An alternative approach

The random variable $M_{n}$ in the above analysis shares a lot with the pricing kernel that we defined in the binomial setting, due to the fact that the future cash flows $W_{n}$ are valued as

$$W_{0}=\mathbf{E}_{P}\left[ M_{n}W_{n}\right]$$   .

Therefore it is natural to look for some relationships with the equilibrium cash flow valuation. We start this analysis by assuming that it the same random shocks affecting the stocks that affect the consumption. Namely we assume that the pricing kernel [which is the normalized marginal rate of substitution, the derivative of the utility function] evolves according to

$$\mathcal{M}_{k} = \mathcal{M}_{k-1}\exp \left\{ -\beta Y_{k}-\alpha -\frac{1}{2}\beta ^{2}\right\}$$

$$\mathcal{M}_{k} = \exp \left\{ -\beta B_{k}-\left( \alpha +%% \frac{1}{2}\beta ^{2}\right) k\right\}$$

Such a setting would emerge if the representative agent has an exponential utility profile, $U(c)=\frac{c^{1-\gamma}}{1-\gamma}$, where the consumption follows a process similar to the stock price, namely $c_{k+1}=c_k \exp \left\{ \alpha_c-\frac{1}{2} \beta_c^2 + \beta_c Y_k \right\}$. One could therefore write

$$\mathcal{M}_{k} = \frac{U_c(c_k)}{U_c(c_0)}$$

in the above form, with $\alpha=\gamma\alpha_c-\frac{1}{2}\gamma(1+\gamma)\beta_c^2$ and $\beta=\gamma\beta_c$. Remember that $\gamma=0$ implies risk neutrality: $\alpha=0$ and $\beta=0$.

The coefficient $\beta$ represents the sensitivity of the marginal rate of substitution with respect to the random shocks. It is therefore a measure of the risk aversion. One can use the bond and stock prices to further identify the parameters $\alpha$ and $\beta$. The bond price as inferred from the money market account has to satisfy

$$e^{-r}=\mathbf{E}_{P}\left[\frac{\mathcal{M}_{k}}{\mathcal{M}_{k-1}}\vert\mathscr{F}_{k-1}\right]$$,

which will give

$$e^{-r}=\mathbf{E}_{P}\exp \left\{ -\beta Y_{k}-\alpha -\frac{1}{2}\beta ^{2}\right\} \Rightarrow \alpha =r$$.

The stock price will be given by the relevant Euler equation, namely

$$S_0=\mathbf{E}_{P}\left[\mathcal{M}_k S_k \right]$$.

This expression will give after some algebra

$$\beta=\frac{\mu-r}{\sigma}$$

One can observe that $\beta$ as inferred above is once again the price per unit of volatility risk. In addition, one can also observe that discounted with the marginal rate of substitution prices form by construction martingales.

In addition, any $\mathscr{F}_{n}$-measurable future cash flows $W_{n}$ are valued today as dictated by the Euler equation, as

$$W_{0} = \mathbf{E}_{P}\left[\mathcal{M}_{n}W_{n}%% \right]$$

$$= \frac{1}{\sqrt{2\pi n}}\int_{-\infty }^{+\infty }W_{n}(x)\exp \le... ...eta x-\left( \alpha +\frac{1}{2}\beta ^{2}\right) n-\frac{x^{2}}{2n}\right\} dx$$

$$= e^{-rn}\frac{1}{\sqrt{2\pi n}}\int_{-\infty }^{+\infty }W_{n}(x)\exp \left\{ -\frac{\left[ x+\beta n\right] ^{2}}{2n}\right\} dx$$

$$= e^{-rn}\mathbf{E}_{Q}W_{n}$$

where the measure $Q$ is defined as the one where $Y_{k}+\beta$ are independent standardized normals.

To recap, we investigated two different approaches that one can utilize in order to retrieve the Black-Scholes option pricing formula in a discrete time setting. The first one is based on the principle of no-arbitrage, and uses the powerful Girsanov-Cameron-Martin theorem to establish an equivalent risk neutral probability measure. The transition from the objective to the risk adjusted measure is carried out using the methodology that is most tractable computationally. The second approach is based on the Euler conditions that underlie every economy where agents are utility maximizers. It is shown that the Black-Scholes formula is consistent with a setting where agents have utilities of the power form.