# The minimal entropy martingale measure for insurance-linked securities pricing

Suppose that we have a CAT bond contract that pays coupons at discrete points in time as well as a principal at maturity time $$T$$ if no triggering event happens during the term of the contract. More precisely, we have that

At times $$\{0=t_0, t_1, t_2, ..., t_n=T\}$$ with time intervals $$t_i - t_{i-1} = \Delta$$ for $$i=1,2, ..., n$$ the CAT coupons are paid.

The trigger mechanism is characterized by:

1-Threshold level $$D$$

2-Compound Poison process $$L_t = \sum_{j=1}^{N(t)}X_j$$, which models the aggregate losses, where $$X_j\stackrel{iid}{\sim}F$$ independent of $$N(t)\sim Poisson(\lambda(t))$$.

3-$$\tau = inf\{ t : L_t \geqslant D\}$$

Triggering event happens for a given interval $$[t_{i-1}, t_i]$$, if $$\tau \leqslant t_i$$.

The Payment process is given by:

For $$i = 1, 2, \cdots , n-1$$: $$\begin{eqnarray*} Pay_{CAT}(t_i)=\bigg(\text{coupon}_i\bigg)\Delta M I_{\{\tau > t_i\}} + \zeta M I_{\{\tau \leqslant t_i\}} \end{eqnarray*}$$ For $$i=n$$: $$\begin{eqnarray*} \begin{gathered} Pay_{CAT}(t_i)=\left[\bigg(\text{coupon}_i\bigg )\Delta M +M\right] I_{\{\tau > t_i\}} + \zeta M I_{\{\tau \leqslant t_i\}} \end{gathered} \end{eqnarray*}$$ where $$M$$ is the principal, and $$\zeta\in(0, 1)$$ is the recovery rate. Uisng the arbitrage-free pricing theory, we know that the present price at time zero is given by the expectation of disocunted cashflow under a risk-neutral martingale measure $$\mathbb{Q}$$, i.e.,

$$$$P^{cat}(0, T) = \mathbb{E}^{\mathbb{Q}}\left[\sum_{i=1}^{n}D(0, t_i) Pay_{CAT}(t_i)\right]$$$$

where $$D(0, t_i) = exp\big(\int_{0}^{t_i}r(u)du\big)$$ is called the discount factor and that the stochastic process $$r(u)$$ stands for the risk-free interest rate following a CIR model. We can easily compute the above expectation under some assumptions. As the market of the CAT bond is not complete (due to the fact that the claim under consideration is not perfectly hedgeable), the pricing measure $$\mathbb{Q}$$ is not a unique measure.

My question is here:

Is it possible to apply the minimal entropy martingale measure for the above model? We know that the minimal entropy measure is chosen such that it has a minimum distance (in some sense) from the real-world measure $$\mathbb{P}$$. How we can derive the minimal entropy martingale measure for such a contract and use it as the pricing measure to find the CAT bond price?