From Positive Pricing Densities to Discounted-Price Martingales
Summary
The document asks whether a strictly positive density that assigns nonpositive expectation to every bounded loss-adjusted simple-strategy payoff is enough to make discounted stock prices martingales under the measure it defines. The questioner notes that zero expectation for every simple-strategy gain would imply the martingale property, and seeks to extend that reasoning to the stated inequality condition.
The included answer instead sketches a separate route: model the stock and savings account under a risk-free measure, apply Itô’s lemma to their ratio, and infer that a driftless discounted price is a martingale. It gives no derivation connecting the original inequality to the required zero conditional expectations. It also leaves regularity conditions unspecified, so the answer does not resolve the measure-theoretic question posed.
Key ideas
- A martingale claim concerns conditional expected changes in discounted prices.
- The question proposes using simple-strategy gains to establish that property.
- The answer describes an Itô calculation for a stock price divided by a savings account.
- The response does not show how the stated inequality yields the martingale condition.
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Full text
# Proving the discounted stock price is martingale
# Proving the discounted stock price is martingale
Let $\mathcal{K}_s$ be $$ \mathcal{K}_s=\{\tilde{V}_t(\theta):0\leq t<\infty,\,\theta\text{ a simple strategy}\},$$ where $\tilde{V}_t(\theta)$ is the discounted value process of the self financing strategy $\theta$, $$ \tilde{V}_t(\theta)=x+\sum_{j=1}^k\sum_{i=0}^{m-1} a^j_{t_i}(\tilde{S}^j_{t_{i+1}\wedge t}-\tilde{S}^j_{t_i\wedge t}).$$
Now let $\mathcal{U}$ be $$ \mathcal{U}=\{f-h:f\in\mathcal{K}_s,\;h\in L_+^\infty\},$$ where $$L_+^\infty(P) = \{ Z\in L^\infty(P):P(Z\geq0)=1\}. $$
Suppose there exists $g \in \mathcal L^q$ with $P(g>0)=1$ such that $$\int fgdP\leq0\hspace{1cm}\forall f\in\mathcal{U}.$$
I want to show that this implies that the discounted stock prices $(\tilde S_t^1,\tilde S_t^2,....,\tilde S_t^k)$ are $Q$-martingales where $dQ=gdP$.
My attempt: I know that if $\int fg dP=0$ for all $f$ in $\mathcal K_s$ then $(\tilde S_t^1,\tilde S_t^2,....,\tilde S_t^k)$ are $Q$-martingales because $1_A (\tilde S_t^i-\tilde S_s^i) \in \mathcal {K}_s$ for $A\in \mathcal G_s$ and $0 \leq s \leq t$. Can anyone please help me with the above case.
## Answer by Lech (score 1)
https://quant.stackexchange.com/a/66528
Simple steps:
- Take a stochastic process for $S(t)$ and the money-savings account $M(t)$. Both under the risk-free measure Q.
- Apply Ito's lemma to find the dynamics of $\frac{S(t)}{M(t)}$.
- You will find that the dynamics $d\left(\frac{S(t)}{M(t)}\right)$ has no drift thus discounted stock process $\frac{S(t)}{M(t)}$ is a martingale.
I skipped here all the regularity conditions but that should be enough for you to follow that path. Good luck.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.