Pricing Catastrophe Options with Credit Risk in a Regime-Switching Model
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Abstract:
In this paper, we consider the price of catastrophe options with credit risk in a regime-switching model. We assume that the macroeconomic states are described by a continuous-time Markov chain with a finite state space. By using the measure change technique, we derive the price expressions of catastrophe put options. Moreover, we conduct some numerical analysis to demonstrate how the parameters of the model affect the price of the catastrophe put option.
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Keywords:
- pricing /
- catastrophe option /
- credit risk /
- regime-switching /
- measure change
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1. Introduction
There have been many severe catastrophes in the past few decades, which have attracted increasing attention to catastrophe derivatives pricing. Cox and Pedersen[1] examined the price of catastrophe bonds by briefly discussing the equilibrium price and its relationship to the standard arbitrage-free pricing framework. Cox et al.[2] assumed that the price process of the asset is driven by a geometric Brownian motion with additional downward jumps of a prespecified size in the event of a catastrophe. They applied this model to price catastrophe options. Jaimungal and Wang[3] generalized the results of Cox et al.[2]. They assumed that the losses follow a compound Poisson process and that the drop in asset price depends on the total loss level. They obtained the closed-form formulae for the price of catastrophe put options. Jiang et al.[4] presented a catastrophe put option pricing model that accounts for interest rate uncertainty. Xu and Wang[5] assumed that the interest rate process and the default intensity process are modeled through the Vasicek model and provided the price expressions of catastrophe put options with credit risk.
In recent years, the pricing frameworks with a regime-switching model have been used by many researchers in modern financial economics, see, for example, Elliott et al.[6], Wang and Wang[7], and Wang et al.[8]. In the pricing frameworks with a regime-switching model, the market is assumed to be in different states depending on the state of the economy. A regime shift from one economic state to another may occur due to various financial factors, such as changes in business conditions, management decisions and other macroeconomic conditions. In this paper, we incorporate the model of Jiang et al.[4] into the pricing framework with a regime-switching model, where the issuing company's share price process, the loss process and the price process of the asset are all related to the macroeconomic states. We study the price of the catastrophe put option with credit risk in the proposed model.
The rest of this paper is organized as follows. In Section 2, we present the basic assumptions and the dynamics of the issuing company's share price process, the loss process and the price process of the asset. In Section 3, we adopt a measure change to determine an equivalent martingale probability measure for pricing catastrophe put options. In Section 4, we obtain some closed-form results for pricing catastrophe put options with credit risk in a regime-switching model. We present some numerical analysis to examine how the parameters of the model affect the price of the catastrophe put option in Section 5.
2. Modeling assumptions
Given a filtered complete probability space $ (\Omega,{\cal{F}},({\cal{F}}_t)_{t=0}^T,P) $, where $ {\cal{F}}={\cal{F}}_T $ and P is a real-word probability measure. In this paper, the macroeconomic states are described by a continuous-time irreducible Markov chain $ \{X(t)\}_{t\geq 0} $ with a finite state space $ D=\{{\bf{e}}_{\bf{1}},{\bf{e}}_{\bf{2}},\cdots,{\bf{e}}_{\bf{N}}\}, $ $ {\bf{e}}_{\bf{i}}=(0,\cdots,0,1,0,\cdots,0)'\in R^N $, where $ ' $ denotes the transpose of a vector or a matrix. As in Elliott et al.[9], the process $ \{X(t)\}_{t\geq 0} $ has the following decomposition:
$$ X(t)=X(0)+\int_0^tAX(u){\mathrm{d}}u+M(t), $$ (1) where $ A=(a_{ij})_{i,j=1,2,\cdots,N} $ is the generator of the process $ X(t) $ and $ M(t) $ is a martingale with respect to the natural filtration generated by $ \{X(t)\}_{t\geq 0} $.
Let $ \{r(t):t\geq 0\} $ be the risk-free short rate process, which is defined by
$$ r(t)=<{\bf{r}},X(t)>, $$ (2) where $ {\bf{r}}=(r_1,r_2,\cdots,r_N)'\in R^N $ with $ r_i>0 $ for each $ i=1,2,\cdots,N, $ and $ <\cdot,\cdot> $ denotes the inner product in $ R^N $.
Let $ \{S(t):t\geq 0\} $ be the share price process of the issuing company, $ \{L(t):t\geq 0\} $ be the loss process of the the issuing company, and $ \{V(t):t\geq 0\} $ be the price process of the asset of the issuing company. We assume that the dynamics of $ S(t), L(t) $ and $ V(t) $ are given by
