# Closed-Form Optimal Strategies of Continuous-Time Options with Stochastic Differential Equations

Authors: Wei Yan
Journal: Complexity
 (2017)

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## Abstract

A continuous-time portfolio selection with options based on risk aversion utility function in financial market is studied. The different price between sale and purchase of options is introduced in this paper. The optimal investment-consumption problem is formulated as a continuous-time mathematical model with stochastic differential equations. The prices processes follow jump-diffusion processes (Weiner process and Poisson process). Then the corresponding Hamilton-Jacobi-Bellman (HJB) equation of the problem is represented and its solution is obtained in different conditions. The above results are applied to a special case under a Hyperbolic Absolute Risk Aversion (HARA) utility function. The optimal investment-consumption strategies about HARA utility function are also derived. Finally, an example and some discussions illustrating these results are also presented.


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## Body

## 1. Introduction

Pricing theory of options is an important research field. In 1973, Black and Scholes published an innovative paper [1]. In this paper, a theoretical valuation formula is derived. The well-known Black-Scholes formula is deduced. This formula’s creative point does not lie in the investor’s preferences but a risk-neutral method with a risk-free asset. It is applied in many fields extensively. However, the varieties of options become more and more. The theory of options pricing must be improved. Up to now, the most options pricing models are extended based on the Black-Scholes formula. Merton [2], Jeanblanc-Picque and Pontier [3], and Lin and Ye [4] added the jump process to the option pricing formula; Barron and Jensen [5] spread the formula while the interest rate has difference between the deposition and loan. In addition, there are a lot of other pricing models, such as Cox et al. [6].Nowadays, the objective is to earn wealth with portfolio selection although the options pricing is still a central research topic. We could regard the options as a risk asset, like stocks. As a risk asset, options catch more and more attention in financial market. There exists options trading in many exchanges all over the world, such as CBOE (Chicago Board Option Exchange), PHLX (Philadelphia Stock Exchange), AMEX (American Stock Exchange), PSE (Pacific Stock Exchange), NYSE (New York Stock Exchange). In most options exchange, they follow the market maker rules. The price of purchasing or selling an options contract is equivalent to the price at which the market maker wants to sell or purchase it. Therefore, the options have the same position with other underlying assets. However, for the general investment-consumption problem analysis approach, there has been few literatures involving options trading. In this paper, the options contract can be added to the optimal investment-consumption problem.One of the frequent questions in finance is how to allocate a certain amount of money in different assets and at what time instant. The earliest approach to consider the optimal portfolio problem is so called mean-variance criterion. It was pioneered by Markowitz [7] and has been playing a critical role in the theory of portfolio selection. It has also gained widespread acceptance as a practical tool for portfolio optimization. However, it is a single-period model which makes one-off decision at the beginning of the period and holds on until the end of the period. Gradually, researchers extended the above single-period model to continuous-time models [8, 9]. An optimal consumption-investment problem has been formulated. By applying results from stochastic control theory to the optimal portfolio problem, explicit solutions have been obtained for some special cases. Many researches about continuous-time portfolio selection appeared [10–15].In general literatures, continuous-time investment-consumption model is modeled by assuming a market in which the evolution of assets prices is described by a continuous process. However, as pointed by Merton [2, 16] and Jarrow and Rudd [17], the actual price of stock is not continuous because there is some external information affecting it. In fact, in addition to the continuous process based on Brown motion, the analysis of price evolution does reveal sudden and rare breaks logically accounted for by exogenous events on information. Such a behavior from probabilistic point of view is naturally modeled by a point process. This process governed by Brownian motion and point process is called jump-diffusion process. It is a discontinuous price process. Some concrete results [2, 18] prove that there are some jump processes in reality. A stochastic exchange rate with jump-diffusion processes is concerned in the mean-variance portfolio selection [19]. However, it is a linear-quadratic model. The calculation process and results may not be complicated. We hope that the objective function has a general form like a Hyperbolic Absolute Risk Aversion (HARA) utility function of this paper. Moreover, there is a nondifferentiable point in the capital process in the following paper. These lead to the calculation process and results that are more complicated. We want to obtain some more general results in a sense.In financial markets, the price of options has difference between selling and purchasing. Different from the classical Merton model, we make a hypothesis on the European call option trading price that the purchasing price is usually bigger than selling price. This simple hypothesis is only a constraint between purchasing and selling. The investor can add some other constraints. However, more complicated constraints would prevent the derivation of closed-form analytical solutions. Therefore, we add this simple constraint to the unconstrained classical Merton model.This paper is concerned with the optimal investment-consumption portfolio selection of options with different price between sale and purchase based on a risk aversion utility function in financial markets. We study how the different prices affect the strategies of investment-consumption. So the problem is formulated as a continuous-time mathematical model. Considering the difference price of the options, the paper is more complex in the price process than traditional Merton model. Besides, these pricing processes follow jump-diffusion processes (Weiner process and Poisson process). The model contains discontinuous price processes. Then, the corresponding Hamilton-Jacobi-Bellman (HJB) equation of the problem is represented and its solutions are obtained in different conditions. Finally, the above results are applied to a special case in which the objective function is a Hyperbolic Absolute Risk Aversion (HARA) utility function. Under the above objective function, we can obtain a closed-form solution. We can avoid the computational complexity in numerical calculation, getting the investment strategy more effectively and concisely.


