Pricing a Spread Option on Two Correlated Assets
Summary
The document explains how to value a European option with payoff max(S1 − K·S2, 0) when both assets follow geometric Brownian motion under Black–Scholes assumptions. It presents two equivalent viewpoints: treat the payoff as a rainbow option on S1 and K·S2, or express it as an option on the asset ratio and value it in units of S2.
In the ratio formulation, the relative volatility is derived from both assets’ volatilities and their correlation. The forward ratio and that volatility can then be used in a Black–Scholes calculation, with the result converted back into currency using the second asset’s forward and a discount factor. The document gives a conceptual formula rather than a numerical example or empirical validation. Its assumptions include constant rates, no dividends, and lognormal asset dynamics; practical pricing may require adjustments when these assumptions do not hold.
Key ideas
- A spread payoff on two assets can be represented as a rainbow option on the first asset and a scaled second asset.
- The same payoff can be written as an option on the ratio of the two asset prices, valued in units of the denominator asset.
- The ratio’s variance depends on both volatilities and their correlation.
- The ratio-option value is converted to currency using the second asset’s forward and the discount factor.
- The approach relies on Black–Scholes-style dynamics and the stated assumptions about rates and dividends.
Tags
Full text
# multi asset option pricing # multi asset option pricing Assuming option on each single asset can be priced by Black Scholes, i.e. both S1 and S2 follow GBM. The correlation between vol of S1 and that of S2 is rho. Assuming constant interest rate, no dividend, what would be the formula to price an European option with this payoff C=max(S1-K*S2,0)? K is the strike. ## Answer by Valometrics.com (score 4) https://quant.stackexchange.com/a/63620 This is a rainbow option with two assets $S_1$ and $S_3=KS_2$. $S_3$ also follows the Black & Scholes stochastic equation with initial value $KS_2(0)$ and the same other parameters as $S_2$. You can find in section 3 (The Result of Margrabe) of the following article the formula to price these kind of products: http://finmod.co.za/Pricing%20Rainbow%20Options.pdf ## Answer by Peter A (score 0) https://quant.stackexchange.com/a/63621 You can re-write the payout as $C = S_2 \, max( S_1 / S_2 - K, 0)$ and then value the option in units of $S_2$ at first. Say $S_2$ is the IBM share price, then we would value in units of IBM shares. In that case it is a simple option with payout $max( S_1 / S_2 - K, 0)$. If the volatilities are $\sigma_1$ and $\sigma_2$ and the correlation is $\rho$, the volatility of $S_1 / S_2$ is $\sigma_3$ where $\sigma_3^2 = \sigma_1^2 + \sigma_2^2 - 2\rho \sigma_1 \sigma_2$. Then the forward value is given by the Black--Scholes formula $FV = BS(F_1/F_2, K, \sigma_3)$ in units of $S_2$, and in USD the present value is $PV = D_f\, F_2\, BS(F_1/F_2, K, \sigma_3)$ where $D_f$ is the USD discount factor, $F_1$ and $F_2$ are the forward levels for $S_1$ and $S_2$.
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.