Pricing dynamic fund protection under hidden Markov models

Pricing dynamic fund protection under hidden Markov models Abstract In this article, we discuss the pricing of dynamic fund protection when the value process of the investment fund is governed by a geometric Brownian motion with parameters modulated by a continuous-time, finite-state hidden Markov chain. Under a risk-neutral probability measure, selected by the Esscher transform, we adopt the partial differential equation approach to value the dynamic fund protection. Using the estimated sequence of the hidden Markov chain, we apply the Baum–Welch algorithm and the Viterbi algorithm to derive the maximum likelihood estimates of the parameters. Numerical examples are provided to illustrate the practical implementation of the model. 1. Introduction Compared with upside profits, investors often place greater emphasis on downside risk. Partly due to this reason, academic researchers and industry practitioners have innovated a variety of financial and insurance products to deal with the downside risk, including traditional put options. However, put options can only provide investors with protection at the exercise moment. Although innovative investment protection plans are introduced to overcome this disadvantage, many of them are too sophisticated. Among different investment protection plans, dynamic fund protections (DFPs) have been a popular type of investment plans. Compared with traditional put options, DFP plans protect investors for the whole investment period. This feature may explain why the dynamic fund protection plans have attracted both academic researchers’ and investors’ attention. There are different forms of DFP plans. The most fundamental one is that the investment fund value will be upgraded when the fund value falls below a certain protection level. This kind of protection plan was pioneered by Gerber & Shiu 1998. Since the work of Gerber & Shiu 1998, much attention has been paid to the DFP valuation when the value process of the underlying investment fund is governed by various stochastic models, such as the classical Black–Scholes model and the constant elasticity of variance (CEV) process (see Gerber & Shiu, 1999, 2003a,b; Gerber & Pafumi, 2000; Imai & Boyle, 2001; Fung & Li, 2003; Chu & Kwok, 2004). Indeed, DFP plans are products with lookback option features, which were shown in Imai & Boyle (2001); Chu & Kwok (2004) and Wong & Chan (2007). Wong & Chan (2007) considered the valuation of DFPs under a multiscale stochastic volatility model and derived semi-analytical formulas using asymptotic techniques. They also established the relationship between a fixed strike lookback call option and a DFP. In their paper, the parity equation between a fixed strike lookback call option and a floating strike lookback put option, derived in Wong & Kwok (2003), was used. Apart from pricing the DFPs, Tse et al. (2008) investigated the hedging of discrete-DFPs. Wong & Lam (2010) assumed that the investment fund follows a Lévy process and solved the valuation problem. Regime-switching models are one of the important classes of models in financial econometrics. This class of model can incorporate the impacts of changes in economic conditions on financial and economic dynamics. Hamilton (1989) popularized the use of regime-switching models in financial econometrics. There are many works on valuing options under regime-switching models. Some examples of early works on this topic are Naik (1993), Guo (2001), Buffington & Elliott (2002a, b), Elliott et al. (2005) and Siu (2005), amongst others. Due to the presence of an additional source of uncertainty described by the modulating Markov chain, the market in a regime-switching model is, in general, incomplete. Consequently, there is more than one equivalent martingale measure for valuation. Gerber & Shiu (1994) introduced the use of the Esscher transform to price options. The Esscher transform has been used to select an equivalent martingale measure in a regime-switching market (see, for example, Elliott et al., 2005; Siu, 2005, 2008, 2011; Boyle & Draviam, 2007; Yuen & Yang, 2010; Siu et al., 2011; etc.). It seems that the mainstream of the literature on option valuation under regime-switching models considered the situation where the Markov chain is completely observable while, in practice, the states of the chain (or the states of the economy) may not be directly observable. Hidden Markov models present a natural choice for modelling transitions in hidden states of an economy. Assuming that the Markov chain is not observable, in a discrete-time regime-switching model, Ishijima & Kihara (2005) studied an option valuation problem using a locally risk-neutral valuation relationship. A higher-order hidden Markov model (HHMM) was considered in Ching et al. (2007). They assumed the dynamics of the asset price follow the HHMM and investigated the valuation of exotic options, including Asian options, barrier options and lookback options. Liew & Siu (2010) considered option valuation under a discrete-time hidden Markov regime-switching Gaussian model. They adopted both the Esscher transform and the extended Girsanov’s principle to select a pricing kernel. There are some works in the literature concerning option valuation in continuous-time hidden Markov regime-switching models. Some examples are Elliott & Siu (2014) for hidden Markov-modulated pure jump processes, Siu (2013, 2014) for hidden Markov-modulated jump-diffusion models and Elliott & Siu (2015) for a hidden Markov-modulated jump-diffusion model with jumps attributed to trading volumes, amongst others. These works mainly focused on valuing financial options. In this article, we discuss the valuation of the DFP plans when the dynamics of the investment fund are governed by a geometric Brownian motion whose coefficients are modulated by a hidden Markov chain. We assume that the appreciation rate and the volatility are modulated by a continuous-time, finite-state hidden Markov chain. The states of the chain represent hidden states of an economy. The Esscher transform is employed to select a pricing kernel for the valuation of the DFP. Considering the lookback option feature in the DFP, we adopt a partial differential equation (PDE) approach to price the floating strike lookback put options. Then, the prices of the corresponding fixed strike lookback call options can be calculated according to the put-call parity derived in Wong & Kwok (2003). In a recent paper by Jin et al. (2016), the pricing of DFP was also studied in a Markovian regime-switching modelling framework. Compared with Jin et al. (2016), our paper uses a PDE approach and assumes the chain is hidden, while Jin et al. (2016) uses a Laplace transform approach and assumes the chain is observable. To estimate the parameters of the HMM, we adopt the Baum–Welch algorithm, based on the discretization of continuous-time smoother. Then, the Viterbi algorithm is applied to obtain the most-probable path of the hidden Markov chain. The maximum likelihood estimates of model parameters are derived using the most probable path of the hidden Markov chain. To illustrate the valuation of DFP, we give numerical examples based on real financial data. The rest of the article is structured as follows: Section 2 presents the hidden Markov-modulated investment fund model. In Section 3, we select a pricing kernel by the Esscher transform. Section 4 presents the PDE approach to value the DFP. In Section 5, we discuss how to estimate the states and transition matrix of the HMM. The maximum likelihood estimates of the model parameters are also presented. Some numerical examples are given in Section 6. The final section summarizes the article. 2. The model dynamics In this section, we consider a continuous-time economy with two investment securities, namely, a risk-free bond $$B$$ and an investment fund $$S$$, which are continuously tradable on the time horizon $$\mathcal{T}$$. Here, $$\mathcal {T} := [0,T]$$, where $$T<\infty$$. We assume that there are no transaction costs and taxes involved in trading. Furthermore, all dividends are reinvested in the fund and any fractional units of the index can be traded. Let $$(\Omega, \mathcal {F}, \mathcal {P})$$ be a complete probability space, where $$\mathcal {P}$$ is a real-world probability measure. We adopt bold-face letters to denote matrices (or vectors), and write $$\mathbf{y}'$$ for the transpose of a matrix, or a vector $$\mathbf{y}$$. Let $$\mathbb{1}_{E}$$ denote the indicator function of an event $$E$$. We assume that the hidden states of an economy are modelled by a continuous-time, finite-state Markov chain $${\bf X} := \{ {\bf X}_t | t \in \mathcal {T} \}$$. The Markov chain is defined on $$(\Omega, \mathcal {F}, \mathcal {P})$$ with the finite state space $$\mathcal {S} : = \{ \mathbf{s}_1, \mathbf{s}_2, \ldots , \mathbf{s}_N \}$$. Without loss of generality, following Elliott et al. (1995), the state space of $${\bf X}$$ is identified with a finite set of standard unit vectors $$\mathcal {E}:= \{\mathbf{e}_1, \mathbf{e}_2, \ldots, \mathbf{e}_N \}$$, where $${\bf e}_i \in \mathfrak{R}^N$$ are column vectors and the $$j$$th component of $${\bf e}_i$$ is the Kronecker delta $$\delta_{ij}$$, for each $$i, j = 1,2,\ldots, N$$. We call $$\mathcal {E}$$ the canonical state space of the chain $${\bf X}$$. We define $${\bf A} := ( a_{ij} )_{i, j = 1, 2, \ldots, N}$$ as the generator of the chain $${\bf X}$$ under $$\mathcal {P}$$, where $$a_{i i} = - \sum_{j=0, j \neq i}^N a_{i j}, i=0,1,\ldots, N$$. Note that the transition probability, from time $$t$$ to time $$t+\Delta t$$, is $$\boldsymbol{I}+{\bf A}\Delta t$$, where $$\boldsymbol{I}$$ denotes the identity matrix.1 Elliott et al. (1995) obtained the following semi-martingale dynamics for the chain:   Xt=X0+∫0tAXsds+Mt . where $$\{\mathbf{M}_t | t \in \mathcal {T} \}$$ is an $$\mathfrak{R}^N$$-valued square-integrable martingale under $$\mathcal{P}$$ with respect to the natural filtration generated by $$\mathbf{X}$$. The price dynamics of the risk-free bond and the investment fund are now defined. Firstly, we assume that the bond price process $$\{ B_t | t \in \mathcal {T} \}$$ is governed by   dBt=rBtdtt∈T,B0=1 . Here, the instantaneous market interest rate of the bond is assumed to be a positive constant, i.e. $$r > 0$$. In general, one may consider a stochastic interest rate which depends on the chain $${\bf X}_t$$. In this case, filtering of interest rate is involved, and this complicates the modelling and pricing issues. To simplify the matter, we consider a relatively simpler situation that the interest rate is a constant and only focus on the volatility risk. Let $$W := \{W_t | t \in \mathcal {T}\}$$ be a standard Brownian motion on $$(\Omega, \mathcal {F}, \mathcal {P})$$ with respect to its right-continuous, $$\mathcal {P}$$-complete, natural filtration. We suppose that $$W$$ and $${\bf X}$$ are stochastically independent under $$\mathcal {P}$$.2 The dynamics of the investment fund $$S:=\{S_t | t \in \mathcal {T}\}$$ is governed by a Markovian regime-switching geometric Brownian motion:   dSt=μtStdt+σtStdWt,t∈T,S0=s. The appreciation rate $$\mu_t$$ and the volatility $$\sigma_t$$ are modulated by