Why Black–Scholes Uses an Asset-Price Process Despite Risk-Neutral Pricing
Summary
The document explains why Black–Scholes specifies a geometric Brownian motion (GBM) for the underlying even though option valuation is commonly expressed using risk-neutral probabilities. Its central point is that risk-neutral pricing does not remove the role of the asset’s real-world behavior: a replicating strategy is adjusted over time as the underlying moves, so the model’s assumptions about those movements matter to the cost and feasibility of hedging.
Under the model’s completeness assumption, specifying the physical process supports a unique hedge cost, while the risk-neutral measure provides a convenient mathematical way to calculate that cost. The explanation is conceptual rather than a derivation. It does not develop the conditions under which replication works, discuss model limitations such as jumps or transaction costs, or compare alternative pricing frameworks; its claims apply to the idealized setting assumed by Black–Scholes.
Key ideas
- Dynamic hedging responds to the underlying’s realized movements, so pricing assumptions cannot be wholly detached from asset dynamics.
- Risk-neutral probabilities provide a calculation framework for valuing the hedge rather than eliminating the role of hedging assumptions.
- In a complete-market model, the assumed dynamics support a uniquely determined replication cost.
- The explanation is intuitive and does not examine cases where Black–Scholes assumptions fail.
Tags
Full text
# Why Black Scholes model needs to assume S_t follows GBM if physical probabilities do not matter? # Why Black Scholes model needs to assume S_t follows GBM if physical probabilities do not matter? Beginner here learning about black scholes looking for a high level/intuitive explanation. So I've learned that the physical/"true" probabilities of S_t (or whatever underlying asset) do not matter for the option values, and that one should use risk-neutral probabilities to ensure no arbitrage. At the same time, it seems one of the assumptions of black scholes is an explicit physical distribution for stock price (from assuming its process follows a GBM). Why does it need to assume a physical property of the underlying if only the risk-neutral probabilities matter? ## Answer by Arshdeep (score 4) https://quant.stackexchange.com/a/66100 You are correct, in that you need not (explicitly) specify real world dynamics to calculate option prices. Indeed in many rates derivatives models, you simply assume a unique risk neutral measure exists (completeness), specify the dynamics under the risk neutral measure (risk neutral probabilities) and price your options. At the same time, it is important to note that any model cannot give a price without explicitly/implicitly saying (implying) something about the real world physical process: Recall that ultimately every model is giving you a cost of (real world) dynamic hedging. Since you hedge dynamically (reactively), you are obviously exposed to the real world dynamics of the underlying. You therefore cannot have a model that does not have a comment (implicitly or explicitly) on the true dynamics of the underlying. Ultimately, risk neutral probabilities are a mathematical convenience to find out the cost of this real world hedging strategy. So when you say only risk neutral probabilities matter, you are actually saying that 'only the cost of this real world dynamic hedging strategy matters'. You cannot completely separate the two. Black Scholes assumes a physical specification because as soon as that is done, completeness guarantees that the cost of this hedging strategy is determined. Risk neutral probabilities are just a mathematical utility to calculate what this cost is.
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.