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Martingale Pricing and Discounted Asset Values

Article Quant Q&A · Author: Ulysses
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Summary

The document examines why discounted traded asset prices are martingales under a risk-neutral pricing measure. It raises a potential confusion in an argument that compares holding an asset until a future date with selling it now and investing the proceeds at the risk-free rate. The two strategies do not have identical realized terminal payoffs: the asset price is random, while the invested cash grows deterministically.

The answer explains the intended logic through valuation. Both positions have the same value at the current time, so their prices under expectation-based valuation must agree when discounted expectations are taken under the pricing measure. This yields the martingale pricing condition for the asset. The exchange offers a short conceptual clarification rather than a full proof: the key equality concerns present values under the pricing measure, not equality of future outcomes path by path. The argument presumes the applicable pricing and no-arbitrage framework.

Key ideas

  • An asset held to a future date and cash invested at the risk-free rate generally have different realized terminal payoffs.
  • Martingale pricing equates their current values through discounted expectations under a pricing measure.
  • The martingale condition is an expectation statement, not an assertion that terminal outcomes match almost surely.
  • The explanation sketches the intuition but does not provide a complete formal derivation.

Tags

Derivatives pricingStatisticsRisk management
Full text
# Pricing rule shall be a martingale measure


# Pricing rule shall be a martingale measure












In the book "Financial Modelling with jump processes" by Cont and Tankov there is a chapter that explains martingale pricing principles. It is not extremely formal, but gives the idea underlying the method. There he shows that any linear positive pricing rule can be associated with a probability measure (or expectation operator defined on a linear subspace of random variables representing the contingent claims). That's perfectly clear, and argument is rather simple.

The next step is to show that under some non-arbitrage argument, the discounted prices of traded securities must be martingales w.r.t. to the latter measure. The argument is as follows:

> Consider now an asset $S^i$ traded at price $S^i_t$. This asset can be held until $T$, generating a terminal payoff $S^i_T$, or be sold for $S^i_i$: the resulting sum invested at the interest rate $r$ will then generate a terminal wealth of $\mathrm e^{r(T-t)}S^i_t$. These two buy-and-hold strategies are self-financing and have the same terminal payoff so they should have the same value at $t$.

The bold part in the quote is unclear to me: how do we know that terminal payoffs are the same? I even think, that in the simplest case of BS model they won't be: if $S^i_t$ is a GBM with drift $\mu$ and volatility $\sigma$, then $S^i_T \neq \mathrm e^{r(T-t)}S^i_t$ under the physical measure, and since we are looking for the equivalent measures, neither payoffs will be same (a.s.) under such measures.

Can somebody clarify, whether this is indeed a mistake in the argument, or am I missing something here?

## Answer by Mark Joshi (score 1)

https://quant.stackexchange.com/a/15524

I am guessing that the argument is as follows.

They certainly have the same value at time t since they are both worth $S_t$ then. If they have the same value at $t$ they should have the same value at time $0.$

So if we are pricing by expectation our measure has to give the same discounted expectation price to both portfolios. So we must have

$$ e^{-rT} E( S_T) = e^{-rT} E(e^{r(T-t)}S_t) $$

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.