## 2. Mathematical Model

In this section, we formulate a mathematical model. There are 3 assets being traded on a finite horizon[0,T]. They are one bond, one risky asset,and its corresponding options. In this paper, we only consider European call option contracts like other kinds of similar risky assets.One asset is a bond, whose priceS0(t) evolves according to the differential equation (1)dS0t=S0trtdt,t∈0,T,S00=s0,where r(t) is the interest rate of the bond.Similar to Merton [16], the price process F(t) of the risk asset should satisfy the following stochastic differential equation governed by a Weiner process and a Poisson process: (2)dFt=Ft-ϖtdt+θtdWt+ζtdNt,t∈0,T,F0=f0>0,where ϖ(t) is the appreciation rate and θ(t) and ζ(t) are the volatility coefficient;t− is s<t while s→t. We consider a financial market which is subject to uncertainty that enters through a one-dimensional Weiner process W(t) on its canonical space ΩW,LW,PW and a one-dimensional Poisson process N(t) on its canonical space (ΩN,LN,PN) with intensity equal to λ(t). By Jeanblanc-Picque and Pontier [3], the compensated Poisson process defined by Mt=Nt-∫0Tλsdst∈0,T is a martingale. We denote the PW-augmentation of σWs:0≤s≤t by LtW and the PN-augmentation of σNs:0≤s≤t by LtN. Let (Ω,L,P)=ΩW⊗ΩN,LW⊗LN,PW⊗PN denote the product space. On this space, W(t) and N(t) are independent. Moreover, the coefficients r(t), ϖ(t), θ(t), ζ(t), and λ(t) in the stochastic differential equations are deterministic, Borel-measurable, and bounded.Applying Ito’s formula [20] to the purchasing price Y(t,F) of the corresponding European call options, we obtain (3)dYt,F=Yt+FtϖtYF+12Ft2θt2YFFdt+FtθtYFdWt+Yt,Ft+Ftζt-Yt,FtdNt,t∈0,T,YT,FT=FT-q+,where Yt=∂Y(t,F)/∂t, YF=∂Y(t,F)/∂F, YFF=∂2Yt,F/∂F2, FT-q+=FT-q,ifFT≥q;0,ifF(T)<q, and q is strike price (the rest of the partial deferential symbols of this paper are the same as Yt). Y(t,F) is actually a function of futures prices F(t) and time t. F(t) should be a unitary part in Y(t,F). When we apply Ito’s formula, we should apply it to the corresponding process Yt,Ft, which is defined in terms of the function Y and the process F(t).We assume that the price of options is different when selling or purchasing. The purchasing price is usually bigger than selling price. We set the selling price asG(t,F)=α(t)Y(t,F), where 0<α(t)<1. Moreover, for simplicity it is assumed that α(t) is a monotone nondecreasing function differentiable with respect to time t. Then G(t,F) satisfies the following equation: (4)dGt,F=Gt+FtϖtGF+12Ft2θt2GFFdt+FtθtGFdWt+Gt,Ft+Ftζt-Gt,FtdNt,t∈0,T,GT,FT=αTFT-q+.Substituting G(t,F)=αtYt,F into the above equation, we obtain (5)dGt,F=α˙tYt,F+αtYt+αtFtϖtYF+12αtFt2θt2YFFdt+αtFtθtYFdWt+αtYt,Ft+Ftζt-Yt,FtdNt,t∈0,T,GT,FT=αTFT-q+.Letπ0(t), π1(t), and π2(t) be the amount allocated to the bond, risky asset, and options, respectively, which may be positive or negative. It is assumed that ω0(t), ω1(t), and ω2(t) are the shares of the above derivatives mentioned, respectively, and then π0(t)=ω0(t)S0(t), π1(t)=ω1(t)F(t), π2(t)=ω2(t)Y(t,F), or π2(t)=ω2(t)G(t,F).It is assumed that an investor has initial capitalx(t)=x0(t)(t=0).When an investor is purchasing option contracts (π2(t)>0), we can get the differential equation as follows by considering the dynamic process of investment and consumption {c(t),0≤t≤T}:(6)dxt=ω0tdS0t+ω1tdFt+ω2tdYt,F-ctdt=xtrt+π1tϖt-rt+π2tYt+FtϖtYF+1/2F2θt2YFFYt,Ft-rt-ctdt+π1tθt+π2tFtθtYFYt,FtdWt+π1tζt+π2tYt,Ft+Ftζt-Yt,FtYt,FtdNt,t∈0,T,x0=x0.Setb1(t)=ϖ(t),  b2(t)=Yt+F(t)ϖ(t)YF+1/2F2θt2YFF/Y(t,F(t)),  σ1(t)=θ(t),  σ2(t)=F(t)θ(t)YF/Y(t,F(t)),  φ1(t)=ζ(t),  φ2(t)=Y(t,F(t)+F(t)ζ(t))-Y(t,F(t))/Y(t,F(t)). Then, it can be obtained that(7)dxt=xtrt+π1tb1t-rt+π2tb2t-rt-ctdt+π1tσ1t+π2tσ2tdWt+π1tφ1t+π2tφ2tdNt,t∈0,T,x0=x0.When an investor is selling an option contract (π2(t)≤0), we can get the similar differential equation(8)dxt=ω0tdS0t+ω1tdFt+ω2tdGt,F-ctdt=xtrt+π1tb1t-rt+π2tb2t-rt+α˙tαt-ctdt+π1tσ1t+π2tσ2dWt+π1tφ1t+π2tφ2dNt,t∈0,T,x0=x0.Remark 1.
In this paper, we regard options as general risky assets just like stocks. Besides, it is assumed thatα(t) is not constant, but a function of time t.Merging (7) and (8), it can be obtained that(9)dxt=xtrt+π1tb1t-rt+π2tb2t-rt-α˙tαtπ2t--ctdt+π1tσ1t+π2tσ2dWt+π1tφ1t+π2tφ2dNt,t∈0,T,x0=x0,where π2t-=0,ifπ2t>0;-π2(t),ifπ2(t)≤0.Then, (9) can be transformed into the following equation:(10)dxt=xtrt+πt′bt-rt12-α˙tαtπ2t--ctdt+πt′σtdWt+πt′φtdNt,t∈0,T,x0=x0,where 12 denotes a 2-dimensional column vector with every entry being 1; b(t)=b1t,b2t′,π(t)=π1t,π2t′,σ(t)=σ1t,σ2t′,φ(t)=φ1t,φ2t′. (M′denotes the transpose of any matrix or vector M.)The objective of an investor is to maximize the utility of consumption and terminal expected wealth.(11)Jx,c,π1,π2=E∫0TU1t,ctdt+U2T,xT,where the utility functions U1(·) and U2(·) are risk aversion functions.


