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Why Option Prices Are Not Martingales Before Discounting

Article Quant Q&A · Author: econmajorr
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Summary

The document asks whether the undiscounted price of a European call under the Bachelier model is a martingale. Its answer uses conditional expectation under the risk-neutral measure and deterministic discount factors to relate the expected future option value to its current value. With interest rates, the undiscounted option price generally does not satisfy the martingale property.

Instead, discounting the future option value back over the interval makes the conditional expectation equal to the current price. The reasoning uses the tower property of conditional expectations and the fact that discount factors multiply across adjacent periods. The answer presents this as a general no-arbitrage pricing result rather than a feature specific to Bachelier dynamics. Its stated setup assumes deterministic rates; it does not address extensions with stochastic rates or other complications.

Key ideas

  • An undiscounted option price generally is not a martingale when rates are nonzero.
  • Under deterministic rates, discounting the future option price produces the martingale property.
  • The result follows from risk-neutral valuation and iterated conditional expectations.
  • The argument is model independent within its stated pricing assumptions.

Tags

OptionsDerivatives pricingRisk management
Full text
# Is the undiscounted value process of a Euro call option under Bachelier model a Martingale?


# Is the undiscounted value process of a Euro call option under Bachelier model a Martingale?












Assume that $c_t$ is the UNDISCOUNTED price process for a European call option in Bachelier model. In Bachelier model call option pricing formula the formulas is discussed. The undiscounted value process is $c_t = (S_t-K)\Phi( \frac{S_t-K}{\sigma\sqrt{T-t}})+\sigma\sqrt{T-t}\phi( \frac{S_t-K}{\sigma\sqrt{T-t}})$.

Is $c_t$ a martingale process?

My personal guess is YES, because of the first fundamental theorem of asset pricing. Am I correct?

## Answer by Daneel Olivaw (score 2)

https://quant.stackexchange.com/a/46100

Let $c_t$ be the price of an European call with maturity $T$ and $D_{t,T}$ the discount factor from $T$ to $t$. We assume deterministic rates. Then note that for $s<t\leq T$: $$\begin{align} E^Q_s\left(c_t\right)&=E^Q_s\left(E^Q_t\left(D_{t,T}(S_T-K)^+\right)\right) \\[3pt] &=E^Q_s\left(D_{t,T}(S_T-K)^+\right) \\[3pt] &=E^Q_s\left(\frac{D_{s,t}}{D_{s,t}}D_{t,T}(S_T-K)^+\right) \\ &=\frac{c_s}{D_{s,t}}\end{align}$$ because $D_{s,t}D_{t,T}=D_{s,T}$. The second inequality stems from the fact that: $$E^Q_s(E^Q_t(\cdot))=E^Q_s(\cdot)$$ if $s<t$, this is the Law of Iterated Expectations. From the last equation you see $c_t$ is not a martingale, however rearranging: $$E^Q_s\left(D_{s,t}c_t\right)=D_{s,s}c_s=c_s$$ Thus the discounted call price is a martingale. As you can see this is a model-free result.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.