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Risk-Neutral Measure Changes in Black–Scholes Pricing

Article Quant Q&A · Author: k b
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Summary

The document asks why a stock-price model with drift μ can be expressed under a different probability measure with drift equal to the risk-free rate r, while retaining the same volatility. This is the risk-neutral change of measure used in derivative pricing: under appropriate assumptions, discounted asset prices behave as martingales, allowing arbitrage-free prices to be calculated as discounted expected payoffs under the risk-neutral measure.

The response points to arbitrage-free pricing and standard introductory references as the motivation. It offers no derivation, conditions for the measure change, or discussion of the distinction between real-world and risk-neutral probabilities. The sketch is therefore useful as a conceptual prompt, but readers need a fuller treatment of the assumptions and change-of-measure argument to apply it correctly.

Key ideas

  • A change of probability measure can replace the stock’s drift with the risk-free rate for pricing.
  • The risk-neutral measure supports arbitrage-free derivative valuation under model assumptions.
  • The volatility term remains in the stock-price dynamics after the drift adjustment.
  • The document names the motivation but leaves the mathematical derivation and assumptions unspecified.

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Derivatives pricingOptionsStatistics
Full text
# Why changing measure is necessary?


# Why changing measure is necessary?












I want to understand the logic for why this is:

We have our model for the stock price behaviour:

$$d{S_t} = \mu {S_t}dt + \sigma {S_t}d{\tilde W_t}$$

It describes the development of a stock price over time using the risk-adjusted expected return $\mu$ and the real uncertainty in the stochastic term. We want to change the probability measure in such a way that the stochastic process remains a Brownian motion but with a drift of r instead of $\mu$. .... To repeat the manner of speaking, we want the process to change gear from an instantaneous increase of $\mu$ to r and leave the rest as before.

$$d{S_t} = r{S_t}dt + \sigma {S_t}d{\tilde W_t}$$

## Answer by mbison (score 1)

https://quant.stackexchange.com/a/33821

they change the measure to get the arbitrage free price. Check the link below, or have a look at Hull for an easy introduction.

https://en.wikipedia.org/wiki/Risk-neutral_measure

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