$$ S(t)= S(0){\mathrm{exp}}\left\{\int_0^t\left[\mu_1(u)-\frac{1}{2}\sigma_1^2(u)\right]{\mathrm{d}}u+\int_0^t\sigma_1(u){\mathrm{d}}\overline{W}_1(u)-\alpha L(t)+\beta(t)\right\}, $$ (3) $$ \begin{split} L(t)&= \textstyle\sum\limits_{j=1}^{N(t)}Y_j, \ \ \ \beta(t)=\displaystyle\int_0^t\int_0^\infty(1-\mathrm{e}^{-\alpha y})\lambda(u) f_L(y){\mathrm{d}}y{\mathrm{d}}u,\\ V(t)&= V(0){\mathrm{exp}}\left\{\int_0^t\left[\mu_2(u)-\frac{1}{2}\sigma_2^2(u)\right]{\mathrm{d}}u+\displaystyle\int_0^t\sigma_2(u){\mathrm{d}}\overline{W}_2(u)\right\}, \end{split}$$ (4) where $ \overline{W}_1(t) $ and $ \overline{W}_2(t) $ are standard Brownian motions under P with
$$ \mathsf{Cov}\left(\mathrm{d} W_1(t), \mathrm{d} \overline{W}_2(t)\right)=\rho \mathrm{d} t, \rho \in R; $$ the appreciation rate $ \mu_1(t), \mu_2(t) $ and the volatility $ \sigma_1(t), \sigma_2(t) $ depend on $ \{X(t)\}_{t\geq 0} $, which are defined by
$$ \mu_1(t)=<\text{μ}_{\bf{1}},X(t)>,\sigma_1(t)=<\text{σ}_{\bf{1}},X(t)>, $$ $$ \mu_2(t)=<\text{μ}_{\bf{2}},X(t)>,\sigma_2(t)=<\text{σ}_{\bf{2}},X(t)>, $$ where
$$ \text{μ}_{\bf{1}}=(\mu_{11},\mu_{12},\cdots,\mu_{1N})'\in R^N,\ \ \text{σ}_{\bf{1}}=(\sigma_{11},\sigma_{12},\cdots,\sigma_{1N})'\in R^N, $$ $$ \text{μ}_{\bf{2}}=(\mu_{21},\mu_{22},\cdots,\mu_{2N})'\in R^N,\ \ \text{σ}_{\bf{2}}=(\sigma_{21},\sigma_{22},\cdots,\sigma_{2N})'\in R^N, $$ with $ \mu_{1i}>0, \mu_{2i}>0, \sigma_{2i}>0 $ and $ \sigma_{2i}>0 $ for each $ i=1,2,\cdots,N $; $ \{Y_j : j = 1,2,\cdots\} $ are i.i.d. random variables representing the size of the i-th loss with p.d.f. $ f_L(y) $ and mean l, and $ \{N(t):t\geq 0\} $ is a doubly stochastic Poisson process with arrival rate $ \lambda(t) $, which is defined by
$$ \lambda(t)=<\text{λ},X(t)>, $$ where $ \text{λ}=(\lambda_1,\lambda_2,\cdots,\lambda_N)'\in R^N $ with $ \lambda_i>0 $ for each $ i=1,2,\cdots,N. $ Since $ N(t) $ is a doubly stochastic Poisson process, there exists a standard Poisson process $ \overline{N}(t) $ which is independent of the intensity process $ \lambda(t) $, such that $ N(t) = \overline{N}(\Lambda_t) $, where
$$ \Lambda_t=\int_0^t\lambda(s)\mathrm{d}s. $$ See Grandell[10]. We also suppose that $ X(t) $ is independent of $ \overline{W}_1(t), \overline{W}_2(t) $ and $ \overline{N}(t) $.
Denote
$$ {\cal{F}}_t={\cal{F}}_t^X\textstyle\bigvee{\cal{F}}_t^{\bar{W}_1}\textstyle\bigvee{\cal{F}}_t^{\bar{W}_2}\textstyle\bigvee{\cal{F}}_t^L, $$ (5) where
$$ \begin{split} {\cal{F}}_t^X&=\sigma(X(s):0\leq s\leq t),\\ {\cal{F}}_t^{\overline{W}_1}&=\sigma\left(\overline{W}_1(s):0\leq s\leq t\right),\\ {\cal{F}}_t^{\overline{W}_2}&=\sigma\left(\overline{W}_2(s):0\leq s\leq t\right),\\ {\cal{F}}_t^L&=\sigma(L(s):0\leq s\leq t) \end{split} $$ are the natural filtrations generated by $ X(t),\overline{W}_1(t),\overline{W}_2(t) $ and $ L(t) $, respectively.
3. Equivalent martingale probability measure
In this section, we will illustrate an equivalent martingale probability measure. As in Cox et al.[2], if a liquid market for catastrophe options exists, then an equivalent martingale probability measure Q exists by standard derivative pricing theory, not necessarily unique, under which the discounted relative price processes $ \left\{\mathrm{e}^{-\int_0^tr(s)\mathrm{d}s}\psi(t):t\geq 0\right\} $ are martingales, for all contingent capitals $ \psi(t) $. We follow Cox et al.[2], Elliott et al.[6], and adopt Merton’s[11] assumption that the jumps are systematic and non-diversifiable. So the arrival rate and distribution of catastrophic events are not altered by measure changes.
Proposition 1 Let $ \eta_T^Q $ be the Radon-Nikodym process
$$ \begin{split} \eta_T^Q=\frac{{\mathrm{d}}Q}{{\mathrm{d}}P}= &{\mathrm{exp}}\Bigg\{\int_0^Tl_1(u)\sigma_1(u){\mathrm{d}}\overline{W}_1(u)+\int_0^Tl_2(u)\sigma_2(u){\mathrm{d}}\overline{W}_2(u) \\& -\frac{1}{2}\int_0^T[l_1^2(u)\sigma_1^2(u)+l_2^2(u)\sigma_2^2(u)+2\rho l_1(u)\sigma_1(u)l_2(u)\sigma_2(u)]{\mathrm{d}}u\Bigg\}, \end{split} $$ (6) where
$$ l_1(u)=\frac{\rho[\mu_2(u)-r(u)]\sigma_1(u)-[\mu_1(u)-r(u)]\sigma_2(u)}{(1-\rho^2)\sigma_1^2(u)\sigma_1(u)}, $$ $$ l_2(u)=\frac{\rho[\mu_1(u)-r(u)]\sigma_2(u)-[\mu_2(u)-r(u)]\sigma_1(u)}{(1-\rho^2)\sigma_1(u)\sigma_1^2(u)}. $$ Then $ W_1(t) $ and $ W_2(t) $ defined by
$$ W_1(t)=\overline{W}_1(t)-\int_0^tl_1(u)\sigma_1(u){\mathrm{d}}u-\rho\int_0^tl_2(u)\sigma_2(u){\mathrm{d}}u $$ (7) and
$$ W_2(t)=\overline{W}_2(t)-\rho\int_0^tl_1(u)\sigma_1(u){\mathrm{d}}u-\int_0^tl_2(u)\sigma_2(u){\mathrm{d}}u $$ (8) are standard Brownian motions under Q with instantaneous correlation ρ, i.e.,
$$ \mathsf{Cov}({\mathrm{d}}W_1(t),{\mathrm{d}}W_2(t))=\rho {\mathrm{d}}t. $$ Finally, let $ D(t,T)=\exp\{-\int_t^Tr(u){\mathrm{d}}u\} $, and then $ \{D(0,t)S(t):t\geq 0\} $ and $ \{D(0,t)V(t): t\geq 0\} $ are martingales under Q, respectively.
