Portfolio Optimization under Local-Stochastic Volatility: Coefficient Taylor Series Approximations & Implied Sharpe Ratio

# Portfolio Optimization under Local-Stochastic Volatility: Coefficient Taylor Series Approximations & Implied Sharpe Ratio

Matthew Lorig Department of Applied Mathematics, University of Washington, Seattle, WA, USA. Work partially supported by NSF grant DMS-0739195.    Ronnie Sircar ORFE Department, Princeton University, Princeton, NJ, USA. Work partially supported by NSF grant DMS-1211906.
July 11, 2019
###### Abstract

We study the finite horizon Merton portfolio optimization problem in a general local-stochastic volatility setting. Using model coefficient expansion techniques, we derive approximations for the both the value function and the optimal investment strategy. We also analyze the ‘implied Sharpe ratio’ and derive a series approximation for this quantity. The zeroth-order approximation of the value function and optimal investment strategy correspond to those obtained by Merton (1969) when the risky asset follows a geometric Brownian motion. The first-order correction of the value function can, for general utility functions, be expressed as a differential operator acting on the zeroth-order term. For power utility functions, higher order terms can also be computed as a differential operator acting on the zeroth-order term. We give a rigorous accuracy bound for the higher order approximations in this case in pure stochastic volatility models. A number of examples are provided in order to demonstrate numerically the accuracy of our approximations.

## 1 Introduction

The continuous time portfolio optimization problem was first studied by Merton (1969), where he considers a market that contains a riskless bond, which grows at a fixed deterministic rate, and multiple risky assets, each of which is modeled as a geometric Brownian motion with constant drift and constant volatility. In this setting, Merton obtains an explicit expression for the value function and optimal investment strategy of an investor who wishes to maximize expected utility when the utility function has certain specific forms. However, much empirical evidence suggests that volatility is stochastic and is driven by both local and auxiliary factors, and so it is natural to ask how an investor would change his investment strategy in the presence of stochastic volatility.

There have been a number of studies in this direction, a few of which, we now mention. Darius (2005) studies the finite horizon optimal investment problem in a CEV local volatility model. Chacko and Viceira (2005) examine the infinite-horizon optimal investment problem in a Heston-like stochastic volatility model. While both studies provide an explicit expression for an investor’s value function and optimal investment strategy, the results are specific to the models studied in these two papers and for power utility functions.

Approximation methods, which have been extensively used for option pricing and related problems, have been adapted for the portfolio selection problem, allowing for a wider class of volatility models and utility functions. The Merton problem for power utilities under fast mean-reverting stochastic volatility was analyzed by asymptotic methods in (Fouque et al., 2000, Chapter 10), and the related partial hedging stochastic control problem in Jonsson and Sircar (2002b, a) using asymptotic analysis for the dual problem. More recently, Fouque et al. (2012) consider a general class of multiscale stochastic volatility models and general utility functions. Here, volatility is driven by one fast-varying and one slow-varying factor. The separation of time-scales allows the authors to obtain explicit approximations for the investor’s value function and optimal control, by combining singular and regular perturbation methods on the primal problem. These methods were previously developed to obtain explicit price approximations for various financial derivatives, as described in the book Fouque et al. (2011).

Here we study the Merton problem in a general local-stochastic volatility (LSV) setting. The LSV setting encompasses both local volatility models (e.g., CEV and quadratic) and stochastic volatility models (e.g., Heston and Hull-White) as well as models that combine local and auxiliary factors of volatility (e.g., SABR and -SABR). As explicit expressions for the value function and optimal investment strategy are not available in this very general setting, we focus on obtaining approximations for these quantities. Specifically, we will obtain an approximation for the solution of a nonlinear Hamilton-Jacobi-Bellman partial differential equation (HJB PDE) by expanding the PDE coefficients in a Taylor series. The Taylor series expansion method was initially developed in Pagliarani and Pascucci (2012) to solve linear pricing PDEs under local volatility models, and is closely related to the classical parametrix method (see, for instance, Corielli et al. (2010) for applications in finance). The method was later extended in Lorig et al. (2014b) to include more general polynomial expansions and to handle multidimensional diffusions. Additionally, the technique has been applied to models with jumps; see Pagliarani et al. (2013), Lorig et al. (2014c) and Lorig et al. (2014a). We remark that the PDEs that arise in no-arbitrage pricing theory are linear, whereas the HJB PDE we consider here is fully nonlinear.