## 3. HJB Equation

In this section, we will deduce the HJB function corresponding to (10) and (11) similar to Sage and White [21].Firstly, we set the value function as follows:(12)Vt,x=maxc,π⁡Et,x∫tTU1s,cds+U2T,x.Then, it can be obtained as follows by Bellman optimal principle that(13)Vt,x=maxc,π⁡Et,x∫tt+ΔtU1s,cds+∫t+ΔtTU1s,cds+U2T,x=maxc,π⁡Et,x∫tt+ΔtU1s,cds+Vt+Δt,x+Δx,where Et,x denotes the conditional expectation when the initial condition is given by (t,x).It can be obtained that(14)Vt+Δt,x+Δx=Vt,x+VtΔt+VxΔx+12VttΔt2+12VxxΔx2+VtxΔtΔx,Δxt=xtrt+πt′bt-rt12-α˙tαtπ2t--ctΔt+πt′σtΔWt+πt′φtΔNt=xtrt+πt′bt-rt12+λtφt-α˙tαtπ2t--ctΔt+πt′σtΔWt+πt′φtΔMt.Substituting the above two equations into (13), we can get(15)Vt,x=maxc,π⁡Et,xU1t,cΔt+Vt,x+VtΔt+VxΔx+12VttΔt2+12VxxΔx2+VtxΔtΔx=maxc,π⁡Et,xU1t,cΔt+Vt,x+VtΔt+Vxxrt+π′bt-rt12+λtφt-α˙tαtπ2--cΔt+π′σtΔWt+π′φtΔMt+12VttΔt2+12Vxxxrt+π′bt-rt12+λtφt-α˙tαtπ2--cΔt+π′σtΔWt+π′φtΔMt2+VtxΔtxrt+π′bt-rt12+λtφt-α˙tαtπ2--cΔt+π′σtΔWt+π′φtΔMt.By Karatzas and Shreve [22], it can be obtained that dW(t)·dW(t)=dt,dW(t)·dt=0,dM(t)·dM(t)=λ(t)dt,dM(t)·dt=0.After substituting them into the above equation, we subtract from both sides of the equationV(t,x) divided by Δt. Finally, by letting Δt→0, we obtain(16)Vt+Vxxrt+maxcU1t,c-cVx+maxπVxπ′bt-rt12+λtφt-α˙tαtπ2-+12Vxxπ′σtσt′π+λtπ′φtφt′π=0.SetB(t)=b(t)-r(t)12+λ(t)φ(t),D(t)=σtσt′+λtφtφt′. It is assumed that D(t) is positive definite. Substituting them into (16), it is obtained that(17)Vt+Vxxrt+maxcU1t,c-cVx+maxπVxπ′Bt-α˙tαtπ2-+12Vxxπ′Dtπ=0,with V(T,x)=U2(T,x). Equation (17) is the HJB equation corresponding to (10) and (11).