the chain $${\bf X}$$ as follows:   μt:=⟨μ,Xt⟩,σt:=⟨σ,Xt⟩,t∈T. Here $$\left< \cdot, \cdot \right>$$ denotes the scalar product in $$\mathfrak{R}^N$$. Moreover, $$\boldsymbol{\mu}=( \mu_1, \mu_2, \ldots, \mu_N )' \in \mathfrak{R}^N$$ and $$\boldsymbol{\sigma}=( \sigma_1, \sigma_2, \ldots, \sigma_N )' \in \mathfrak{R}^N$$ for each $$t \in \mathcal {T}$$, where $$\sigma_i>0$$ for all $$i=1,2,\ldots,N$$. Let $$\{Y_t | t \in \mathcal{T}\}$$ be the logarithmic return process, where $$Y_t := \ln (S_t/S_0 )$$ for each $$t \in \mathcal {T}$$. Then, the underlying fund dynamics can be written as:   St=S0eYt,t∈T, where   Yt=Y0+∫0t(⟨μ,Xs⟩−12⟨σ2,Xs⟩)ds+∫0t⟨σ,Xs⟩dWs,t∈T. Here, $$\boldsymbol{\sigma}^2 := (\sigma_1^2, \sigma_2^2, \ldots, \sigma_N^2)^\prime \in \mathfrak{R}^N$$. For notational simplicity, write $$\eta_i = \mu_i - \frac{1}{2} \sigma_i^2$$ for $$i = 1, 2, \ldots, N$$ and $$\boldsymbol{\eta} = (\eta_1, \eta_2, \ldots, \eta_N) ^ \prime$$. Then, the dynamics of $$Y$$ can be rewritten as follows:   Yt=Y0+∫0t⟨η,Xs⟩ds+∫0t⟨σ,Xs⟩dWs,t∈T. (2.1) Let $$\mathcal {F}^Y =\{\mathcal {F}^Y_t | t \in \mathcal {T} \}$$ and $$\mathcal {F}^S =\{\mathcal {F}^S_t | t \in \mathcal {T} \}$$ be the natural filtrations generated by $$\{Y_t| t \in \mathcal {T}\}$$ and $$\{S_t | t \in \mathcal {T}\}$$, respectively. Also, we assume that the filtrations given above are right continuous and complete. Since $$\mathcal {F}^Y$$ and $$\mathcal {F}^S$$ are equivalent, either of them could be used as the observed information structure. Here, we adopt $$\mathcal {F}^Y$$ as the observed information structure. We define the filtration $$\mathcal{G} = \{ \mathcal{G}_{t,s}| 0\leq s\leq t \in \mathcal{T} \}$$ by letting the double indexed $$\sigma$$-field $$\mathcal{G}_{t,s} := \mathcal{F}^{{\bf X}}_t \bigvee \mathcal{F}^Y_s$$, for any $$s,t \in \mathcal{T}$$ with $$s\leq t$$, where $$\mathcal{A} \bigvee \mathcal{B}$$ is the minimal $$\sigma-$$algebra containing the $$\sigma-$$algebras $$\mathcal{A}$$ and $$\mathcal{B}$$. We write $$\mathcal{G}_t = \mathcal{G}_{t,t}$$, for all $$t \in \mathcal{T}$$, and $$\mathcal{G} =\{ \mathcal{G}_t | t \in \mathcal {T}\}$$. Then $$\mathcal {F}^Y$$, $$\mathcal {F}^{{\bf X}}$$ and $$\mathcal {G}$$ represent the flows of observable information, hidden information and full information, respectively. 3. The Esscher transform The developments in this section follow that in Elliott et al. (2005). As in Elliott et al. (2005), let $$\theta_t$$ be the regime switching Esscher parameter, which can be written as   θt=⟨θ,Xt⟩, where $$\boldsymbol{\theta} = ( {\theta_1, \theta_2, \ldots, \theta_N} ) ^\prime$$. Write $$(\theta \cdot Y)_t := \int_0^t \theta_s {\rm d} Y_s$$, for each $$t \in \mathcal {T}$$. The regime switching Esscher transform $$\mathcal {Q}_\theta \sim \mathcal {P}$$ on $${\mathcal {G}}_t$$ with respect to a family of parameters $$\{\theta_s | 0\leq s\leq t\}$$ is given by   dQθdP|Gt=e(θ⋅Y)tE[e(θ⋅Y)t|FtX], where $$\mbox{E}[\cdot]$$ is the expectation taken under $$\mathcal {P}$$. Since   E[e(θ⋅Y)t|FtX]=exp⁡[∫0t(⟨μ,Xs⟩−12⟨σ2,Xs⟩)θsds+12∫0t⟨σ2,Xs⟩θs2ds], the Radon-Nikodym derivative of the regime switching Esscher transform is given by   dQθdP|Gt=exp⁡{∫0t⟨σ,Xs⟩θsdWs−12∫0t⟨σ2,Xs⟩θs2ds}. Let $$\{ \theta^*_t | t \in \mathcal{T} \}$$ be a family of risk-neutral regime switching Esscher parameters. Consider the following martingale condition:   S0=EQθ∗[e−rtSt|FtX],t∈T. Using a version of Bayes’ rule, the martingale condition can be rewritten as:   ∫0t(⟨μ,Xs⟩−r+θs∗⟨σ2,Xs⟩)ds=0. Elliott et al. (2005) shows that $$\theta^*_t$$ can be determined uniquely as:   θt∗=r−⟨μ,Xt⟩⟨σ2,Xt⟩, Then, the Radon-Nikodym derivative can be written by:   dQθ∗dP|Gt=exp⁡{∫0t(r−⟨μ,Xs⟩⟨σ,Xs⟩)dWs−12∫0t(r−⟨μ,Xs⟩⟨σ,Xs⟩)2ds}. Using Girsanov’s theorem, $$\widehat{W}_t = W_t + \int_0^t \left(\frac{r- \mu_s}{ \sigma_s} \right) d s$$ is a standard Brownian motion with respect to $$\mathcal {G}$$ under $$\mathcal {Q}_{\theta^*}$$. Hence, under the probability measure $$\mathcal {Q}_{\theta^*}$$, the stock price dynamics can be written as   dSt=rStdt+⟨σ,Xt⟩StdW^t. (3.1) 4. Pricing the DFP During the whole investment period, investors who own the investment fund are guaranteed a predetermined protection level under the DFP. Let $$K$$ denote the constant protection level. As indicated by Imai & Boyle (2001) and Wong & Chan (2007), the amount received by the holder of the fund is then given by:   STmax{1,max0≤τ≤TKSτ}, where $$S_t$$ is the value of the fund without the protection. Hence, the terminal payoff for the DFP should be:   DFPT =STmax{1,max0≤τ≤TKSτ}−ST =STmax{0,max0≤τ≤TKSτ−1}. Write   Ft=K/Stand Nt=max0≤τ≤tFτ. Then   DFPT=STmax(NT−1,0). As noted in Wong & Chan (2007), the payoff function of a fixed strike lookback call is:   cfix(T,FT,NT,1)=max(NT−1,0). According to Wong & Chan (2007), if we view $$S$$ as an exchange rate and hence $$F$$ as an asset trading in the foreign currency world, this option can simply be valued as a fixed strike lookback call in ‘the foreign currency world’ followed by multiplying the exchange rate $$S_t$$. Note that the fixed strike price of the lookback call is one unit. Hence, a model-independent formula is written as:   DFPt=St×cfix(t,Ft,Nt). To value the DFP, we focus on the process of $$F$$ in the foreign currency world. The dynamics of $$F_t$$ under the risk-neutral probability $$\mathcal {Q}_{\theta^*}$$ is given by:   dFt=−(r−σt2)Ftdt−σtFtdW^t. (4.1) We now price the European-type fixed strike lookback call under a hidden Markov-modulated regime-switching model. Under the risk-neutral probability measure $$\mathcal {Q}_{\theta^*}$$, a price at time $$t$$ with payoff $$V_T$$ at maturity $$T$$ is given by:   Vt=EQθ∗[e−r(T−t)VT|Gt]. (4.2) Here $${\mbox E}_{{\mathcal{Q}}_{\theta^*}} [\cdot]$$ is the expectation taken with respect to $${\mathcal{Q}}_{\theta^*}$$. Defining $$D_t :=\max\limits_{0 \leq u \leq t} S_u$$, Shreve (2004) showed that the pair of the processes $$(S_t,D_t)$$ has the Markov property. Similarly, we can show that the triplet of the processes $$({\bf X}_t, S_t, D_t)$$ has the joint Markov property. Denote   St=S0eW^t, where $$\hat{W_t}=\int_0^t \bigg(r - \frac{1}{2} \sigma^2_s \bigg) {\rm d} s + \int_0^t \sigma_s {\rm d} W_s$$, and define $$\hat{M}_t=\max\limits_{0 \leq u \leq t} \hat{W}_u$$. Then, $$D_t :=\max\limits_{0 \leq u \leq t} S_u = S_0 e^{\hat{M}_t}$$. Since $$\hat{W}_T - \hat{W}_t = \int_t^T \bigg( r- \frac{1}{2} \sigma^2_s \bigg) {\rm d} s + \int_t^T \sigma_s {\rm d} W_s$$ and $$\sup\limits_{t \leq u \leq T} (\hat{W}_u - \hat{W}_t)$$ are independent of $$\mathcal{G}_t$$, and   DT = max0≤u≤TSu =S0eM^T =S0esupt≤u≤TW^u1{M^t≤supt≤u≤TW^u}+S0eM^t1{M^t>supt≤u≤TW^u} =Stesupt≤u≤T(W^u−W^t)1{DtSt≤exp⁡(supt≤u≤T(W^u−W^t))}+Dt1{DtSt>exp⁡(supt≤u≤T(W^u−W^t))}. Then   EQθ∗[f(XT,ST,DT)|Gt] =EQθ∗[f(XT,SteW^T−W^t,Stesupt≤u≤T(W^u−W^t)1{DtSt≤exp⁡(supt≤u≤T(W^u−W^t))} +Dt1{DtSt>exp⁡(supt≤u≤T(W^u−W^t))})|Xt,St,Dt] is a function of $$({\bf X}_t, S_t, D_t)$$. Thus, we can see that the triplet of the processes $$({\bf X}_t, S_t, D_t)$$ has the joint Markov property. Similarly, we could show the triplet of the processes $$({\bf X}_t, F_t, N_t)$$ also has the joint Markov property. There exists a measurable function $$v(t,{\bf x},y,z)$$ such that $$V_t=v(t, {\bf X}_t, F_t, N_t )$$. The following theorem is standard. It is derived under the assumption that $$v (t, {\bf x}, y, z) \in {\mathcal{C}}^{1,2,1}$$, where $${\mathcal{C}}^{1, 2, 1}$$ is the space of all functions $$v (t, {\bf x}, y, z)$$ which is continuously differentiable in $$(t, z)$$ and twice continuously with respect to $$y$$ for each $${\bf x} \in {\mathcal{E}}$$. Theorem 4.1 Let $$v (t, {\bf x}, y, z )$$ denote the price of the fixed strike lookback option at time $$t$$ under the assumption that $${\bf X}_t={\bf x}, F_t=y, \ \mbox{and} \ N_t=z$$. Note that $$\frac{\partial v}{\partial t}$$, $$\frac{\partial v}{\partial y}$$ and $$\frac{\partial^2 v}{\partial y^2}$$ represent the first derivative of the function with respect to $$t$$ and $$y$$ and the second derivative with respect to $$y$$, respectively. Then the price of the option is a solution of the following PDE:    −rv(t,x,y,z)−∂v∂y(t,x,y,z)(r−σt2)y+∂v∂t(t,x,y,z) +12∂2v∂y2(t,x,y,z)σt2y2+⟨v,Ax⟩=0 (4.3) in the region of $$\{ (t,{\bf x},y,z); 0 \leq t < T, 0 \leq y \leq z \}$$. Here, $$\boldsymbol{v} = (v_1, v_2, \ldots, v_N)^\prime$$. Similarly to a PDE for a lookback option, Eq. (4.3) does not involve the derivative of $$v$$ with respect to $$z$$ though $$v$$ is a function of $$z$$, where $$z$$ is the maximum price up to and including time $$t$$. The variable $$z$$ only plays the role for specifying the boundary condition of $$v$$. Proof. The proof is standard. Applying It$$\hat{o}'$$s differentiation rule to $$e^{-rt} v (t, {\bf X}_t, F_t, N_t )$$ gives:   d(e−rtv(t,x,y,z)) =e−rt[−rv(t,x,y,z)dt+∂v∂y(t,x,y,z)dFt+∂v∂t(t,x,y,z)dt +∂v∂z(t,x,y,z)dNt+12∂2v∂y2(t,x,y,z)d⟨Fc,Fc⟩t +⟨v,Ax⟩dt+⟨v,dMt⟩] =e−rt[−rv(t,x,y,z)−∂v∂y(t,x,y,z)(r−σt2)y+∂v∂t(t,x,y,z) +12∂2v∂y2(t,x,y,z)σt2y2+⟨v,Ax⟩]dt +e−rt∂v∂z(t,x,y,z)dNt+e−rt⟨v,dMt⟩ −e−rtσty∂v∂y(t,x,y,z)dW^t. Since the discounted value process is a martingale, the drift term must be zero, i.e. the coefficients of the d$$t$$ term must be zero. □ The system of PDEs needs to be solved subject to particular terminal conditions and the boundary conditions. It is not easy to determine the boundary conditions for the fixed strike lookback call. Similarly to Wong & Chan (2007), we first focus on the lookback options with linear homogeneous payoffs, i.e. the floating strike lookback options. They stated all existing lookback options could be valued through the pricing formulas of floating strike lookback options. Besides, Wong & Kwok (2003) showed put-call parity relations between a fixed strike lookback call and a floating strike lookback put option. Then we could derive the price of fixed lookback call options. The corresponding terminal payoff of the floating strike lookback put is given by   v(T,x,y,z)=z−y. Following Wilmott et al. (1997) and Boyle & Draviam (2007), the boundary conditions are   v(t,x,0,z)=ze−r(T−t), and   ∂v∂z(t,x,y=z)=0. 