An expansion in the model space in the context of utility maximization

Abstract

In the framework of an incomplete financial market where the stock price dynamics are modeled by a continuous semimartingale (not necessarily Markovian), an explicit second-order expansion formula for the power investor’s value function—seen as a function of the underlying market price of risk process—is provided. This allows us to provide first-order approximations of the optimal primal and dual controls. Two specific calibrated numerical examples illustrating the accuracy of the method are also given.

Appendices

Appendix A: Details on the Kim–Omberg model

The following result summarizes the main properties in [23].

Theorem A.1

(Kim and Omberg 1996)

Let the market price of risk process be defined by (5.3), $M:=B$, and let $p<0$. Then there exist continuously differentiable functions $b,c:[0,\infty)\to{\mathbb{R}}$ such that for $t\in[0,T)$, we have

$$ \textstyle\begin{array}{rlrl} b'(t) &= -\alpha_{4}\, b(t)- \alpha_{1}\, c(t) + \alpha_{2}\, b(t) c(t), \qquad & b(T)&=0, \\ c'(t) &= q - 2 \alpha_{4}\, c(t) + \alpha_{2}\, c^{2}(t), & c(T)&=0, \end{array} $$

where $\alpha_{1}:=\theta\kappa$, $\alpha_{2} := (1+q) \beta^{2}+ \gamma^{2}$, $\alpha_{3} := \beta^{2}+\gamma^{2}$ and $\alpha_{4} := q \beta- \kappa$. Furthermore, the primal optimizer is given by

$$\begin{aligned} \hat{\pi}^{\mathit{KO}}_{t} = \frac{b(t)\beta+ (c(t)\beta-1) \lambda^{\mathit{KO}}_{t}}{p-1},\quad t\in[0,T]. \end{aligned}$$

For $p<0$, the above Riccati equation describing $c$ has a “normal non-exploding solution” as defined in the Appendix of [23]. Therefore, all three functions $a$, $b$ and $c$ are bounded on any finite subinterval $[0,T]$ of $[0,\infty)$.

Lemma A.2

Let $(\lambda,\lambda')$ be defined by (5.4) and (5.5) and let $p<0$. For the basis model $\lambda$, the primal and dual optimizers are given by

$$\begin{aligned} \hat{\pi}^{(0)}_{t} = \frac{\lambda_{t}}{1-p}, \qquad \hat{\nu}^{(0)}_{t} = \gamma\big(b(t)+c(t)\lambda_{t}\big),\quad \qquad t\in[0,T]. \end{aligned}$$

Furthermore, the processes $(\gamma^{B},\gamma^{W})$ appearing in the representation (3.6) of $\varPhi$ are given by (5.6) and (5.7), where the functions $C_{j}$ satisfy the ODEs

$$ \renewcommand{\arraystretch}{1.3} \textstyle\begin{array}{rlrl} -C_{1}'(t)&= \tilde{b}(t) \, C_{4}(t)+ \gamma^{2} \, C_{5}(t), \qquad & C_{1}(T)&=0, \\ -C_{2}'(t)&= \tilde{b}(t) \, C_{6}(t) - \kappa\, C_{2}(t),& C_{2}(T)&=0, \\ -C_{4}'(t) &= q\, C_{2}(t) - \tilde{c}(t) \, C_{4}(t) + 2\tilde{b}(t) \, C_{5}(t), \qquad & C_{4}(T)&=0, \\ -C_{5}'(t) &= q\, C_{6}(t) -2 \tilde{c}(t) \, C_{5}(t),& C_{5}(T)&=0, \\ -C'_{6}(t) &= - \big(\kappa+\tilde{c}(t)\big) \, C_{6}(t) -1,& C_{6}(T)&=0 \end{array} $$

on $[0,T)$, with $(a,b,c)$ as in Theorem  A.1 with $\beta:=0$, $\tilde{b}(t):=\kappa\theta-\gamma^{2} b(t)$ and $\tilde{c}(t) := \kappa+\gamma^{2} c(t)$. Furthermore, for the measure $\tilde{\mathbb{P}}^{(0)}$ defined by (2.6) and for all $T>0$, we have

$$\begin{aligned} \Delta^{(0)} &:= {\mathbb{E}}^{\tilde{\mathbb{P}}^{(0)}}\left[ \int _{0}^{T} \lambda'_{s}\hat{\pi}_{s}^{(0)} ds\right] = -\frac{1}{1-p} \big(C_{1}(0) + C_{4}(0) \lambda_{0} +C_{5}(0) \lambda_{0}^{2}\big). \end{aligned}$$

Proof

The first part follows from Theorem A.1 applied to the case $\beta:=0$. To find the representation (3.6), we define the function

$$f(t,x,\lambda):= \frac{x^{p}}{p}e^{-a(t) - b(t)\lambda- \frac{1}{2} c(t)\lambda^{2}},\quad t\in[0,T], x>0, \lambda\in{\mathbb{R}}, $$

where the functions $(b,c)$ are as in Theorem A.1 and $a$ is given by

$$ a'(t) = -\alpha_{1}\, b(t) - \frac{1}{2}\alpha_{3}\, c(t) + \frac{1}{2}\alpha_{2}\, b^{2}(t), \qquad a(T)=0. $$