## 4. Optimal Investment-Consumption Strategies

In this section, the optimal investment-consumption strategies will be derived.


### 4.1. Optimal Consumption Strategies

From the HJB equation (17), and by applying a first-order optimality condition, we derive that(18)∂U1t,c-cVx∂c=0.That is to say, c∗ satisfies(19)∂U1t,c∂c=Vx.


### 4.2. Optimal Investment Strategies

We will also deduce the optimal investment strategies from HJB function (17). Set(20)fπ=fπ1,π2=Vxπ′Bt-α˙tαtπ2-+12Vxxπ′Dtπ,f1π=f1π1,π2=Vxπ′Bt+12Vxxπ′Dtπ,f2π=f2π1,π2=Vxπ′Bt+π2α˙tαt+12Vxxπ′Dtπ.By settingB1(t)=B(t) and B2(t)=B(t)+0,α˙t/αt′, the above three equations can be written as follows:(21)fπ=fπ1,π2=Vxπ′B1t-α˙tαtπ2-+12Vxxπ′Dtπ,(22)f1π=f1π1,π2=Vxπ′B1t+12Vxxπ′Dtπ,(23)f2π=f2π1,π2=Vxπ′B2t+12Vxxπ′Dtπ,thus getting the following equation(24)fπ=fπ1,π2=f1π1,π2,ifπ2>0,f2π1,π2,ifπ2≤0.From (22), by applying the first-order unconstrained optimality condition, we obtain(25)∂f1π∂π=0⟹π⌢=-VXVXXDt-1B1t,and f1π⌢=-VX2/2VXXB1t′Dt-1B1(t).Similarly, from (23) we obtain(26)∂f2π∂π=0⟹π˘=-VXVXXDt-1B2t,and f2(π˘)=-VX2/2VXXB2t′Dt-1B2(t).Next, let us compareπ⌢ with π˘. We have(27)π⌢-π˘=VXVXXDt-10,α˙tαt′.By Merton [16], V(t,x) is concave function because the utility functions U1(·) and U2(·) are risk aversion functions. Then, Vx≥0,Vxx<0, and D(t) is positive definite. So π⌢2-π˘2≤0. We will consider three cases.(1)
π
⌢
2
>
0. For any π≤0, we have(28)f2π=Vxπ′B2t+12Vxxπ′Dtπ≤Vxπ′B1t+12Vxxπ′Dtπ=f1π≤f1π⌢.Therefore, max⁡f(π)=max⁡f1(π)=f1π⌢=-VX2/2VXXB1t′Dt-1B1(t), and(29)π∗=π⌢=-VXVXXDt-1B1t.The HJB function (17) can be written as follows:(30)Vt+Vxxrt+U1t,c∗-c∗Vx-VX22VXXB1t′Dt-1B1t=0,(31)VT,x=U2T,x.(2)
π
⌢
2
≤
0
≤
π
˘