5. Estimation of the hidden Markov model parameters Since the unknown parameters depend on the hidden state of the economy, the state of the HMM $${\bf X}_t$$ plays a vital role in the estimation of the model. For option valuation we discussed earlier, we consider a risk-neutral probability $$\mathcal {Q}_{\theta^*}$$ selected by the Esscher transform. Siu (2011) showed that an optimal equivalent martingale measure selected by the minimal relative entropy approach does not price the regime-switching risk. Mathematically, the probability law of the chain $${\bf X}$$ remains the same after the measure change. It is not unreasonable to consider the hidden states of the Markov chain under the real-world probability $$\mathcal {P}$$ instead of the risk-neutral probability. In practice, the return process is observed under $$\mathcal {P}$$ instead of under $$\mathcal {Q}_{\theta^*}$$. Consequently, it is reasonable to discuss the estimation of the Markov chain parameters under $$\mathcal {P}$$ rather than under $$\mathcal {Q}_{\theta^*}$$. Note that real data is sampled discretely over time. To estimate the parameters of the hidden Markov chain, algorithms based on discretization of continuous-time filters and smoothers are needed. Here, we adopt the Baum–Welch algorithm, based on the discretization of continuous-time smoothers.3 5.1. Estimating states of the HMM The smoother-based estimators facilitate the application of the Expectation Maximum (EM) algorithm, which can be regarded as an extension of the discrete-time Baum–Welch algorithm. However, it is known that the Baum–Welch algorithm only finds a local maximum, rather than a global maximum, in the parameter space. Here, we will give a brief introduction to the Baum–Welch algorithm. Let $$\boldsymbol{P} = [p_{ij}]_{i,j = 1,2, \ldots, N}$$ be the transition probability matrix, where $$p_{ij}= P({\bf X}_t = {\bf e}_j | {\bf X}_{t-1} = {\bf e}_i)$$ and $$\pi_i = P({\bf X}_1={\bf e}_i)$$ represent the initial state distribution. In addition, the corresponding observation sequence is assumed to be $${Y_1=y_1, \ldots, Y_T = y_T}$$. Let $$c_j(y_t) = P(Y_t = y_t | {\bf X}_t={\bf e}_j)$$ be the probability of a particular observation at time $$t$$ when the Markov chain is in the state $${\bf e}_j$$ and $$\boldsymbol{C}(\cdot) = (c_1(\cdot), c_2(\cdot), \ldots, c_N(\cdot))$$. Specially, we partition the observation sequence into $$L$$ groups, i.e. $$O = \{o_1, \ldots, o_L\}$$. As assumed in Bilmes (1998), we write the complete set of HMM parameters as $$\boldsymbol{\psi} = (\boldsymbol{P}, \boldsymbol{C}, \boldsymbol{\pi})$$.4 Step 1: Firstly, we give a brief introduction to the forward and backward procedures given in Bilmes (1998). Define the following forward variable $$\alpha_i(t)$$ and backward variable $$\beta_i(t)$$:   αi(t) =P(Y1=y1,…,Yt=yt,Xt=ei|ψ),βi(t) =P(Yt+1=yt+1,…,YT=yT|Xt=ei,ψ). Here, $$\alpha_i(t)$$ is the joint probability of observing $$(y_1, \ldots, y_t)$$ and that the state of the Markov chain is $${\bf e}_i$$ at time $$t$$. $$\beta_i(t)$$ is the probability of observing $$(y_{t+1}, \ldots, y_T)$$ given that the state of the Markov chain is $${\bf e}_i$$ at time $$t$$. Then, as in Bilmes (1998), an efficient induction of the pair $$\alpha_i(t)$$ and $$\beta_i(t)$$ is as follows:   αi(1) =πici(y1),αj(t+1) =[∑i=1Nαi(t)pij]cj(yt+1),P(Y|ψ) = ∑i=1Nαi(T), and   βi(T) =1 ,βi(t) = ∑i=1Npijcj(yt+1)βj(t+1),P(Y|ψ) = ∑i=1Nβi(1)πici(y1). Step 2: Then, two other random variables are defined. The first is the probability of the chain being in state $${\bf e}_i$$ at time $$t$$ given the observation sequence $$Y$$:   γi(t)=P(Xt=ei|Y,ψ)=P(Y,Xt=ei|ψ)P(Y|ψ)=P(Y,Xt=ei|ψ)∑j=1NP(Y,Xt=ej|ψ). Note that   αi(t)βi(t)=P(y1,…,yt,Xt=ei|ψ)P(yt+1,…,yT|Xt=ei,ψ)=P(Y,Xt=ei|ψ), so $$\gamma_i(t)$$ can be rewritten as   γi(t)=αi(t)βi(t)∑j=1Nαj(t)βj(t). The second one is the probability of the chain being in state $${\bf e}_i$$ at time $$t$$ while being in state $${\bf e}_j$$ at time $$t+1$$:   ξij(t) =P(Xt=ei,Xt+1=ej|Y,ψ) =P(Xt=ei,Xt+1=ej,Y|ψ)P(Y|ψ) =αi(t)pijcj(yt+1)βj(t+1)∑i=1N∑j=1Nαi(t)pijcj(yt+1)βj(t+1). It is worth noting that the sums of the above two variables have practical interpretations: i. $$\sum_{t=1}^T \gamma_i(t)$$: the expected number of times in state $${\bf e}_i$$; ii $$\sum_{t=1}^{T-1} \xi_{ij} (t)$$: the expected number of transitions from state $${\bf e}_i$$ to state $${\bf e}_j$$ for $$Y$$. Step 3: Then, the informal version of the Baum–Welch algorithm5 is given by i. $$\bar{\pi}_i = \gamma_i(1)$$: the expected relative frequency spent in state $${\bf e}_i$$ at time 1; ii. $$\bar{a}_{ij} = \frac{\sum_{t=1}^{T-1} \xi_{ij}(t)}{\sum_{t=1}^{T-1} \gamma_{i}(t) }$$: the expected number of transitions from state $${\bf e}_i$$ to state $${\bf e}_j$$ relative to the expected total number of transitions away from state $${\bf e}_i$$; iii. $$\bar{b}_{i}(k) = \frac{\sum_{t=1}^{T} \mathbb{1}_{y_t = o_k}\gamma_{i}(t) }{\sum_{t=1}^{T} \gamma_{i}(t) }$$: the expected number of times the output observations have been equal to $$o_k$$ while in state $${\bf e}_i$$ relative to the expected total number of times in state $${\bf e}_i$$. Step 4: The Viterbi algorithm is used to determine the most likely path, i.e. the path $$(\boldsymbol{q}_1, \ldots, \boldsymbol{q}_T)$$ which maximizes the likelihood $$p(\boldsymbol{q}_1, \boldsymbol{q}_2, \ldots, \boldsymbol{q}_T | Y, \boldsymbol{\psi})$$. Let $$\delta_i(t)$$ be the path which is in state $${\bf e}_i$$ at time $$t$$ with the highest probability, i.e.   δi(t)=maxq1,q2,…,qt−1P(q1,q2,…,qt=ei,y1,y2,…,yt|ψ), then   δj(t+1)=maxi[δi(t)pij]cj(yt+1). The recursive form of the variable is   δi(1) =πici(y1),δj(t) = max1≤i≤Nδi(t−1)pijcj(yt), and the most probable path satisfies the following conditions   qt=ei∗,t=T,T−1,T−2,…,1, where   i∗={argmax1≤i≤Nδi(T),t=T,argmax1≤i≤Nδi(t)P(Xt+1=qt+1|Xt=ei),t=T−1,T−2,…,1.  After the most-probable path through the HMM states is defined using the Viterbi approach, we can estimate the parameters of the dynamics of the investment fund. 5.2. Maximum likelihood estimations of parameters When we obtain the most-probable path, we further use the maximum likelihood estimation approach to estimate the appreciation rate and the volatility in the dynamics of asset price. It is easy to see that the probability density function of the observation process $$Y$$ is given by   p(Yt=yt|η1,η2,…,ηN,σ1,σ2,…,σN)=∑j=1N12πσj2exp⁡(−(yt−ηj)22σj2)1{qt=ej}. Then the likelihood function is   L(y1,…,yT;η1,η2,…,ηN,σ1,…,σN)=∏t=1T∑j=1N12πσj2exp⁡(−(yt−ηj)22σj2)1{qt=ej}. Let the derivatives of the logarithm likelihood function with respect to $$\eta_j$$ and $$\sigma_j$$$$(j=1,\ldots,N)$$ be zeros. The maximum likelihood estimations of the parameters can be derived.   ηj^ =∑t=1Tyt1{qt=ej}∑t=1T1{qt=ej},σj^2 =∑t=1T(yt−η^j)21{qt=ej}∑t=1T1{qt=ej}forj=1,2,…,N. Remark 5.1 One may argue that the smoother-based estimation equations are recursive and can be implemented by the standard method of discretization. However, when the states of the hidden Markov chain and the observation sequences are discrete, forward-backward estimates are easier to be calculated under the Baum–Welch method. Furthermore, the EM algorithm concerning the smoother-based estimation requires initial estimates of both the transition matrix of the hidden Markov chain and the volatilities of the observation process. The convergence of the algorithm may depend on the selection of the initial estimates. Here the transition matrix and the states of the hidden Markov chain are first determined. Then, given the estimated sequence of the HMM, the appreciation rates and volatilities of the investment fund are estimated via the maximum likelihood estimation. Remark 5.2 Baum & Petrie (1966) gave consistency and asymptotic normality of the maximum-likelihood estimator when observations take values in a finite set. In Bickel et al. (1998), the asymptotic normality of the maximum likelihood estimator for general HMMs was established under mild conditions. Bickel et al. (1998) also proved that the observed information matrix is a consistent estimator of the Fisher information. Remark 5.3 Here we adopt a two-stage estimation method, where the first stage involves the use of the Viterbi algorithm to determine the most likely path of the hidden Markov chain and the second stage involves the use of the maximum likelihood estimation to estimate the means and volatilities over different states of the chain given the most likely path of the chain. It seems that the Baum–Welch method fits well to the Viterbi algorithm in the first stage of the estimation. It may not be convenient to use the Kalman filter in the first stage. Furthermore, the Baum–Welch algorithm involves the use of smoothed estimates which can incorporate more information than the filtered estimates. 6. Numerical examples When we obtain the filtered estimates of the hidden Markov chain, we replace the corresponding hidden quantities by their filtered estimates. Then, the PDE (4.3) is given as follows:    −rv(t,x,y,z)−∂v∂y(t,x,y,z)(r−σ^t2)y+∂v∂t(t,x,y,z) +12∂2v∂y2(t,x,y,z)σ^t2y2+⟨v,A^x⟩=0. Here, we adopt a similar dimension reduction technique as that in Wilmott et al. (1997) and Boyle & Draviam (2007). Let $$u=\frac{y}{z}$$, $$G_i(t, u)=\frac{1}{z} v(t, {\bf x}={\bf e}_i, y, z)$$. Then the PDE becomes    −∂Gi∂u(t,u)(r−σ^t2)u+12∂2Gi∂u2(t,u)σ^t2u2+∂Gi∂t(t,u)+⟨Gi,A^ei⟩−rGi(t,u)=0. The corresponding terminal conditions become the following form:   Gi(t=T,u)=max(1−u,0). The boundary conditions are   Gi(t,u=0)=e−r(T−t), and when the maximum is measured continuously, the boundary condition at $$F=N$$ becomes a boundary condition at $$u=1$$:   ∂Gi∂u|u=1=Gi(t,u=1). It is easy to solve the above PDE and we could obtain the prices of the floating lookback put options. According to the put-call parity in Wong & Kwok (2003),   cfix(t,q,F,Nt,;1)=pfl(t,q,F,max(Nt,1))+F−e−r(T−t), we can derive the prices of the fixed strike lookback call options and the DFP contract. Here, we shall give two numerical examples to illustrate the valuation of a dynamic fund protection when the dynamics of the investment fund follow a Geometric Brownian motion with parameters being modulated by a hidden Markov chain. For simplicity, we assume that the Markov chain has two states. The paper by Taylor (1999) provided some empirical evidence for the use of a Markov chain model with two states for volatility. The real data examples we present here use data sets of daily closing prices of IBM (International Business Machines Corporation) and Apple Inc., from May 2008 to April 2010, retrieved from Yahoo Finance. There are 484 observations in each data set. During the period, the economy was influenced by the financial crisis. Consequently, it is of practical interest to investigate the valuation problem when the economy has substantial changes. On the other hand, the stock prices of financial industries have not been selected since the whole industry was affected significantly by the crisis. As mentioned earlier, the Baum–Welch algorithm is adopted. The estimates of relevant parameters are given by   P^IBM=(0.77940.22060.18290.8171),P^Apple=(0.79820.20180.19820.8018). Here $$\hat{P}_{IBM}$$ and $$\hat{P}_{Apple}$$ represent the transition probability matrices of the IBM example and the Apple Inc. example, respectively. From the given data, we can easily calculate the following estimates:   η^IBM=(−0.06780.2148) ,η^Apple=(0.08280.6253),  σ^IBM2=(0.01010.0013) ,σ^Apple2=(0.03410.0100). Clearly, in the two examples, the estimators of the volatilities $$\sigma_1 > \sigma_2$$, indicate that State 1 and State 2 represent a ‘Bad’ economy and a ‘Good’ one, respectively. The estimation results are consistent with those in Liew & Siu (2010). The differences between the estimated volatilities in the two states may be attributed to structural changes in data due to the financial crisis and the rebounds after the crisis. Note that the estimated drift parameter for a ‘Bad’ economy in a regime-switching model is not uncommon, which is similar with the numerical example in Liew & Siu (2010). After we obtain the estimates, we can solve the valuation problem using the PDE approach. The values for parameters, used in the discretization of the PDEs, are assumed to be:   Δt=1/250(year). Also, we assume the risk-free interest rate is:   r=0.05(1/year). Using finite difference methods, we calculate the values of the DFP