The rest of the paper proceeds as follows. In Section 2, we introduce a general class of local-stochastic volatility models, define a representative investor’s value function and write the associated HJB PDE. Section 3 presents the first order approximation and formulas for the principal LSV correction to the value function and the optimal investment strategy. Motivated by the notion of Black-Scholes implied volatility, we also develop the notion of implied Sharpe ratio, that provides a greater intuition about the resulting formulas, which are summarized in Section 3.7. We discuss higher order terms in Section 4, and show that power utilities are particularly amenable to obtaining explicit formulas for further terms in the approximation. In Section 5, we provide explicit results for power utility. In particular, we derive rigorous error bounds for the value function in a stochastic volatility setting. In Section 6, we provide two numerical examples, illustrating the accuracy and versatility of our approximation method. Section 7 concludes.

## 2 Merton Problem under Local-Stochastic Volatility

We consider a local-stochastic volatility model for a risky asset :

 dStSt =~μ(St,Yt)dt+~σ(St,Yt)dB(1)t (1) dYt =~c(St,Yt)dt+~β(St,Yt)dB(2)t,

where and are standard Brownian motions under a probability measure with correlation coefficient : . The log price process is, by Itô’s formula, described by the following :

 dXt =b(Xt,Yt)dt+σ(Xt,Yt)dB(1)t (2) dYt =c(Xt,Yt)dt+β(Xt,Yt)dB(2)t, (3)

where , and similarly from , and we have defined

 b(Xt,Yt)=μ(Xt,Yt)−12σ2(Xt,Yt).

The model coefficient functions are smooth functions of and are such that the Markovian system (2) admits a unique strong solution.

### 2.1 Utility Maximization and HJB Equation

We denote by the wealth process of an investor who invests units of currency in at time and invests units of currency in a riskless money market account. For simplicity, we assume that the risk-free rate of interest is zero, and so the wealth process satisfies

 dWt =πtStdSt=πtμ(Xt,Yt)dt+πtσ(Xt,Yt)dB(1)t. (4)

The investor acts to maximize the expected utility of portfolio value, or wealth, at a fixed finite time horizon : , where is a smooth, increasing and strictly concave utility function satisfying the “usual conditions” and .

We define the investor’s value function by

 V(t,x,y,w):=supπ∈Π\mathdsE{U(WT)∣Xt=x,Yt=y,Wt=w}, (5)

where is the set of admissible strategies , which are non-anticipating and satisfy

 \mathdsE{∫T0π2tσ2(Xt,Yt)dt}<∞. (6)

and a.s.

We assume that . The Hamilton-Jacobi-Bellman partial differential equation (HJB-PDE) associated with the stochastic control problem (5) is

 (∂∂t+A)V+maxπ∈\mathdsRAπV =0, V(T,x,y,w) =U(w), (7)

where the operators and are given by

 A =12σ2(x,y)∂2∂x2+ρσ(x,y)β(x,y)∂2∂x∂y+12β2(x,y)∂2∂y2+b(x,y)∂∂x+c(x,y)∂∂y, (8) Aπ =12π2σ2(x,y)∂2∂w2+π(σ2(x,y)∂2∂x∂w+ρσ(x,y)β(x,y)∂2∂y∂w+μ(x,y)∂∂w). (9)

We refer to the books Fleming and Soner (1993) and Pham (2009) for technical details.

The optimal strategy is given (in feedback form) by

 π∗=−(σ2(x,y)Vxw+ρσ(x,y)β(x,y)Vyw+μ(x,y)Vw)σ2(x,y)Vww, (10)

where subscripts indicate partial derivatives.

