# Pricing options under a specific framework

I have a specific framework in mind and I would like to value options under this framework. I am not sure whether a closed form solution exists or Monte Carlo methods would work. The framework I have in mind is the one from Lettau and Wachter 2007 (paper here).

In summary this is the framework:

Let $\epsilon_{t+1}$ denote a 3 × 1 vector of independent standard normal shocks that are independent of variables observed at time t.

Let $D_t$ denote the aggregate dividend in the economy at time t, and $d_t = ln D_t$. The aggregate dividend is assumed to evolve according to:

$\Delta d_{t+1} = g + z_t + \sigma_d \epsilon_{t+1}$.

where: $z_{t+1} = \phi_z +\sigma_z \epsilon_{t+1}$

Also assume that the stochastic discount factor is driven by a single state variable $x_t$ where:

$x_{t+1} = (1-\phi_x) \bar{x} + \phi_x x_t + \sigma_x \epsilon_{t+1}$.

$\sigma_d, \sigma_x, \sigma_z$ are all 1x3 vectors.

The stochastic discount factor is exogenously defined as: $M_{t+1} = exp ( -r^f - \frac{1}{2} x_t^2 - x_t \epsilon_{d,t+1})$ where:

where: $\epsilon_{d,t+1} = \frac{sigma_d}{\lVert \sigma_d \rVert} \epsilon_{t+1}$

It is quite straight forward to show that the price-dividend ratio of the equity claim the sum of all the claims to future dividends:

$\frac{P_t^m}{D_t} = \sum_{n=1}^\infty \frac{P_{nt}}{D_t} = \sum_{n=1}^\infty exp(A(n) + B_x(n) x_t + B_z(n) z_t)$ where $A,B_x,B_z$ are solved in closed form solution.

Now what I am looking is a method to calculate the value of a call option, with strike $K$ and maturity $\tau$ under this framework, meaning:

$C(t,\tau,K) = E_t[M_{t,\tau}max(P^m_t-K,0)]$

Not sure whether a closed form solution exists... if not would Monte Carlo, work?

• why would monte carlo not work? you are using a stochastic discount factor so no need to change any measures. You can just discretize the processes that you have and discount using the stoch discount factor. Nov 28 '15 at 12:10
• I agree. That might probably be the most straightforward solution to do this ... Nov 28 '15 at 17:33

1 - The Payoff function up to Maturity is known = max(P_t_m - K, 0)
2 - The stochastic Discount factor has also been explicitly stated (M_t_tau)