Proof It can be easily seen that $ \eta_T^Q > 0 $, P-a.s. and $ \eta_T^Q $ is a Radon-Nikodym process which induces the measure change from P to Q. From Girsanov's theorem, we know that $ W_1(t) $ and $ W_2(t) $ are standard Brownian motions under Q with $ \mathsf{Cov}({\mathrm{d}}W_1(t), {\mathrm{d}}W_2(t))=\rho {\mathrm{d}}t. $
By Bayes' rule, for all $ t\geq u $, we have
$$\begin{split} &\mathsf{E}^Q[D(0,t)S(t)|{\cal{F}}_u] =\dfrac{\mathsf{E}^P\left[D(0,t)S(t)\dfrac{{\mathrm{d}}Q}{{\mathrm{d}}P}|{\cal{F}}_u\right]}{\mathsf{E}^P\left[\dfrac{{\mathrm{d}}Q}{{\mathrm{d}}P}|{\cal{F}}_u\right]}\\ = &\dfrac{\mathsf{E}^P\left[D(0,t)S(t)\mathsf{E}^P(\dfrac{{\mathrm{d}}Q}{{\mathrm{d}}P}|{\cal{F}}_t)|{\cal{F}}_u\right]}{\mathsf{E}^P\left[\dfrac{{\mathrm{d}}Q}{{\mathrm{d}}P}|{\cal{F}}_u\right]} =\mathsf{E}^P\left[D(0,t)S(t)\dfrac{\eta_t^Q}{\eta_u^Q}|{\cal{F}}_u\right]\\ =&D(0,u)S(u)\mathsf{E}^P\bigg\{\exp\bigg\{\int_u^t[l_1(s)+1]\sigma_1(s){\mathrm{d}}\overline{W}_1(s)+\int_u^tl_2(s)\sigma_2(s){\mathrm{d}}\overline{W}_2(s) -A(u,t)\\& +\int_u^t\bigg[\mu_1(s)-r(s) -\dfrac{1}{2}\sigma_1^2(s) -\dfrac{1}{2}l_1^2(s)\sigma_1^2(s)-\dfrac{1}{2}l_2^2(s)\sigma_2^2(s)\\& -\rho l_1(s)\sigma_1(s)l_2(s)\sigma_2(s)\bigg]{\mathrm{d}}s\bigg\}|{\cal{F}}_u\bigg\}\\ =&D(0,u)S(u)\exp\left\{\int_u^t[\mu_1(s)-r(s)+l_1(s)\sigma_1^2(s)+\rho\sigma_1(s)l_2(s)\sigma_2(s)]\mathrm{d}s\right\}\\ =&D(0,u)S(u), \end{split}$$ (9) where $ A(u,t)=\alpha[L(t)-L(u)]-[\beta(t)-\beta(u)], $ and
$$\begin{split} & \mathsf{E}^Q[D(0,t)V(t)|{\cal{F}}_u] =\dfrac{\mathsf{E}^P\left[D(0,t)V(t)\dfrac{{\rm{d}}Q}{{\rm{d}}P}|{\cal{F}}_u\right]}{E^P\left[\dfrac{{\rm{d}}Q}{{\rm{d}}P}|{\cal{F}}_u\right]}\\ = &\dfrac{\mathsf{E}^P\left[D(0,t)V(t)\mathsf{E}^P\left(\dfrac{{\rm{d}}Q}{{\rm{d}}P}|{\cal{F}}_t\right)|{\cal{F}}_u\right]}{\mathsf{E}^P\left[\dfrac{{\rm{d}}Q}{{\rm{d}}P}|{\cal{F}}_u\right]} =\mathsf{E}^P\left[D(0,t)V(t)\dfrac{\eta_t^Q}{\eta_u^Q}|{\cal{F}}_u\right]\\ =&D(0,u)V(u)\mathsf{E}^P\bigg\{\exp\bigg\{\int_u^tl_1(s)\sigma_1(s){\rm{d}}\overline{W}_1(s)+\int_u^t[l_2(s)+1]\sigma_2(s){\rm{d}}\overline{W}_2(s)\\& +\int_u^t\bigg[\mu_2(s)-r(s)-\dfrac{1}{2}\sigma_2^2(s)-\dfrac{1}{2}l_1^2(s)\sigma_1^2(s)-\dfrac{1}{2}l_2^2(s)\sigma_2^2(s)\\& -\rho l_1(s)\sigma_1(s)l_2(s)\sigma_2(s)\bigg]\mathrm{d}s\bigg\}|{\cal{F}}_u\bigg\}\\ =&D(0,u)V(u)\exp\left\{\int_u^t(\mu_2(s)-r(s)+l_2(s)\sigma_2^2(s)+\rho l_1(s)\sigma_1(s)\sigma_2(s))\mathrm{d}s\right\}\\ = & D(0,u)V(u). \end{split}$$ (10) So $ \{D(0,t)S(t):t\geq 0\} $ and $ \{D(0,t)V(t):t\geq 0\} $ are martingales under Q, respectively.
Remark 1 From (3), (4), (7) and (8), we know
$$ S(t)= S(0){\mathrm{exp}}\left\{\int_0^t\left[r(u)-\frac{1}{2}\sigma_1^2(s)\right]{\mathrm{}}{\mathrm{d}}u+\int_0^t\sigma_1(s){\mathrm{d}}W_1(u)-\alpha L(t)+\beta(t)\right\}, $$ (11) $$ V(t)= V(0){\mathrm{exp}}\left\{\int_0^t\left[r(u)-\frac{1}{2}\sigma_2^2(s)\right]{\mathrm{d}}u+\int_0^t\sigma_2(s){\mathrm{d}}W_2(u)\right\}. $$ (12) 4. Pricing catastrophe put options
To reflect the influence of the macroeconomic states on the price of a catastrophe put option with credit risk in a regime-switching model, We assume that the catastrophe put options whose promised payoff is $ (K-S(T))^+ $ and actual payoff is $ (K- S(T)) ^+\frac{(1-\omega)V(T)}{D} $ in the event of a default, where ω denotes the value of the deadweight costs associated with bankruptcy and is expressed as a percentage of the value of the asset of the issuing company, and D denotes the total amount of claims. Hence, letting $ p(0,T) $ denote the price of the catastrophe put option with credit risk at time 0, which matures at time T, we have
$$\begin{split} p(0,T)= &\mathsf{E}^Q\left[D(0,T)I_{\{L(T)-L(0)>L\}}(K-S(T))^+I_{\{V(T)\geq D^*\}}\right]\\&+\mathsf{E}^Q\left[D(0,T)I_{\{L(T)-L(0)>L\}}(K-S(T))^+\frac{(1-\omega)V(T)}{D}I_{\{V(T)<D^*\}}\right], \end{split} $$ (13) where the parameter L is the trigger level of losses above which the catastrophe put option becomes in-the-money, K is the strike price at which the issuing company is obligated to purchase unit share if losses exceed L, and $ D^* $ is the default boundary of the issuing company.
