Overview
The market price of risk shows that teh value of
{% \frac{\alpha -r}{\sigma} %}
is the same for all derivatives with the same underlying.
This discussion follows that found in Bjork
Setup
We start given two derivatives, the price of which will be denoted {% F %} and {% G %}. Each is a derivative of a single underlying asset, whose price is denoted by {% X %}.
Recalling the Ito Lemma
{% \displaystyle df(t, X(t)) = ( \frac{\partial f}{\partial t} + \mu \frac{\partial f}{\partial x} + \frac{1}{2} \sigma^2 \frac{\partial ^2 f}{\partial x^2} )dt
+ \sigma \frac{\partial f}{\partial x} dW(t)
%}
we then write the price evolution of each derivative as
{% dF = \alpha_F F dt + \sigma_F F dW %}
{% dG = \alpha_G G dt + \sigma_G G dW %}
{% \alpha_F = \frac{F_t + \mu F_x + \frac{1}{2}\sigma^2 F_{XX}}{F} %}
where here we write:
- {% F_t = \frac{\partial F}{\partial t} %}
- {% F_{XX} = \frac{\partial^2 F}{\partial X^2} %}
Risk Free Portfolio
Next, we construct portfolio of the two derivatives. The value {% V %} of the portfolio follows the dynamics given
{% dV = V[w_F \frac{dF}{F} + w_G \frac{dG}{G}] %}
where we write the weight of {% F %} in the portfolio as {% w_F %}. In addition, the portfolio weights should sum to one.
{% w_F + w_G = 1 %}
Then we have that
{% dV = V[w_F \alpha_F + w_G \alpha_G] dt + V[w_F \sigma_F + w_G \sigma_G] dW %}
Next, we choose {% w_F %} and {% w_G %} such that the portfolio is riskless
This is given by the following weights
{% w_F = \frac{-\sigma_G}{\sigma_F - \sigma_G} %}
{% w_G = \frac{\sigma_F}{\sigma_F - \sigma_G} %}
{% dv = V \frac{\alpha_G \sigma_F - \alpha _F \sigma_G}{\sigma_F - \sigma_G} dt %}
Because this portfolio has no risk, the drift must equal the risk free rate
{% \frac{\alpha_G \sigma_F - \alpha _F \sigma_G}{\sigma_F - \sigma_G} = r %}
This implies that
{% \frac{\alpha_F -r}{\sigma_F} = \frac{\alpha_G -r}{\sigma_G} %}
Capital Asset Pricing Model