Optimal Bitcoin trading with inverse futures

Abstract

We consider an optimal trading problem for an investor who trades Bitcoin spot and Bitcoin inverse futures, plus a risk-free asset. The investor seeks an optimal strategy to maximize her expected utility of terminal wealth. We obtain explicit solutions to the investor’s optimal strategies under both exponential and power utility functions. Empirical studies confirm that optimal strategies perform well in terms of Sharpe ratio and Sortino ratio and beat the long-only strategy in Bitcoin spot.

Appendices

Appendix A: information on Bitcoin and Bitcoin futures

See Tables 3, 4, 5 and 6.

Table 3 Ten largest cryptocurrencies by market capitalization

Table 4 Facts of standard Bitcoin futures contracts at CBOE and CME

Table 5 Facts of Bitcoin inverse futures contracts at BitMEX

Table 6 Top 5 exchanges by 24 h Bitcoin futures trading volumes

Appendix B: Technical derivations of methodology

1.1 Appendix B.1: Technical derivations in the HJB method

By a measurable selection argument, we show that the value function V to Problem (3.1) satisfies the dynamic programming principle (DPP), also called the Bellman’s principle, which reads as

$$\begin{aligned} V(t,x) = \sup _{ u} \; {\mathbb {E}}_{t,x} \big [ V(t+\tau , X^*_{t+\tau })) \big ], \end{aligned}$$

where $X^*$ is the optimal wealth process and $\tau$ is any stopping time over [t, T]. By letting $\tau \rightarrow t$ and noting $V \in \mathrm {C}^{1,2}$, we obtain the HJB equation in (3.2).

Proof

(Proof of Theorem 1) Let us take an arbitrary admissible control $u \in {\mathcal {A}}(t,x)$. By applying Itô’s formula to $v(t, X_t^u)$ (noting $v \in \mathrm {C}^{1,2}$) and taking conditional expectation, we obtain

$$\begin{aligned} {\mathbb {E}}_{t,x}\big [v(T, X^u_T) \big ] = v(t,x) + {\mathbb {E}}_{t,x}\left[ \int _t^T \, \left( v_t(s, X^u_s) + {\mathcal {L}}^u \, v(s, X^u_s) \right) \, \mathrm {d}s \right] . \end{aligned}$$

Using the HJB Eq. (3.2) and the boundary condition, we get

$$\begin{aligned} {\mathbb {E}}_{t,x}\big [U(X_T^u )\big ] = {\mathbb {E}}_{t,x}\big [v(T, X^u_T) \big ] \le v(t,x), \quad \forall \, u \in {\mathcal {A}}(t,x). \end{aligned}$$

If $u^*$ solves the optimization problem in (3.3) and $u^* \in {\mathcal {A}}(t,x)$, then the above inequality becomes an equality for $u^*$. The proof is now complete. $\square$

Suppose there exists a classical solution V to Problem (3.1), and V is concave w.r.t. the argument x. By Theorem 1, we solve the optimization problem in (3.3) and obtain (the candidate of) an optimal strategy $u^*=(\theta ^*, \delta ^*)$ by

$$\begin{aligned} \theta ^*(t,x)&= - \dfrac{\mu _1 \nu _2 - ( \mu _2 - \nu _2) \nu _3}{\nu _1 \nu _2 - \nu _3^2} \, \frac{V_x(t,x)}{ V_{xx}(t,x)} = - {\mathcal {C}}_1 \, \frac{V_x(t,x)}{ V_{xx}(t,x)} , \end{aligned}$$

(B.1)

$$\begin{aligned} \delta ^*(t,x)&= - \dfrac{ (\mu _2 - \nu _2) \nu _1 - \mu _1 \nu _3 }{\nu _1 \nu _2 - \nu _3^2} \, \frac{V_x(t,x)}{ V_{xx}(t,x)} = - {\mathcal {C}}_2 \, \frac{V_x(t,x)}{ V_{xx}(t,x)}, \end{aligned}$$

(B.2)

where constants $\nu _i$, $i=1,2,3$, are given in (2.4) and ${\mathcal {C}}_i$, $i=1,2$, are defined by (4.3).

