Expected Call and Put Payoffs for a Normally Distributed Log Price

Summary

The document gives a closed-form expectation for a call or put payoff when the log of the underlying price follows a normal distribution. A sign indicator selects the option type, while the expression uses the normal cumulative distribution function to account for the chance that the option finishes in the money and the expected underlying value in that region.

The result follows from integrating the log-normal density and includes the Black–Scholes formula as a special case, with its mean and variance parameters set by the initial price, interest rate, volatility, and time to expiry. This makes the expression useful when the log-price distribution is normal but its parameters do not match the usual Black–Scholes setup. The document does not discuss transaction costs, early exercise, or departures from normality, so the formula’s applicability depends on those modeling assumptions.

Key ideas

• A normal log-price distribution implies a log-normal underlying price.
• A single closed-form expectation covers both call and put payoffs.
• The option type is selected by a sign indicator.
• Black–Scholes is a special case with specific mean and variance parameters.

Tags

# Black-Scholes formula given arbitrary value of $S_{T}$


# Black-Scholes formula given arbitrary value of $S_{T}$












Is there a formula for Black and Scholes when we have expected payoff $\mathbb{E}[\max(se^{X}-K,0)]$ for $X$ having any normal distribution?

## Answer by Kevin (score 6, accepted)

https://quant.stackexchange.com/a/57659

Let $X\sim N(m,v^2)$ be normally distributed. Then, for all strikes $K>0$ and $\omega\in\{-1,1\}$, \begin{align*} \mathbb{E}[\max\{\omega(e^X-K),0\}]=\omega e^{m+\frac{1}{2}v^2}\Phi\left(\omega\frac{m-\ln(K)+v^2}{v}\right)-\omega K\Phi\left(\omega\frac{m-\ln(K)}{v}\right), \end{align*} where $\Phi$ is the standard normal cdf. The indicator $\omega$ is used to differentiate whether you have a call option ($\omega=1$) or a put option ($\omega=-1$).

It follows directly from integrating the log-normal density. Brigo and Mercurio call this ``a useful calculation''.

The Black-Scholes formula is a special case of this equation where $m=\ln(S_0)+\left(r-\frac{1}{2}\sigma^2\right)T$ and $v^2=\sigma^2T$.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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