Inserting the optimal strategy into the HJB-PDE (7) yields

 (∂∂t+A)V+N(V),V(T,x,y,w)=U(w), (11)

where is a nonlinear term, which is given by

 N(V)=−(σ(x,y)Vxw+ρβ(x,y)Vyw+λ(x,y)Vw)22Vww, (12)

and we have introduced the Sharpe ratio

 λ(x,y)=μ(x,y)σ(x,y). (13)

### 2.2 Constant Parameter Merton Problem

We review and introduce notation that will be used later for the constant parameter Merton problem, that is, when and in (1) are constant, and so therefore and are constant and the stock price follows the geometric Brownian motion

 dStSt=μdt+σdB(1)t.

Then the Merton value function for the investment problem for this stock, whose constant Sharpe ratio is , is the unique smooth solution of the HJB PDE problem

 Mt−12λ2M2wMww=0,M(T,w)=U(w), (14)

on and . Smoothness of given a smooth utility function (as assumed above), as well as differentiability of in is easily established by the Legendre transform, which converts (14) into a linear constant coefficient parabolic PDE problem for the dual. Regularity results for the latter problem are standard.

It is convenient to introduce the Merton risk tolerance function

 R(t,w;λ):=−MwMww(t,w;λ), (15)

and the operator notation

 Dk:=(R(t,w;λ))k∂k∂wk,k=1,2,⋯. (16)

We recall also the Vega-Gamma relationship taken from (Fouque et al., 2012, Lemma 3.2):

###### Lemma 2.1.

The Merton value function satisfies the “Vega-Gamma” relation

 ∂M∂λ=−(T−t)λD2M, (17)

where is defined in (16).

Thus the derivative of the value function with respect to the Sharpe ratio (analogous to an option price’s derivative with respect to volatility, its Vega) is proportional to its negative “second derivative” (which is analogous to the option price’s second derivative with respect to the stock price, its Gamma). This result will be used repeatedly in deriving the implied Sharpe ratio in Section 3.5 and the approximation to the optimal portfolio in Section 3.6.

## 3 Coefficient Expansion & First Order Approximation

For general and , there is no closed form solution of (11). The goal of this section is to find series approximations for the value function

 V=V(0)+V(1)+V(2)+⋯,

and the optimal investment strategy

 π∗=π∗0+π∗1+π∗2+⋯,

using model coefficient (Taylor series) expansions. This approach is developed for the linear European option pricing problem in a general LSV setting in Lorig et al. (2014b), where explicit approximations for option prices and implied volatilities are obtained by expanding the coefficients of the underlying diffusion as a Taylor series. Note that, here the HJB-PDE (11) is fully nonlinear. Our first order approximation formulas are summarized in Section 3.7.

### 3.1 Coefficient Polynomial Expansions

We begin by fixing an point . For any function that is analytic in a neighborhood of , we define the following family of functions indexed by :

 χa(x,y):=∞∑n=0anχn(x,y), (18)

where

 χn(x,y):=n∑k=0χn−k,k⋅(x−¯x)n−k(y−¯y)k,χn−k,k:=1(n−k)!k!∂n−kx∂kyχ(¯x,¯y), (19)

and we note that is a constant. Observe that is the Taylor series of about the point . Here, is an accounting parameter that will be used to identify successive terms of our approximation.

In the PDE (11), we will replace each of the coefficient functions

 χ∈{μ,c,σ2,β2,λ2,σβ,βλ}

by , for some , and then use the series expansion (18) for . Another way of saying this is that we assume the coefficients are of the form

 χ(¯x+a(x−¯x),¯y+a(y−¯y)),

whose exact Taylor series is given by (18), and we are interested in the case when .

