Equilibrium Framework

Contents (hide)

  1. 1. The Euler equation
  2. 2. Power utility and the Black-Scholes model
    1. 2.1 The setting
    2. 2.2 Determining the parameters from observed contracts
    3. 2.3 Derivative prices

1.  The Euler equation

In equilibrium theory it is assumed that economic agents have a utility function \fs4 U(W) which relates to their wealth \fs4 W. Sometimes consumption is used instead, but in principle, whatever the utility measures, it will be a state variable of the economy. Assets are priced by these agents in a way that maximizes their utility.

What really determines the value of an asset is its correlation with the state variable. Assets that are negatively correlated with the state variable will be in high demand, as holding them will produce profits when wealth is low, and will therefore boost utility in bad times. This demand will result in higher prices for these assets. This is in fact analogous to the beta coeficients in the CAPM.

It turns out that all assets will obey the so called Euler equation, which states that their value must be discounted with the marginal rate of substitution (MRS). In particular, if the asset offers a random payoff \fs4 X_T at time \fs4 T, then its fair value at time zero is given by

\fs4 X_0 = E \left[ \frac{U_W(W_T)}{U_W(W_0)} \cdot X_T \right]

where the subscripts denote derivatives.

The shape of the utility function can be seen as a measure of the risk aversion of the investors. Typical utility function are increasing and concave, which a feature that is manifested by a positive and decreasing first derivative. The MRS \fs4 \mathcal{M}_t = U_W(W_t)/U_W(W_0) is therefore a measure of the change in utility as the state variable changes.

If \fs4 X_T and \fs4 W_T are positively correlated, then large values (gains) in \fs4 X_T are weighted by MRS values which are lower than one, while small values (losses) are weighted by MRS values that are higher than one. This will produce a fair value \fs4 X_0 which will be less than its expected payoff value \fs4 e^{-rT}EX_T, indicating a positive risk premium.

2.  Power utility and the Black-Scholes model

2.1  The setting

It is instructive to view the pricing of options in this framework. In particular, we will assume a wealth variable which follows a geometric Brownian motion

\fs4 dW_t = a W_t dt + b W_t dB_t

We will also assume a power utility function

\fs4 U(W) = \frac{W^{1-\gamma}}{1-\gamma}

The coefficient \fs4 \gamma can be viewed as a risk aversion coefficient, as outlined in the previous section. This will in turn produce a MRS process that satisfies

\fs4 \mathcal{M}_t = \exp\left\{ -\gamma \left( a - \frac{1}{2}b^2 \right)t - \gamma b B_t \right\}

2.2  Determining the parameters from observed contracts

In the above expression we have the dynamics parameters for the state variable which are not directly observed. But we observe the fair prices for bonds and stocks, and we can determine these parameters based on that information

The risk free bond promises to deliver \fs4 X_T = 1, and therefore the Euler equation will give its fair price as

\fs4 \exp \left\{ -r T\right\} = E \left[ \mathcal{M}_T \cdot 1 \right] = \exp \left\{ -\gamma \alpha T + \frac{1}{2} b^2 \gamma (1+\gamma) T\right\}

which will give the first constraint

\fs4 r= \gamma \alpha - \frac{1}{2} b^2 \gamma (1+\gamma)

We also observe the underlying stock price, which has dynamics

\fs4 dS_t = \mu S_t dt + \sigma S_t dZ_t \Rightarrow S_t = S_0 \exp\left\{ \left( \mu - \frac{1}{2} \sigma^2 \right) t + \sigma Z_t \right\}

We will assume that the correlation between the stock and the state variable is given by \fs4 \rho. Then, the fair price of the stock will imply (after some algebra) the second constraint

\fs4 S_0 = E \left[ \mathcal{M}_T \cdot S_T \right] \Rightarrow b = \frac{\mu - r}{\gamma \sigma \rho}

For future reference we can write the expression for \fs4 b in terms of the Sharpe ratio \fs4 \omega = \frac{\mu - r}{\sigma} as \fs4 b = \frac{\omega}{\gamma\rho}

2.3  Derivative prices

We now have established the dynamics of the state variable, and we can in principle price any contingent claim. Say that we investigate a generic claim with payoff \fs4 f(S_T). By invoking the decomposition \fs4 B_T = \rho Z_T + \sqrt{1-\rho^2} Z^\ast_T for an independent Gaussian \fs4 Z^\ast_T \sim N(0,T), we can write the value \fs4 f_0 at time zero

\fs4 f_0 = E \left[ \mathcal{M}_T \cdot f(S_T) \right] = e^{-rT} E \left[ \exp\left\{-\omega Z_T - \frac{1}{2}\omega^2 T \right\} \cdot f(S_T) \right]
\fs4 \quad\quad\quad = e^{-rT} E \left[ \exp\left\{-\omega Z_T - \frac{1}{2}\omega^2 T \right\} \cdot f(S(Z_T)) \right]

That is the value of the payoff is the discounted expectation of the payoff which is weighted by a martingale that depends on the risk premium. This martingale is the Radon-Nikodym derivative, used by Girsanov theorem to produce equivalent probability measures. We can actually write this expectation in integral form, since \fs4 Z_T\sim N(0,T)

\fs4 f_0 = e^{-rT} \frac{1}{\sqrt{2\pi T}}\int_{\mathbb{R}} \exp\left\{-\omega z - \frac{1}{2}\omega^2 T \right\} \cdot f(S(z)) \cdot \exp\left\{-\frac{z^2}{2T} \right\} \cdot dz
\fs4 \quad \quad \quad = e^{-rT} \frac{1}{\sqrt{2\pi T}}\int_{\mathbb{R}} \exp\left\{-\frac{(z+\omega T)^2}{2T} \right\} \cdot f(S(z)) \cdot dz

In the above expression we have of course that \fs4 S(z) = S_0 \exp\left\{ \left( \mu - \frac{1}{2} \sigma^2 \right) t + \sigma z \right\}. By substituting \fs4 \tilde{z} = z + \omega T we can write the integral as

\fs4 f_0 = e^{-rT} \frac{1}{\sqrt{2\pi T}}\int_{\mathbb{R}} \exp\left\{-\frac{{\tilde{z}}^2}{2T} \right\} \cdot f(S(\tilde{z})) \cdot d{\tilde{z}}

where now \fs4 S(\tilde{z}) = S_0 \exp\left\{ \left( r - \frac{1}{2} \sigma^2 \right) t + \sigma \tilde{z} \right\}. It is now straightforward to verify that the above integral is an expectation, which we can write in the form

\fs4 f_0 = e^{-rT} E \left[ f(S({\tilde{Z}}_T)) \right]

In the above form we can see the risk adjusted pricing, since the value is expressed as the discounted expected payoff of an asset that has a drift equal to the risk free rate \fs4 r rather than the true drift \fs4 \mu.

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