Both ${\mathcal {C}}_1$ and ${\mathcal {C}}_2$ are well defined due to the assumption (2.3), i.e., $\nu _1 \nu _2 - \nu _3^2 \ne 0$. By plugging $u^*=(\theta ^*, \delta ^*)$ in (B.1) and (B.2) into the HJB Eq. (3.2), we obtain

$$\begin{aligned} V_t(t,x) - \frac{1}{2} \, \frac{|| (\mu _2 - \nu _2) \sigma _1 - \mu _1 \sigma _2 ||^2}{\nu _1 \nu _2 - \nu _3^2} \, \frac{V_x^2(t,x)}{ V_{xx}(t,x)} = 0 , \end{aligned}$$

(B.3)

where $|| \cdot ||$ denotes the standard Euclidean norm. We cannot derive a closed form solution to (B.3) for a general utility function U. When U is given by an exponential or a power utility function, we are able to solve (B.3) explicitly.

1.2 Appendix B.2: Technical derivations in the martingale method

Suppose there exists an admissible control $u^*$ satisfying the condition (3.4) in Lemma 1. Since $u^* \in {\mathcal {A}}(x)$, we have ${\mathbb {E}}\left[ |X_T^*|^2\right] < \infty$ and thus $0<{\mathbb {E}}\left[ U'(X_T^{*}) \right] <\infty$. Now we define a new probability measure ${\mathbb {Q}}$ by

$$\begin{aligned} \frac{\mathrm {d}{\mathbb {Q}}}{\mathrm {d}{\mathbb {P}}} = \dfrac{U'(X_T^{*})}{{\mathbb {E}}\left[ U'(X_T^{*}) \right] } := H_T \end{aligned}$$

and the Radon–Nikodym derivative process $H=(H_t)_{0 \le t \le T}$ by $H_t := {\mathbb {E}}_t[H_T]$. By definition, H is a ${\mathbb {P}}$-martingale, and by the martingale representation theorem, has the representation form of $\mathrm {d}H_t = - H_t \, \lambda _t^\top \, \mathrm {d}W_t$, or equivalently,

$$\begin{aligned} H_t = H_0 \cdot \exp \left( -\int _0^t \lambda _s^\top \, \mathrm {d}W_s - \frac{1}{2} \int _0^t \lambda _s^\top \lambda _s \, \mathrm {d}s\right) , \end{aligned}$$

(B.4)

where $H_0=1$ and $\lambda =(\lambda _1,\lambda _2,\ldots , \lambda _n)^\top$ is an n-dimensional stochastic process in space ${\mathbb {L}}^2[0,T]$, often called the market price of risk. Using the Girsanov theorem, we assert that $W^{\mathbb {Q}}$, defined by

$$\begin{aligned} W^{\mathbb {Q}}_t = W_t + \int _0^t \, \lambda _s^\top \, \mathrm {d}s, \quad \forall \, t \in [0,T], \end{aligned}$$

(B.5)

is an n-dimensional Brownian motion under the new measure ${\mathbb {Q}}$. By the above change of measure and (2.10), we rewrite the optimality condition (3.4) as

$$\begin{aligned} {\mathbb {E}}^{\mathbb {Q}}\left[ X_T^u\right]&= {\mathbb {E}}^{\mathbb {Q}}\left[ \int _0^T \theta _t \left( \mu _1 \, \mathrm {d}t + \sigma _1^\top \, \mathrm {d}W_t\right) +\delta _t \left( (\mu _2 - \nu _2) \, \mathrm {d}t + \sigma _2^\top \mathrm {d}W_t\right) \right] \nonumber \\&\quad \text { is a constant}, \end{aligned}$$

(B.6)

for all $u \in {\mathcal {A}}(x)$, where ${\mathbb {E}}^{\mathbb {Q}}$ denotes taking expectation under ${\mathbb {Q}}$.

