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I'm currently taking a Mathematical Finance module at University and one of the recommended texts is “An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation” by D.J. Higham. One of the chapters in this book is about using Monte Carlo Methods to value an option and approximating Greeks.

In this chapter, he defines $V(S,t)$ to be the time t value of a European option with payoff $F(S_{T})$, when the underlying stock's price is S, and we aim to approximate the partial derivative of V with respect to S (the delta) at time zero. Using finite differences, we can say that:

$$ \frac{\partial V}{\partial S} \approx \frac{V(S+h,t)-V(S,t)}{h} $$

Hence, we can use the risk neutral valuation formula:

$$ V(S_{0},0) = e^{-rT}\mathbb{E}_{\mathbb{Q}}(F(S_{T})) $$

However, Higham goes on to write that we can hence approximate the time zero delta by computing Monte Carlo estimates of the two expected values in:

$$ e^{-rT}\frac{\mathbb{E}_{\mathbb{Q}}(F(S_{T}) \mid S(0)=S_{0})-\mathbb{E}_{\mathbb{Q}}(F(S_{T}) \mid S(0)=S_{0}+h)}{h} $$

I really don't understand why this is the case - shouldn't the numerator be the other way round? I'd have passed it off as a printing error but all the examples in that chapter (e.g. writing out a Monte Carlo algorithm) following the above expression are consistent with it. Could someone explain to me why the numerator isn't the other way round i.e. multiplied by -1?

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You are right, this does not make sense. Your intuition is the correct one.

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  • $\begingroup$ Remark: @buffalo, unless he directly expresses the delta as the quantity of stocks you need to be delta neutral (so if delta of the derivative is positive, he expresses the quantity of stocks to hold as minus delta). $\endgroup$
    – Quantuple
    May 17 at 6:07

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