Let $ J_i=\int_0^T<{\bf{e}}_{\bf{i}},X(u)>\mathrm{d}u $ be the amount of time $ \{X(t)\}_{t\geq 0} $ has spent in state i ($ i=1,2,\cdots,N $) over the time interval $ [0,T] $, and then $ J_1+J_2+\cdots +J_N=T $. So we only consider $ J_1,J_2,\cdots,J_{N-1} $. Define
$$ {\bf{J}} =(J_1, J_2,\cdots, J_{N-1}), $$ (14) $$ R_T({\bf{J}}) =\int_0^Tr(u){\mathrm{d}}u=\textstyle\sum\limits_{i=1}^{N-1}(r_i-r_N)J_i+r_NT, $$ (15) $$ \lambda_T({\bf{J}}) =\int_0^T\lambda(u){\mathrm{d}}u=\textstyle\sum\limits_{i=1}^{N-1}(\lambda_i-\lambda_N)J_i+\lambda_NT, $$ (16) $$ U^1_T({\bf{J}}) =\int_0^T\sigma_1^2(u){\mathrm{d}}u=\textstyle\sum\limits_{i=1}^{N-1}(\sigma_{1i}^2-\sigma_{1N}^2)J_i+\sigma_{1N}^2T, $$ (17) $$ U^2_T({\bf{J}}) =\int_0^T\sigma_2^2(u){\mathrm{d}}u=\textstyle\sum\limits_{i=1}^{N-1}(\sigma_{2i}^2-\sigma_{2N}^2)J_i+\sigma_{2N}^2T, $$ (18) $$ U^{12}_T({\bf{J}}) =\int_0^T\sigma_1(u)\sigma_2(u){\mathrm{d}}u =\textstyle\sum\limits_{i=1}^{N-1}(\sigma_{1i}\sigma_{2i}-\sigma_{1N}\sigma_{2N})J_i+\sigma_{1N}\sigma_{2N}T, $$ (19) and
$$ \begin{split} p(0,T,{\bf{J}}) =&\mathsf{E}^Q[D(0,T)I_{\{L(T)-L(0)>L\}}(K-S(T))^+I_{\{V(T)\geq D^*\}}|{\cal{F}}^X_T]+\\& +\mathsf{E}^Q\left[D(0,T)I_{\{L(T)-L(0)>L\}}(K-S(T))^+\frac{(1-\omega)V(T)}{D}I_{\{V(T)<D^*\}}\right]|{\cal{F}}^X_T]. \end{split} $$ (20) Then we can rewrite $ p(0,T) $ as
$$ p(0,T)= \mathsf{E}^Q[p(0,T,{\bf{J}})] =\mathsf{E}^Q[\mathsf{E}_1({\bf{J}})-\mathsf{E}_2({\bf{J}})+\mathsf{E}_3({\bf{J}})-\mathsf{E}_4({\bf{J}})], $$ (21) where
$$ \mathsf{E}_1({\bf{J}}) =\mathsf{E}^Q[D(0,T)I_{\{L(T)-L(0)>L\}}K I_{\{K\geq S(T)\}}I_{\{V(T)\geq D^*\}}|{\cal{F}}^X_T], $$ (22) $$ \mathsf{E}_2({\bf{J}}) =\mathsf{E}^Q[D(0,T)I_{\{L(T)-L(0)>L\}}S(T)I_{\{K\geq S(T)\}}I_{V(T)\geq D^*}|{\cal{F}}^X_T], $$ (23) $$ \mathsf{E}_3({\bf{J}}) =\mathsf{E}^Q\left[D(0,T)I_{\{L(T)-L(0)>L\}}K \frac{(1-\omega)V(T)}{D}I_{\{K\geq S(T)\}}I_{\{V(T)<D^*\}}|{\cal{F}}^X_T\right], $$ (24) $$ \mathsf{E}_4({\bf{J}}) =\mathsf{E}^Q\left[D(0,T)I_{\{L(T)-L(0)>L\}}S(T)\frac{(1-\omega)V(T)}{D}I_{\{K\geq S(T)\}}I_{\{V(T)<D^*\}}|{\cal{F}}^X_T\right]. $$ (25) Proposition 2 Let $ E_1({\bf{J}}) $ and $ E_2({\bf{J}}) $ be determined by (22) and (23). Then we have
$$ \mathsf{E}_1({\bf{J}})= K\mathrm{e}^{-\lambda_T({\bf{J}})}\textstyle\sum\limits_{n=1}^\infty\frac{(\lambda_T({\bf{J}}))^n}{n!} \displaystyle\int_L^{\infty}D(0,T,{\bf{J}})N_2(d_1(y,{\bf{J}}),d_2({\bf{J}}),-\hat{\rho}({\bf{J}}))f_L^{*n}(y){\mathrm{d}}y, $$ (26) $$ \mathsf{E}_2({\bf{J}})=\mathrm{e}^{-\lambda_T({\bf{J}})} \textstyle\sum\limits_{n=1}^\infty\frac{(\lambda_T({\bf{J}}))^n}{n!} \displaystyle\int_L^{\infty}S(0)\mathrm{e}^{-\alpha y+\beta(T)}N_2(d_3(y,{\bf{J}}),d_4({\bf{J}}),-\hat{\rho}({\bf{J}}))f_L^{*n}(y){\mathrm{d}}y, $$ (27) where
$$\qquad\, \begin{split} d_1(y,{\bf{J}})&= \dfrac{\ln\dfrac{K}{S(0)}-R_T({\bf{J}})+\dfrac{1}{2}U^1_T({\bf{J}})+\alpha y-\beta(T)}{\sqrt{U^1_T({\bf{J}})}},\\ d_2({\bf{J}})&=\dfrac{\ln\dfrac{V(0)}{D^*}+R_T({\bf{J}})-\dfrac{1}{2}U^2_T({\bf{J}})}{\sqrt{U^2_T({\bf{J}})}}, \, \hat{\rho}({\bf{J}})=\dfrac{\rho U^{12}_T({\bf{J}})}{\sqrt{U^1_T({\bf{J}})U^2_T({\bf{J}})}}, \end{split} $$ $$ \begin{split} d_3(y,{\bf{J}})&=\dfrac{\ln\dfrac{K}{S(0)}-R_T({\bf{J}})-\dfrac{1}{2}U^1_T({\bf{J}})+\alpha y-\beta(T)}{\sqrt{U^1_T({\bf{J}})}},\\ d_4({\bf{J}})&=\dfrac{\ln\dfrac{V(0)}{D^*}+R_T({\bf{J}})-\dfrac{1}{2}U^2_T({\bf{J}})+\rho U^{12}_T({\bf{J}})}{\sqrt{U^2_T({\bf{J}})}}. \end{split} $$ $ N_2(\cdot,\cdot,\cdot) $ is the bivariate normal cumulative distribution function and $ f_L^{*n}(y) $ represents the $ n$-fold convolution of the probability density function $ f_L(y) $.