We consider two special admissible strategies $u_1 = (\theta _1, \delta _1)$ and $u_2=(\theta _2, \delta _2)$, which are given by

$$\begin{aligned} \theta _{1,t} = x \cdot {\mathbb {I}}_{\{t \le \tau \}}, \quad \delta _{1,t} = 0, \quad \text {and} \quad \theta _{2,t} = 0, \quad \delta _{2,t} = {\mathbb {I}}_{\{t \le \tau \}}, \end{aligned}$$

where $\tau \in [0, T]$ is an arbitrary stopping time and ${\mathbb {I}}$ is an indicator function. It is clear that both $u_1$ and $u_2$ are admissible strategies. By (B.6) and the arbitrariness of stopping time $\tau$, we obtain

$$\begin{aligned} \left( \sigma _1^\top \, W_t + \mu _1 t \right) _{0 \le t \le T} \quad \text {and} \quad \left( \sigma _2^\top \, W_t + (\mu _2 -\nu _2) t\right) _{0 \le t \le T} \end{aligned}$$

are both ${\mathbb {Q}}$-martingales. In consequence, this result leads to

$$\begin{aligned} \sigma _1^\top \lambda _t = \mu _1 \quad \text {and} \quad \sigma _2^\top \lambda _t = \mu _2 - \nu _2. \end{aligned}$$

(B.7)

If the market is complete ($n=2$), there exists a unique solution to (B.7); otherwise, we have infinitely many choices for $\lambda$ (see Karatzas and Shreve 1998).

Given the dynamics of X in (2.10), we apply Itô’s formula to $U'(X_T^*)$ and get

$$\begin{aligned} \mathrm {d}U'(X_t^*) = - U'(X_t^*) \left[ \text {(diffusion term)} \, \mathrm {d}W_t + \text {(drift term)} \, \mathrm {d}t \right] . \end{aligned}$$

By matching the drift term in the above equation with the one in (B.4), we obtain an equality that allows us to express $\lambda$ using $\theta ^*$ and $\delta ^*$. We solve the two equations in (B.7) and obtain the optimal strategy $(\theta ^*, \delta ^*)$.

Appendix C: Technical proofs

Proof

(Proof of Theorem 2 using the HJB method) With U being an exponential utility, we make an educated guess of an ansatz in the form of

$$\begin{aligned} V(t,x) = -\frac{1}{\gamma } \, e^{-\gamma x} \cdot f(t), \end{aligned}$$

where f is positive for all t and satisfies $f(T) = 1$. Let us denote

$$\begin{aligned} \epsilon := \frac{1}{2} \, \frac{|| (\mu _2 - \nu _2) \sigma _1 - \mu _1 \sigma _2 ||^2}{\nu _1 \nu _2 - \nu _3^2} > 0. \end{aligned}$$

(C.1)

Plugging the above ansatz into the HJB (B.3), we obtain

$$\begin{aligned} f'(t) - \epsilon \, f(t) = 0, \quad f(T) = 1, \end{aligned}$$

which leads to the unique solution given by

$$\begin{aligned} f(t) = e^{-\epsilon (T - t)}. \end{aligned}$$

The value function to Problem (3.1) is then obtained by

$$\begin{aligned} V(t,x) = - \frac{1}{\gamma } \, e^{-\epsilon (T - t) - \gamma x}, \end{aligned}$$

(C.2)

where $\epsilon$ is defined in (C.1).

Next, we substitute the value function into (B.1) and (B.2), and obtain the (candidate) optimal strategy $u^*= (\theta ^*, \delta ^*)$ by

$$\begin{aligned} \theta _t^* = \frac{{\mathcal {C}}_1}{\gamma } \quad \text {and} \quad \delta _t^* = \frac{{\mathcal {C}}_2}{\gamma }, \quad \forall \, t \in [0,T], \end{aligned}$$

where ${\mathcal {C}}_1$ and ${\mathcal {C}}_2$ are defined in (4.3). Using the relation (2.8), we find the optimal number of inverse futures contracts by

$$\begin{aligned} N_t^* = \frac{\delta _t^*}{K Z_t} = \frac{{\mathcal {C}}_2}{\gamma K} \, \frac{F_t}{S_t}, \quad \forall \, t \in [0,T]. \end{aligned}$$

