Deriving the First Strike Derivative in the Breeden–Litzenberger Formula

Summary

The document explains an intermediate step in deriving the Breeden–Litzenberger relationship between option prices and the distribution of the underlying at expiry. A call price is expressed as the discounted expected payoff, integrated over terminal prices above the strike. The question is why differentiating this expression with respect to strike removes the payoff term at the lower integration boundary.

The answer applies the Leibniz integral rule. The boundary contribution is zero because the payoff factor equals zero when terminal price equals strike; differentiating the payoff inside the integral then gives the negative probability density integrated above the strike. The discount factor is treated as constant with respect to strike. This is a focused calculus explanation of the first strike derivative, rather than a full derivation of the second-derivative result or a discussion of assumptions such as discounting conventions and market conditions.

Key ideas

• The call value can be written as a discounted integral of the terminal payoff over prices above the strike.
• Differentiating an integral with a moving lower bound requires the Leibniz integral rule.
• The boundary term vanishes because the call payoff is zero when terminal price equals strike.
• The first strike derivative is the negative discounted tail probability under the stated setup.

Tags

# How to derive the Breeden-Litzenberger formula?


# How to derive the Breeden-Litzenberger formula?












I was going through a proof of the Breeden-Litzenberger formula and I was stuck on one of the intermediate steps.

The pdf for the prices of the underlying at expiry is defined as $f(x)$ and $S_T$ is the price of the underlying at expiry and $K$ is the strike price so the probability that the underlying expires at a price higher than the strike is: Moreover, the fair value of the option is the expected payout of the option at expiry, discounted by the risk-free rate till expiry, or mathematically as So now, if I want to calculate the fair value of the option, Now, if I differentiate the call option value with respect to the strike, how do I arrive at this answer shown below and how does $(x-K)$ just disappear?

## Answer by phdstudent (score 5, accepted)

https://quant.stackexchange.com/a/60252

You need to use the leibniz integral rule. More details here: https://en.wikipedia.org/wiki/Leibniz_integral_rule

But in a nuthshell (ignoring the $e^{-r\tau}$ which is a constant):

$$\frac{\partial C}{\partial K} = -(K-K)f(x)\frac{\partial K}{\partial K} + \int^\infty_K \frac{\partial}{\partial K}(x-K)f(x)dx = -\int^\infty_K f(x)dx$$

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