Differentiating a European Call Payoff with Respect to Spot

Summary

The document examines the derivative of a call option payoff, defined as the positive part of terminal asset price minus strike. The question proposes a piecewise expression for the quantity formed by multiplying the payoff derivative by spot and subtracting the payoff, and asks whether the payoff can be differentiated as if it were simply the positive part of spot.

The answer corrects that step: differentiating the call payoff with respect to its underlying price gives the price multiplied by an indicator that the option finishes above the strike. The response flags an ambiguity in the question about whether the derivative is with respect to terminal price or initial spot. It does not work through the resulting piecewise values or discuss a pricing model, so readers must keep the differentiation variable and the payoff's exercise region explicit when extending the calculation.

Key ideas

• A call payoff is zero below the strike and increases linearly above it.
• The payoff derivative with respect to terminal underlying price is zero below the strike and one above it.
• Multiplying that derivative by spot produces spot times an indicator for finishing above the strike.
• The derivative's meaning depends on whether the variable is terminal price or initial spot.

Tags

# Differentiating a Payoff


# Differentiating a Payoff












Okay this is probably going to be an extremely easy/straightforward question but I thought I should post it here just to double check. Suppose I have a payoff $\Phi = (S_{T}-K)^{+}$. Now let's say I now have an equation: $u = s\partial_{s}\Phi - \Phi$, this means that given a payoff $\Phi$ as given above then, substituting this payoff into the equation and assuming $S_T = S_{0}\exp((r-1/2)T+\sigma\sqrt{T}Z_i))$ then I should get:

$u = max(S_{T},0) - max(S_{T}-K,0)$, right?

And from this equation, the possible solutions should be:

If $S_{T} > K$, $u = K$, if $S_{T} < K$ and $S_{T} > 0$, $u = S_{T}$, and if $S_{T} < K$ and $S_{T} < 0$ then $u = 0$.

Is all of this correct? I know this is really trivial but I just thought I should check...

## Answer by Mark Joshi (score 0, accepted)

https://quant.stackexchange.com/a/25941

$$s\partial_{s}\Phi = S_T I_{S_T>K}.$$ so no.

(I am not absolutely sure whether you want to differentiate wrt S_T or S_0 however.)

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