Variance Rate in the Black–Scholes Geometric Brownian Motion Model

Summary

The document explains how to interpret the variance-rate assumption in the original Black–Scholes model. For a diffusion process, the instantaneous variance rate is the square of the coefficient multiplying the Brownian motion. In geometric Brownian motion for a stock, that coefficient is proportional to the stock price, so the stock-price variance rate scales with the square of its level.

The return variance rate is constant when the diffusion coefficient of proportional returns is constant. The answer connects this interpretation to quadratic variation, whose rate of accumulation is the squared diffusion coefficient. This resolves the apparent tension between a stock-price variance rate that depends on the stock price and a constant return variance rate. The brief exchange does not provide the full option-pricing derivation or identify the requested Merton paper, so its scope is limited to clarifying terminology.

Key ideas

• In a diffusion, the variance rate is the square of the coefficient on the Brownian motion term.
• For geometric Brownian motion, the stock-price diffusion coefficient scales with the stock price.
• The resulting variance rate for the stock price therefore scales with the square of its level.
• A constant proportional diffusion coefficient gives a constant variance rate for returns.
• Quadratic variation accumulates at the instantaneous variance rate.

Tags

# Original Black-Scholes paper assumptions -- "variance rate"


# Original Black-Scholes paper assumptions -- "variance rate"












In the 5th page of Black and Scholes' original paper on option pricing formulas, they write the following assumption:

> b) The stock price follows a random walk in continuous time with a variance rate proportional to the square of the stock price. Thus the distribution of possible stock prices at the end of any finite interval is log- normal. The variance rate of the return on the stock is constant.

I'm a bit confused about the first sentence and how it relates to the GBM assumption $$dS_t = S_t[ \mu dt + \sigma dW_t],$$ especially since the last sentence says that the "variance rate of return" (which I assume is $\sigma^2$) is constant over time. In particular, I don't see where $S_t^2$ comes into play.

Also, I read somewhere that Merton cleaned up the derivation and put it in more mathematically rigorous language. Where can I find that paper?

--

Answer: if $dX_t = \mu(t,X_t,\dots) dt + \sigma(t,X_t, \dots) dW_t$, then $\mu$ is called the "drift rate" and $\sigma^2$ is called the variance rate. The connection to quadratic variation is that $d\langle X \rangle_t = \sigma(t,X_t,\dots)^2 dt$, i.e. the drift rate of the quadratic variation of $X_t$ is in fact the variance rate of $X_t$.

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