Under $u^*= (\theta ^*, \delta ^*)$, we solve the SDE (2.10) and obtain the optimal wealth process $X^*$ by

$$\begin{aligned} X_t^*&= x + \frac{1}{\gamma } \big ( \mu _1 {\mathcal {C}}_1 + (\mu _2 - \nu _2) {\mathcal {C}}_2 \big ) t + \frac{1}{\gamma } ({\mathcal {C}}_1 \sigma _1 + {\mathcal {C}}_2 \sigma _2)^\top \, W_t, \nonumber \\&= x + \frac{2 \epsilon }{\gamma } t + \frac{1}{\gamma } ({\mathcal {C}}_1 \sigma _1 + {\mathcal {C}}_2 \sigma _2)^\top \, W_t, \qquad \forall \, t \in [0,T], \end{aligned}$$

(C.3)

where $\epsilon$ is defined in (C.1) and the second equality is derived by using (2.4) and (4.3). The strategy $u^*= (\theta ^*, \delta ^*)$ found above is admissible, and hence is optimal to Problem (2.11) by Theorem 1. The proof is now complete. $\square$

Proof

(Proof of Theorem 2using the martingale method) Suppose there exists an admissible strategy $u^*=(\theta ^*, \delta ^*)$ that satisfies the optimality condition (3.4), and denote the corresponding wealth process by $X^*$. Recall $U'(x) = e^{-\gamma x}$. By applying Itô’s formula to $U'(X^*)$, we derive

$$\begin{aligned} \frac{\mathrm {d}U'(X_t^*)}{ U'(X_t^*) }&= - \gamma \left( \theta _t^* \sigma _1 + \delta _t^* \sigma _2\right) ^\top \, \mathrm {d}W_t -\gamma \left( \mu _1 \theta _t^* + (\mu _2 - \nu _2) \delta _t^* \right) \mathrm {d}t\nonumber \\&\quad + \frac{1}{2} \gamma ^2 ||\theta _t^* \sigma _1 + \delta _t^* \sigma _2 ||^2 \, \mathrm {d}t. \end{aligned}$$

(C.4)

By matching the diffusion term in (B.4) and (C.4), we obtain

$$\begin{aligned} \lambda _t = \gamma (\theta _t^* \sigma _1 + \delta _t^* \sigma _2), \quad \forall \, t \in [0,T]. \end{aligned}$$

We next plug the above $\lambda _t$ into (B.7), and derive the system of equations for $\theta ^*$ and $\delta ^*$ as follows (noting the definitions of $\nu _i$’s in (2.4))

$$\begin{aligned} {\left\{ \begin{array}{ll} \nu _1 \cdot \theta _t^* + \nu _3 \cdot \delta ^* = \dfrac{\mu _1}{\gamma }, \\ \nu _3 \cdot \theta _t^* + \nu _2 \cdot \delta ^* = \dfrac{\mu _2 - \nu _2}{\gamma }. \end{array}\right. } \end{aligned}$$

Due to (2.3), the above system bears a unique solution $(\theta ^*, \delta ^*)$, which is given by (4.2).

The proof is then complete once we verify (1) $u^*$ is admissible (which is already done in the previous proof), (2) the two means of computing $H_T$, (B.4) and (C.4), are consistent, and (3) the optimality condition (3.4) holds.

To verify the second claim, we solve from (C.4) and get

$$\begin{aligned} U'(X_T^*)&= U'(x) \, \exp \left( - (\mu _1 {\mathcal {C}}_1 + (\mu _2 - \nu _2) {\mathcal {C}}_2 ) T - \int _0^T \, \lambda _t^\top \, \mathrm {d}W_t \right) , \\ {\mathbb {E}}\left[ U'(X^*_T) \right]&= U'(x) \, \exp \left( - (\mu _1 {\mathcal {C}}_1 + (\mu _2 - \nu _2) {\mathcal {C}}_2 ) T + \frac{1}{2} \int _0^T \, \lambda _t^\top \lambda _t \, \mathrm {d}t \right) . \end{aligned}$$

We then recall (B.4) and obtain

$$\begin{aligned} \frac{U'(X^*_T)}{{\mathbb {E}}\left[ U'(X^*_T) \right] } = \exp \left( -\int _0^T \, \lambda _t^\top \, \mathrm {d}W_t - \frac{1}{2} \int _0^T\lambda _t^\top \lambda _t \, \mathrm {d}t \right) = H_T, \end{aligned}$$

which confirms $H_T$ (the new measure ${\mathbb {Q}}$) is well defined.

