When Risk-Neutral Discounted Expectations Price an Option
Summary
The document explains the conditions behind pricing a contingent claim as its discounted conditional expected payoff under an equivalent martingale measure. One route is to extend the market to include the claim and require that discounted prices in the enlarged market remain martingales. Another is to show that the claim is attainable: a self-financing dynamic portfolio replicates its payoff, and the claim’s theoretical price is then the cost of setting up that hedge.
The answers emphasize that the expectation formula is not justified for every payoff in every market simply because an equivalent martingale measure exists for the traded assets. Replicability is central to the hedging argument; without it, the document does not establish a unique price from that argument alone. The discussion is conceptual and gives no worked model, proof details, or treatment of incomplete-market pricing choices. It also refers to the money account as the numeraire for discounting, linking the formula to the chosen short-rate process.
Key ideas
- A discounted conditional expectation under a martingale measure prices a claim when the relevant assumptions are satisfied.
- A replicating, self-financing strategy gives a hedge-based price for an attainable claim.
- The discounted value of a suitable portfolio is a martingale under the pricing measure.
- An equivalent martingale measure for existing assets alone does not establish the claim’s unique price without further conditions.
- The money account provides the discounting numeraire in the stated setup.
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Full text
# Option pricing, origin of formula $\Pi( t,X)= E^{\mathbb{Q}}\left[e^{-\int_{t}^{T}r_s\,ds} X| \mathcal{F}_t\right]$
# Option pricing, origin of formula $\Pi( t,X)= E^{\mathbb{Q}}\left[e^{-\int_{t}^{T}r_s\,ds} X| \mathcal{F}_t\right]$
Imagine a model with stock prices and dividends of these stocks, as well as a market bond with associated short rate process. It is known that this model is arbitrage-free if there exists an equivalent martingale measure $Q$.
It is then asserted that the price of a call option at time $t$ is the discounted conditional expectation under the equivalent martingale measure $Q$ of its payoff.
Question: Why is this true? The way I think about it is that if we imagined that the call option is a new stock which we introduce to the market, then we can consider it as a stock with no dividends up until expiration date where the final dividend is then its payoff. If we imagine "adding" this stock to our model, then it would remain arbitrage-free if and only if this "new stock" was priced so that $Q$ remains an equivalent martingale measure, and this precisely meanas that the call option price at time $t$ needs to be given as that conditional expectation (discounted).
Is this reasoning correct?
## Answer by user16651 (score 2)
https://quant.stackexchange.com/a/27471
Generally we consider this issue for every $T$-claim contingent $\Pi(t,X)$. However, there are two main approach in this context. As you mentioned, for first approach we should demand that the extended market $\Pi(.,X)\,,\,S_0\,,S_1,...,S_N$ is free of arbitrage possibilities. Indeed we demand that there should exist a martingale measure $Q$ for the extended market. Applying the definition of a martingale measure we obtain $$\frac{\Pi(t,X)}{S_0(t)}=E^{\mathbb{Q}}\left[\frac{\Pi(T,X)}{S_0(T)}|\,\mathcal{F}_t\right]$$ In particular we assume that $S_0(t)$ is the money account: $$S_0(t)=S_0(0)=\exp\left(\int_{t}^{T}r_sds\right)$$ For second approach, if the claim is attainable, with hedging portfolio $h$, then the only reasonable price is given by $\Pi(t,X) = V (t, h)$.
## Answer by user29970 (score 2)
https://quant.stackexchange.com/a/27472
It's only true if the claim can be replicated by dynamically hedging with the tradeable assets. So any proof should certainly refer to that property.
My proof would be:
- There is a dynamic portfolio that replicates the claim, i.e. which is self-financing, pre-visible, and has terminal value equal to the value of the call option
- The value of any portfolio, with any trading strategy that doesn't involve peeking into the future, is a (discounted) martingale under Q
- The present value of the dynamic portfolio is the expected value of the final value which is the expected value of the call payoff under the martingale measure
- The theoretical value of the call option is the cost of setting up a perfect hedge, which is the initial value of the replicating portfolioShown 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.