2. The two optimization problems are(32)maxf1ππ2≥0,t∈0,T(33)maxf2ππ2≤0,t∈0,T.Solving (32), by introducing the Lagrange multiplier l, we have(34)L1π1,π2,l=Vxπ′B1t+12Vxxπ′Dtπ-l·π2,where l≥0. Then(35)∂L1π1,π2,l∂π1=VxB1t+VxxDtπ=0.Consideringπ⌢2≤0≤π˘2 and the constraint of (32), it is obtained that the optimal solution of (32) satisfies π-2=0. Substituting it into (35), we can get the following equation:(36)Vxbt-rt+λtφ1t+VXXσ12t+λtφ12tπ-1=0.Then π-1=-VX/VXX·b(t)-r(t)+λ(t)φ1(t)/σ12(t)+λ(t)φ12(t).Following the same line of argument, the solution of (33) is π-2=0,π-1=-VX/VXX·b(t)-r(t)+λ(t)φ1(t)/σ12(t)+λ(t)φ12(t).As the above discussion mentioned, it is shown that(37)π∗=π-=-VXVXX·b1t-rt+λtφ1tσ12t+λtφ12t0.Therefore,max⁡f(π)=f1(π-)=-(VX)2/2VXX·bt-rt+λtφ1t2/σ12(t)+λ(t)φ12(t).SettingH(t)2=b1t-rt+λtφ1t2/σ12(t)+λ(t)φ12(t), the above equation can be written as follows:(38)max⁡fπ=f1π-=-VX22VXXHt2.The HJB function (17) can be written as follows:(39)Vt+Vxxrt+U1t,c∗-c∗Vx-VX22VXXHt2=0,and V(T,x)=U2(T,x).(3)
π
˘
2
<
0. For any π≤0, we have(40)f1π=Vxπ′B1t+12Vxxπ′Dtπ≤Vxπ′B2t+12Vxxπ′Dtπ=f2π≤f2π˘.Therefore, max⁡f(π)=max⁡f2(π)=f1(π˘)=-VX2/2VXXB2t′Dt-1B2(t), and(41)π∗=π˘=-VXVXXDt-1B2t.The HJB function (17) can be written as follows:(42)Vt+Vxxrt+U1t,c∗-c∗Vx-VX22VXXB2t′Dt-1B2t=0,and V(T,x)=U2(T,x).Next, we will discuss the conditions which lead to the above classifications.(43)π⌢=-VxVxxDt-1B1t=-VxVxxσtσt′+λtφtφt′-1bt-rt12+λtφt=-VxVxxσ12t+λtφ12tσ1tσ2t+λtφ1tφ2tσ1tσ2t+λtφ1tφ2tσ22t+λtφ22t-1·b1t-rt+λtφ1tb2t-rt+λtφ2t=-VxVxxι1-ι2λtσ1tφ2t-λtσ2tφ1t2γ1-γ2λtσ1tφ2t-λtσ2tφ1t2,π˘=-VxVxxDt-1B2t=-VxVxxσtσt′+λtφtφt′-1bt-rt12+λtφt+0,α˙tαt′=-VxVxxσ12t+λtφ12tσ1tσ2t+λtφ1tφ2tσ1tσ2t+λtφ1tφ2tσ22t+λtφ22t-1·b1t-rt+λtφ1tb2t-rt+λtφ2t+α˙tαt=-VxVxxι1-ι2+σ1tσ2t+λtφ1tφ2tα˙t/αtλtσ1tφ2t-σ2tφ1t2γ1+σ12t+λtφ12tα˙t/αt-γ2λtσ1tφ2t-σ2tφ1t2,where(44)ι1=σ12t+λtφ22tb1t-rt+λtφ1t,ι2=σ1tσ2t+λtφ1tφ2tb2t-rt+λtφ2t,λ1=σ12t+λtφ12tb2t-rt+λtφ2t,λ2=σ1tσ2t+λtφ1tφ2tb1t-rt+λtφ1t.Then the optimal investment strategies of (10) and (11) can be written as follows: (45)π∗=π⌢,ifγ1≥γ2;π-,ifγ1<γ2≤γ1+σ12t+λtφ12tα˙tαt;π˘,ifγ1+σ12t+λtφ12tα˙tαt≤γ2.