Proof By the property of conditional expectation, we have
$$ \mathsf{E}_1({\bf{J}})=K\mathsf{E}^Q\{D(0,T)I_{\{L(T)-L(0)>L\}}\mathsf{E}^Q[I_{\{K\geq S(T)\}}I_{\{V(T)\geq D^*\}}|{\cal{F}}^X_T\textstyle \bigvee{\cal{F}}^L_T]|{\cal{F}}^X_T\}. $$ (28) From $ (11) $ and $ (12) $, we know
$$ S(T)= S(0){\mathrm{exp}}\left\{\int_0^T\left[r(u)-\frac{1}{2}\sigma_1^2(u)\right]{\mathrm{d}}u+\int_0^T\sigma_1(u){\mathrm{d}}W_1(u)-\alpha L(T)+\beta(T)\right\}, $$ (29) $$ V(T)= V(0){\mathrm{exp}}\left\{\int_0^T\left[r(u)-\frac{1}{2}\sigma_2^2(u)\right]{\mathrm{d}}u+\int_0^T\sigma_2(u){\mathrm{d}}W_2(u)\right\}. $$ (30) So
$$\begin{split} &\mathsf{E}^Q\left[I_{\{K\geq S(T)\}}I_{\{V(T)\geq D^*\}}|{\cal{F}}^X_T \textstyle \bigvee {\cal{F}}^L_T\right]\\ =&Q\bigg(\int_0^T\sigma_1(u){\mathrm{d}}W_1(u)\leq \ln\frac{K}{S(0)}-R_T({\bf{J}})+\frac{1}{2}U^1_T({\bf{J}})+\alpha L(T)-\beta(T),\\& \int_0^T\sigma_2(u){\mathrm{d}}W_2(u)\geq \ln\frac{D^*}{V(0)}-R_T({\bf{J}})+\frac{1}{2}U^2_T({\bf{J}})|{\cal{F}}^X_T\textstyle \bigvee {\cal{F}}^L_T\bigg)\\ =&N_2(d_1(L(T),{\bf{J}}),d_2({\bf{J}}),-\hat{\rho}({\bf{J}})). \end{split}$$ By computing the expectation over the observed losses $ L(T) $, we get
$$ \begin{split} & \mathsf{E}_1({\bf{J}})=K\mathsf{E}^Q\{D(0,T)I_{\{L(T)>L+L(0)\}}N_2(d_1(L(T),{\bf{J}}),d_2({\bf{J}}),-\hat{\rho}({\bf{J}}))|{\cal{F}}^X_T\}\\ = &K\mathrm{e}^{-\lambda_T({\bf{J}})} \textstyle\sum\limits_{n=1}^\infty\frac{(\lambda_T({\bf{J}}))^n}{n!} \int_L^{\infty}D(0,T,{\bf{J}})N_2(d_1(y,{\bf{J}}),d_2({\bf{J}}),-\hat{\rho}({\bf{J}}))f_L^{*n}(y) {\mathrm{d}}y, \end{split} $$ (31) where $ D(0,T,{\bf{J}})=D(0,T) $.