by varying the guarantee level $$K$$ and the time-to-maturity $$T$$. Here, we assume the value of $$K$$ is less than the value of the naked fund on May 1 2008. It is not unreasonable to make this assumption. The organizations issuing the DFP will never design such investment plans, if so, they will upgrade the fund immediately after the DFP issued. Table 1 and Table 2 present numerical results for the price of a DFP. Table 1 DFP prices for IBM versus different K and T  Protection level  $$T=2$$  $$T=3$$  ($${}$$)  State 1  State 2  State 1  State 2  95  27.4092  7.1824  22.0819  4.2292  100  33.6141  11.0479  27.7742  8.0946  105  39.8190  14.9134  33.4664  11.9601  110  46.0239  18.7789  39.1586  15.8256  115  52.2288  22.7863  44.8509  19.7899  120  58.4337  27.7111  50.5431  24.3925  Protection level  $$T=2$$  $$T=3$$  ($${}$$)  State 1  State 2  State 1  State 2  95  27.4092  7.1824  22.0819  4.2292  100  33.6141  11.0479  27.7742  8.0946  105  39.8190  14.9134  33.4664  11.9601  110  46.0239  18.7789  39.1586  15.8256  115  52.2288  22.7863  44.8509  19.7899  120  58.4337  27.7111  50.5431  24.3925  Table 2 DFP prices for Apple Inc. versus different K and T  Protection level  $$T=2$$  $$T{=}3$$  ($${}$$)  State 1  State 2  State 1  State 2  160  103.7537  22.5602  107.7473  26.7511  170  115.8936  29.6254  119.8609  33.8025  180  128.0334  36.6907  131.9745  40.8538  190  140.1732  43.7559  144.0882  47.9052  200  152.3131  50.8212  156.2018  54.9566  210  164.4529  57.8864  168.3154  62.0079  Protection level  $$T=2$$  $$T{=}3$$  ($${}$$)  State 1  State 2  State 1  State 2  160  103.7537  22.5602  107.7473  26.7511  170  115.8936  29.6254  119.8609  33.8025  180  128.0334  36.6907  131.9745  40.8538  190  140.1732  43.7559  144.0882  47.9052  200  152.3131  50.8212  156.2018  54.9566  210  164.4529  57.8864  168.3154  62.0079  From Table 1 and Table 2, we note that the prices of DFP decrease when the maturity time is longer in the IBM example, while the opposite trend presents in the Apple Inc. example. However, as calculated earlier, the other parameters are similar in the two examples. This is also another reason why we chose the stock prices of the two companies. Note that in the Black–Scholes formula, the Greek ‘Theta’ is defined to measure the change in the option price when the time to maturity decreases. Our results verify that ‘Theta’ can be either positive or negative. The prices of the DFP in State 1 are significantly higher than the corresponding prices in State 2. This is intuitively clear since higher risk premiums are required to compensate the risk attributed to a ‘bad’ economy. This also provides some empirical evidence for the use of the regime-switching model. In addition, we simulate the price of the DFP under the Black–Scholes model. Goldman et al. (1979) derived the pricing formula for a floating strike look back put option. Then we calculated the prices of the corresponding DFPs. Here, we only give the comparison results in the IBM example. Similar results can be simulated in the Apple Inc. example. Fig. 1. View largeDownload slide DFP prices corresponding to different protection levels when $$T=2$$ and $$T=3$$ Fig. 1. View largeDownload slide DFP prices corresponding to different protection levels when $$T=2$$ and $$T=3$$ In Fig. 1, two solid lines indicate values arising from our approach while two dashed lines correspond to the BS model, and the lines with $$*$$ represent the prices of DFP that are derived when the economy state is ‘Good’. Besides, from the numerical results, the effect of different economy states cannot be ignored in the valuation process of DFP. To highlight the effect of regime-switching parameters in the DFP valuation, we construct Table 3 that compares the BS price and our price numerically. Here, we simply take the prices when $$T=2$$ as an example. The prices from the BS model are computed using Monte Carlo simulations. In the BS model (State 1), a geometric Brownian motion with a constant volatility $$\sigma_1$$ is assumed while in the BS model (State 2), a geometric Brownian motion with a constant volatility $$\sigma_2$$ is assumed. As seen from Table 3, when volatility changes, the prices of DFP change significantly. In other words, the price of a DFP will significantly change if the state of the economy changes. Thus, it is not difficult to imagine the trace off faced by insurance companies. If an investment plan is sold at a lower price, insurance companies may face larger losses when the economy is ‘bad’. Consequently, insurance companies would like to charge a higher price to hedge the volatility risk. However, on the other hand, the DFP with a higher price would be less attractive to an investor. To some extent, this explains that the effect of HMM cannot be ignored in the valuation process. Table 3 DFP prices for IBM under different models with T=2  Protection level  BS model  Our model  ($${}$$)  State 1  State 2  State 1  State 2  95  24.3810  6.0101  27.4092  7.1824  100  31.5274  10.8379  33.6141  11.0479  105  38.7840  15.8445  39.8190  14.9134  110  46.1506  20.8948  46.0239  18.7789  115  53.6273  25.9613  52.2288  22.7863  120  61.2141  31.0896  58.4337  27.7111  Protection level  BS model  Our model  ($${}$$)  State 1  State 2  State 1  State 2  95  24.3810  6.0101  27.4092  7.1824  100  31.5274  10.8379  33.6141  11.0479  105  38.7840  15.8445  39.8190  14.9134  110  46.1506  20.8948  46.0239  18.7789  115  53.6273  25.9613  52.2288  22.7863  120  61.2141  31.0896  58.4337  27.7111  Fig. 2. View largeDownload slide Relative differences $$\frac{DFP_{bs}-\widehat{DFP}}{DFP_{bs}}$$ versus $$K$$ Fig. 2. View largeDownload slide Relative differences $$\frac{DFP_{bs}-\widehat{DFP}}{DFP_{bs}}$$ versus $$K$$ To make a comparison between the estimates of the DFP price $$\widehat{DFP}$$ and the Black–Scholes price $$DFP_{bs}$$, we plot the relative difference between the Black–Scholes price and our estimation. Figure 2 is the plot of relative difference $$\frac{DFP_{bs}-\widehat{DFP}}{DFP_{bs}}$$ versus $$K$$, for two fixed values for $$T$$. In Figure 2, the two solid lines represent the relative differences when the maturity time is $$T=2$$ and the two dash lines depict the relative differences when $$T=3$$. The lines with $$*$$ represent that the economic state is ‘Good’. From the figure, it is clear that the differences between the estimated dynamic fund protection values and the BS ones are comparatively small, say between 0 and 0.5% most of the time. Moreover, the higher the protection level, the smaller the relative differences are. 7. Conclusions This article considered the valuation of the DFP. We assumed that the investment fund depends upon a continuous-time, finite-state, hidden Markov chain. Firstly, the Esscher transform was applied to select a pricing kernel. Then, we adopted the PDE approach to price the DFP. To estimate the states and parameters of the HMM, Baum–Welch algorithm, based on the discretization of continuous-time smoothers, was applied. After we obtained the most-likely path of the HMM by the Viterbi algorithm, we derived the maximum likelihood estimation of the model parameters. In the numerical examples, we calculated the prices of the DFP for two companies, IBM and Apple Inc. Moreover, we compared results from our model with those obtained from the pricing formula of Goldman et al. (1979). The numerical results reveal that regime shifts have a pronounced effect on the prices of the DFP. Acknowledgements The authors wish to thank two anonymous referees and the editors for their valuable comments, which have been greatly helpful in improving the paper. Funding The authors wish to thank two anonymous referees and the editors for their helpful comments. K. F. and R. W. would like to acknowledge the 111 Project (B14019), National Natural Science Foundation of China (11501211, 11571113), Shanghai Subject Chief Scientist (14XD1401600), Shanghai Pujiang Program (15PJC026), Shanghai Philosophy Social Science Planning Office Project (2015EJB002), China Postdoctoral Science Foundation (2015M581564) and Shanghai Chenguang Plan (15CG22). T. K. S. would like to acknowledge a Discovery Grant from the Australian Research Council (ARC) (Project No.: DP130103517). Y. S. would like to acknowledge financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC)(Project No.: RGPIN-2016-05677). References Baum, L.E. & Petrie, T. 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Google Scholar CrossRef Search ADS   Footnotes 1 As explained in Boyle & Draviam (2007), if at time $$t$$, $${\bf X} (t) = {\bf e}_i$$, then the state at $$t+\Delta t$$ will be $${\bf e}_j$$ with probability $$a_{ij} \Delta t$$, and the probability of remaining in the state is $$1-\sum_{j=0,j\neq i}^M a_{i j} \Delta t = 1+a_{i i}\Delta t$$. 2 This assumption ensures that the regime switching PDE can be obtained when pricing the dynamic fund protection. 3 As discussed in earlier works, compared to filtered estimates, the smoothed estimates can incorporate the extra information obtained from the observations between time $$t$$ and time $$T$$. Furthermore, to apply the robust discretization of continuous-time filters, the dynamics of the observation process should not include stochastic integrals. This is also the reason why Elliott et al. (2003) made approximations to the observation process. For details, interested readers can refer to Elliott et al. (1995, 2003); James et al. (1996) and Malcolm & Elliott (2010). 4 Let $$\boldsymbol{\Psi}$$ be an open subset of $$\mathfrak{R}^n$$. Suppose that to each $$\boldsymbol{\psi} \in \boldsymbol{\Psi}$$, we have a smooth assignment $$\boldsymbol{\psi} \rightarrow (P(\boldsymbol{\psi}), B(\boldsymbol{\psi}), \pi(\boldsymbol{\psi}))$$. Under these assumptions, for each fixed $$y_1, y_2, \ldots, y_T$$, $$P_{ y_1 y_2 \ldots y_T}(\boldsymbol{\psi}) = P_{y_1 y_2 \ldots y_T}$$, $$P_{ y_1 y_2 \ldots y_T}(P(\boldsymbol{\psi}), B(\boldsymbol{\psi}), \pi(\boldsymbol{\psi}))$$ is a smooth function of $$\boldsymbol{\psi}$$. Given a fixed $$Z-$$sample $$y = y_1, \ldots, y_T$$, a parameter value $$\boldsymbol{\psi}^*$$ which maximizes the likelihood $$P_y (\boldsymbol{\psi}) = P_{y_1 \ldots y_T}(\boldsymbol{\psi})$$ can be found. To maximize the likelihood function, Baum et al. (1970) defined a continuous transformation $$\mathcal{T}$$ mapping $$\boldsymbol{\Psi}$$ into itself with the property that $$P_{y_1 y_2 \ldots y_T} ( \mathcal{T} (\boldsymbol{\psi})) > P_{y_1 y_2 \ldots y_T} (\boldsymbol{\psi})$$ unless $$\boldsymbol{\psi}$$ is a critical point of $$P_{y_1 y_2 \ldots y_T} ( \boldsymbol{\psi})$$. 5 A more general case is that the underlying state sequence is assumed to be hidden or unobserved. In this case, the $$Q$$ function is introduced. Details can refer to Baum et al. (1970). © The authors 2016. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Management Mathematics Oxford University Press

Pricing dynamic fund protection under hidden Markov models

, Volume 29 (1) – Jan 1, 2018
19 pages

/lp/ou_press/pricing-dynamic-fund-protection-under-hidden-markov-models-DDTBW7qT0V
Publisher
Oxford University Press
© The authors 2016. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
ISSN
1471-678X
eISSN
1471-6798
D.O.I.
