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Monte Carlo Pricing of a Call with a Conditional Second-Asset Payout

Article Quant Q&A · Author: J.Doe
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Summary

The document describes a European call whose exercise condition depends only on one commodity futures contract, while exercise also grants the holder a second futures contract. The payoff is the usual positive excess of the first asset over the strike, plus the value of the second asset whenever the first asset finishes at or above the strike. The question is how to value this conditional two-asset payout.

The response proposes Monte Carlo valuation under risk-neutral dynamics: simulate paths or terminal outcomes for both underlyings, calculate the specified payoff in each scenario, average the payoffs, and discount the average to today. It suggests geometric Brownian motion with risk-free drift and each asset’s volatility. This is a general outline, not a complete model specification. A practical implementation also needs suitable assumptions for the joint distribution, including dependence between the assets, as well as inputs for discounting and contract settlement; the brief answer does not discuss calibration or model risk.

Key ideas

  • The exercise condition depends on the first underlying crossing the strike.
  • Exercise delivers the second underlying in addition to the first asset’s call value.
  • Monte Carlo valuation can estimate the payoff by simulating both underlyings jointly under risk-neutral dynamics.
  • The simulated payoffs are averaged and discounted to obtain a present value estimate.
  • A usable model needs assumptions about dependence between the two assets.

Tags

CommoditiesFuturesOptionsDerivatives pricing
Full text
# Pricing a European call option that has one underlying asset to compare with strike but 2 underlyings as payout


# Pricing a European call option that has one underlying asset to compare with strike but 2 underlyings as payout












This is a real world problem and not a research one.

We are being proposed to buy an option that has to be exercised on a specific date T. So it is a European option.

This option has a strike price of K = 50. We are ITM if and only if the price of asset S1 is above K. So it is a call option.

However, if we choose to exercise the option we will receive S1 at a price of 50 - like for any call option, but also another asset S2 for free. This asset can then be sold on the market to make a profit. S1 and S2 are both future contracts for physical delivery of 2 different commodities.

So the payout of the call option would be assuming we are ITM:

P = (S1 - K) + S2

and so the price of the option should be the discounted expected payoff and so:

C = exp(-rT) E( max(S1 - K; 0) + S2 x Id)

Id is 1 when S1 >= K and 0 if not.

How can I price this option knowing that S1 alone is compared to the strike price and I will get S2 if I am ITM regardless of its price ?

I really have never seen this before.

## Answer by Randor (score 0)

https://quant.stackexchange.com/a/60701

do a montecarlo simulation of both underlyings each having a gemetric brownian motion distribution with drift = risk free rate, and vol=vol of the underlying. in such a way, you get values of each underlying in many scenarios. then in each scenario, calculate your payoff then average the payoffs then discount that to the present and that's your option price

hope that helps!

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.