Risk-Neutral Pricing and Risk Premia in Exchange Options
Summary
The document asks why an exchange option is priced under the risk-neutral measure when the two assets may have different expected returns in the real world. The answer explains that real-world pricing would require estimating the relevant risk premium, which can vary across investors and is difficult to quantify. Risk-neutral valuation instead changes the probability measure so assets are modeled as earning the risk-free rate in expectation, allowing the discounted expected payoff to provide a common pricing framework.
This is a brief conceptual explanation rather than a derivation of the exchange-option formula. It does not discuss the conditions needed for risk-neutral valuation, market completeness, or how the assets’ covariance and volatilities enter the option price. The cited recommendation to consult standard quantitative finance texts is not accompanied by supporting calculations or empirical evidence.
Key ideas
- Risk-neutral valuation prices an option using discounted expected payoff under a pricing measure.
- Under that measure, expected asset returns are set to the risk-free rate for pricing purposes.
- Real-world valuation would require specifying risk premia for the assets.
- The explanation does not derive the exchange-option formula or discuss its assumptions in detail.
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Full text
# Magrabe Exchange Option: not equal drifts
# Magrabe Exchange Option: not equal drifts
I need to calculate the price of exchange option between 2 assets $S_1$ and $S_2$ The formula is given here Wiki: Magrabe formula or here Quant Stack Exchange. In the derivation of the formula it is assumed, that price is the discounted expecation of future claim under risk-neutral measure $Q$: $$p=e^{-r_{f}}\mathbb{E}_{Q}(S_1-S_2)^{+}$$. and under risk-neutral measure it is assumed, that: $$dS_{i,t}=\mu_{i}dt+\sigma_{i}dW^{i}_{t}$$ $$\mu_{i}=r_{f}\text{, for }i=1,2$$.
My question is:
Why do we calculate expectation under risk-neutral measure $Q$ and not real-world measure $P$ ? In real world return on each asset can be different, i.e. $\mu_1\neq\mu_2\neq r_f$. Then it would make sence to use real-world measure, under which this condition is fullfilled.
## Answer by SmallChess (score 1, accepted)
https://quant.stackexchange.com/a/16367
You can essentially get your answer from any quantitative finance book. I recommend Shreve's book. Note that this has nothing to do with pricing an exchange option.
You could price in the real-world measure, but that's very difficult if impossible because you'd need to know the risk premium for an instrument. By changing the probability measure to risk-neutral, you can assume all investors demand only the risk-free rate and thus you could discount your expected price with the risk-free rate.
Please read risk neutral measure on Wikipedia. In particular, focus on "Unfortunately, the discount rates would vary between investors and an individual's risk preference is difficult to quantify."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.