Pricing options under a specific framework

Question

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?

Answer 1

You can always estimate the expectation by simulating the distribution. In your case if it is a European option, you can just calculate the average simulated payoff. If it is an American option you can use a tree model to determine at each point whether exercising or not based on the maximum of exercise and continuation value.

Answer 2

Unless I am mistaken, the solution is embedded within the post. In a nutshell, I would have used numerical methods such as a mere binomial tree to the extent that:

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)

Consequently, from expiry (if not open-ended) going backward at each node, check whether or not it's worth exercising the option (depending on its Style) and discount it back to finally end up with a fair value (price).

Hope it helps