Black–Scholes Derivation and the Role of Risk-Neutral Measures
Summary
The question asks whether one can derive the Black–Scholes partial differential equation without invoking Girsanov’s theorem, and whether a stock with nonzero drift is a martingale. The answer distinguishes the original replication argument from later probability-based pricing frameworks. In the replication approach, a dynamically hedged portfolio leads to the pricing equation without first changing probability measures. The brief response notes that the original Black–Scholes work followed this route.
Later arbitrage-pricing theory frames derivative values as expectations under a risk-neutral measure; Girsanov’s theorem is relevant to that change of measure, with Novikov’s condition among the technical tools used to justify it. These methods provide different theoretical routes to pricing rather than implying that the stock itself must be a martingale under the real-world measure. The document does not present a full derivation, correct the question’s informal stochastic setup, or spell out the assumptions behind replication, hedging, or the measure change. It is an orientation to the conceptual distinction, not a complete mathematical treatment.
Key ideas
- The Black–Scholes equation can be derived through dynamic replication without first changing probability measures.
- Risk-neutral pricing expresses derivative values as expectations under a pricing measure.
- Girsanov’s theorem supports changing measures in the probability-based framework.
- A nonzero real-world drift does not by itself contradict risk-neutral pricing.
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# Black Scholes model without using Girsanov's theorem? It might happen?
# Black Scholes model without using Girsanov's theorem? It might happen?
We can calculate the stock price by the equation: $\frac{dS_t}{dt} = \mu dt + \sigma dB_t$,where $B_t$ is a Brownian motion.
First i create a portfolio that consists of $\Phi$ units of stock share and $\phi$ units of cash. Denote the amount of share and cash at time t as $\Phi_t$,$\phi_t$.Then, the value of the portfolio at time t $(V_t)$ will be the sum of the value of stock share $(φ_t*S_t)$ and the amount of real interest that can be earned by possessing the cash for dt amount of time $(rP dt)$ so that $V_t = \Phi_t S_t + \phi_t r P dt$. I do the calculations without using Girsanov's theorem and i get the Black-Scholes equation:$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S_t^2\frac{\partial^2 V}{\partial x^2} + r S_t \frac{\partial V}{\partial x} - r V = 0 $. For $\mu \ne 0$, the process $S_t$ is not a martingale, right? In many bibliographies authors uses the Girsanov's and Novikov's theorem something that i didn't use. I can 't understand the difference between my solution and thw other way. Can somebody help me; I hope i didn't confuse you.
## Answer by Ivan (score 5)
https://quant.stackexchange.com/a/54180
A very interesting topic ! Black-Scholes originally did not make use of the Girsanov theorem and arrived at the equation the way you described.
Later theoretical work on arbitrage pricing uncovered the concepts of the risk-neutral measure and derivatives pricing as an expectation under that measure. That work relies on stochastic calculus far more and one could argue is a more “satisfying” approach. This is where the Girsanov theorem comes into play. See Harrison, Kreps, Pliska. Here is an overview: https://www.fields.utoronto.ca/programs/scientific/09-10/finance/courses/pliska2.pdfShown 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.