The martingale properties of $(f(t,\hat{X}^{(0)}_{t},\lambda_{t}))$ and $(\hat{X}^{(0)}_{t}\hat{Y}^{(0)}_{t})$ as well as the proportionality property $(\hat{X}^{(0)}_{T})^{p}\propto\hat{X}^{(0)}_{T} \hat{Y} ^{(0)}_{T}$ produce

$$pf(t, \hat{X}^{(0)}_{t},\lambda_{t})=p{\mathbb{E}}[f(T, \hat{X}^{(0)} _{T},\lambda_{T})|{\mathcal{F}}_{t}]\propto{\mathbb{E}}[ \hat{Y}^{(0)} _{T}\hat{X}^{(0)}_{T}|{\mathcal{F}}_{t}] = \hat{X}^{(0)}_{t}\hat{Y} ^{(0)}_{t}. $$

By computing the dynamics of the left-hand side, we see from Girsanov’s theorem that the two processes (see (4.13))

$$\begin{aligned} \begin{aligned} dB^{\tilde{\mathbb{P}}^{(0)}}_{t} &= - q\lambda_{t}dt + dB_{t}, \\ \quad dW^{\tilde{\mathbb{P}}^{(0)}}_{t} &= \big(b(t)+c(t)\lambda_{t} \big)\gamma dt + dW_{t}, \end{aligned} \end{aligned}$$

are independent Brownian motions under $\tilde{\mathbb{P}}^{(0)}$. These dynamics and Itô’s lemma ensure that

$$\begin{aligned} N_{t} := \int_{0}^{t} &\lambda'_{s}\lambda_{s} ds -C_{1}(t) -C_{2}(t) \lambda'_{t}- C_{4}(t) \lambda_{t} -C_{5}(t) \lambda_{t}^{2}-C_{6}(t) \lambda_{t}\lambda'_{t} \end{aligned}$$

is a $\tilde{\mathbb{P}}^{(0)}$-local martingale. Because the processes $(\lambda,\lambda')$ remain Ornstein–Uhlenbeck processes under $\tilde{\mathbb{P}}^{(0)}$ and the functions $C_{1},\dots,\! C_{6}$ are bounded, $N$ is indeed a $\tilde{\mathbb{P}}^{(0)}$-martingale. Furthermore, thanks to the zero terminal conditions imposed on $C_{1},\dots,C_{6}$, we see that

$$\begin{aligned} \varPhi=\frac{1}{1-p}\int_{0}^{T} \!\! \lambda_{t} \lambda'_{t}dt &= \frac{1}{1-p}N_{T} = \frac{1}{1-p}N _{0} +\int_{0}^{T} \!\! \gamma^{B}_{t} dB^{\tilde{{\mathbb{P}}}^{(0)}}_{t} +\int_{0}^{T} \! \! \gamma^{W}_{t} dW^{\tilde{{\mathbb{P}}}^{(0)}}_{t} \end{aligned}$$

for $(\gamma^{B},\gamma^{W})$ defined by (5.6) and (5.7). □

Appendix B: Details on the extended affine models

The following result is from [25].

Theorem B.1

(Kraft 2005)

Let $p<0$ and consider the model (5.8) with $\lambda_{t}:=1$. The primal optimal control is $\hat{\pi}^{(0)}_{t} = \frac{f(t)-1}{p-1}$, the dual optimal control corresponding to $W$ is $\hat{\nu}^{(0)}_{t} =f(t) \sqrt{F_{t}}$, and the deterministic function $f$ is given by

$$\begin{aligned} f'(t) &= \frac{-f(t)(f(t)(\beta^{2}+\gamma^{2})+2\kappa)+p(f(t)^{2} \gamma^{2}-1+2f(t)(\beta+\kappa))}{2 (p-1)}, \\ f(T) &=0. \end{aligned}$$

For the model (5.10) with $\lambda_{t}:=\lambda_{21}$, the primal optimal control is $\hat{\pi}^{(0)}_{t} = \frac{\lambda_{21}}{1-p}$, the dual optimal controls corresponding to $(W^{(1)},W^{(3)})$ are

$$\hat{\nu}^{(0)}_{t} =\Big(f(t) \sqrt{F^{(1)}_{t}},0\Big), $$

and the deterministic function $f$ is given by

$$ f'(t)=\frac{1}{2} \bigg(-2 b_{11} f(t) + f(t)^{2} + \frac{\lambda_{21} ^{2} p}{1 - p}\bigg), \qquad f(T)=0. $$

We then turn to the representation (3.6) of $\varPhi:= \int _{0}^{T} \hat{\pi}^{(0)}_{t} \lambda_{t}' \sigma_{t}^{2} dt$. For the model (5.8), this is trivial because $\int_{0}^{T} \hat{\pi}^{(0)}_{t} dt$ is deterministic (implying that $\gamma^{B}= \gamma^{W}=0$). The next lemma provides the representation for the model in (5.11).