Let
$$ \eta_T^S=\frac{{\mathrm{d}}Q^S}{{\mathrm{d}}Q}= {\mathrm{exp}}\left\{\int_0^T\sigma_1(u){\mathrm{d}}W_1(u)-\frac{1}{2}\int_0^T\sigma_1^2(u){\mathrm{d}}u\right\}. $$ (32) By Girsanov's theorem, $ W^S_1(t)=W_1(t)-\int_0^t\sigma_1(u){\mathrm{d}}u $ and $ W^S_2(t)=W_2(t)-\rho\int_0^t\sigma_1(u){\mathrm{d}}u $ are two standard Brownian motions with $ \mathsf{Cov}({\mathrm{d}}W^S_1(t),{\mathrm{d}}W^S_2(t))=\rho {\mathrm{d}}t $ under $ Q^S $. So
$$ S(T)=S(0){\mathrm{exp}}\left\{\int_0^T\left[r(u)+\frac{1}{2}\sigma_1^2(u)\right]{\mathrm{d}}u+\int_0^T\sigma_1(u){\mathrm{d}}W^S_1(u)-\alpha L(T)+\beta(T)\right\}, $$ (33) $$ V(T)=V(0){\mathrm{exp}}\left\{\int_0^T\left[r(u)-\frac{1}{2}\sigma_2^2(u)\right]{\mathrm{d}}u+\int_0^T\sigma_2(u){\mathrm{d}}W^S_2(u)+\int_0^T\rho\sigma_1(u)\sigma_2(u){\mathrm{d}}u\right\}. $$ (34) By the property of conditional expectation, we get
$$ \mathsf{E}_2({\bf{J}}) = \mathsf{E}^Q\left\{D(0,T,{\bf{J}})I_{\{L(T)-L(0)>L\}}E^Q[S(T)I_{\{K\geq S(T)\}}I_{\{V(T)\geq D^*\}}|{\cal{F}}^X_T\bigvee {\cal{F}}^L_T]|{\cal{F}}^X_T\right\}. $$ (35) From Bayes' rule, we have
$$\begin{split} &\mathsf{E}^Q\left[S(T)I_{\{K\geq S(T)\}}I_{\{V(T)\geq D^*\}}|{\cal{F}}^X_T\textstyle \bigvee {\cal{F}}^L_T\right]\\ =&\mathsf{E}^Q\left[\eta_T^S|{\cal{F}}^X_T\textstyle \bigvee {\cal{F}}^L_T\right]\mathsf{E}^{Q^S}\left[\frac{S(T)}{\eta_T^S}I_{\{K\geq S(T)\}}I_{\{V(T)\geq D^*\}}|{\cal{F}}^X_T\textstyle \bigvee{\cal{F}}^L_T\right]\\ =&\mathsf{E}\left[\exp\left\{\int_0^T\sigma_1(u){\mathrm{d}}W_1(u)-\frac{1}{2}\int_0^T\sigma_1^2(u){\mathrm{d}}u\right\}\right] \mathsf{E}^{Q^S}\left[\frac{S(T)}{\eta_T^S}I_{\{K\geq S(T)\}}I_{\{V(T)\geq D^*\}}|{\cal{F}}^X_T\textstyle \bigvee {\cal{F}}^L_T\right]\\ =&S(0)D(0,T,{\bf{J}})\mathrm{e}^{-A(0,T)}\mathsf{E}^{Q^S}\left[I_{\{K\geq S(T)\}}I_{\{V(T)\geq D^*\}}|{\cal{F}}^X_T\textstyle \bigvee {\cal{F}}^L_T\right]\\ =&S(0)D(0,T,{\bf{J}})\mathrm{e}^{-A(0,T)}Q^S\bigg[\int_0^T\sigma_1(u){\mathrm{d}}W^S_1(u)\leq \ln\frac{K}{S(0)}-R_T({\bf{J}})-\frac{1}{2}U^1_T({\bf{J}})+A(0,T),\\& \int_0^T\sigma_2(u){\mathrm{d}}W^S_2(u)\geq \ln\frac{D^*}{V(0)}-R_T({\bf{J}})+\frac{1}{2}U^2_T({\bf{J}})-\rho U^{12}_T({\bf{J}})|{\cal{F}}^X_T \textstyle \bigvee {\cal{F}}^L_T\bigg]\\ =&S(0)D(0,T,{\bf{J}})\mathrm{e}^{-A(0,T)}N_2(d_3(L(T),{\bf{J}}),d_4({\bf{J}}),-\hat{\rho}({\bf{J}})). \\[-2pt] \end{split} $$ (36) Substituting (36) into (35), we get
$$ \mathsf{E}_2({\bf{J}})= \mathrm{e}^{-\lambda_T({\bf{J}})} \textstyle\sum\limits_{n=1}^\infty\frac{(\lambda_T({\bf{J}}))^n}{n!} \displaystyle\int_L^{\infty}S(0)\mathrm{e}^{-\alpha y+\beta(T)}N_2(d_3(y,{\bf{J}}),d_4({\bf{J}}),-\hat{\rho}({\bf{J}}))f_L^{*n}(y){\mathrm{d}}y. $$ (37) The proof is completed. □
Proposition 3 Let $ \mathsf{E}_3({\bf{J}}) $ and $ \mathsf{E}_4({\bf{J}}) $ be determined by (24) and (25). Then we have
$$ \mathsf{E}_3({\bf{J}})= K\frac{(1-\omega)}{D}\mathrm{e}^{-\lambda_T({\bf{J}})}\textstyle\sum\limits_{n=1}^\infty\dfrac{(\lambda_T({\bf{J}}))^n}{n!}\displaystyle\int_L^{\infty}V(0)f_L^{*n}(y) N_2(d_5(y,{\bf{J}}),d_6({\bf{J}}),\hat{\rho}({\bf{J}})){\mathrm{d}}y, $$ (38) $$ \begin{split} \mathsf{E}_4({\bf{J}})= & \frac{(1-\omega)}{D}\mathrm{e}^{-\lambda_T({\bf{J}})}\textstyle\sum\limits_{n=1}^\infty\dfrac{(\lambda_T({\bf{J}}))^n}{n!}\\&\times\int_L^{\infty}\dfrac{S(0)V(0)}{D(0,T,{\bf{J}})} \mathrm{e}^{-\alpha y+\beta(T)}f_L^{*n}(y) N_2(d_7(y,{\bf{J}}),d_8({\bf{J}}),\hat{\rho}({\bf{J}})){\mathrm{d}}y,\qquad\qquad\;\; \end{split} $$ (39) where
