Market Price of Risk
One Factor
Suppose the derivatives only dependents on $\theta$ and assume the process of $\theta$ is:
\[\frac{d\theta}{\theta} = mdt + sdw\]m and d only depends on $\theta$ and t.
Suppose the $f$ is the price of derivative which only dependt on $\theta$ and t. Suppose the process followed by $f$ is:
\[\frac{df}{f} = \mu dt + \sigma dw\]The $\mu$ and $\sigma$ should always satisfy the following formula: \(\frac{\mu - r}{\sigma} = \lambda\)
And the $\lambda$ is known as the market price of risk of $\theta$. If the $\lambda$ equals to 0, then it’s tranditional Risk-Neutral world.
Extend to Multi Factors
Suppose there are multiple $\theta_i$ follow the process:
\[\frac{d\Theta_i}{\Theta_i} = m_idt + s_idw_i\]And the the derivative price $f$ can be written as:
\[\frac{df}{f} = \mu dt + \sum_{i=1}^n\sigma_i dw_i\]Then they should satisfy the following equation:
\[\mu - r = \sum_{i=1}^n\sigma_i\lambda_i\]Where $\lambda_i$ is the market price of risk for $\theta_i$.
The Equavalent Martingale Measure Result
Suppose there are 2 traded securities $f$ and $g$ dependent on a single source of uncertainty. And define $\theta$ as follows: \(\theta = \frac{f}{g}\)
By define the Market Price of Risk $\lambda$ as the volatility $\sigma_g$ of security g. $\theta$ can be proved as Martingale. Below is the prove
\[df = (r+\sigma_g \sigma_f)fdt + \sigma_f f dw\]
\[dg = (r+\sigma_g^2)gdt + \sigma_g g dw\]
Then \(\ln f = (r + \sigma_g \sigma_f -\sigma_f^2/2)dt + \sigma_f dw\)
\[\ln g = (r + \sigma_g^2/2)dt + \sigma_g dw\]
Subtract $\ln g$ from $\ln f$
\[d(\ln f - \ln g) = d\ln\frac{f}{g} = -\frac{(\sigma_f - \sigma_g)^2}{2}dt + (\sigma_f - \sigma_g)dw\]
Assume $X = \frac{f}{g}$ and use Ito’s lemma
\[d(\frac{f}{g}) = (\sigma_f - \sigma_g)\frac{f}{g}dz\]
By proving that $\theta$ is martingle. This could be used to calculate the price of security $f$. \(\frac{f_0}{g_0} = E_g(\frac{f_T}{g_T})\)
where $E_g$ denotes the expected value in a world defined by numeraire $g$.
So the problem right now is how to choose $g$.
Money Market Account as the Numeraire (It’s also traditional Risk-Neutral world)
Suppose numeraire g follow the below process: \(dg = rgdt\) and
\[g_0 = 1\]So the security $g$ can be calculated as below:
\[f_0 = \hat{E}(e^{-\int_0^Trdt}f_T)\]where $\hat{E}$ is risk-neutral expectation. Because $\lambda = 0$ as defined in $g$’s process.
If we assume the interest rate is constant, it can be rewritten as below:
\[f_0 = e^{-rT}\hat{E}(f_T)\]Zero-Coupon Bond Price as Numeraire
Define $g_t = P(t,T)$ as the price at time t of a risk-free zero-coupon bond that pays off $1 at time T.
So, $g_0 = P(0,T)$ and $g_T = P(T,T) = 1$
Apply this to the formula, we can get the equation of $f_0$ as below:
\[f_0 = P(0,T)E_g(\frac{f_T}{g_T}) =P(0,T)E_g(f_T)\]Also, if security $f$ is a forward contract and $\theta$ is the underlying Asset. The value at time T would be:
\[f_T = \theta_T - F\]put this back to formula .
\[f_0 = P(0,T)E_g(\theta_T-F)=P(0,T)(E_g(\theta)-F) = 0\]where $F$ is the forward price set at time 0 to make the value of forward contract to be 0.
