Option Pricing Limits Require Risk-Neutral Drift and Itô’s Lemma

Summary

The document corrects an attempted derivation of an option price limit under the Black–Scholes–Merton framework. It explains that an option’s price is an expectation under the risk-neutral measure, where the stock’s drift is the risk-free rate, rather than the physical-measure drift used to describe expected real-world returns. This distinction matters when deriving theoretical option values.

It also identifies a missing volatility correction in the stock-price solution. Applying Itô’s lemma to the logarithm of geometric Brownian motion gives a log-price drift reduced by half the variance rate; omitting this term leads to an incorrect terminal stock expression. The reply recommends deriving the log process first and then obtaining the terminal value. The document does not include the full exam question or its image, nor does it complete the option-limit calculation, so it provides the essential correction rather than a full solution.

Key ideas

• Option valuation uses expectations under the risk-neutral measure.
• Under that measure, the stock’s drift is the risk-free rate rather than its real-world expected return.
• Itô’s lemma introduces a negative half-variance term in the drift of the log stock price.
• Deriving the log-price process first helps avoid errors in the terminal stock-price expression.
• The answer corrects the setup but does not complete the limiting-price calculation.

Tags

# How might I answer this past exam question relating to the limiting price of an option?


# How might I answer this past exam question relating to the limiting price of an option?












The following image shows a past exam question that I am attempting to answer (for which I do not have a mark scheme):

I believe that under the BMS model, the payoff of a stock at maturity $T$ is given by $$ S_T = S_0 \exp \left( \mu T + \sigma W_T \right) $$

Thus, the payoff of the stock in the question would be given by $$ S_T = \exp \left( T + W_T \right) $$

Therefore, I would expect the payoff of the option to be $$ V_0 (T) = \left| \frac{T}{2} + T + W_T \right| = \left| \frac{3T}{2} + W_T \right| $$

However, $$ \lim_{T \rightarrow \infty} \frac{\left| \frac{3T}{2} + W_T \right|}{\sqrt{T}} = \lim_{T \rightarrow \infty} \left| \frac{3\sqrt{T}}{2} + \frac{W_T}{\sqrt{T}} \right| = \infty $$

What am I doing wrong?

## Answer by byouness (score 1, accepted)

https://quant.stackexchange.com/a/39899

Two things:

- The price of the option is the expectation under the risk-neutral measure. Under this measure, the stock price's drift is $r$ and not $\mu$: $$dS_t = S_t r dt + S_t \sigma dW_t$$

- When you integrate to get $S_T$, you have made a mistake: $$S_T = S_0\exp\left(\left(r-\frac{\sigma^2}{2} \right)T + \sigma W_T \right)$$

Start by writing out the expression of $d\ln(S_t)$ using Itô's lemma, then deduce $\ln(S_T)$.

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.