To show the third assertion, we notice that the optimality condition (3.4) is the same as ${\mathbb {E}}^{\mathbb {Q}}\left[ X_T^u \right]$ is a constant for all $u \in {\mathcal {A}}(x)$. Given any admissible strategy u, we obtain from (2.10), (B.5) and (B.7) that

$$\begin{aligned} \mathrm {d}X_t^u&= \big (\theta _t \mu _1 + \delta _t (\mu _2 - \nu _2 )\big ) \mathrm {d}t + \left( \theta _t \sigma _1^\top + \delta _t \sigma _2^\top \right) \left( \mathrm {d}W_t^{\mathbb {Q}}- \lambda _t^\top \mathrm {d}t \right) \\&= \left( \theta _t \sigma _1^\top + \delta _t \sigma _2^\top \right) \mathrm {d}W_t^{\mathbb {Q}}, \qquad X_0^u = x, \end{aligned}$$

which shows ${\mathbb {E}}^{\mathbb {Q}}\left[ X_T^u \right] = x$. The proof is now complete. $\square$

Proof

(Proof of Theorem 3using the HJB and the martingale methods)

• (1) The HJB Method.

  Under the HJB method, we try the following ansatz

  $$\begin{aligned} V(t,x) = \frac{1}{\alpha } x^\alpha \cdot g(t), \end{aligned}$$

  where g is positive for all t and satisfies $g(T) = 1$.

  Plugging the above ansatz into the HJB (B.3) leads to

  $$\begin{aligned} g'(t) + \frac{\alpha \, \epsilon }{1 - \alpha } \, g(t) = 0, \quad g(T)=1, \end{aligned}$$

  where $\epsilon$ is defined in (C.1). The unique solution is then given by

  $$\begin{aligned} g(t) = e^{\frac{\alpha \, \epsilon }{1 - \alpha } (T-t)}, \end{aligned}$$

  and the value function to Problem (3.1) is obtained by

  $$\begin{aligned} V(t, x) =\frac{1}{\alpha } \, e^{\frac{\alpha \, \epsilon }{1 - \alpha } (T-t)} \, x^\alpha . \end{aligned}$$

  By plugging the above V into (B.1) and (B.2), we obtain the optimal strategy $u^*=(\theta ^*, \delta ^*)$ in (4.6). The SDE (2.10) under $u^*$ becomes

  $$\begin{aligned} \mathrm {d}X_t^* = X_t^* \, \frac{{\mathcal {C}}_1 \mu _1 + {\mathcal {C}}_2 (\mu _2 - \nu _2)}{1 - \alpha } \, \mathrm {d}t + X_t^* \, \frac{{\mathcal {C}}_1 \sigma _1^\top + {\mathcal {C}}_2 \sigma _2^\top }{1 - \alpha } \, \mathrm {d}W_t, \quad X_0^* = x, \end{aligned}$$

  which clearly admits a unique positive solution. In addition, we have $X^* \in {\mathbb {L}}^2[0,T]$, and hence $\theta ^*, \delta ^* \in {\mathbb {L}}^2[0,T]$, proving the admissibility of $u^*=(\theta ^*, \delta ^*)$.

• (2) The Martingale Method.