Consider now the following family of HJB-PDE problems

 (∂∂t+Aa)Va+Na(Va)=0,Va(T,x,y,w)=U(w), (20)

where, for , and are obtained from and in (8) and (12) by making the change

 {μ,c,σ2,β2,λ2,σβ,βλ} ↦{μa,ca,(σ2)a,(β2)a,(λ2)a,(σβ)a,(βλ)a}. (21)

The linear operator in the PDE (20) can therefore be written as

 Aa=∞∑n=0anAn,

where we define

 An :=(12σ2)n(x,y)∂2∂x2+(ρσβ)n(x,y)∂2∂x∂y+(12β2)n(x,y)∂2∂y2+bn(x,y)∂∂x+cn(x,y)∂∂y, (22)

and the expansion of the nonlinear term is a more involved computation.

We construct a series approximation for the function as a power series in :

 Va(t,x,y,w) =∞∑n=0anV(n)(t,x,y,w). (23)

Note that the functions are not constrained to be polynomials in , and in general they will not be. Our approximate solution to (11), which is the problem of interest, will then follow by setting .

### 3.2 Zeroth & First Order Approximations

We insert (23) into (20) and collect terms of like powers of . At lowest order we obtain

 (∂∂t+A0)V(0)−(σ0V(0)xw+ρβ0V(0)yw+λ0V(0)w)22V(0)ww=0,V(0)(T,x,y,w)=U(w), (24)

where the linear operator , found from (22), has constant coefficients:

 A0=12σ20∂2∂x2+ρσ0β0∂2∂x∂y+12β20∂2∂y2+b0∂∂x+c0∂∂y. (25)

As a consequence, the solution of (24) is independent of and : , and we have

 V(0)t−12λ20(V(0)w)2V(0)ww=0,V(0)(T,w)=U(w). (26)

We observe that (26) is the same as the PDE problem (14) that arises when solving the Merton problem assuming the underlying stock has a constant drift , diffusion coefficient and so constant Sharpe ratio . Therefore, we have

 V(0)(t,w)=M(t,w;λ0).

The PDE (26) can be solved either analytically (for certain utility functions ), or numerically.

Recall the definition of the risk tolerance function in (15) and the operators in (16), where now we take in those formulas the Sharpe ratio :

 R(t,w;λ0)=−V(0)wV(0)ww(t,w;λ),Dk=(R(t,w;λ0))k∂k∂wk,k=1,2,⋯. (27)

Proceeding to the order terms in (20), we obtain

 (∂∂t+A0)V(1)+12λ20D2V(1)+λ20D1V(1)+ρβ0λ0D1∂∂yV(1)+μ0D1∂∂xV(1)=−(12λ2)1D1V(0). (28)

We can re-write this more compactly as

 (∂∂t+A0+B0)V(1)+H1=0,V(1)(T,x,y,w)=0, (29)

where the linear operator and the source term are given by

 B0 =12λ20D2+λ20D1+ρβ0λ0D1∂∂y+μ0D1∂∂x, (30) H1(t,x,y,w) =(12λ2)1(x,y)D1V(0)(t,w). (31)

### 3.3 Transformation to Constant Coefficient PDEs

Next, we apply a change of variable such that can be found by solving a linear PDE with constant coefficients. We begin with the following lemma.

###### Lemma 3.1.

Let be the solution of (26) and let and be as given in (22) and (31), respectively. Then satisfies the following PDE problem

 (∂∂t+A0+B0)V(0)=0,V(0)(T,w)=U(w). (32)
###### Proof.

This follows directly from observing that the nonlinear term in (26) can be written

 (V(0)w)2V(0)ww=(−V(0)wV(0)ww)2V(0)ww=D2V(0), or (V(0)w)2V(0)ww=−(−V(0)wV(0)ww)V(0)w=−D1V(0). (33)

Therefore, from (26), we have

 (∂∂t+12λ20D2+λ20D1)V(0)=0, (34)

and (32) follows from the fact that does not depend on , while and the last two terms in the expression (30) for take derivatives in those variables. ∎

Next, it will be helpful to introduce the following change of variables.

