Differentiating the Risk-Neutral Call Price by Strike

Summary

The document asks how to differentiate the risk-neutral value of a call option with respect to its strike. It starts from the discounted expected payoff and restricts the integral to terminal prices above the strike, then cites the familiar results: the first strike derivative is the negative discounted upper-tail probability, and the second derivative is the discounted terminal-price density at the strike.

It raises use of Leibniz's rule despite the infinite upper integration bound, but does not supply the derivation or resolve the notation issue in the displayed integral. The relationship is useful for connecting call prices across strikes to the risk-neutral distribution, though it assumes a suitable density and a correctly specified pricing measure. This is a question rather than a complete tutorial, so readers seeking the explicit differentiation steps will need another source.

Key ideas

• The first strike derivative of a call price is the negative discounted probability that the terminal asset price exceeds the strike.
• The second strike derivative is the discounted risk-neutral density evaluated at the strike.
• The document asks how Leibniz's rule applies when the integral has an infinite upper bound.
• The stated derivative relationships connect option prices across strikes to the terminal risk-neutral distribution.

Tags

# Differentiating the Risk Neutral Price of a Call Option


# Differentiating the Risk Neutral Price of a Call Option












I know that the risk-neutral price of a call option with strike price $K$ is given by:

$$C(K,S_T) = e^{-rT}\int_{0}^{\infty }(S_T-K)^+g(S_T)dS_t$$

Since a payoff is only valid when $ S_T >K $ this turns into:

$$C(K,S_T) = e^{-rT}\int_{K}^{\infty }(S_T-K)\times g(S_T)dS_t$$

Now I wanted to differentiate the call formula once and twice with respect to K, so $\frac{\partial C}{\partial K}$ and $\frac{\partial ^2C}{\partial K^2}$ but this is where I am a little stuck. Would I use Leibniz integral rule where $f(K,S_T) = (S_T-K)\times g(S_T)$? If so there is the issue where one of the bounds is infinity which won't work out. I've seen books and papers calculating the first and second derivative but they do not show the explicit steps.

So in the Hull book, $\frac{\partial C}{\partial K} = -e^{-rT}\int_{K}^{\infty } g(S_T)dS_t$ and $\frac{\partial^2 C}{\partial K^2} = e^{-rT}g(K)$, but how were these derivates actually calculated? A brief explanation would be greatly appreciated.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.