Black-Scholes and Fundamentals

So basically

$dS_t=\mu S_tdt+\sigma S_tdWt$

and

$\mu=r-\frac12\sigma^2$

I have just been thinking about this later equation. This is very interesting because it ties together risk-free rate, volatility and asset drift. I always like and try to look at equation from some simple perspective, for example assuming that something is huge or very small or 0, and trying to watch how it impacts other variables. This is good approach to remember some dependencies.

So looking at this later equation, first thing to note is the negative sign of volatility. This is OK when trying to explain why VIX is index of fear and that "investors" don't like increase in volatilities. But increasing risk-free rate in macroeconomics theory translates to increased demand for bonds and decrease in demand for stocks, so their prices drop - this assumption is quite real in today's market - when US Treasuries yields rise stocks go down and vice versa.

So this is not in agreement with this also fundamental assumption $\mu=r-\frac12\sigma^2$.

How do you interpret this fact?

• Generally from a purely empirical standpoint over many different market cycles, there is an inverse relationship in the demand for fixed income instruments and equities. Demand and supply for different asset classes is one thing, the purely technical relationship between yields and bond prices an entirely different. – Matt Mar 23 '13 at 9:06

I think you are interpreting too much into the matter. The $-\frac12\sigma^2$ is just a correction term that comes from Jensen's inequality.

You need this when switching from supposedly symmetric returns (normal distribution) to the skewed price process (log-normal distribution).

I think there are no deeper truths to be found here.

• Agree with that notion. – Matt Mar 23 '13 at 9:07
• Agreed, it's just a convexity correction, there is no economic meaning. – experquisite Mar 23 '13 at 18:07
• yes, but again: mi is asset drift, it is mi from first equation which has known interpretation in physics if this equation has to describe asset move. but, OK, this is not simply average rate of move – 4pie0 Mar 23 '13 at 19:13
• @vonjd Jensen inequality is one thing and this relationship is other. you don't need Jensen ineq to derive this, this comes straight from the first assumption and from the fact that $E(S_T)=S_0e^{rT}$ – 4pie0 Mar 24 '13 at 12:05

One thing to keep in mind here is that the world of risk-free/arbitrage-free models is not necessarily the real world. Specifically, this equation

$$\mu = r - \frac{1}{2}\sigma^2$$

occurs not because this is the way stocks behave in reality (they don't! For S&P 500, long-run $\mu$ is closer to 6-9%, if I recall correctly), but because using any other number in the pricing formula would create arbitrage.

• yes, but still mi is asset drift, it is mi from first equation which has known interpretation in physics if this equation has to describe asset move – 4pie0 Mar 23 '13 at 19:09