Pricing a Positive Spread Between Correlated Lognormal Assets

Summary

The document asks how to calculate the expected positive difference between two correlated lognormal asset prices. It identifies the payoff as the positive part of the spread and connects the problem, when the assets are stocks, to a Margrabe exchange option.

The suggested method changes the numeraire to the second asset, expressing the value through the ratio of the first asset to the second under a measure where that ratio is a martingale. The response says the ratio’s lognormal distribution makes the expectation straightforward. No derivation, numerical example, or empirical evidence is provided, and the explanation assumes the stock and pricing-measure framing; it does not spell out parameter requirements or address cases outside that setup.

Key ideas

• The positive part of the difference between two lognormal asset prices is an exchange-option payoff.
• Changing the numeraire to the second asset reframes the calculation in terms of the price ratio.
• Under the associated measure, the asset-price ratio is treated as a martingale.
• The response relies on the ratio being lognormally distributed to evaluate the expectation.

Tags

# Expected value of bivariate lognormal spread


# Expected value of bivariate lognormal spread












I don´t know how to derivate the Expected Value for the following problem:

Suppose that the random vector `(S_1, S_2)` has a bivariate lognormal distribution with parameter vector `(u_1, u_2, v_1, v_2, p)` such that vector `(U_1, U_2)=[(ln{S_1}-u_1)/v_1, (ln{S_2}-u_2)/v_2]` has a standard bivariate normal distribution with correlation p

... Now, how would you derivate expected positive difference of the bivariate lognormal spread, t.i. : `E[max(S_1 - S_2, 0)]`?

## Answer by Mark Joshi (score 2)

https://quant.stackexchange.com/a/19403

if they are stocks, this problem is called pricing a Margrabe option and it is generally solved by change of numeraire. Take $S_2$ to be the numeraire. Then the value of the option is $$ S_2(0) \mathbb{E}_{S_2}( (S_1(T)/S_2(T)-1)_+) $$ where the expectation is taken in the measure that has $S_1/S_2$ as a martingale. Since it's a martingale and log-normal at time T the expectation is easy to compute and you are done.

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