###### Definition 3.2.

We define the co-ordinate by the transformation

 z(t,w) =−logV(0)w(t,w)+12λ20(T−t). (35)

We have the following change of variables formula, as used also in (Fouque et al., 2012, Section 2.3.2).

###### Lemma 3.3.

For a smooth function , define by

 ˆV(t,x,y,w)=q(t,x,y,z(t,w)).

Then we have

 (∂∂t+A0+B0)ˆV=(∂∂t+A0+C0)q, (36)

where the operator is given by

 C0 =12λ20∂2∂z2+ρβ0λ0∂2∂y∂z+μ0∂2∂x∂z. (37)
###### Proof.

We shall use the shorthand . From (35), we have that , and so, differentiating (43), we find

 ˆVt=qt−⎛⎝V(0)twV(0)w+12λ20⎞⎠qz,D1ˆV=qz,D2ˆV=qzz−R(0)wqz. (38)

Then, using the first expression in (33) to write the PDE (26) for as , and differentiating this with respect to gives

 V(0)tw=12λ20(R(0))2V(0)www+λ20R(0)R(0)wV(0)ww.

But from , we have , and so

 V(0)tw=12λ20(R(0)w+1)V(0)w−λ20R(0)wV(0)w,

which gives that

 V(0)twV(0)w=−12λ20(R(0)w−1). (39)

Therefore, we have

 (∂∂t+12λ20D2+λ20D1)ˆV=qt+(12λ20(R(0)w−1)−12λ20)qz+12λ20(qzz−R(0)wqz)+λ20qz,

which establishes that

 (∂∂t+12λ20D2+λ20D1)ˆV=(∂∂t+12λ20∂2∂z2)q. (40)

More directly, we have

 (ρβ0λ0D1∂∂y+μ0D1∂∂x)ˆV=(ρβ0λ0∂2∂y∂z+μ0∂2∂x∂z)q,

which, combined with (40), leads to (36). ∎

We define by . Then the PDE (32) for is transformed to the (constant coefficient) backward heat equation for :

 (∂∂t+12λ20∂2∂z2)q(0)=0,q(0)(T,z)=U((U′)−1(e−z)), (41)

but of course the transformation (35) depends on the solution itself. Again, as does not depend on , while and the last two terms in the expression (37) for take derivatives in those variables, we can write

 (∂∂t+A0+C0)q(0)=0. (42)

Now let be defined from by

 V(1)(t,x,y,w) =q(1)(t,x,y,z(t,w)), (43)

using the transformation (35). Then, using Lemma 3.3, we see that the PDE (29) for , which has -dependent coefficients through the dependence of in (30) on , is transformed to the constant coefficient equation for :

 (∂∂t+A0+C0)q(1)+Q1=0,q(1)(T,x,y,z)=0. (44)

The source term is found from , where, from (31), we have

 Q1(t,x,y,z)=(12λ2)1(x,y)q(0)z. (45)

### 3.4 Explicit expression for V(1)

In this section, we will show that , solution of (29), can be written as a differential operator acting on . First, we look at the PDE problem

 Hq+Q=0,q(T,x,y,z)=0, (46)

where is the constant coefficient linear operator

 H=∂∂t+A0+C0. (47)

We also suppose that the source term is of the following special form:

 Q(t,x,y,z)=∑k,l,n(T−t)n(x−¯x)k(y−¯y)lv(t,x,y,z), (48)

where the sum is finite and is a solution of the homogeneous equation

 Hv=0. (49)

We first define (the commutator) by

 H((x−¯x)v)=(x−¯x)Hv+LXv, (50)

and so a direct calculation using the expressions (25) and (37) for and respectively shows that

 LX=(μ0−12σ20)I+σ20∂∂x+ρσ0β0∂∂y+μ0∂∂z, (51)

where is the identity operator. Similarly defining (the commutator) by

 H((y−¯y)v)=(y−¯y)Hv+LYv,

 LY =c0I+β20∂∂y+ρσ0β0∂∂x+ρβ0λ0∂∂z, (52)

We next introduce the following operators indexed by :

 MX(s)=(x−¯x)I+(s−t)LX,MY(s)=(y−¯y)I+(s−t)LY, (53)

Then we have the following result by construction of these operators.