  Under the martingale method, we have

  $$\begin{aligned} \mathrm {d}U'(X_t^*) = - (1-\alpha ) U'(X_t^*) \left( \frac{\theta _t^*}{X_t^*} \sigma _1^\top + \frac{\delta _t^*}{X_t^*} \sigma _2^\top \right) \, \mathrm {d}W_t + \text {(drift term)} \, \mathrm {d}t, \end{aligned}$$

  and, as a result,

  $$\begin{aligned} \lambda _t = (1-\alpha ) \left( \frac{\theta _t^*}{X_t^*} \sigma _1 + \frac{\delta _t^*}{X_t^*} \sigma _2 \right) , \quad \forall \, t \in [0,T]. \end{aligned}$$

  Using (B.7), we obtain the linear system of $(\theta ^*, \delta ^*)$ by

  $$\begin{aligned} {\left\{ \begin{array}{ll} \nu _1 \cdot \theta _t^* + \nu _3 \cdot \delta _t^* = \dfrac{\mu _1}{1 - \alpha } \, X_t^* \\ \nu _3 \cdot \theta _t^* + \nu _2 \cdot \delta _t^* = \dfrac{\mu _2 - \nu _2}{1 - \alpha } \, X_t^* \end{array}\right. } \end{aligned}$$

  which leads to (4.6). The verification processes are the same as in the exponential utility case, and are omitted. The proof is now complete. $\square$

Appendix D: Discussions on $V=V(t,x)$

In Problem (3.1), we directly assign a function V(t, x) of arguments time t and state x to the supremum problem, and call it the value function of the problem. However, if we directly investigate Problem (2.11) under the original setup and strategies $u = (\theta , N)$, one may question the claim $V = V(t,x)$. After all, the state process X in this case is given by (2.7), in which the ratio process $Z = S / F$ appears explicitly. Should the value function V be in the form of V(t, x, z)? Such a question is also raised by the anonymous referee.

We have provided explanations and references to why $V=V(t,x)$ in Remark 2, which is convincing after we introduce the fictitious asset ${\bar{F}}$ to replace Bitcoin inverse futures, and the new control variable $\delta$ in (2.8) to replace the number of inverse contracts N. Here we show that $V=V(t,x)$ holds as claimed using two different methods. We only focus on the exponential utility case, and note that the power utility case follows as well.

• (1) In the first method, we directly compute ${\mathbb {E}}_{t,x}\left[ X_T^*\right]$ and show that it is a function of t and x. Notice that we do not make any assumption on the value function when we obtain the optimal strategy $u^*$ in (4.2) by the martingale method. Under strategy $u^*$, the optimal wealth process $X^*$ is given by (C.3) (the martingale method would lead to the exactly same equation for $X^*$). Using this result, we obtain

  $$\begin{aligned} {\mathbb {E}}_{t,x}\left[ X_T^*\right]&= {\mathbb {E}}_{t,x}\left[ -\frac{1}{\gamma } \exp \left( - \gamma x - 2 \epsilon (T-t) - ({\mathcal {C}}_1 \sigma _1^\top + {\mathcal {C}}_2 \sigma _2^\top ) (W_T- W_t)\right) \right] \\&= -\frac{1}{\gamma } \exp \left( - \gamma x - \epsilon (T-t) \right) , \end{aligned}$$

  where we have used $\frac{1}{2} || {\mathcal {C}}_1 \sigma _1 + {\mathcal {C}}_2 \sigma _2 ||^2 = \epsilon$ to derive the second equality. The above result coincides with the value function found in (C.2) using the HJB method.

• (2) In the second method, we revisit Problem (3.1) under the original choice of strategy $u = (\theta , N)$, where $\theta$ is the amount invested in Bitcoin and N is the number of the inverse futures contracts. Recall the dynamics of X under $u = (\theta , N)$ is given by (2.7). In such a setup, we make the following modifications to the admissibility condition in Definition 1: $\theta \in L^2[t,T]$ and $Z N \in L^2[t,T]$, where $Z=S/F$ is defined in (2.6).

For any $u \in {\mathcal {A}}(t,x)$ and $\varphi \in \mathrm {C}^{1,2,2}$, define the following operators

$$\begin{aligned} {\mathcal {L}}_1 \varphi (t,x,z)&:= \left( \mu _1 - \mu _2 + \nu _2 - \nu _3 \right) z \, \varphi _z + \frac{1}{2} ||\sigma _1- \sigma _2||^2 z^2 \, \varphi _{zz}, \\ {\mathcal {L}}^u_2 \varphi (t,x,z)&:= \left( \mu _1 \theta + K(\mu _2 - \nu _2)z N\right) \varphi _x + \frac{1}{2} || \theta \sigma _1 + KzN \sigma _2 ||^2 \varphi _{xx} \\&\quad + \left( \sigma _1 - \sigma _2 \right) ^\top \left( \theta \sigma _1 + KzN \sigma _2 \right) \, z \, \varphi _{xz}. \end{aligned}$$

