Long-term optimal portfolios with floor

Abstract

Long-term risk-sensitive portfolio optimization is studied with floor constraint. A simple rule to characterize its solution is mentioned under a general setting. Following this rule, optimal portfolios are constructed in several ways, using the optimal portfolio without floor constraint, combined with ideas of dynamic portfolio insurance, such as CPPI (constant proportion portfolio insurance), OBPI (option-based portfolio insurance), and DFP (dynamic fund protection). In addition, examples are presented with explicit computations of solutions.

Appendices

Appendix A: Proof of Proposition 5.1

First, for a given T∈ℝ_>0 and x∈ℝ_>0, we solve the CRRA utility maximization

where

$$u_\gamma(x):=\frac{1}{\gamma} x^{\gamma} \quad\text{with}\ \gamma\in(-\infty,0)\cup(0,1)$$

is the CRRA utility function,

is the space of admissible investment strategies, and $\bar{\mathbb{E}}_{T}[\cdot]:=\int_{\bar{\varOmega}}(\cdot)Z_{T}^{-1} d\tilde{\mathbb{P}}\otimes d{\mathbb{P}}^{0} |_{\bar{\mathcal{F}}_{T}}$ with

$$Z_T^{-1}:={\mathcal{E}}\bigl(-\theta^\top w\bigr)_T^{-1} ={\mathcal{E}} \bigl(\theta^\top \tilde{w}\bigr)_T.$$

Recall that $(S^{0}_{T})^{\gamma} \in L^{1}({\mathbb{P}}^{0})$ since we assume (5.1). So we apply the $({\mathbb{P}}^{0}, {\mathcal{F}}^{0}_{t})$-Brownian martingale representation theorem to define an m-dimensional $({\mathcal{F}}^{0}_{t})$-progressively measurable process $\eta^{(T)}:=(\eta_{t}^{(T)})_{t\in[0,T]}$ that satisfies

$${\mathbb{E}}^0 \bigl[ \bigl(S^0_T\bigr)^{\gamma} | {\mathcal{F}}_t^0 \bigr] = {\mathbb{E}}^0 \bigl[ \bigl(S^0_T\bigr)^{\gamma}\bigr] {\mathcal{E}} \biggl( \int\bigl(\eta^{(T)}\bigr)^\top \,dz\biggr)_t$$

for any t∈[0,T]. Using the above, we define the “state-price density” process by

$$H_t:= {\mathbb{E}}^0 \bigl[ \bigl(S^0_T\bigr)^{\gamma} \bigr] \bigl(S^0_t\bigr)^{-1} {\mathcal{E}} \bigl( -\theta^\top w\bigr)_t {\mathcal{E}} \biggl( \int\bigl(\eta^{(T)}\bigr)^\top \,dz \biggr)_t, \quad t\in[0,T].$$

Here, recalling

$$H_T = u'_\gamma\bigl(S^0_T\bigr) {\mathcal{E}} \bigl( -\theta^\top w \bigr)_T,$$

we have

$$ I_\gamma(yH_T) =S^0_TI_\gamma(y) I_\gamma \bigl( {\mathcal{E}} \bigl( -\theta^\top w \bigr)_T \bigr)$$

(A.1)

for y∈ℝ_>0, where we define

$$I_\gamma(y):= \bigl(u'_\gamma\bigr)^{-1}(y) = y^{-1/(1-\gamma)}.$$

Further, note that for any ,

$$\bar{\mathbb{E}}_T\bigl[H_T X^{x,\pi}_T\bigr] =x \int_{\bar{\varOmega}} {\mathcal{E}} \biggl(\int \pi^\top \,dR \biggr)_T d\tilde{\mathbb{P}}\otimes d{\mathbb{P}}^0 \le x$$

since ${\mathcal{E}} (\int\pi^{\top}dR )$ is a $\bar{\mathbb{P}}$-supermartingale. We now deduce for any and x,y∈ℝ_>0 that

(A.2)

where we use the relation

$$\max_{x>0} \bigl\{ u_\gamma(x)-y x \bigr\} = u_\gamma \bigl(I_\gamma(y) \bigr) -y I_\gamma(y).$$

Let

$$\hat{y}_T(x):= u'_\gamma \biggl(\frac{x}{\bar{\mathbb{E}}_TH_T I_\gamma(H_T)} \biggr).$$

We see that

(A.3)

where we use the relation (A.1) and definition (5.2). So, in (A.2), let $y:=\hat{y}_{T}(x)$, recall relation (A.3), and use the $\bar{\mathbb{P}}$-martingale property of ${\mathcal{E}} ( \int\hat{\pi}^{\top}\,dR )$. We now deduce the optimality of , i.e.,

$$\bar{\mathbb{E}}_T u_\gamma \bigl( X^{x,\pi}_T\bigr) \le \bar{\mathbb{E}}_T u_\gamma \bigl(X^{x,\hat{\pi}1_{[0,T]}}_T \bigr)$$

for any , or equivalently

$$ \frac{1}{\gamma} \log \bar{\mathbb{E}}_T \bigl(X^{x,\pi}_T \bigr)^\gamma \le \frac{1}{\gamma} \log \bar{\mathbb{E}}_T \bigl( X^{x,\hat{\pi}1_{[0,T]}}_T\bigr)^\gamma $$

(A.4)

for any .

