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Using Itô’s Lemma to Resolve the Risk-Neutral Stock Price Expectation

Article Quant Q&A · Author: kchpchan
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Summary

The question examines an apparent conflict between the risk-neutral geometric Brownian motion for a stock and the expected terminal price implied by no-arbitrage pricing. The resolution is to apply Itô’s lemma to the logarithm of the stock price. The log process has drift equal to the risk-free rate minus half the variance rate, rather than simply the risk-free rate.

Exponentiating this corrected expression gives a terminal stock price whose expected value under the risk-neutral measure grows at the risk-free rate. The variance correction in log space offsets the exponential term arising from the expectation of a normally distributed Brownian increment. This is a concise mathematical correction, not an analysis of whether the model assumptions fit actual markets; it assumes the stated constant-drift, constant-volatility diffusion and risk-neutral measure.

Key ideas

  • Taking the logarithm of a geometric Brownian motion requires Itô’s lemma.
  • The log-price drift under the risk-neutral measure includes a negative half-variance adjustment.
  • The exponential of the log process has expected terminal value growing at the risk-free rate.
  • The result depends on the assumed diffusion model with constant volatility.

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Derivatives pricingStatisticsOptions
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# Why do we need $dS_t=r S_tdt+\sigma S_tdW_t^Q$?


# Why do we need $dS_t=r S_tdt+\sigma S_tdW_t^Q$?












Suppose $S_t$ is the stock price and follows the dynamics $$dS_t=\mu S_tdt+\sigma S_tdW_t$$. According to Girsanov, we can apply change of measure and obtain $dS_t=r S_tdt+\sigma S_tdW_t^Q$, this implies $\ln S_T = \ln S_0+rT+\sigma W_T^Q$, and therefore $$\mathbb{E}^Q[S_T]=S_0 e^{rT+\frac{1}{2}\sigma^2T}$$, however $\mathbb{E}^Q[S_T]=S_0e^{rT}$ by Fundamental Theorem of Asset Pricing, which is contradicted. Please correct me.

## Answer by Mark Joshi (score 4, accepted)

https://quant.stackexchange.com/a/16412

It doesn't imply

$$ \ln S_T=\ln S_0+rT+σW^Q_T,$$

it implies

$$ \ln S_T=\ln S_0+(r-0.5\sigma^2)T+σW^Q_T.$$

Look up Ito's lemma.

This is covered in just about any book on financial maths including my own Concepts etc.

## Answer by wsw (score 1)

https://quant.stackexchange.com/a/16413

If $dS_t = r S_t \, dt + \sigma S_t \, dW_t^Q$, $$S_T = S_0 \, e^{\sigma W_T^Q + \left( r - \frac{1}{2} \sigma^2\right) T}\, .$$

Hence $\mathbb{E}\left[ S_T \right] = S_0 \, e^{rT} \,.$

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.