## 4.1. Optimal Consumption Strategies

From the HJB equation (17), and by applying a first-order optimality condition, we derive that(18)∂U1t,c-cVx∂c=0.That is to say, c∗ satisfies(19)∂U1t,c∂c=Vx.


## 4.2. Optimal Investment Strategies

We will also deduce the optimal investment strategies from HJB function (17). Set(20)fπ=fπ1,π2=Vxπ′Bt-α˙tαtπ2-+12Vxxπ′Dtπ,f1π=f1π1,π2=Vxπ′Bt+12Vxxπ′Dtπ,f2π=f2π1,π2=Vxπ′Bt+π2α˙tαt+12Vxxπ′Dtπ.By settingB1(t)=B(t) and B2(t)=B(t)+0,α˙t/αt′, the above three equations can be written as follows:(21)fπ=fπ1,π2=Vxπ′B1t-α˙tαtπ2-+12Vxxπ′Dtπ,(22)f1π=f1π1,π2=Vxπ′B1t+12Vxxπ′Dtπ,(23)f2π=f2π1,π2=Vxπ′B2t+12Vxxπ′Dtπ,thus getting the following equation(24)fπ=fπ1,π2=f1π1,π2,ifπ2>0,f2π1,π2,ifπ2≤0.From (22), by applying the first-order unconstrained optimality condition, we obtain(25)∂f1π∂π=0⟹π⌢=-VXVXXDt-1B1t,and f1π⌢=-VX2/2VXXB1t′Dt-1B1(t).Similarly, from (23) we obtain(26)∂f2π∂π=0⟹π˘=-VXVXXDt-1B2t,and f2(π˘)=-VX2/2VXXB2t′Dt-1B2(t).Next, let us compareπ⌢ with π˘. We have(27)π⌢-π˘=VXVXXDt-10,α˙tαt′.By Merton [16], V(t,x) is concave function because the utility functions U1(·) and U2(·) are risk aversion functions. Then, Vx≥0,Vxx<0, and D(t) is positive definite. So π⌢2-π˘2≤0. We will consider three cases.(1)
π
⌢
2
>
0. For any π≤0, we have(28)f2π=Vxπ′B2t+12Vxxπ′Dtπ≤Vxπ′B1t+12Vxxπ′Dtπ=f1π≤f1π⌢.Therefore, max⁡f(π)=max⁡f1(π)=f1π⌢=-VX2/2VXXB1t′Dt-1B1(t), and(29)π∗=π⌢=-VXVXXDt-1B1t.The HJB function (17) can be written as follows:(30)Vt+Vxxrt+U1t,c∗-c∗Vx-VX22VXXB1t′Dt-1B1t=0,(31)VT,x=U2T,x.(2)
π
⌢
2
≤
0
≤
π
˘
2. The two optimization problems are(32)maxf1ππ2≥0,t∈0,T(33)maxf2ππ2≤0,t∈0,T.Solving (32), by introducing the Lagrange multiplier l, we have(34)L1π1,π2,l=Vxπ′B1t+12Vxxπ′Dtπ-l·π2,where l≥0. Then(35)∂L1π1,π2,l∂π1=VxB1t+VxxDtπ=0.Consideringπ⌢2≤0≤π˘2 and the constraint of (32), it is obtained that the optimal solution of (32) satisfies π-2=0. Substituting it into (35), we can get the following equation:(36)Vxbt-rt+λtφ1t+VXXσ12t+λtφ12tπ-1=0.Then π-1=-VX/VXX·b(t)-r(t)+λ(t)φ1(t)/σ12(t)+λ(t)φ12(t).Following the same line of argument, the solution of (33) is π-2=0,π-1=-VX/VXX·b(t)-r(t)+λ(t)φ1(t)/σ12(t)+λ(t)φ12(t).As the above discussion mentioned, it is shown that(37)π∗=π-=-VXVXX·b1t-rt+λtφ1tσ12t+λtφ12t0.Therefore,max⁡f(π)=f1(π-)=-(VX)2/2VXX·bt-rt+λtφ1t2/σ12(t)+λ(t)φ12(t).SettingH(t)2=b1t-rt+λtφ1t2/σ12(t)+λ(t)φ12(t), the above equation can be written as follows:(38)max⁡fπ=f1π-=-VX22VXXHt2.The HJB function (17) can be written as follows:(39)Vt+Vxxrt+U1t,c∗-c∗Vx-VX22VXXHt2=0,and V(T,x)=U2(T,x).(3)
π
˘
2
<
0. For any π≤0, we have(40)f1π=Vxπ′B1t+12Vxxπ′Dtπ≤Vxπ′B2t+12Vxxπ′Dtπ=f2π≤f2π˘.Therefore, max⁡f(π)=max⁡f2(π)=f1(π˘)=-VX2/2VXXB2t′Dt-1B2(t), and(41)π∗=π˘=-VXVXXDt-1B2t.The HJB function (17) can be written as follows:(42)Vt+Vxxrt+U1t,c∗-c∗Vx-VX22VXXB2t′Dt-1B2t=0,and V(T,x)=U2(T,x).Next, we will discuss the conditions which lead to the above classifications.(43)π⌢=-VxVxxDt-1B1t=-VxVxxσtσt′+λtφtφt′-1bt-rt12+λtφt=-VxVxxσ12t+λtφ12tσ1tσ2t+λtφ1tφ2tσ1tσ2t+λtφ1tφ2tσ22t+λtφ22t-1·b1t-rt+λtφ1tb2t-rt+λtφ2t=-VxVxxι1-ι2λtσ1tφ2t-λtσ2tφ1t2γ1-γ2λtσ1tφ2t-λtσ2tφ1t2,π˘=-VxVxxDt-1B2t=-VxVxxσtσt′+λtφtφt′-1bt-rt12+λtφt+0,α˙tαt′=-VxVxxσ12t+λtφ12tσ1tσ2t+λtφ1tφ2tσ1tσ2t+λtφ1tφ2tσ22t+λtφ22t-1·b1t-rt+λtφ1tb2t-rt+λtφ2t+α˙tαt=-VxVxxι1-ι2+σ1tσ2t+λtφ1tφ2tα˙t/αtλtσ1tφ2t-σ2tφ1t2γ1+σ12t+λtφ12tα˙t/αt-γ2λtσ1tφ2t-σ2tφ1t2,where(44)ι1=σ12t+λtφ22tb1t-rt+λtφ1t,ι2=σ1tσ2t+λtφ1tφ2tb2t-rt+λtφ2t,λ1=σ12t+λtφ12tb2t-rt+λtφ2t,λ2=σ1tσ2t+λtφ1tφ2tb1t-rt+λtφ1t.Then the optimal investment strategies of (10) and (11) can be written as follows: (45)π∗=π⌢,ifγ1≥γ2;π-,ifγ1<γ2≤γ1+σ12t+λtφ12tα˙tαt;π˘,ifγ1+σ12t+λtφ12tα˙tαt≤γ2.