###### Lemma 3.4.

Recall that solves the homogeneous equation (49). Then

 HMkX(s)MlY(s)v=0, (54)

for integers .

###### Proof.

We first calculate

 HMXv=MXHv+LXv−LXv+(s−t)HLXv=(s−t)HLXv,

where we have used (50). But since and are constant coefficient operators which commute, we have using (49). Therefore, given a solution of the homogeneous equation, also solves the homogeneous equation, namely . Iterating we have that for integers . Similarly for integers , and so the result (54) follows. ∎

From this we can exploit the special structure of the source to obtain the following formula.

###### Proposition 3.5.

The solution to (46) where the source is of the form (48) is given by

 q(t,x,y,z)=∑k,l,n∫Tt(T−s)nMkX(s)MlY(s)v(t,x,y,z)ds. (55)
###### Proof.

Due to the linearity of the problem, it suffices to consider a single term of the polynomial:

 Q(t,x,y,z)=(T−t)n(x−¯x)k(y−¯y)lv(t,x,y,z).

Then, we check that the solution is given by

 q(t,x,y,z)=∫Tt(T−s)nMkX(s)MlY(s)v(t,x,y,z)ds

by computing

 Hq =−(T−t)nMkX(t)MlY(t)v(t,x,y,z)+∫Tt(T−s)nHMkX(s)MlY(s)v(t,x,y,z)ds (56) =−(T−t)n(x−¯x)k(y−¯y)lv(t,x,y,z) (57) =−Q, (58)

using Lemma 3.4 for the second term. The formula (55) in the general polynomial case follows, and clearly the zero terminal condition is satisfied by (55). ∎

We can now solve for the first correction in the series expansion.

###### Proposition 3.6.

The solution to (44) is given by

 q(1)(t,x,y,z)=(T−t)λ0A(t,x,y)q(0)z(t,z)+12(T−t)2λ0Bq(0)zz(t,z), (59)

where

 A(t,x,y) =λ1,0[(x−¯x)+12(T−t)(μ0−12σ20)]+λ0,1[(y−¯y)+12(T−t)c0], B =λ1,0μ0+λ0,1ρβ0λ0. (60)
###### Proof.

We observe that since satisfies the homogeneous PDE from (42), so does , which follows from differentiating the constant coefficient PDE for . Then applying Proposition 3.5 with , and and substituting the definitions (53) for and leads to

 q(1)(t,x,y,z) =[(12λ2)1,0((T−t)(x−¯x)+12(T−t)2LX) +(12λ2)0,1((T−t)(y−¯y)+12(T−t)2LY)]q(0)z(t,z).

Finally, substituting for and from (51) and (52) and using that does not depend on leads to (59). ∎

In the original variables, this leads to

 V(1)(t,x,y,w)=(T−t)λ0A(t,x,y)D1V(0)(t,w)+12(T−t)2λ0BD21V(0)(t,w). (61)

### 3.5 Implied Sharpe Ratio

In an analogy to implied volatility, for a fixed maturity and utility function , one can define the Merton implied Sharpe ratio111The authors thank Jean-Pierre Fouque for a number of fruitful discussions, from which the concept of the Merton implied Sharpe ratio arose. corresponding to value function of Section 2.2 as the unique positive solution of

 Va(t,x,y,w) =M(t,w;Λa). (62)

The existence and uniqueness of the implied Sharpe ratio follows from the fact that (i) the function satisfies , since an investor with initial wealth can always obtain a terminal utility by investing all of his money in the riskless bank account, and (ii) the function is strictly increasing in . Since a higher implied Sharpe ratio is indicative of a better investment opportunity, we are interested to know how local stochastic volatility model parameters affect the implied Sharpe ratio.

Using our first order approximation