Option pricing, origin of formula $\Pi( t,X)= E^{\mathbb{Q}}\left[e^{-\int_{t}^{T}r_s\,ds} X| \mathcal{F}_t\right]$

Question

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 1

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 2

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:

1. 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

2. The value of any portfolio, with any trading strategy that doesn't involve peeking into the future, is a (discounted) martingale under Q

3. 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

4. The theoretical value of the call option is the cost of setting up a perfect hedge, which is the initial value of the replicating portfolio