## 5. HARA Utility Function

Generally speaking, the analytical solutions to (30), (39), and (42) can not be obtained easily. However, the analytical solutions to the above HJB equations can be derived when the utility function is a HARA function [16]. Moreover, the optimal investment-consumption strategies can be given. Therefore, we assume that both utility functions of (11) are HARA utility functions; that is,(46)U1t,y=U2t,y=e-ρtyκκ,where 0<κ<1,ρ>0 is a discounted factor.We can get the optimal consumption strategies from (19):(47)c∗=eρtVx1/κ-1.Then(48)maxcU1t,c-cVx=U1t,c∗-c∗Vx=1κe-ρteρtVxκ/κ-1-eρtVx1/κ-1Vx=1-κκeρt/κ-1Vxκ/κ-1.Next, we will derive the optimal investment strategies.(1)
π
⌢
≥
0. The HJB function (30) can be written as follows:(49)Vt+Vxxrt+1-κκeρt/κ-1Vxκ/κ-1-VX22VXXB1t′Dt-1B1t=0,and V(T,x)=e-ρTxκ/κ.By Merton [16], letting V(t,x)=β1(t)xκ/κ and substituting it into (49), we can obtain(50)β1t+κrt+121-κB1t′Dt-1B1tβ1t+1-κeρt/κ-1β1tκ/κ-1=0,β1T=e-ρT.Setting(51)m1t=κrt+121-κB1t′Dt-1B1t,nt=1-κeρt/κ-1and substituting them into (50), we can obtain(52)β1t+m1tβ1t+ntβ1tκ/κ-1=0,β1T=e-ρT.Settingη1(t)=β1(t)e-∫tTm1(s)ds and substituting it into (52), we can get(53)η1t+ntη1tκ/κ-1e1/κ-1∫tTm1sds=0,η1T=e-ρT.By solving it, we can obtain(54)η1t=eρT/κ-1+11-κ∫tTnse1/κ-1∫sTm1vdvds1-κ.Then (55)β1t=e∫tTm1sdseρT/κ-1+11-κ∫tTnse1/κ-1∫sTm1vdvds1-κ.SubstitutingV(t,x)=β1(t)xκ/κ into (29) and (47), we can get the optimal investment-consumption strategies as follows:(56)π∗=π⌢=-Dt-1B1tκ-1x,c∗=eρt/κ-1β1t1/κ-1x.(2)
π
⌢
≤
0
≤
π
˘. The HJB function (39) can be written as follows:(57)Vt+Vxxrt+1-κκeρt/κ-1Vxκ/κ-1-VX22VXXHt2=0,andV(T,x)=e-ρTxκ/κ.Similar to (1), we can obtain by letting V(t,x)=β1(t)xκ/κ and substituting it into (57)(58)β1t+κrt+121-κHt2β1t+1-κeρt/κ-1β1tκ/κ-1=0,β1T=eρT.Setting(59)m2t=κrt+121-κHt2and substituting it into (58), we can obtain(60)β2t+m2tβ2t+ntβ2tκ/κ-1=0,β2T=e-ρT.Settingη2(t)=β2(t)e-∫tTm2(s)ds and substituting it into (52), we can get(61)η2t+ntη2tκ/κ-1e1/κ-1∫tTm2sds=0,η2T=e-ρT.By solving it, we can obtain(62)η2t=eρT/κ-1+11-κ∫tTnse1/κ-1∫sTm2vdvds1-κ.Then (63)β2t=e∫tTm2sdseρT/κ-1+11-κ∫tTnse1/κ-1∫sTm2vdvds1-κ.By substitutingV(t,x)=β2(t)xκ/κ into (37) and (47), we can get the optimal investment-consumption strategies as follows:(64)π∗=π⌢=-b1t-rt+λtφ1tσ12t+λtφ12t·xκ-10,c∗=eρt/κ-1β2t1/κ-1x.(3)
π
˘
≤
0. The HJB function (42) can be written as follows:(65)Vt+Vxxrt+1-κκeρt/κ-1Vxκ/κ-1-VX22VXXB2t′Dt-1B2t=0,and V(T,x)=e-ρTxκ/κ.Similar to (1), we can obtain by letting V(t,x)=β3(t)xκ/κ and substituting it into (64)(66)β3t+κrt+121-κB2t′Dt-1B2tβ3t+1-κeρt/κ-1β3tκ/κ-1=0,β3T=e-ρT.Setting(67)m3t=κrt+121-κB2t′Dt-1B2tand substituting them into (50), we can obtain(68)β3t+m3tβ3t+ntβ3tκ/κ-1=0,β3T=e-ρT.Settingη3(t)=β3(t)e-∫tTm3(s)ds and substituting it into (68), we can get(69)η3t+ntη3tκ/κ-1e1/κ-1∫tTm3sds=0,η3T=e-ρT.By solving it, we can obtain(70)η3t=eρT/κ-1+11-κ∫tTnse1/κ-1∫sTm3vdvds1-κ.Then(71)β3t=e∫tTm3sdseρT/κ-1+11-κ∫tTnse1/κ-1∫sTm3vdvds1-κ.SubstitutingV(t,x)=β3(t)xκ/κ into (41) and (47), we can get the optimal investment-consumption strategies as follows:(72)π∗=π˘=-Dt-1B2tκ-1x,c∗=eρt/κ-1β3t1/κ-1x.As the above discussion mentioned, we can get the optimal investment-consumption strategies of (11) with HARA utility function as pointed by (46) (73)π∗=-Dt-1B1tκ-1x,ifγ1≥γ2;-b1t-rt+λtφ1tσ12t+λtφ12t·xκ-10,ifγ1<γ2≤γ1+σ12t+λtφ12tα˙tαt;-Dt-1B2tκ-1x,ifγ1+σ12t+λtφ12tα˙tαt≤γ2,(74)c∗=eρt/κ-1β1t1/κ-1x,ifγ1≥γ2;eρt/κ-1β2t1/κ-1x,ifγ1<γ2≤γ1+σ12t+λtφ12tα˙tαt;eρt/κ-1β3t1/κ-1x,ifγ1+σ12t+λtφ12tα˙tαt≤γ2.From (73) and (74), it can be seen that the results are closed-form solutions. We can avoid solving the partial differential equation difficulty and may obtain the optimal investment-consumption policies corresponding capital x directly.


