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Need some help in understanding the author's simplification here (from Emanuel Derman's The Volality Smile). The equation is set to $V(S, t)$, which represents the value of a derivative security at time $t$ with the asset at price $S$. See below: enter image description here

Here, $B$ is a constant and $B>0$, $\delta(\cdot)$ is the Dirac delta function, $C(S, K)$ is the price of a call option with an underlying asset at price $S$ with strike $K$, and $H(\cdot)$ is the Heaviside function.

I believe he simplifies the second term of the equation using the fact that the lower bound begins at $B$ and therefore the Heaviside function will always be 1 when integrating over $K$(though do please let me know if that is incorrect). However, I really am not quite sure where to begin on the first integral. I know there are some tricks when integrating over the Dirac delta function, but I'm not sure it applies in this case given the bounds of the integral.

If anyone has a few moments to look into this I would very thankful!

Saul5813
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  • The integral is the same as integrating over $\mathbb{R}$ ($\delta$ is supported at the origin, so the normal computation process applies). Do you know what $\int_{\mathbb{R}}f(y)\delta(y-x)dy$ equals? – yona Sep 10 '22 at 04:50
  • Hi @yona. Should be $f(x)$ I assume. In my case, I guess I don't know how to deal with both $K$ and $C(S, K)$ in that integral. – Saul5813 Sep 10 '22 at 04:55
  • That's correct. To apply that here, fix $S$ and consider f(K)=K*C(S,K). – yona Sep 10 '22 at 04:57
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    Ah now I see. Thanks @yona – Saul5813 Sep 10 '22 at 05:00

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Think I got it figured out thanks to yona:

Using the property that $$\int_{\mathbb{R}}\ \ f(u)\delta(u-a)du \ = \ f(a)$$

and considering $f(K) = K \times C(S, K)$, we can say

$$\int_B^{\infty}\ \ f(K)\delta(K - B)dK \ = \ f(B) \ = \ BC(S,B)$$

Saul5813
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  • Mathematically, this is not correct, there is a problem if interpretation when the Dirac delta is evaluated at the boundary of an integral, see e.g. https://math.stackexchange.com/questions/4427947/what-is-the-value-of-the-integral-int-inftya-deltax-a-dx-and-relate/4427952#4427952 – LL 3.14 Sep 10 '22 at 09:14
  • Thanks @LL3.14 - good to know. So I suppose then that the author has introduced ambiguity by way of his initial notation? Either way I assume the result above is probably what the author was going for. – Saul5813 Sep 10 '22 at 15:28