Pricing a Payoff on Two Correlated Geometric Brownian Motions
Summary
The document considers the expected positive part of a linear combination of a geometric Brownian motion observed at two different times, less a strike. It notes that a sum of lognormal variables generally lacks a closed-form distribution, so treating the sum as lognormal is an approximation rather than an exact solution. A cited response describes moment matching as one way to approximate sums of independent lognormals, while warning that the approximation may not capture behavior near zero.
For the two-time payoff, the response factors the later value into the earlier value multiplied by an independent lognormal increment. Conditioning on the earlier value turns the inner expectation into a Black–Scholes-type calculation; the remaining expectation over the earlier value can then be integrated numerically using its lognormal density. The answer provides a computational approach, not a closed-form final expression or numerical validation. The alternative moment-matching discussion concerns independent lognormals and does not directly resolve the dependence structure of the two observations.
Key ideas
- A sum of lognormal variables generally does not have a closed-form distribution.
- Moment matching can approximate a lognormal sum, but may misrepresent its lower tail.
- Factor the later geometric Brownian motion value into an earlier value and an independent increment.
- Conditioning on the earlier value yields a Black–Scholes-type expectation.
- Numerical integration over the earlier value completes the proposed valuation method.
Tags
Full text
# Linear combination of geometric Brownian motion
# Linear combination of geometric Brownian motion
Let $X_t= e^{\left(\mu-\sigma^2/2 \right)t+\sigma W_t}$ be a geometric Brownian motion with drift $\mu$ and volatility $\sigma$. I am trying to find an analytical solution to
$$\mathbb{E}\left[ \max(a X_T + b X_S -K,0)\right],$$ where $a$, $b$ and $K$ are constants and $0<S<T$.
My objective is to find the critical point as from which $aX_T + bX_S$ will be greater than $K$ so that I can disregard the maximum function and evaluate the expectation.
Am I right to say that if I had only $Y= X_T + X_S$, I could use the relation $X_T + X_S=2X_S + X_T - X_S$ to find its mean and variance and subsequently find the critical point?
Is there any way to proceed in the same way for my original problem?
## Answer by Mark Joshi (score 2)
https://quant.stackexchange.com/a/19222
You can write
$$\mathbb{E}\left[ \max(a X_T + b X_S -K,0)\right] = \mathbb{E}\left[ \max(a X_S Y_{S,T} + b X_S -K,0)\right],$$
with $Y_{S,T} = X_T/X_S.$
For a given value of $X_S$ we can write $$\mathbb{E}\left[ \max(a X_S Y_{S,T} + b X_S -K,0)\right] = X_S \mathbb{E}\left[ \max(a Y_{S,T} + b -K/X_s,0)\right],$$ since $Y_{S,T}$ is log-normal this can be evaluated by a BS type formula.
We then integrate numerically over the value of $X_S$ with a log-normal density.
## Answer by user16891 (score 0)
https://quant.stackexchange.com/a/19217
- let $X$ be a log-normal random with mean $\mu$ and variance $\sigma^2$ then $aX$ is said to have a scaled log-normal distribution with mean $a\mu$ and and variance $a^2\sigma^2$.
- Let $X_j$ be a independent log-normally distributed variables with mean $\mu_j$ and variance $\sigma_j^2$ and $Y=\sum_{j=1}^{n}X_j$.The distribution of $Y$ has no closed-form expression, but can be reasonably approximated by another log-normal distribution $Z$ at the right tail. Its probability density function at the neighborhood of $0$ has been characterized and it does not resemble any log-normal distribution.A commonly used approximation due is obtained by matching the mean and variance of another log-normal distribution: \begin{align} &\mu_Z=\ln\left(\sum_{j=1}^{n}e^{2\mu_j+\sigma_j^2/2}\right)-\frac{1}{2}\sigma^2_Z\\ &\sigma^2_Z=\ln\left(\frac{\sum_{j=1}^{n}{e^{2\mu_j+\sigma_j^2}}(e^{\sigma_j^2}-1)}{\left[\sum_{j=1}^{n}e^{2\mu_j+\sigma_j^2}\right]^2}+1\right) \end{align}
- for more details, you can download these Asymptotic Behavior of Tail Density for Sum of Correlated Log-normal Variables Fitting The Log Skew Normal To The Sum Of Independent Log-normals Distribution.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.