The HJB equation to Problem (3.1) then reads as

$$\begin{aligned} v_t(t,x,z) + {\mathcal {L}}_1 v(t,x,z) + \sup _{u \in {\mathbb {R}}^2} \; {\mathcal {L}}_2^u \, v(t,x,z) = 0, \end{aligned}$$

(D.1)

along with the boundary condition

$$\begin{aligned} v(T,x,z) = U(x) . \end{aligned}$$

Solving the supremum problem in (D.1) gives

$$\begin{aligned} \theta ^*(t,x,z) = - {\mathcal {C}}_1 \, \frac{V_x}{ V_{xx}} - z \, \frac{V_{xz}}{V_{xx}} \quad \text {and} \quad N^*(t,x,z) = - \frac{{\mathcal {C}}_2 }{K \, z} \frac{V_x}{ V_{xx}}+ \frac{1}{K}\, \frac{V_{xz}}{V_{xx}}, \end{aligned}$$

where ${\mathcal {C}}_1$ and ${\mathcal {C}}_2$ are defined in (4.3). By plugging the above $u^*=(\theta ^*, N^*)$ back into (D.1), we obtain after simplifications that

$$\begin{aligned} V_t + \epsilon _1 \left( V_{zz} - \frac{V_{xz}^2}{V_{xx}}\right) \, z^2 + \left( \epsilon _2 \, V_z - \epsilon _3 \, \frac{V_x V_{xz}}{V_{xx}}\right) \, z - \epsilon _4 \, \frac{V_x^2}{V_{xx}}=0, \end{aligned}$$

(D.2)

where, to ease notations, we define $\epsilon _i$, $i=1,2,3,4$, by

$$\begin{aligned} \begin{aligned} \epsilon _1 := \frac{||\sigma _1 - \sigma _2||^2}{2}, \quad \epsilon _2 := \mu _1 - \mu _2 + \nu _2 - \nu _3, \quad \epsilon _3 := \epsilon _2 + \nu _3, \quad \epsilon _4 := \epsilon \text { in } (C.1). \end{aligned} \end{aligned}$$

Note that the simplified HJB (D.2) holds for a general utility U.

In the next step, given the exponential utility in (4.1), we guess the ansatz in the form of (with slight abuse of notation we still use f here)

$$\begin{aligned} V(t,x,z) = U(x) \cdot f(t,z), \quad f(t,z) > 0 \quad \text {and}\quad f(T,z) = 1. \end{aligned}$$

Using (D.2), we derive the non-linear PDE of f by⁸

$$\begin{aligned} f_t + \epsilon _1 \left( f_{zz} - \frac{f_z^2}{f} \right) z^2 + (\epsilon _2 - \epsilon _3) f_z z - \epsilon _4 f = 0, \quad f(T,z) = 1. \end{aligned}$$

(D.3)

Since f is required to be a positive function, we consider the following transformation

$$\begin{aligned} f(t,z) = e^{-\epsilon _4(T-t) + h(t,z)}, \end{aligned}$$

for some $h \in \mathrm {C}^{1,2}$ with $h(T,z)=0$. We then derive the PDE of h from (D.3) by

$$\begin{aligned} h_t + \epsilon _1 \, z^2 \, h_{zz} - (\epsilon _2 - \epsilon _3) \, z \, h_z = 0, \quad h(T,z)= 0. \end{aligned}$$

(D.4)

By Feynman–Kac formula, we obtain $h(t,z)=0$. Since $\epsilon _1 >0$, the PDE (D.4) is uniformly parabolic and the standard PDE theory then implies $h(t,z)=0$ is the unique solution to (D.3). As a result, we obtain

$$\begin{aligned} f(t,z) = e^{-\epsilon _4(T-t)}, \end{aligned}$$

which is independent of z and leads to the same value function V(t, x) in (C.2).