10.1093/imaman/dpw014
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Abstract

Abstract In this article, we discuss the pricing of dynamic fund protection when the value process of the investment fund is governed by a geometric Brownian motion with parameters modulated by a continuous-time, finite-state hidden Markov chain. Under a risk-neutral probability measure, selected by the Esscher transform, we adopt the partial differential equation approach to value the dynamic fund protection. Using the estimated sequence of the hidden Markov chain, we apply the Baum–Welch algorithm and the Viterbi algorithm to derive the maximum likelihood estimates of the parameters. Numerical examples are provided to illustrate the practical implementation of the model. 1. Introduction Compared with upside profits, investors often place greater emphasis on downside risk. Partly due to this reason, academic researchers and industry practitioners have innovated a variety of financial and insurance products to deal with the downside risk, including traditional put options. However, put options can only provide investors with protection at the exercise moment. Although innovative investment protection plans are introduced to overcome this disadvantage, many of them are too sophisticated. Among different investment protection plans, dynamic fund protections (DFPs) have been a popular type of investment plans. Compared with traditional put options, DFP plans protect investors for the whole investment period. This feature may explain why the dynamic fund protection plans have attracted both academic researchers’ and investors’ attention. There are different forms of DFP plans. The most fundamental one is that the investment fund value will be upgraded when the fund value falls below a certain protection level. This kind of protection plan was pioneered by Gerber & Shiu 1998. Since the work of Gerber & Shiu 1998, much attention has been paid to the DFP valuation when the value process of the underlying investment fund is governed by various stochastic models, such as the classical Black–Scholes model and the constant elasticity of variance (CEV) process (see Gerber & Shiu, 1999, 2003a,b; Gerber & Pafumi, 2000; Imai & Boyle, 2001; Fung & Li, 2003; Chu & Kwok, 2004). Indeed, DFP plans are products with lookback option features, which were shown in Imai & Boyle (2001); Chu & Kwok (2004) and Wong & Chan (2007). Wong & Chan (2007) considered the valuation of DFPs under a multiscale stochastic volatility model and derived semi-analytical formulas using asymptotic techniques. They also established the relationship between a fixed strike lookback call option and a DFP. In their paper, the parity equation between a fixed strike lookback call option and a floating strike lookback put option, derived in Wong & Kwok (2003), was used. Apart from pricing the DFPs, Tse et al. (2008) investigated the hedging of discrete-DFPs. Wong & Lam (2010) assumed that the investment fund follows a Lévy process and solved the valuation problem. Regime-switching models are one of the important classes of models in financial econometrics. This class of model can incorporate the impacts of changes in economic conditions on financial and economic dynamics. Hamilton (1989) popularized the use of regime-switching models in financial econometrics. There are many works on valuing options under regime-switching models. Some examples of early works on this topic are Naik (1993), Guo (2001), Buffington & Elliott (2002a, b), Elliott et al. (2005) and Siu (2005), amongst others. Due to the presence of an additional source of uncertainty described by the modulating Markov chain, the market in a regime-switching model is, in general, incomplete. Consequently, there is more than one equivalent martingale measure for valuation. Gerber & Shiu (1994) introduced the use of the Esscher transform to price options. The Esscher transform has been used to select an equivalent martingale measure in a regime-switching market (see, for example, Elliott et al., 2005; Siu, 2005, 2008, 2011; Boyle & Draviam, 2007; Yuen & Yang, 2010; Siu et al., 2011; etc.). It seems that the mainstream of the literature on option valuation under regime-switching models considered the situation where the Markov chain is completely observable while, in practice, the states of the chain (or the states of the economy) may not be directly observable. Hidden Markov models present a natural choice for modelling transitions in hidden states of an economy. Assuming that the Markov chain is not observable, in a discrete-time regime-switching model, Ishijima & Kihara (2005) studied an option valuation problem using a locally risk-neutral valuation relationship. A higher-order hidden Markov model (HHMM) was considered in Ching et al. (2007). They assumed the dynamics of the asset price follow the HHMM and investigated the valuation of exotic options, including Asian options, barrier options and lookback options. Liew & Siu (2010) considered option valuation under a discrete-time hidden Markov regime-switching Gaussian model. They adopted both the Esscher transform and the extended Girsanov’s principle to select a pricing kernel. There are some works in the literature concerning option valuation in continuous-time hidden Markov regime-switching models. Some examples are Elliott & Siu (2014) for hidden Markov-modulated pure jump processes, Siu (2013, 2014) for hidden Markov-modulated jump-diffusion models and Elliott & Siu (2015) for a hidden Markov-modulated jump-diffusion model with jumps attributed to trading volumes, amongst others. These works mainly focused on valuing financial options. In this article, we discuss the valuation of the DFP plans when the dynamics of the investment fund are governed by a geometric Brownian motion whose coefficients are modulated by a hidden Markov chain. We assume that the appreciation rate and the volatility are modulated by a continuous-time, finite-state hidden Markov chain. The states of the chain represent hidden states of an economy. The Esscher transform is employed to select a pricing kernel for the valuation of the DFP. Considering the lookback option feature in the DFP, we adopt a partial differential equation (PDE) approach to price the floating strike lookback put options. Then, the prices of the corresponding fixed strike lookback call options can be calculated according to the put-call parity derived in Wong & Kwok (2003). In a recent paper by Jin et al. (2016), the pricing of DFP was also studied in a Markovian regime-switching modelling framework. Compared with Jin et al. (2016), our paper uses a PDE approach and assumes the chain is hidden, while Jin et al. (2016) uses a Laplace transform approach and assumes the chain is observable. To estimate the parameters of the HMM, we adopt the Baum–Welch algorithm, based on the discretization of continuous-time smoother. Then, the Viterbi algorithm is applied to obtain the most-probable path of the hidden Markov chain. The maximum likelihood estimates of model parameters are derived using the most probable path of the hidden Markov chain. To illustrate the valuation of DFP, we give numerical examples based on real financial data. The rest of the article is structured as follows: Section 2 presents the hidden Markov-modulated investment fund model. In Section 3, we select a pricing kernel by the Esscher transform. Section 4 presents the PDE approach to value the DFP. In Section 5, we discuss how to estimate the states and transition matrix of the HMM. The maximum likelihood estimates of the model parameters are also presented. Some numerical examples are given in Section 6. The final section summarizes the article. 2. The model dynamics In this section, we consider a continuous-time economy with two investment securities, namely, a risk-free bond $$B$$ and an investment fund $$S$$, which are continuously tradable on the time horizon $$\mathcal{T}$$. Here, $$\mathcal {T} := [0,T]$$, where $$T<\infty$$. We assume that there are no transaction costs and taxes involved in trading. Furthermore, all dividends are reinvested in the fund and any fractional units of the index can be traded. Let $$(\Omega, \mathcal {F}, \mathcal {P})$$ be a complete probability space, where $$\mathcal {P}$$ is a real-world probability measure. We adopt bold-face letters to denote matrices (or vectors), and write $$\mathbf{y}'$$ for the transpose of a matrix, or a vector $$\mathbf{y}$$. Let $$\mathbb{1}_{E}$$ denote the indicator function of an event $$E$$. We assume that the hidden states of an economy are modelled by a continuous-time, finite-state Markov chain $${\bf X} := \{ {\bf X}_t | t \in \mathcal {T} \}$$. The Markov chain is defined on $$(\Omega, \mathcal {F}, \mathcal {P})$$ with the finite state space $$\mathcal {S} : = \{ \mathbf{s}_1, \mathbf{s}_2, \ldots , \mathbf{s}_N \}$$. Without loss of generality, following Elliott et al. (1995), the state space of $${\bf X}$$ is identified with a finite set of standard unit vectors $$\mathcal {E}:= \{\mathbf{e}_1, \mathbf{e}_2, \ldots, \mathbf{e}_N \}$$, where $${\bf e}_i \in \mathfrak{R}^N$$ are column vectors and the $$j$$th component of $${\bf e}_i$$ is the Kronecker delta $$\delta_{ij}$$, for each $$i, j = 1,2,\ldots, N$$. We call $$\mathcal {E}$$ the canonical state space of the chain $${\bf X}$$. We define $${\bf A} := ( a_{ij} )_{i, j = 1, 2, \ldots, N}$$ as the generator of the chain $${\bf X}$$ under $$\mathcal {P}$$, where $$a_{i i} = - \sum_{j=0, j \neq i}^N a_{i j}, i=0,1,\ldots, N$$. Note that the transition probability, from time $$t$$ to time $$t+\Delta t$$, is $$\boldsymbol{I}+{\bf A}\Delta t$$, where $$\boldsymbol{I}$$ denotes the identity matrix.1 Elliott et al. (1995) obtained the following semi-martingale dynamics for the chain:   Xt=X0+∫0tAXsds+Mt . where $$\{\mathbf{M}_t | t \in \mathcal {T} \}$$ is an $$\mathfrak{R}^N$$-valued square-integrable martingale under $$\mathcal{P}$$ with respect to the natural filtration generated by $$\mathbf{X}$$. The price dynamics of the risk-free bond and the investment fund are now defined. Firstly, we assume that the bond price process $$\{ B_t | t \in \mathcal {T} \}$$ is governed by   dBt=rBtdtt∈T,B0=1 . Here, the instantaneous market interest rate of the bond is assumed to be a positive constant, i.e. $$r > 0$$. In general, one may consider a stochastic interest rate which depends on the chain $${\bf X}_t$$. In this case, filtering of interest rate is involved, and this complicates the modelling and pricing issues. To simplify the matter, we consider a relatively simpler situation that the interest rate is a constant and only focus on the volatility risk. Let $$W := \{W_t | t \in \mathcal {T}\}$$ be a standard Brownian motion on $$(\Omega, \mathcal {F}, \mathcal {P})$$ with respect to its right-continuous, $$\mathcal {P}$$-complete, natural filtration. We suppose that $$W$$ and $${\bf X}$$ are stochastically independent under $$\mathcal {P}$$.2 The dynamics of the investment fund $$S:=\{S_t | t \in \mathcal {T}\}$$ is governed by a Markovian regime-switching geometric Brownian motion:   dSt=μtStdt+σtStdWt,t∈T,S0=s. The appreciation rate $$\mu_t$$ and the volatility $$\sigma_t$$ are modulated by the chain $${\bf X}$$ as follows:   μt:=⟨μ,Xt⟩,σt:=⟨σ,Xt⟩,t∈T. Here $$\left< \cdot, \cdot \right>$$ denotes the scalar product in $$\mathfrak{R}^N$$. Moreover, $$\boldsymbol{\mu}=( \mu_1, \mu_2, \ldots, \mu_N )' \in \mathfrak{R}^N$$ and $$\boldsymbol{\sigma}=( \sigma_1, \sigma_2, \ldots, \sigma_N )' \in \mathfrak{R}^N$$ for each $$t \in \mathcal {T}$$, where $$\sigma_i>0$$ for all $$i=1,2,\ldots,N$$. Let $$\{Y_t | t \in \mathcal{T}\}$$ be the logarithmic return process, where $$Y_t := \ln (S_t/S_0 )$$ for each $$t \in \mathcal {T}$$. Then, the underlying fund dynamics can be written as:   St=S0eYt,t∈T, where   Yt=Y0+∫0t(⟨μ,Xs⟩−12⟨σ2,Xs⟩)ds+∫0t⟨σ,Xs⟩dWs,t∈T. Here, $$\boldsymbol{\sigma}^2 := (\sigma_1^2, \sigma_2^2, \ldots, \sigma_N^2)^\prime \in \mathfrak{R}^N$$. For notational simplicity, write $$\eta_i = \mu_i - \frac{1}{2} \sigma_i^2$$ for $$i = 1, 2, \ldots, N$$ and $$\boldsymbol{\eta} = (\eta_1, \eta_2, \ldots, \eta_N) ^ \prime$$. Then, the dynamics of $$Y$$ can be rewritten as follows:   Yt=Y0+∫0t⟨η,Xs⟩ds+∫0t⟨σ,Xs⟩dWs,t∈T. (2.1) Let $$\mathcal {F}^Y =\{\mathcal {F}^Y_t | t \in \mathcal {T} \}$$ and $$\mathcal {F}^S =\{\mathcal {F}^S_t | t \in \mathcal {T} \}$$ be the natural filtrations generated by $$\{Y_t| t \in \mathcal {T}\}$$ and $$\{S_t | t \in \mathcal {T}\}$$, respectively. Also, we assume that the filtrations given above are right continuous and complete. Since $$\mathcal {F}^Y$$ and $$\mathcal {F}^S$$ are equivalent, either of them could be used as the observed information structure. Here, we adopt $$\mathcal {F}^Y$$ as the observed information structure. We define the filtration $$\mathcal{G} = \{ \mathcal{G}_{t,s}| 0\leq s\leq t \in \mathcal{T} \}$$ by letting the double indexed $$\sigma$$-field $$\mathcal{G}_{t,s} := \mathcal{F}^{{\bf X}}_t \bigvee \mathcal{F}^Y_s$$, for any $$s,t \in \mathcal{T}$$ with $$s\leq t$$, where $$\mathcal{A} \bigvee \mathcal{B}$$ is the minimal $$\sigma-$$algebra containing the $$\sigma-$$algebras $$\mathcal{A}$$ and $$\mathcal{B}$$. We write $$\mathcal{G}_t = \mathcal{G}_{t,t}$$, for all $$t \in \mathcal{T}$$, and $$\mathcal{G} =\{ \mathcal{G}_t | t \in \mathcal {T}\}$$. Then $$\mathcal {F}^Y$$, $$\mathcal {F}^{{\bf X}}$$ and $$\mathcal {G}$$ represent the flows of observable information, hidden information and full information, respectively. 3. The Esscher transform The developments in this section follow that in Elliott et al. (2005). As in Elliott et al. (2005), let $$\theta_t$$ be the regime switching Esscher parameter, which can be written as   θt=⟨θ,Xt⟩, where $$\boldsymbol{\theta} = ( {\theta_1, \theta_2, \ldots, \theta_N} ) ^\prime$$. Write $$(\theta \cdot Y)_t := \int_0^t \theta_s {\rm d} Y_s$$, for each $$t \in \mathcal {T}$$. The regime switching Esscher transform $$\mathcal {Q}_\theta \sim \mathcal {P}$$ on $${\mathcal {G}}_t$$ with respect to a family of parameters $$\{\theta_s | 0\leq s\leq t\}$$ is given by   dQθdP|Gt=e(θ⋅Y)tE[e(θ⋅Y)t|FtX], where $$\mbox{E}[\cdot]$$ is the expectation taken under $$\mathcal {P}$$. Since   E[e(θ⋅Y)t|FtX]=exp⁡[∫0t(⟨μ,Xs⟩−12⟨σ2,Xs⟩)θsds+12∫0t⟨σ2,Xs⟩θs2ds], the Radon-Nikodym derivative of the regime switching Esscher transform is given by   dQθdP|Gt=exp⁡{∫0t⟨σ,Xs⟩θsdWs−12∫0t⟨σ2,Xs⟩θs2ds}. Let $$\{ \theta^*_t | t \in \mathcal{T} \}$$ be a family of risk-neutral regime switching Esscher parameters. Consider the following martingale condition:   S0=EQθ∗[e−rtSt|FtX],t∈T. Using a version of Bayes’ rule, the martingale condition can be rewritten as:   ∫0t(⟨μ,Xs⟩−r+θs∗⟨σ2,Xs⟩)ds=0. Elliott et al. (2005) shows that $$\theta^*_t$$ can be determined uniquely as:   θt∗=r−⟨μ,Xt⟩⟨σ2,Xt⟩, Then, the Radon-Nikodym derivative can be written by:   dQθ∗dP|Gt=exp⁡{∫0t(r−⟨μ,Xs⟩⟨σ,Xs⟩)dWs−12∫0t(r−⟨μ,Xs⟩⟨σ,Xs⟩)2ds}. Using Girsanov’s theorem, $$\widehat{W}_t = W_t + \int_0^t \left(\frac{r- \mu_s}{ \sigma_s} \right) d s$$ is a standard Brownian motion with respect to $$\mathcal {G}$$ under $$\mathcal {Q}_{\theta^*}$$. Hence, under the probability measure $$\mathcal {Q}_{\theta^*}$$, the stock price dynamics can be written as   dSt=rStdt+⟨σ,Xt⟩StdW^t. (3.1) 4. Pricing the DFP During the whole investment period, investors who own the investment fund are guaranteed a predetermined protection level under the DFP. Let $$K$$ denote the constant protection level. As indicated by Imai & Boyle (2001) and Wong & Chan (2007), the amount received by the holder of the fund is then given by:   STmax{1,max0≤τ≤TKSτ}, where $$S_t$$ is the value of the fund without the protection. Hence, the terminal payoff for the DFP should be:   DFPT =STmax{1,max0≤τ≤TKSτ}−ST =STmax{0,max0≤τ≤TKSτ−1}. Write   Ft=K/Stand Nt=max0≤τ≤tFτ. Then   DFPT=STmax(NT−1,0). As noted in Wong & Chan (2007), the payoff function of a fixed strike lookback call is:   cfix(T,FT,NT,1)=max(NT−1,0). According to Wong & Chan (2007), if we view $$S$$ as an exchange rate and hence $$F$$ as an asset trading in the foreign currency world, this option can simply be valued as a fixed strike lookback call in ‘the foreign currency world’ followed by multiplying the exchange rate $$S_t$$. Note that the fixed strike price of the lookback call is one unit. Hence, a model-independent formula is written as:   DFPt=St×cfix(t,Ft,Nt). To value the DFP, we focus on the process of $$F$$ in the foreign currency world. The dynamics of $$F_t$$ under the risk-neutral probability $$\mathcal {Q}_{\theta^*}$$ is given by:   dFt=−(r−σt2)Ftdt−σtFtdW^t. (4.1) We now price the European-type fixed strike lookback call under a hidden Markov-modulated regime-switching model. Under the risk-neutral probability measure $$\mathcal {Q}_{\theta^*}$$, a price at time $$t$$ with payoff $$V_T$$ at maturity $$T$$ is given by:   Vt=EQθ∗[e−r(T−t)VT|Gt]. (4.2) Here $${\mbox E}_{{\mathcal{Q}}_{\theta^*}} [\cdot]$$ is the expectation taken with respect to $${\mathcal{Q}}_{\theta^*}$$. Defining $$D_t :=\max\limits_{0 \leq u \leq t} S_u$$, Shreve (2004) showed that the pair of the processes $$(S_t,D_t)$$ has the Markov property. Similarly, we can show that the triplet of the processes $$({\bf X}_t, S_t, D_t)$$ has the joint Markov property. Denote   St=S0eW^t, where $$\hat{W_t}=\int_0^t \bigg(r - \frac{1}{2} \sigma^2_s \bigg) {\rm d} s + \int_0^t \sigma_s {\rm d} W_s$$, and define $$\hat{M}_t=\max\limits_{0 \leq u \leq t} \hat{W}_u$$. Then, $$D_t :=\max\limits_{0 \leq u \leq t} S_u = S_0 e^{\hat{M}_t}$$. Since $$\hat{W}_T - \hat{W}_t = \int_t^T \bigg( r- \frac{1}{2} \sigma^2_s \bigg) {\rm d} s + \int_t^T \sigma_s {\rm d} W_s$$ and $$\sup\limits_{t \leq u \leq T} (\hat{W}_u - \hat{W}_t)$$ are independent of $$\mathcal{G}_t$$, and   DT = max0≤u≤TSu =S0eM^T =S0esupt≤u≤TW^u1{M^t≤supt≤u≤TW^u}+S0eM^t1{M^t>supt≤u≤TW^u} =Stesupt≤u≤T(W^u−W^t)1{DtSt≤exp⁡(supt≤u≤T(W^u−W^t))}+Dt1{DtSt>exp⁡(supt≤u≤T(W^u−W^t))}. Then   EQθ∗[f(XT,ST,DT)|Gt] =EQθ∗[f(XT,SteW^T−W^t,Stesupt≤u≤T(W^u−W^t)1{DtSt≤exp⁡(supt≤u≤T(W^u−W^t))} +Dt1{DtSt>exp⁡(supt≤u≤T(W^u−W^t))})|Xt,St,Dt] is a function of $$({\bf X}_t, S_t, D_t)$$. Thus, we can see that the triplet of the processes $$({\bf X}_t, S_t, D_t)$$ has the joint Markov property. Similarly, we could show the triplet of the processes $$({\bf X}_t, F_t, N_t)$$ also has the joint Markov property. There exists a measurable function $$v(t,{\bf x},y,z)$$ such that $$V_t=v(t, {\bf X}_t, F_t, N_t )$$. The following theorem is standard. It is derived under the assumption that $$v (t, {\bf x}, y, z) \in {\mathcal{C}}^{1,2,1}$$, where $${\mathcal{C}}^{1, 2, 1}$$ is the space of all functions $$v (t, {\bf x}, y, z)$$ which is continuously differentiable in $$(t, z)$$ and twice continuously with respect to $$y$$ for each $${\bf x} \in {\mathcal{E}}$$. Theorem 4.1 Let $$v (t, {\bf x}, y, z )$$ denote the price of the fixed strike lookback option at time $$t$$ under the assumption that $${\bf X}_t={\bf x}, F_t=y, \ \mbox{and} \ N_t=z$$. Note that $$\frac{\partial v}{\partial t}$$, $$\frac{\partial v}{\partial y}$$ and $$\frac{\partial^2 v}{\partial y^2}$$ represent the first derivative of the function with respect to $$t$$ and $$y$$ and the second derivative with respect to $$y$$, respectively. Then the price of the option is a solution of the following PDE:    −rv(t,x,y,z)−∂v∂y(t,x,y,z)(r−σt2)y+∂v∂t(t,x,y,z) +12∂2v∂y2(t,x,y,z)σt2y2+⟨v,Ax⟩=0 (4.3) in the region of $$\{ (t,{\bf x},y,z); 0 \leq t < T, 0 \leq y \leq z \}$$. Here, $$\boldsymbol{v} = (v_1, v_2, \ldots, v_N)^\prime$$. Similarly to a PDE for a lookback option, Eq. (4.3) does not involve the derivative of $$v$$ with respect to $$z$$ though $$v$$ is a function of $$z$$, where $$z$$ is the maximum price up to and including time $$t$$. The variable $$z$$ only plays the role for specifying the boundary condition of $$v$$. Proof. The proof is standard. Applying It$$\hat{o}'$$s differentiation rule to $$e^{-rt} v (t, {\bf X}_t, F_t, N_t )$$ gives:   d(e−rtv(t,x,y,z)) =e−rt[−rv(t,x,y,z)dt+∂v∂y(t,x,y,z)dFt+∂v∂t(t,x,y,z)dt +∂v∂z(t,x,y,z)dNt+12∂2v∂y2(t,x,y,z)d⟨Fc,Fc⟩t +⟨v,Ax⟩dt+⟨v,dMt⟩] =e−rt[−rv(t,x,y,z)−∂v∂y(t,x,y,z)(r−σt2)y+∂v∂t(t,x,y,z) +12∂2v∂y2(t,x,y,z)σt2y2+⟨v,Ax⟩]dt +e−rt∂v∂z(t,x,y,z)dNt+e−rt⟨v,dMt⟩ −e−rtσty∂v∂y(t,x,y,z)dW^t. Since the discounted value process is a martingale, the drift term must be zero, i.e. the coefficients of the d$$t$$ term must be zero. □ The system of PDEs needs to be solved subject to particular terminal conditions and the boundary conditions. It is not easy to determine the boundary conditions for the fixed strike lookback call. Similarly to Wong & Chan (2007), we first focus on the lookback options with linear homogeneous payoffs, i.e. the floating strike lookback options. They stated all existing lookback options could be valued through the pricing formulas of floating strike lookback options. Besides, Wong & Kwok (2003) showed put-call parity relations between a fixed strike lookback call and a floating strike lookback put option. Then we could derive the price of fixed lookback call options. The corresponding terminal payoff of the floating strike lookback put is given by   v(T,x,y,z)=z−y. Following Wilmott et al. (1997) and Boyle & Draviam (2007), the boundary conditions are   v(t,x,0,z)=ze−r(T−t), and   ∂v∂z(t,x,y=z)=0. 