$$ \begin{split} d_5(y,{\bf{J}})&=\dfrac{\ln\dfrac{K}{S(0)}-R_T({\bf{J}})+\dfrac{1}{2}U^1_T({\bf{J}})-\rho U^{12}_T({\bf{J}}) +\alpha y-\beta(T)}{\sqrt{U^1_T({\bf{J}})}},\\ d_6({\bf{J}})&=\dfrac{\ln\dfrac{D^*}{V(0)}-R_T({\bf{J}})-\dfrac{1}{2}U^1_T({\bf{J}})}{\sqrt{U^2_T({\bf{J}})}},\\ d_7(y,{\bf{J}})&=\dfrac{\ln\dfrac{K}{S(0)}-R_T({\bf{J}})-\dfrac{1}{2}U^1_T({\bf{J}})-\rho U^{12}_T({\bf{J}})+\alpha y-\beta(T)}{\sqrt{U^1_T({\bf{J}})}},\\ d_8({\bf{J}})&=\dfrac{\ln\dfrac{D^*}{V(0)}-R_T({\bf{J}})-\dfrac{1}{2}U^2_T({\bf{J}})-\rho U^{12}_T({\bf{J}})}{\sqrt{U^2_T({\bf{J}})}}. \end{split}$$ Proof Define
$$ \eta_T^V =\frac{{\mathrm{d}}Q^V}{{\mathrm{d}}Q}={\mathrm{exp}}\left\{\int_0^T\sigma_2(u){\mathrm{d}}W_2(u)-\frac{1}{2}\int_0^T\sigma_2^2(u){\mathrm{d}}u\right\}. $$ (40) By Girsanov's theorem, $ W^V_1(t)=W_1(t)-\rho\int_0^t\sigma_2(u){\mathrm{d}}u $ and $ W^V_2(t)=W_2(t)-\int_0^t\sigma_2(u){\mathrm{d}}u $ are two standard Brownian motions with $ \mathsf{Cov}({\mathrm{d}}W^V_1(t),{\mathrm{d}}W^V_2(t))=\rho {\mathrm{d}}t $ under $ Q^V $. Similar to the calculation of $ \mathsf{E}_1({\bf{J}}) $ and $ \mathsf{E}_2({\bf{J}}) $, we get
$$ \mathsf{E}_3({\bf{J}})= K\frac{(1-\omega)}{D}\mathrm{e}^{-\lambda_T({\bf{J}})}\textstyle\sum\limits_{n=1}^\infty\frac{(\lambda_T({\bf{J}}))^n}{n!} \displaystyle\int_L^{\infty}V(0)N_2(d_5(y,{\bf{J}}),d_6({\bf{J}}),\hat{\rho}({\bf{J}}))f_L^{*n}(y){\mathrm{d}}y. $$ (41) Define
$$ \eta_T^{SV} =\frac{{\mathrm{d}}Q^{SV}}{{\mathrm{d}}Q^V}={\mathrm{exp}}\left\{\int_0^T\sigma_1(u){\mathrm{d}}W^V_1(u)-\frac{1}{2}\int_0^T\sigma_1^2(u){\mathrm{d}}u\right\}. $$ (42) By Girsanov's theorem, $ W^{SV}_1(t)=W_1^V(t)-\int_0^t\sigma_1(u){\mathrm{d}}u $ and $ W^{SV}_2(t)=W_2^V(t)- \rho\int_0^t\sigma_1(u){\mathrm{d}}u $ are two standard Brownian motions with $ \mathsf{Cov}({\mathrm{d}}W^{SV}_1(t),{\mathrm{d}}W^{SV}_2(t))=\rho {\mathrm{d}}t $ under $ Q^{SV} $. Similar to the calculation of $ \mathsf{E}_1({\bf{J}}) $, $ \mathsf{E}_2({\bf{J}}) $ and $ \mathsf{E}_3({\bf{J}}) $, we get
$$\begin{split} \mathsf{E}_4({\bf{J}})= & \frac{(1-\omega)}{D}\mathrm{e}^{-\lambda_T({\bf{J}})}\textstyle\sum\limits_{n=1}^\infty\dfrac{(\lambda_T({\bf{J}}))^n}{n!}\\&\times\int_L^{\infty}\dfrac{S(0)V(0)}{D(0,T,{\bf{J}})} \mathrm{e}^{-\alpha y+\beta(T)}f_L^{*n}(y) N_2(d_7(y,{\bf{J}}),d_8({\bf{J}}),\hat{\rho}({\bf{J}})){\mathrm{d}}y. \end{split}$$ (43) The proof is completed. □
In order to deduce the closed-form expression for pricing the catastrophe put option with credit risk in a regime-switching model, we need to know the distribution of $ {\bf{J}} $. In particular, we consider $ N=2 $, that is $ X(t) $ switches between only 2 states, where state $ {\bf{e}}_{\bf{1}} $ and state $ {\bf{e}}_{\bf{2}} $ represent a “good” economy and a “bad” economy, respectively. Taylor[14] pointed out that a Markov chain with two states is sufficient to distinguish a normal market from the one experiencing a crisis. Let $ f_1^{k}(s,T) $ be the probability density of $ J_1 $ given $ X(0)=k $, for $ k\in\{1,2\} $. As in Kovchegova et al.[12] and Yoon et al.[13], for all $ 0\leq s\leq T $, $ f_1^k(s,T) $ satisfies
$$ \begin{split} f_1^1(s,T)=&\mathrm{e}^{-\lambda s-\nu (T-s)}[\delta(T-s)+\sqrt{\frac{\lambda\nu s}{T-s}}I_1(2\sqrt{\lambda\nu s(T-s)}+\lambda I_0(2\sqrt{\lambda\nu s(T-s)}],\\ f_1^2(s,T) =&\mathrm{e}^{-\lambda s-\nu (T-s)}[\delta(s)+\sqrt{\frac{\lambda\nu (T-s)}{s}}I_1(2\sqrt{\lambda\nu s(T-s)}+\nu I_0(2\sqrt{\lambda\nu s(T-s)}], \end{split} $$ where $ I_\rho(z) $ is the modified Bessel function defined by
$$ I_\rho(z)=\textstyle\sum\limits_{n=0}^{\infty}\dfrac{1}{n!\Gamma(\rho+n+1)}(\dfrac{z}{2})^{2n+\rho}. $$ Let $ f_1(s,T) $ be the probability density of $ J_1 $, and then
$$ f_1(s,T)=<(f_1^1(s,T),f_1^2(s,T))',X_0>. $$ Therefore, combining Proposition 2 and Proposition 3, the price of the catastrophe put option with credit risk in a regime-switching model is given by the following theorem.
