The second part of the "Does the CAPM price options" question from the 2006 2nd day asks: Why is it that beta appears in the CAPM but volatility appears in the Black-Scholes forula?
The B-S formula takes as given some equilibrium which gives the price process dS/S = mu dt + sigma dz. A slight abuse of notation would take this process to mean the following: mu is the expected instantaneous return to the stock, sigma is the instantaneous vol of the PRICE of th stock and z is the only priced risk factor. So there is a sense in which sigma is the exposure of the return of the stock to the only priced risk factor z. There might be other sources of variance in the return to the stock, but they will not contribute to the volatility of the price process.
On the other hand, the CAPM says that the expected excess return to the stock will only depend on the expected excess return to the market (read: the only priced risk factor) and the stock returns covariance with the market (normalized), or beta. So that beta and sigma can both be thought of as a measure of the exposure of the return on the stock to the priced risk factor.
Once we use the BS model of stock prices to price options it follows from the CAPM that the only parameter that should matter for the price of the option is the stocks exposure to the only priced risk factor. With the additional assumptions of no arb, complete markets, constant r, and constant sigma, we see that after adjusting for risk the only parameter that matters is vol, hence the BS formulas dependence on sigma.
We can also recover the BS formula directly from the CAPM with log utility (Rubinstein 1976).
I think this reasoning is going in the right direction but not entirely fleshed out, please comment!!
Hey guys, Chapter 6 section D has a nice discussion of how to recover beta pricing from the standard continuous time Diffusion model of a stock.
The upshot is
mu - r = (sigma * sigma_pi)/pi
where pi is a state price deflator. If we assume consumptions is the dividend to total wealth and CRRA representative agent, then this is exactly the CCAPM!
Now notice that if we assume a one dimensional BM is the only source of risk, then
2 comments:
The B-S formula takes as given some equilibrium which gives the price process dS/S = mu dt + sigma dz. A slight abuse of notation would take this process to mean the following: mu is the expected instantaneous return to the stock, sigma is the instantaneous vol of the PRICE of th stock and z is the only priced risk factor. So there is a sense in which sigma is the exposure of the return of the stock to the only priced risk factor z. There might be other sources of variance in the return to the stock, but they will not contribute to the volatility of the price process.
On the other hand, the CAPM says that the expected excess return to the stock will only depend on the expected excess return to the market (read: the only priced risk factor) and the stock returns covariance with the market (normalized), or beta. So that beta and sigma can both be thought of as a measure of the exposure of the return on the stock to the priced risk factor.
Once we use the BS model of stock prices to price options it follows from the CAPM that the only parameter that should matter for the price of the option is the stocks exposure to the only priced risk factor. With the additional assumptions of no arb, complete markets, constant r, and constant sigma, we see that after adjusting for risk the only parameter that matters is vol, hence the BS formulas dependence on sigma.
We can also recover the BS formula directly from the CAPM with log utility (Rubinstein 1976).
I think this reasoning is going in the right direction but not entirely fleshed out, please comment!!
Hey guys, Chapter 6 section D has a nice discussion of how to recover beta pricing from the standard continuous time Diffusion model of a stock.
The upshot is
mu - r = (sigma * sigma_pi)/pi
where pi is a state price deflator. If we assume consumptions is the dividend to total wealth and CRRA representative agent, then this is exactly the CCAPM!
Now notice that if we assume a one dimensional BM is the only source of risk, then
(mu - r)/sigma = sigma_pi/pi
is the market price of risk.
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