5. Estimation of the hidden Markov model parameters Since the unknown parameters depend on the hidden state of the economy, the state of the HMM $${\bf X}_t$$ plays a vital role in the estimation of the model. For option valuation we discussed earlier, we consider a risk-neutral probability $$\mathcal {Q}_{\theta^*}$$ selected by the Esscher transform. Siu (2011) showed that an optimal equivalent martingale measure selected by the minimal relative entropy approach does not price the regime-switching risk. Mathematically, the probability law of the chain $${\bf X}$$ remains the same after the measure change. It is not unreasonable to consider the hidden states of the Markov chain under the real-world probability $$\mathcal {P}$$ instead of the risk-neutral probability. In practice, the return process is observed under $$\mathcal {P}$$ instead of under $$\mathcal {Q}_{\theta^*}$$. Consequently, it is reasonable to discuss the estimation of the Markov chain parameters under $$\mathcal {P}$$ rather than under $$\mathcal {Q}_{\theta^*}$$. Note that real data is sampled discretely over time. To estimate the parameters of the hidden Markov chain, algorithms based on discretization of continuous-time filters and smoothers are needed. Here, we adopt the Baum–Welch algorithm, based on the discretization of continuous-time smoothers.3 5.1. Estimating states of the HMM The smoother-based estimators facilitate the application of the Expectation Maximum (EM) algorithm, which can be regarded as an extension of the discrete-time Baum–Welch algorithm. However, it is known that the Baum–Welch algorithm only finds a local maximum, rather than a global maximum, in the parameter space. Here, we will give a brief introduction to the Baum–Welch algorithm. Let $$\boldsymbol{P} = [p_{ij}]_{i,j = 1,2, \ldots, N}$$ be the transition probability matrix, where $$p_{ij}= P({\bf X}_t = {\bf e}_j | {\bf X}_{t-1} = {\bf e}_i)$$ and $$\pi_i = P({\bf X}_1={\bf e}_i)$$ represent the initial state distribution. In addition, the corresponding observation sequence is assumed to be $${Y_1=y_1, \ldots, Y_T = y_T}$$. Let $$c_j(y_t) = P(Y_t = y_t | {\bf X}_t={\bf e}_j)$$ be the probability of a particular observation at time $$t$$ when the Markov chain is in the state $${\bf e}_j$$ and $$\boldsymbol{C}(\cdot) = (c_1(\cdot), c_2(\cdot), \ldots, c_N(\cdot))$$. Specially, we partition the observation sequence into $$L$$ groups, i.e. $$O = \{o_1, \ldots, o_L\}$$. As assumed in Bilmes (1998), we write the complete set of HMM parameters as $$\boldsymbol{\psi} = (\boldsymbol{P}, \boldsymbol{C}, \boldsymbol{\pi})$$.4 Step 1: Firstly, we give a brief introduction to the forward and backward procedures given in Bilmes (1998). Define the following forward variable $$\alpha_i(t)$$ and backward variable $$\beta_i(t)$$:   αi(t) =P(Y1=y1,…,Yt=yt,Xt=ei|ψ),βi(t) =P(Yt+1=yt+1,…,YT=yT|Xt=ei,ψ). Here, $$\alpha_i(t)$$ is the joint probability of observing $$(y_1, \ldots, y_t)$$ and that the state of the Markov chain is $${\bf e}_i$$ at time $$t$$. $$\beta_i(t)$$ is the probability of observing $$(y_{t+1}, \ldots, y_T)$$ given that the state of the Markov chain is $${\bf e}_i$$ at time $$t$$. Then, as in Bilmes (1998), an efficient induction of the pair $$\alpha_i(t)$$ and $$\beta_i(t)$$ is as follows:   αi(1) =πici(y1),αj(t+1) =[∑i=1Nαi(t)pij]cj(yt+1),P(Y|ψ) = ∑i=1Nαi(T), and   βi(T) =1 ,βi(t) = ∑i=1Npijcj(yt+1)βj(t+1),P(Y|ψ) = ∑i=1Nβi(1)πici(y1). Step 2: Then, two other random variables are defined. The first is the probability of the chain being in state $${\bf e}_i$$ at time $$t$$ given the observation sequence $$Y$$:   γi(t)=P(Xt=ei|Y,ψ)=P(Y,Xt=ei|ψ)P(Y|ψ)=P(Y,Xt=ei|ψ)∑j=1NP(Y,Xt=ej|ψ). Note that   αi(t)βi(t)=P(y1,…,yt,Xt=ei|ψ)P(yt+1,…,yT|Xt=ei,ψ)=P(Y,Xt=ei|ψ), so $$\gamma_i(t)$$ can be rewritten as   γi(t)=αi(t)βi(t)∑j=1Nαj(t)βj(t). The second one is the probability of the chain being in state $${\bf e}_i$$ at time $$t$$ while being in state $${\bf e}_j$$ at time $$t+1$$:   ξij(t) =P(Xt=ei,Xt+1=ej|Y,ψ) =P(Xt=ei,Xt+1=ej,Y|ψ)P(Y|ψ) =αi(t)pijcj(yt+1)βj(t+1)∑i=1N∑j=1Nαi(t)pijcj(yt+1)βj(t+1). It is worth noting that the sums of the above two variables have practical interpretations: i. $$\sum_{t=1}^T \gamma_i(t)$$: the expected number of times in state $${\bf e}_i$$; ii $$\sum_{t=1}^{T-1} \xi_{ij} (t)$$: the expected number of transitions from state $${\bf e}_i$$ to state $${\bf e}_j$$ for $$Y$$. Step 3: Then, the informal version of the Baum–Welch algorithm5 is given by i. $$\bar{\pi}_i = \gamma_i(1)$$: the expected relative frequency spent in state $${\bf e}_i$$ at time 1; ii. $$\bar{a}_{ij} = \frac{\sum_{t=1}^{T-1} \xi_{ij}(t)}{\sum_{t=1}^{T-1} \gamma_{i}(t) }$$: the expected number of transitions from state $${\bf e}_i$$ to state $${\bf e}_j$$ relative to the expected total number of transitions away from state $${\bf e}_i$$; iii. $$\bar{b}_{i}(k) = \frac{\sum_{t=1}^{T} \mathbb{1}_{y_t = o_k}\gamma_{i}(t) }{\sum_{t=1}^{T} \gamma_{i}(t) }$$: the expected number of times the output observations have been equal to $$o_k$$ while in state $${\bf e}_i$$ relative to the expected total number of times in state $${\bf e}_i$$. Step 4: The Viterbi algorithm is used to determine the most likely path, i.e. the path $$(\boldsymbol{q}_1, \ldots, \boldsymbol{q}_T)$$ which maximizes the likelihood $$p(\boldsymbol{q}_1, \boldsymbol{q}_2, \ldots, \boldsymbol{q}_T | Y, \boldsymbol{\psi})$$. Let $$\delta_i(t)$$ be the path which is in state $${\bf e}_i$$ at time $$t$$ with the highest probability, i.e.   δi(t)=maxq1,q2,…,qt−1P(q1,q2,…,qt=ei,y1,y2,…,yt|ψ), then   δj(t+1)=maxi[δi(t)pij]cj(yt+1). The recursive form of the variable is   δi(1) =πici(y1),δj(t) = max1≤i≤Nδi(t−1)pijcj(yt), and the most probable path satisfies the following conditions   qt=ei∗,t=T,T−1,T−2,…,1, where   i∗={argmax1≤i≤Nδi(T),t=T,argmax1≤i≤Nδi(t)P(Xt+1=qt+1|Xt=ei),t=T−1,T−2,…,1.  After the most-probable path through the HMM states is defined using the Viterbi approach, we can estimate the parameters of the dynamics of the investment fund. 5.2. Maximum likelihood estimations of parameters When we obtain the most-probable path, we further use the maximum likelihood estimation approach to estimate the appreciation rate and the volatility in the dynamics of asset price. It is easy to see that the probability density function of the observation process $$Y$$ is given by   p(Yt=yt|η1,η2,…,ηN,σ1,σ2,…,σN)=∑j=1N12πσj2exp⁡(−(yt−ηj)22σj2)1{qt=ej}. Then the likelihood function is   L(y1,…,yT;η1,η2,…,ηN,σ1,…,σN)=∏t=1T∑j=1N12πσj2exp⁡(−(yt−ηj)22σj2)1{qt=ej}. Let the derivatives of the logarithm likelihood function with respect to $$\eta_j$$ and $$\sigma_j$$$$(j=1,\ldots,N)$$ be zeros. The maximum likelihood estimations of the parameters can be derived.   ηj^ =∑t=1Tyt1{qt=ej}∑t=1T1{qt=ej},σj^2 =∑t=1T(yt−η^j)21{qt=ej}∑t=1T1{qt=ej}forj=1,2,…,N. Remark 5.1 One may argue that the smoother-based estimation equations are recursive and can be implemented by the standard method of discretization. However, when the states of the hidden Markov chain and the observation sequences are discrete, forward-backward estimates are easier to be calculated under the Baum–Welch method. Furthermore, the EM algorithm concerning the smoother-based estimation requires initial estimates of both the transition matrix of the hidden Markov chain and the volatilities of the observation process. The convergence of the algorithm may depend on the selection of the initial estimates. Here the transition matrix and the states of the hidden Markov chain are first determined. Then, given the estimated sequence of the HMM, the appreciation rates and volatilities of the investment fund are estimated via the maximum likelihood estimation. Remark 5.2 Baum & Petrie (1966) gave consistency and asymptotic normality of the maximum-likelihood estimator when observations take values in a finite set. In Bickel et al. (1998), the asymptotic normality of the maximum likelihood estimator for general HMMs was established under mild conditions. Bickel et al. (1998) also proved that the observed information matrix is a consistent estimator of the Fisher information. Remark 5.3 Here we adopt a two-stage estimation method, where the first stage involves the use of the Viterbi algorithm to determine the most likely path of the hidden Markov chain and the second stage involves the use of the maximum likelihood estimation to estimate the means and volatilities over different states of the chain given the most likely path of the chain. It seems that the Baum–Welch method fits well to the Viterbi algorithm in the first stage of the estimation. It may not be convenient to use the Kalman filter in the first stage. Furthermore, the Baum–Welch algorithm involves the use of smoothed estimates which can incorporate more information than the filtered estimates. 6. Numerical examples When we obtain the filtered estimates of the hidden Markov chain, we replace the corresponding hidden quantities by their filtered estimates. Then, the PDE (4.3) is given as follows:    −rv(t,x,y,z)−∂v∂y(t,x,y,z)(r−σ^t2)y+∂v∂t(t,x,y,z) +12∂2v∂y2(t,x,y,z)σ^t2y2+⟨v,A^x⟩=0. Here, we adopt a similar dimension reduction technique as that in Wilmott et al. (1997) and Boyle & Draviam (2007). Let $$u=\frac{y}{z}$$, $$G_i(t, u)=\frac{1}{z} v(t, {\bf x}={\bf e}_i, y, z)$$. Then the PDE becomes    −∂Gi∂u(t,u)(r−σ^t2)u+12∂2Gi∂u2(t,u)σ^t2u2+∂Gi∂t(t,u)+⟨Gi,A^ei⟩−rGi(t,u)=0. The corresponding terminal conditions become the following form:   Gi(t=T,u)=max(1−u,0). The boundary conditions are   Gi(t,u=0)=e−r(T−t), and when the maximum is measured continuously, the boundary condition at $$F=N$$ becomes a boundary condition at $$u=1$$:   ∂Gi∂u|u=1=Gi(t,u=1). It is easy to solve the above PDE and we could obtain the prices of the floating lookback put options. According to the put-call parity in Wong & Kwok (2003),   cfix(t,q,F,Nt,;1)=pfl(t,q,F,max(Nt,1))+F−e−r(T−t), we can derive the prices of the fixed strike lookback call options and the DFP contract. Here, we shall give two numerical examples to illustrate the valuation of a dynamic fund protection when the dynamics of the investment fund follow a Geometric Brownian motion with parameters being modulated by a hidden Markov chain. For simplicity, we assume that the Markov chain has two states. The paper by Taylor (1999) provided some empirical evidence for the use of a Markov chain model with two states for volatility. The real data examples we present here use data sets of daily closing prices of IBM (International Business Machines Corporation) and Apple Inc., from May 2008 to April 2010, retrieved from Yahoo Finance. There are 484 observations in each data set. During the period, the economy was influenced by the financial crisis. Consequently, it is of practical interest to investigate the valuation problem when the economy has substantial changes. On the other hand, the stock prices of financial industries have not been selected since the whole industry was affected significantly by the crisis. As mentioned earlier, the Baum–Welch algorithm is adopted. The estimates of relevant parameters are given by   P^IBM=(0.77940.22060.18290.8171),P^Apple=(0.79820.20180.19820.8018). Here $$\hat{P}_{IBM}$$ and $$\hat{P}_{Apple}$$ represent the transition probability matrices of the IBM example and the Apple Inc. example, respectively. From the given data, we can easily calculate the following estimates:   η^IBM=(−0.06780.2148) ,η^Apple=(0.08280.6253),  σ^IBM2=(0.01010.0013) ,σ^Apple2=(0.03410.0100). Clearly, in the two examples, the estimators of the volatilities $$\sigma_1 > \sigma_2$$, indicate that State 1 and State 2 represent a ‘Bad’ economy and a ‘Good’ one, respectively. The estimation results are consistent with those in Liew & Siu (2010). The differences between the estimated volatilities in the two states may be attributed to structural changes in data due to the financial crisis and the rebounds after the crisis. Note that the estimated drift parameter for a ‘Bad’ economy in a regime-switching model is not uncommon, which is similar with the numerical example in Liew & Siu (2010). After we obtain the estimates, we can solve the valuation problem using the PDE approach. The values for parameters, used in the discretization of the PDEs, are assumed to be:   Δt=1/250(year). Also, we