Theorem 4 Let $ p(0,T) $ be determined by (21), and then the price of the catastrophe put option with credit risk in a regime-switching model is
$$ \begin{split} p(0,T)&=\int_0^Tp(0,T,s)f_1(s,T){\mathrm{d}}s\\ &=\int_0^T[\mathsf{E}_1(s)-\mathsf{E}_2(s)+\mathsf{E}_3(s)-\mathsf{E}_4(s)]f_1(s,T){\mathrm{d}}s. \end{split} $$ (44) 5. Numerical analysis
In this section, using the explicit formulae obtained in the previous sections, we present some numerical analysis to examine how the parameters of the model affect the price of the catastrophe put option. We assume that the loss size conditional on a loss is fixed at l and that the trigger level is an integer multiple of the loss size, i.e., $ L=Cl $. As explained in Jaimungal and Wang[2], and Jiang et al.[4], the parameter C, which coin the trigger ratio level, represents the ratio of the trigger level to the total expected loss amount. In this predetermined loss case, the probability density function of loss sizes is a Dirac density $ f_L(y)=\delta(y-l) $, and then $ \mathsf{E}_1({\bf{J}}),\mathsf{E}_2({\bf{J}}),\mathsf{E}_3({\bf{J}}) $ and $ \mathsf{E}_4({\bf{J}}) $ can be expressed as follows:
$$ \mathsf{E}_1({\bf{J}})= K\mathrm{e}^{-\lambda_T({\bf{J}})}\textstyle\sum\limits_{n=C}^\infty\dfrac{(\lambda_T({\bf{J}}))^n}{n!}D(0,T,{\bf{J}}) N_2(d_1(nl,{\bf{J}}),d_2({\bf{J}}),-\hat{\rho}({\bf{J}})), $$ (45) $$ \mathsf{E}_2({\bf{J}})= \mathrm{e}^{-\lambda_T({\bf{J}})}\textstyle\sum\limits_{n=C}^\infty\dfrac{(\lambda_T({\bf{J}}))^n}{n!}S(0)\mathrm{e}^{-\alpha nl+\lambda_T({\bf{J}})(1-\mathrm{e}^{-\alpha l})}N_2(d_3(nl,{\bf{J}}),d_4({\bf{J}}),-\hat{\rho}({\bf{J}})), $$ (46) $$ \mathsf{E}_3({\bf{J}})= K\frac{(1-\omega)}{D}\mathrm{e}^{-\lambda_T({\bf{J}})} \textstyle\sum\limits_{n=C}^\infty\dfrac{(\lambda_T({\bf{J}}))^n}{n!}V(0)N_2(d_5(nl,{\bf{J}}),d_6({\bf{J}}),\hat{\rho}({\bf{J}})), $$ (47) $$ \mathsf{E}_4({\bf{J}})=\frac{(1-\omega)}{D}\mathrm{e}^{-\lambda_T({\bf{J}})} \textstyle\sum\limits_{n=C}^\infty\dfrac{(\lambda_T({\bf{J}}))^n}{n!}\dfrac{S(0)V(0)}{D(0,T,{\bf{J}})} \mathrm{e}^{-\alpha nl+\lambda_T({\bf{J}})(1-\mathrm{e}^{-\alpha l})} \cdot N_2(d_7(nl,{\bf{J}}),d_8({\bf{J}}),\hat{\rho}({\bf{J}})). $$ (48) Unless otherwise specified, the following parameters are fixed: $ T=5, V=9, D^*=5, D=8, C=2, \alpha=0.2, l=3, {\bf{r}}=(0.05,0.02)'$, $\text{σ}_{\bf{1}}=(0.1, 0.4)', \text{σ}_{\bf{2}}=(0.1, 0.4)', A= {\left(\begin{array}{*{20}{c}}{-0.4}& {0.4 }\\{0.5} & {-0.5} \end{array}\right)} $.
Fig. 1 presents the relation between the strike price K and the catastrophe put option price p for different ω and X(0). Fixed $ \omega=0.4, \text{λ}=(0.1,0.3)',\rho=0.1, S(0)=80 $, we see that the catastrophe put option price p increases as the strike price K increases. The catastrophe put option price p is higher when we start at the state $ {\bf{e}}_{\bf{2}} $. Fixed the strike price K, we see that the catastrophe put option price p decreases as ω increases. Fig. 2 presents the relation between the share price S and the catastrophe put option price p for different ω and X(0). Fixed $ \omega=0.4, \text{λ}=(0.1,0.3)', \rho=0.1, K=80 $, we see that the catastrophe put option price p decreases as the share price S increases. The catastrophe put option price p is higher when we start at the state $ {\bf{e}}_{\bf{2}} $. Fixed the share price S, we see that the catastrophe put option price p decreases as ω increases.
Fig. 3 presents the relation between the strike price K and the catastrophe put option price p for different $ \text{λ} $ and X(0). Fixed $ \omega=0.4, \text{λ}=(0.1,0.3)',\rho=0.1, S(0)=80 $, we see that the catastrophe put option price p increases as the strike price K increases. The catastrophe put option price p is higher when we start at the state $ {\bf{e}}_{\bf{2}} $. Fixed the strike price K, we see that the catastrophe put option price p increases as λ increases. Fig. 4 presents the relation between the share price S and the catastrophe put option price p for different $ \text{λ} $ and X(0). Fixed $ \omega=0.4,\text{λ}=(0.1,0.3)', \rho=0.1, K=80 $, we see that the catastrophe put option price p decreases as the share price S increases. The catastrophe put option price p is higher when we start at the state $ {\bf{e}}_{\bf{2}} $. Fixed the share price S, we see that the catastrophe put option price p increases as $ \text{λ} $ increases.
Fig. 5 presents the relation between the strike price K and the catastrophe put option price p for different ρ and X(0). Fixed $ \omega=0.4,\text{λ}=(0.1,0.3)',\rho=0.1, S(0)=80 $, we see that the catastrophe put option price p increases as the strike price K increases. The catastrophe put option price p is higher when we start at the state $ {\bf{e}}_{\bf{2}} $. Fixed the strike price K, we see that the catastrophe put option price p decreases as ρ increases. Fig. 6 presents the relation between the share price S and the catastrophe put option price p for different ρ and X(0). Fixed $ \omega=0.4, \text{λ}=(0.1,0.3)', \rho=0.1, K=80 $, we see that the catastrophe put option price p decreases as the share price S increases. The catastrophe put option price p is higher when we start at the state $ {\bf{e}}_{\bf{2}} $. Fixed the share price S, we see that the catastrophe put option price p decreases as ρ increases.
Acknowledgements The authors are very grateful to the referees for valuable comments and criticisms to improve the earlier version of the paper.
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