Risk-Neutral Asset Drifts, Discounted Martingales, and Normal Models
Summary
The document explains how the risk-neutral pricing condition constrains an asset’s drift relative to the money-market account used as numéraire. Under its stated assumptions—Itô diffusion dynamics, a riskless account without stochastic variation, and local martingales that are martingales—the discounted asset price must be a martingale. Applying Itô’s lemma leads to a drift linked to the account’s growth rate and the asset-to-account price ratio.
With an exponentially growing money-market account, this gives the familiar short-rate drift for the asset. The document emphasizes that this drift condition does not require lognormal prices: it cites the Bachelier model, where the asset has an additive Brownian term and a normal price distribution. The conclusion depends on the modeling assumptions; the discussion excludes jumps and does not establish that every tradable asset follows a particular diffusion specification.
Key ideas
- Risk-neutral pricing requires the asset price discounted by the chosen money-market numéraire to be a martingale.
- The implied asset drift depends on the dynamics of the money-market account.
- An exponential account model yields the familiar short-rate drift condition.
- That drift condition does not imply lognormal asset prices.
- The explanation assumes diffusion dynamics and excludes jump modeling.
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Full text
# General Dynamics of a Tradable Asset under the Risk Neutral Measure
# General Dynamics of a Tradable Asset under the Risk Neutral Measure
Is it true that every tradable asset must have a log-normal dynamics under the risk neutral measure where the drift term is the short rate $r$? I.e., is it true that if $X$ is a tradable asset then $$\frac{\mathrm{d}X(t)}{X(t)} = r(t)\mathrm{d}t + \sigma(t, X(t))\mathrm{d}W(t),$$ where $W$ is a Brownian motion under the risk neutral measure for some $\sigma$ (which may or may not be deterministic)?
## Answer by Daneel Olivaw (score 3)
https://quant.stackexchange.com/a/58526
Our market has a tradeable asset $S$ and a risk-less money market account $B$, that is, the numéraire of the risk-neutral measure. We assume the following standard conditions, which are widely applicable to most common models:
- We work in an Itô diffusion setting, and neglect jump modelling: $$\begin{align} & dS_t=\mu(t,S_t)dt+\sigma(t,S_t)dW^S_t \\ & dB_t=r(t,B_t)dt + \varsigma(t,B_t)dW^B_t \end{align}$$
- The money market account $B$ has no stochastic contribution (because it is riskless): $$\varsigma \equiv0$$
- Local martingales are martingales.
The exact requirement is that the discounted asset price is a martingale under the risk-neutral measure, and we want to determine the expression of the drift term $\mu(\cdot)$ $-$ note that all dynamics are expressed under the risk-neutral measure. Per our assumptions, this is equivalent to saying that there exists a function $\eta(\cdot)$ and a Brownian Motion $W$ such that: $$d\left(\frac{S_t}{B_t}\right)=\eta(t,S_t,B_t)dW_t$$ Applying Itô's Lemma: $$\begin{align} d\left(\frac{S_t}{B_t}\right) &=\frac{1}{B_t}dS_t-\frac{S_t}{B_t^2}dB_t+\frac{S_t}{B_t^3}d[B,B]_t-\frac{1}{B_t^2}d[S,B]_t \\ &=\frac{1}{B_t}dS_t-\frac{S_t}{B_t^2}dB_t \end{align}$$ In order to cancel the drift contributions in the above equation, we need to have: $$\begin{align} \mu(t,S_t)=r(t,B_t)\frac{S_t}{B_t} \end{align}$$ That is, the drift of the asset needs to be equal to the drift of the money market account, adjusted by the price ratio between the asset and the MMA.
- Continuously compounded interest model: if $B$ is exponential, that is $r(t,B_t)=rB_t$, then: $$\mu(t,S_t)=rS_t$$
In practice, the money market account is always assumed to have an exponential form because it is the most sensible way to represent mathematically such a security. Because the dynamics of the money market account will restrict the drift of the asset (if we are to ensure the martingale requirement), the drift of the asset will be $rS_t$ in most models. However, the model might not necessarily be log-normal. For example, the Bachelier model is usually specified as follows: $$dS_t=rS_tdt+\sigma dW_t^S$$ which corresponds to a normal distribution for $S$.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.