Conditional Expectations in Risk-Neutral Option Pricing

Summary

The document asks why a European option’s time-t value may be written using an ordinary risk-neutral expectation when a general pricing formula conditions on information available at time t. It sets out the claim’s payoff at maturity and contrasts the two expressions, focusing on whether they produce the same price.

The question invokes arbitrage-free pricing and attainable claims, where discounted portfolio values are martingales under a risk-neutral measure. It does not provide an answer, derivation, market example, or empirical evidence, so it leaves unresolved the assumptions under which the conditional expectation can be replaced by an unconditional one. In particular, it does not explain whether the latter is shorthand for a time-zero valuation or how conditioning relates to information already reflected in the observed spot price. The discussion is useful as a prompt about information sets and option valuation, but it is not a complete pricing guide.

Key ideas

• Risk-neutral pricing expresses a claim’s value using its discounted expected payoff.
• The general time-dependent pricing expression conditions on information available at the valuation time.
• The document asks when an unconditional expectation can represent a conditional one.
• The question gives no derivation or answer establishing equivalence.

Tags

# Should we use the conditional expectation to write the value of an option?


# Should we use the conditional expectation to write the value of an option?












So, I've just started looking into financial mathematics and the following question keeps bugging me. From what I understood, if the market is arbitrage-free and a given contingent claim of value $h$ is attainable, then there is a measure $Q$ such that the discounted asset prices $(\tilde{S}_t)$ form a martingale. Consequently, the discounted values of the portfolio $(\tilde{V}_t)$ also form a martingale. As such, it makes sense to write:

\begin{align} V_t = e^{-r(T-t)} E^Q[V_T|\mathcal{F}_t] = e^{-r(T-t)} E^Q[h|\mathcal{F}_t] \end{align}

And then we consider the fair value of the option at time $t$ to be $V_t$.

Now, I was reading another author and he writes: "The time-$t$ price of a European call on a non-dividend paying stock with spot price $S_t$, when the strike is $K$ and the time to maturity is $\tau = T − t$, is the discounted expected value of the payoff under the risk-neutral measure $Q$." Hence,

\begin{align} C_t = e^{-r\tau}E^Q[h] = e^{-r(T-t)}E^Q[\max(S_T-K,0)] \end{align}

My question is: Why can we take the "usual" expectation when computing $C_t$, instead of using the conditional expectation to $\mathcal{F}_t$? Are they the same? If so, I'm not seeing why... Any help is appreciated. Thanks in advance.

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