Moment Matching for Multi-Asset, Multi-Date Spread Options

Summary

The document addresses pricing a positive-part payoff built from weighted prices of two assets observed at multiple dates. It asks whether a closed-form formula exists when a spread involves more than two prices. The response proposes moment matching: approximate a sum of terms with coefficients of the same sign by a lognormal random variable, or approximate the full combination with a shifted lognormal when all coefficients share a sign.

The approximation parameters can be calibrated by matching moments through the third order, which requires solving a cubic equation. For a simpler approximation, the shift can be set to zero and the first two moments matched. This is an approximation rather than an exact general pricing formula. The discussion does not provide numerical examples, error bounds, or conditions for accuracy, and it leaves the required moments dependent on the model for asset prices and their cross-date and cross-asset dependence.

Key ideas

• A weighted sum of asset prices across assets and dates can be approximated with a lognormal variable for pricing a spread-style payoff.
• When all coefficients have the same sign, a shifted lognormal approximation can represent the full combination.
• Matching moments through the third order requires solving a cubic equation for the approximation parameters.
• Setting the shift to zero yields a simpler approximation based on matching the first two moments.
• The document gives no error bounds or empirical evidence about approximation accuracy.

Tags

# How to price a strategy involving more than 2 different prices?


# How to price a strategy involving more than 2 different prices?












When pricing a spread option with two different prices, one can use Kirk's approximation combined with Margrabe's formula (https://en.wikipedia.org/wiki/Margrabe%27s_formula).

But what if I am pricing an option that involves 3 or 4 different prices ? Is there a closed form formula ? Else how can I price it ?

The payoff of my strategy looks like this:

$$\mathop{\mathbb{E}} \left[\left(\alpha K+\beta F_1(t_1) +\gamma F_1(t_2)+\zeta F_2(t_1) +\lambda F_2(t_2)\right)^+\right]$$ Where $K$ is the strike, $F_1$ and $F_2$ the price of two different assets.

## Answer by Gordon (score 1)

https://quant.stackexchange.com/a/35564

For such problems, you may consider the moment matching approach. For example, you can approximate the combination of terms where the coefficients have the same sign by a log-normal random variable, and then use the approach you mentioned. If all coefficients have the same sign, you can approximate the whole combination by a shifted log-normal random variable. That is, \begin{align*} \beta F_1(t_1) +\gamma F_1(t_2)+\zeta F_2(t_1) +\lambda F_2(t_2) \approx A + B e^{C\xi}, \end{align*} where $A$, $B$ and $C$ are constants, and $\xi$ is a standard normal random variable. Here, the parameters $A$, $B$ and $C$ can be calibrated by matching the moments up to the third order. However, you need to solve a cubic equation. For simplicity, you can assume that $A=0$ to have a second order moment matching.

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