at 21:38
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    $\begingroup$ @MarlonBrando from what I understand, real world drifts aren't computed using the formula you suggested, but rather based on time series of log returns (the simplest model) or other methods. The risk-neutral drift is based on the riskless rate, not a return $\mu$. $\endgroup$
    – KaiSqDist
    Commented May 3 at 13:52
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    $\begingroup$ Thanks! So then in my example 𝜇 would be the risk-neutral drift, while 𝑎 would denote the riskless rate? Sorry for confusing! $\endgroup$
    – Marlon Brando
    Commented May 5 at 14:50
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    $\begingroup$ Yes you are right. $\endgroup$
    – KaiSqDist
    Commented May 5 at 15:40

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