Why Martingale Measures Agree on Attainable Claim Prices
Summary
The note applies the first fundamental theorem of asset pricing to explain why an attainable claim has the same value under every martingale measure. Given a replicating strategy, the theorem expresses its value at any time as the conditional expectation of the terminal payoff discounted by the money market account. Applying that identity under two measures gives the same strategy value, so the measures assign the same price.
It also shows that, if a martingale measure exists, any two strategies replicating the same claim must have identical values at every time: each value equals the same discounted conditional expectation under that measure. The argument is a direct consequence of replication and the stated theorem. It does not establish existence of a martingale measure or address claims that cannot be replicated; the result is limited to attainable claims within the theorem's framework.
Key ideas
- The first fundamental theorem represents the value of a replicating strategy as a discounted conditional expectation.
- Any martingale measures therefore assign the same value to an attainable claim.
- Under one martingale measure, all strategies replicating the same claim have equal values at each time.
- The argument applies to attainable claims and assumes the theorem's conditions hold.
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Full text
# All martingale measures price the attainable claim equally
# All martingale measures price the attainable claim equally
Background Information:
This question is from Lectures on Financial Mathematics: Discrete Asset Pricing.
Theorem 3.2 First Fundamental Theorem of Asset Pricing - Suppose $\nu$ is any measure such that $S/S^{0}$ is a $\nu$-martingale. For an attainable claim $X$ with replicating strategy $\phi$ and $0\leq t\leq T$, we have $$V_t(\phi) = E_{\nu}\left(X\frac{S_t^{0}}{S_T^{0}}|\mathcal{F}_t\right)$$
Question:
> Prove that: All martingale measures price the attainable claim equally, and if there is a martingale measure, then all replicating strategies for a given claim have the same value at all times.
I am sort of confused even where to begin, some guidance or suggestions may help.
## Answer by Gordon (score 2, accepted)
https://quant.stackexchange.com/a/31449
For Question 1, let $\phi$ be a replicating strategy, that is, $V_T(\phi) = X$. Then for any two martingale measures $u$ and $v$, from the First Fundamental Theorem of Asset Pricing, \begin{align*} E_u\left(X\frac{S_t^0}{S_T^0}\mid \mathcal{F}_t\right) = V_t(\phi), \end{align*} and \begin{align*} E_v\left(X\frac{S_t^0}{S_T^0}\mid \mathcal{F}_t\right) = V_t(\phi). \end{align*} That is, all martingale measures price the attainable claim equally.
For Question 2, let $\mu$ be the martingale measure. Moreover, let $\phi$ and $\psi$ be two replicating strategies, that is, $V_T(\phi)= V_T(\psi)=X$. Then, for any time $t$, \begin{align*} V_t(\phi) &= E_{\mu}\left(V_T(\phi)\frac{S_t^0}{S_T^0}\mid \mathcal{F}_t\right)\\ &= E_{\mu}\left(X\frac{S_t^0}{S_T^0}\mid \mathcal{F}_t\right)\\ &= E_{\mu}\left(V_T(\psi)\frac{S_t^0}{S_T^0}\mid \mathcal{F}_t\right)\\ &= V_t(\psi). \end{align*} That is, all replicating strategies for a given claim have the same value at all times.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.