Stochastic Foundations for Finance
Fall 2024
For the money-market account M we have that d\Lambda dM = 0 since M is locally riskless.
Thus, d(\Lambda M) = M d\Lambda + \Lambda dM = M d\Lambda + \Lambda r M dt,
The pricing equation (2) implies that \operatorname{E}d(\Lambda M) = 0, or \operatorname{E}\left(\frac{d \Lambda}{\Lambda}\right) = - r dt
The drift of the SDF in continuous time determines the equilibrium continuously-compounded risk-free rate.
Property 1 (The Fundamental Pricing Equation) Consider an asset S that follows a diffusion \frac{dS}{S} = \mu dt + \sigma dz. If the asset pays a dividend yield q = D / S, and there are no arbitrage opportunities, it must be the case that \mu + q - r = - \frac{1}{dt} \left(\frac{d\Lambda}{\Lambda}\right) \left(\frac{dS}{S}\right). \tag{3} In words, the risk-premium of the asset equals minus the covariance of the SDF and the asset’s returns.
Example 1 Consider a non-dividend paying stock that currently trades for $100. The risk-free rate is 4% per year, continuously compounded and constant for all maturities. The instantaneous volatility of returns is 25% per year.
We saw in a previous example that for an at-the-money call expiring in 9 months we have that C_{S} = 0.5975 and C = 10.05. Thus, the current instantaneous volatility of the call is 0.25 \times \frac{100 \times 0.5975}{10.05} = 148.63\%.
Also, if the beta of the stock with the risk-factor is 1.2, the beta of the call is 1.2 \times \frac{100 \times 0.5975}{10.05} = 7.13.