## 6. A Numerical Example and Discussion

In this section, we will introduce a numerical example to illustrate model (11) with HARA utility function (46). This example is only to show how to obtain the optimal investment-consumption strategies. Some parameters have been simplified to some extent. We can change the assumptions of the parameters to obtain the results which would deliver the different γ1 and γ2.(1)
It is assumed that the initial capital isx0 = 1.19× 105 RMB. We only calculate the capital range [x0,2x0] for illustrating the model. The rest of the parameters are as follows:(75)ρ=0.02,κ=0.5,r=0.025,λ=0.6,ω=0.07,θ=0.39,ζ=0.1.For simplicity, we setT=1. In a given time t=0.5, the futures price F=53 RMB and its corresponding Europe call option price is Y=4.77 RMB. We also suppose that Yt=0.13, YF=0.23, YFF=-0.0123, α=0.85, α˙=1 on this moment. In the above parameters, γ1<γ2≤γ1+(σ12(t)+λ(t)φ12(t))α˙(t)/α(t), the optimal investment-consumption strategies are illustrated as Figure 1 (1.21E+5 means 121,000, the same below). From Figure 1, we can see that we do not invest in options anymore. The investor can only allocate his/her little capital in the futures. A large number capital is invested in the risk-free bond.(2)
We change parametersα˙=0.005 and ζ=0.9 from comparing with (1). In the case γ1+(σ12(t)+λ(t)φ12(t))α˙(t)/α(t)≤γ2, the optimal investment-consumption strategies will change as shown in Figure 2. Figure 2 makes us purchase futures and sell the options.(3)
Ifζ=0.1, YFF=0.005, and ω=0.07, then γ1≥γ2 comparing with (1). The optimal investment-consumption strategies are illustrated in Figure 3, which shows how the investors could sell futures and purchase the options.Figure 1
Optimal strategies underγ1<γ2≤γ1+(σ12(t)+λ(t)φ12(t))α˙(t)/α(t).Figure 2
Optimal strategies underγ1+(σ12(t)+λ(t)φ12(t))α˙(t)/α(t)≤γ2.Figure 3
Optimal strategies underγ1≥γ2.Although this example is simple, it well explains the sense of the model. From the computed results, the optimal investment-consumption strategy is a closed-form solution, thus avoiding solving partial differential equations numerically. We remark that the numerical solution of partial differential equations can be a challenging task. The closed-form analytical solution can greatly improve the calculation speed and reduce the computational complexity. However, you need to make some assumptions of stochastic differential equation parameters if you may also find more details. The focus of this paper is the theoretical results rather than numerical calculations. Therefore, the above example is a simulation which further describes the theoretical results.Comparing with the classical Merton model, the Poisson process is added to the portfolio with Brown motion process. The results are very similar to the traditional Merton solution outwardly. However, the jump process has greatly increased the complexity of the solution process. For utility functions similar to those considered in this paper, relevance of capital between optimal strategies can be obtained after solving the optimal consumption-investment strategies from (73), (74), or the above examples. The investors can differently allocate their capital by adjusting the parameters of utility function.


## 7. Conclusions

We study the optimal investment-consumption portfolio selection of options which have a different price between sale and purchase based on a risk aversion utility function in financial markets. These pricing processes follow a Weiner process and a Poisson process. Then the corresponding HJB equation of the problem is derived and its solutions are obtained in different conditions, respectively. Finally, a special case, when the utility function is HARA, is illustrated. In addition, throughout the paper it is assumed thatα(t) is differentiable with respect to time t and monotone nondecreasing. Future work will be devoted to the collection of a large number of datasets in the real market for calibrating the various parameters of the model.



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*Source: 8734235-2017-07-02.xml*