Lemma B.2

Let $p<0$, let $(\lambda,\lambda')$ be defined by (5.10) and (5.11), and define $\hat{\pi} ^{(0)}_{t} := \frac{\lambda_{21}}{1-p}$. The integrands $(\gamma^{B}, \gamma^{W})$ appearing in the representation (3.6) of

$$\varPhi:= \int_{0}^{T} \hat{\pi}^{(0)}_{t} (\lambda_{20} - \lambda_{21} +\lambda_{22}F_{t}^{(2)} + \lambda_{23}F_{t}^{(3)})dt $$

are given by

$$\begin{aligned} \gamma^{B}_{t} = -C_{2}(t),\quad\gamma^{W}_{t} = -\Big(C_{1}(t) \sqrt{F ^{(1)}_{t}},C_{3}(t) \sqrt{1+F^{(1)}_{t}}\Big), \end{aligned}$$

where the functions $C_{j}$ satisfy on $[0,T]$ the ODEs

$$\begin{aligned} C_{1}'(t) &=-b_{11} C_{1} - b_{31} C_{3}(t)+ C_{1}(t) f(t) - C_{2}(t)(b _{21} -\lambda_{21} + \hat{\pi}^{(0)}_{t}), \\ C_{1}(T) &=0, \\ C_{2}'(t) &= -b_{22} C_{2}(t) - b_{32} C_{3}(t) + \lambda_{22} \hat{\pi}^{(0)}_{t}, \\ C_{2}(T) &=0, \\ C_{3}'(t) &= -b_{23} C_{2}(t) - b_{33} C_{3}(t) + \lambda_{23} \hat{\pi}^{(0)}_{t}, \\ C_{3}(T) &=0. \end{aligned}$$

Proof

This is similar to the proof of Lemma A.2. Because the functions $C_{j}$ satisfy the above ODEs, we have that

$$\begin{aligned} N_{t} &:= \int_{0}^{t} \hat{\pi}^{(0)}_{u} (\lambda_{20} - \lambda _{21} +\lambda_{22}F_{u}^{(2)} + \lambda_{23}F_{u}^{(3)})du \\ & \phantom{=::} - C_{0}(t) - C_{1}(t)F^{(1)}_{t}- C_{2}(t)F^{(2)}_{t}- C_{3}(t)F^{(3)} _{t} \end{aligned}$$

is a $\tilde{{\mathbb{P}}}^{(0)}$-martingale (here we use the $\tilde{\mathbb{P}}^{(0)}$-Brownian motions defined in (4.13)). The function $C_{0}$ is given by

$$\begin{aligned} C_{0}'(t) &= -a_{1} C_{1}(t) - a_{2} C_{2}(t) - a_{3} C_{3}(t) + C_{2}(t) \lambda_{21} - \big(C_{2}(t) - \lambda_{20} + \lambda_{21}\big) \hat{\pi}^{(0)}_{t} \end{aligned}$$

with $C_{0}(T)=0$. Then we have

$$\begin{aligned} {\mathbb{E}}^{\tilde{{\mathbb{P}}}^{(0)}}[\varPhi|{\mathcal{F}}_{t}] = {\mathbb{E}}^{\tilde{{\mathbb{P}}}^{(0)}}[N_{T}|{\mathcal{F}}_{t}] =N _{t}. \end{aligned}$$

Hence, we find the integrands appearing in the representation (3.6) of $\varPhi$ to be as in the statement. □

Finally, we verify the integrability conditions (3.7) and (3.8) in Theorem 3.4, provided that $T>0$ is small. We can define the second-order optimal controls $(\tilde{\pi}, \tilde{\nu})$ by (3.12) and (3.13). We start by considering the model (5.8). Theorem 3.1 in [17] states that the joint Laplace transform of

$$\varLambda:=\int_{0}^{T} \frac{1}{F_{s}}\, ds,\qquad Q:=\int_{0}^{T} F _{s}\, ds $$

is finite for all $T>0$ in some neighborhood of 0 as soon as the strict Feller condition $2\kappa\theta> 1$ holds. This implies that both $\varLambda$ and $Q$ have some finite exponential positive moments (and consequently, $\varLambda$ and $Q$ have all moments). Because the integrand $\gamma^{B}$ appearing in the representation (3.6) of $\varPhi$ is zero in this model, the integrand appearing inside the exponential in (3.8) is deterministic. Consequently, (3.8) holds for all $\varepsilon>0$.

For the model (5.11), we have the estimate

$$\int_{0}^{T} (\lambda'_{t})^{2} (1+F^{(1)}_{t}) dt\le\int_{0}^{T} ( \lambda_{20} - \lambda_{21} +\lambda_{22}F_{t}^{(2)} + \lambda_{23}F _{t}^{(3)})^{2}dt, $$

which shows that the moment requirements in (3.7) hold. Furthermore, by inserting $\lambda'$ defined in (5.11) and $\sigma^{2}_{t} = 1+F^{(1)}_{t}$ into the exponent in (3.8), we see that this exponent is affine in $F^{(1)}$, $F^{(2)}$ and $F^{(3)}$ (with time-dependent coefficients). Consequently, the expectation in (3.8) is finite for small positive values of $\varepsilon$.