Pricing a Quanto Basket Option with a Two-Dimensional Integral

Summary

The document considers a basket call whose components are denominated in different currencies while the payoff is paid in US dollars. It asks how to price and hedge the payoff formed from the sum of the two asset values less a strike, floored at zero. The response explains that the sum of lognormal variables generally lacks a closed-form option price, so approximations may be used.

For valuation, it chooses the US dollar bank account as numeraire and describes the drift adjustment for the euro-denominated asset: its dollar-measured drift includes the instantaneous covariance between that asset and the exchange rate. The expected discounted payoff can then be evaluated as a two-dimensional integral, which the answer says is simpler than Monte Carlo for this setup. The original question mentions simulation and finite-difference hedging, but the response does not give a hedge construction or discuss model calibration, numerical accuracy, or assumptions behind the dynamics.

Key ideas

• A basket payoff involving a sum of lognormal asset values generally has no exact closed-form price.
• Using the dollar bank account as numeraire sets the dollar asset drift to the risk-free rate.
• The foreign-currency asset’s dollar drift includes its covariance with the exchange rate.
• The discounted expected payoff can be evaluated through a two-dimensional integral.
• The response does not provide a detailed hedge method or numerical implementation.

Tags

# How to price a quanto basket option?


# How to price a quanto basket option?












EDIT:

Maybe there is no way to get explicit solutions for basket options (maybe the Black-Scholes differential equation can't be solved directly ??).

Q3: How do you price and hedge ( S1(T) + S2(T) - K )+ at time t. S1 evolves in $ S2 evolves in €, and the flows are in \$ ??

An alternative solution (if S1 and S2 were in $) might be Monte-Carlo simulation of S1 and S2 under risk-free hypothesis. The hedging is done using finite differiation method in the simulations.

Thank you already ;D

Guillaume

## Answer by Mark Joshi (score 1)

https://quant.stackexchange.com/a/31720

well there are approximations for the prices but no exact formula since you have a sum of lognormals.

Take the USD bank account as numeraire. Then the drift of S1 is the drift of the riskless account r. The drift of S2 is $r +C_{f2}$ where $C_{f2}$ is the instantaneous covariance between $S2$ and the FX rate.

Now just compute $$ \mathbb{E} \left( (S_1(T) + S_2(T) - K)_+ \right)e^{-rT}. $$ This is a 2-dim integral so Monte Carlo is overkill. Just use a simple 2d-integral method.

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