assume the risk-free interest rate is:   r=0.05(1/year). Using finite difference methods, we calculate the values of the DFP by varying the guarantee level $$K$$ and the time-to-maturity $$T$$. Here, we assume the value of $$K$$ is less than the value of the naked fund on May 1 2008. It is not unreasonable to make this assumption. The organizations issuing the DFP will never design such investment plans, if so, they will upgrade the fund immediately after the DFP issued. Table 1 and Table 2 present numerical results for the price of a DFP. Table 1 DFP prices for IBM versus different K and T  Protection level  $$T=2$$  $$T=3$$  ($${}$$)  State 1  State 2  State 1  State 2  95  27.4092  7.1824  22.0819  4.2292  100  33.6141  11.0479  27.7742  8.0946  105  39.8190  14.9134  33.4664  11.9601  110  46.0239  18.7789  39.1586  15.8256  115  52.2288  22.7863  44.8509  19.7899  120  58.4337  27.7111  50.5431  24.3925  Protection level  $$T=2$$  $$T=3$$  ($${}$$)  State 1  State 2  State 1  State 2  95  27.4092  7.1824  22.0819  4.2292  100  33.6141  11.0479  27.7742  8.0946  105  39.8190  14.9134  33.4664  11.9601  110  46.0239  18.7789  39.1586  15.8256  115  52.2288  22.7863  44.8509  19.7899  120  58.4337  27.7111  50.5431  24.3925  Table 2 DFP prices for Apple Inc. versus different K and T  Protection level  $$T=2$$  $$T{=}3$$  ($${}$$)  State 1  State 2  State 1  State 2  160  103.7537  22.5602  107.7473  26.7511  170  115.8936  29.6254  119.8609  33.8025  180  128.0334  36.6907  131.9745  40.8538  190  140.1732  43.7559  144.0882  47.9052  200  152.3131  50.8212  156.2018  54.9566  210  164.4529  57.8864  168.3154  62.0079  Protection level  $$T=2$$  $$T{=}3$$  ($${}$$)  State 1  State 2  State 1  State 2  160  103.7537  22.5602  107.7473  26.7511  170  115.8936  29.6254  119.8609  33.8025  180  128.0334  36.6907  131.9745  40.8538  190  140.1732  43.7559  144.0882  47.9052  200  152.3131  50.8212  156.2018  54.9566  210  164.4529  57.8864  168.3154  62.0079  From Table 1 and Table 2, we note that the prices of DFP decrease when the maturity time is longer in the IBM example, while the opposite trend presents in the Apple Inc. example. However, as calculated earlier, the other parameters are similar in the two examples. This is also another reason why we chose the stock prices of the two companies. Note that in the Black–Scholes formula, the Greek ‘Theta’ is defined to measure the change in the option price when the time to maturity decreases. Our results verify that ‘Theta’ can be either positive or negative. The prices of the DFP in State 1 are significantly higher than the corresponding prices in State 2. This is intuitively clear since higher risk premiums are required to compensate the risk attributed to a ‘bad’ economy. This also provides some empirical evidence for the use of the regime-switching model. In addition, we simulate the price of the DFP under the Black–Scholes model. Goldman et al. (1979) derived the pricing formula for a floating strike look back put option. Then we calculated the prices of the corresponding DFPs. Here, we only give the comparison results in the IBM example. Similar results can be simulated in the Apple Inc. example. Fig. 1. View largeDownload slide DFP prices corresponding to different protection levels when $$T=2$$ and $$T=3$$ Fig. 1. View largeDownload slide DFP prices corresponding to different protection levels when $$T=2$$ and $$T=3$$ In Fig. 1, two solid lines indicate values arising from our approach while two dashed lines correspond to the BS model, and the lines with $$*$$ represent the prices of DFP that are derived when the economy state is ‘Good’. Besides, from the numerical results, the effect of different economy states cannot be ignored in the valuation process of DFP. To highlight the effect of regime-switching parameters in the DFP valuation, we construct Table 3 that compares the BS price and our price numerically. Here, we simply take the prices when $$T=2$$ as an example. The prices from the BS model are computed using Monte Carlo simulations. In the BS model (State 1), a geometric Brownian motion with a constant volatility $$\sigma_1$$ is assumed while in the BS model (State 2), a geometric Brownian motion with a constant volatility $$\sigma_2$$ is assumed. As seen from Table 3, when volatility changes, the prices of DFP change significantly. In other words, the price of a DFP will significantly change if the state of the economy changes. Thus, it is not difficult to imagine the trace off faced by insurance companies. If an investment plan is sold at a lower price, insurance companies may face larger losses when the economy is ‘bad’. Consequently, insurance companies would like to charge a higher price to hedge the volatility risk. However, on the other hand, the DFP with a higher price would be less attractive to an investor. To some extent, this explains that the effect of HMM cannot be ignored in the valuation process. Table 3 DFP prices for IBM under different models with T=2  Protection level  BS model  Our model  ($${}$$)  State 1  State 2  State 1  State 2  95  24.3810  6.0101  27.4092  7.1824  100  31.5274  10.8379  33.6141  11.0479  105  38.7840  15.8445  39.8190  14.9134  110  46.1506  20.8948  46.0239  18.7789  115  53.6273  25.9613  52.2288  22.7863  120  61.2141  31.0896  58.4337  27.7111  Protection level  BS model  Our model  ($${}$$)  State 1  State 2  State 1  State 2  95  24.3810  6.0101  27.4092  7.1824  100  31.5274  10.8379  33.6141  11.0479  105  38.7840  15.8445  39.8190  14.9134  110  46.1506  20.8948  46.0239  18.7789  115  53.6273  25.9613  52.2288  22.7863  120  61.2141  31.0896  58.4337  27.7111  Fig. 2. View largeDownload slide Relative differences $$\frac{DFP_{bs}-\widehat{DFP}}{DFP_{bs}}$$ versus $$K$$ Fig. 2. View largeDownload slide Relative differences $$\frac{DFP_{bs}-\widehat{DFP}}{DFP_{bs}}$$ versus $$K$$ To make a comparison between the estimates of the DFP price $$\widehat{DFP}$$ and the Black–Scholes price $$DFP_{bs}$$, we plot the relative difference between the Black–Scholes price and our estimation. Figure 2 is the plot of relative difference $$\frac{DFP_{bs}-\widehat{DFP}}{DFP_{bs}}$$ versus $$K$$, for two fixed values for $$T$$. In Figure 2, the two solid lines represent the relative differences when the maturity time is $$T=2$$ and the two dash lines depict the relative differences when $$T=3$$. The lines with $$*$$ represent that the economic state is ‘Good’. From the figure, it is clear that the differences between the estimated dynamic fund protection values and the BS ones are comparatively small, say between 0 and 0.5% most of the time. Moreover, the higher the protection level, the smaller the relative differences are. 7. Conclusions This article considered the valuation of the DFP. We assumed that the investment fund depends upon a continuous-time, finite-state, hidden Markov chain. Firstly, the Esscher transform was applied to select a pricing kernel. Then, we adopted the PDE approach to price the DFP. To estimate the states and parameters of the HMM, Baum–Welch algorithm, based on the discretization of continuous-time smoothers, was applied. After we obtained the most-likely path of the HMM by the Viterbi algorithm, we derived the maximum likelihood estimation of the model parameters. In the numerical examples, we calculated the prices of the DFP for two companies, IBM and Apple Inc. Moreover, we compared results from our model with those obtained from the pricing formula of Goldman et al. (1979). The numerical results reveal that regime shifts have a pronounced effect on the prices of the DFP. Acknowledgements The authors wish to thank two anonymous referees and the editors for their valuable comments, which have been greatly helpful in improving the paper. Funding The authors wish to thank two anonymous referees and the editors for their helpful comments. K. F. and R. W. would like to acknowledge the 111 Project (B14019), National Natural Science Foundation of China (11501211, 11571113), Shanghai Subject Chief Scientist (14XD1401600), Shanghai Pujiang Program (15PJC026), Shanghai Philosophy Social Science Planning Office Project (2015EJB002), China Postdoctoral Science Foundation (2015M581564) and Shanghai Chenguang Plan (15CG22). T. K. S. would like to acknowledge a Discovery Grant from the Australian Research Council (ARC) (Project No.: DP130103517). Y. S. would like to acknowledge financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC)(Project No.: RGPIN-2016-05677). References Baum, L.E. & Petrie, T. 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Google Scholar CrossRef Search ADS   Footnotes 1 As explained in Boyle & Draviam (2007), if at time $$t$$, $${\bf X} (t) = {\bf e}_i$$, then the state at $$t+\Delta t$$ will be $${\bf e}_j$$ with probability $$a_{ij} \Delta t$$, and the probability of remaining in the state is $$1-\sum_{j=0,j\neq i}^M a_{i j} \Delta t = 1+a_{i i}\Delta t$$. 2 This assumption ensures that the regime switching PDE can be obtained when pricing the dynamic fund protection. 3 As discussed in earlier works, compared to filtered estimates, the smoothed estimates can incorporate the extra information obtained from the observations between time $$t$$ and time $$T$$. Furthermore, to apply the robust discretization of continuous-time filters, the dynamics of the observation process should not include stochastic integrals. This is also the reason why Elliott et al. (2003) made approximations to the observation process. For details, interested readers can refer to Elliott et al. (1995, 2003); James et al. (1996) and Malcolm & Elliott (2010). 4 Let $$\boldsymbol{\Psi}$$ be an open subset of $$\mathfrak{R}^n$$. Suppose that to each $$\boldsymbol{\psi} \in \boldsymbol{\Psi}$$, we have a smooth assignment $$\boldsymbol{\psi} \rightarrow (P(\boldsymbol{\psi}), B(\boldsymbol{\psi}), \pi(\boldsymbol{\psi}))$$. Under these assumptions, for each fixed $$y_1, y_2, \ldots, y_T$$, $$P_{ y_1 y_2 \ldots y_T}(\boldsymbol{\psi}) = P_{y_1 y_2 \ldots y_T}$$, $$P_{ y_1 y_2 \ldots y_T}(P(\boldsymbol{\psi}), B(\boldsymbol{\psi}), \pi(\boldsymbol{\psi}))$$ is a smooth function of $$\boldsymbol{\psi}$$. Given a fixed $$Z-$$sample $$y = y_1, \ldots, y_T$$, a parameter value $$\boldsymbol{\psi}^*$$ which maximizes the likelihood $$P_y (\boldsymbol{\psi}) = P_{y_1 \ldots y_T}(\boldsymbol{\psi})$$ can be found. To maximize the likelihood function, Baum et al. (1970) defined a continuous transformation $$\mathcal{T}$$ mapping $$\boldsymbol{\Psi}$$ into itself with the property that $$P_{y_1 y_2 \ldots y_T} ( \mathcal{T} (\boldsymbol{\psi})) > P_{y_1 y_2 \ldots y_T} (\boldsymbol{\psi})$$ unless $$\boldsymbol{\psi}$$ is a critical point of $$P_{y_1 y_2 \ldots y_T} ( \boldsymbol{\psi})$$. 5 A more general case is that the underlying state sequence is assumed to be hidden or unobserved. In this case, the $$Q$$ function is introduced. Details can refer to Baum et al. (1970). © The authors 2016. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

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IMA Journal of Management MathematicsOxford University Press

Published: Jan 1, 2018

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