Next, from (A.4), we see that for any and T∈ℝ_>0,

This implies the optimality of $\hat{\pi}$ for $(\tilde{\mathrm{ RS}})$ with and the relation (5.3). □

Appendix B: Proof of Theorem 5.2

First of all, let us check the integrability conditions stated in Conditions 4.5 and 4.10. Note that

$$\frac{K_t}{S^0_t} = K_0 {\mathrm{e}}^{|\alpha| (\tilde{\nu}_t - c_1 t)}$$

where c ₁:=|α|/2+β/|α| and $\tilde{\nu}:=\alpha^{\top}\tilde{w}/|\alpha|$ is a $\tilde{\mathbb{P}}$-Brownian motion. So, we see

Similarly, we deduce that

$$\frac{K_s}{\hat{X}_s} = K_0 {\mathrm{e}}^{c_2 (\hat{\nu}_s - c_3 s)}.$$

Here, c ₂:=|α−θ/(1−γ)| and c ₃:=c ₂/2+β/c ₂ are constants, and using the $\hat{\mathbb{P}}$-Brownian motion $\hat{w}:=(\hat{w}_{t})_{t\ge0}$, where

$$\hat{w}_t := \tilde{w}_t -\int_0^t\sigma_u^\top\hat{\pi}_u \,du =\tilde{w}_t -\frac{1}{1-\gamma}\theta t,$$

we define a 1-dimensional $\hat{\mathbb{P}}$-Brownian motion $\hat{\nu}:= (\alpha-\theta/(1-\gamma) )^{\top}\hat{w}/c_{2}$. So we see that

We are now in a position to show the assertions (1)–(4).

(1) $\hat{V}_{\mathrm{ OBPI}}$ and $\hat{V}_{\mathrm{ DFP}}$ can be derived in several ways. For $\hat{V}_{\mathrm{ OBPI}}$, one derivation is introduced in Sect. 8 of Gerber and Shiu [17]. Here, we apply the computation for the American perpetual call option to derive it as follows. Recalling

from (OBPI′), we deduce that

(B.1)

where we define

$$P_t(x) := x \exp \biggl[ \bigl(\alpha-\sigma^\top\hat{\pi}\bigr)^\top\hat{w}_t + \biggl\{ -\beta -\frac{1}{2}\bigl|\alpha-\sigma^\top\hat{\pi}\bigr|^2 \biggr\}t \biggr].$$

In (B.1), we apply the computation of the American perpetual call option price, regarding P _t (x ₁/x ₂) as the Black–Scholes stock price at time t with initial value x ₁/x ₂, volatility $|\alpha-\sigma^{\top}\hat{\pi}|$, dividend rate β>0 and zero interest rate. Recall that for the asset price process

$$P_t^{(r)}(x) := x \exp \biggl[ a z_t + \biggl\{r-\beta-\frac{1}{2}a^2 \biggr\}t \biggr] \quad(t\ge0),$$

where a>0, z is a 1-dimensional $(\hat{\mathbb{P}}, \hat{\mathcal{F}}_{t})$-Brownian motion, β is a dividend rate and r is an interest rate with 0<β<r, we have

(B.2)

where

$$k:=\frac{1}{a} \bigl(\sqrt{\ell^2 +2r} -\ell \bigr)-1 \quad \text{and}\quad \ell:=\frac{r-\beta}{a}-\frac{a}{2}$$

(see e.g. Sect. 2.6 of [28]). The computation to derive (B.2) can be straightforwardly extended to deduce

(B.3)

(formally, (B.3) is obtained by letting r=0 in (B.2)). Thus, the expression for $\hat{V}_{\mathrm{ OBPI}}$ follows.

For $\hat{V}_{\mathrm{ DFP}}$, one derivation is introduced in Remark (a) in Sect. 2 of [18]. Here, we directly compute it as follows. Recalling (4.12) in Remark 4.8, we deduce that

where

$$U_\mathrm{ DFP}(y_1,y_2) := \hat{\mathbb{E}} \Bigl[\sup_{s\in[0,\infty)} P_s(y_1) \vee y_2\Bigr].$$

Furthermore, letting

$$a:=\bigl|\alpha-\sigma^\top\hat{\pi}\bigr|, \qquad b:=\frac{\beta}{a}+\frac{a}{2} >0$$

and using a 1-dimensional $(\hat{\mathbb{P}}, \hat{\mathcal{F}}_{t})$-Brownian motion z, we see that

Thus, the expression for $\hat{V}_{\mathrm{ DFP}}$ follows (or we can utilize computations for the Russian perpetual option to derive $\hat{V}_{\mathrm{ DFP}}$, letting the discounting rate equal to 0 in the Russian option pricing formula in Shepp and Shiryaev [46], Pedersen [37], and [44]).

(2) Recall that

$$\hat{V}_\mathrm{ OBPI}(x_1,x_2) =x_2 \biggl \{ U_\mathrm{ OBPI} \biggl( \frac{x_1}{x_2} \biggr)+1 \biggr\},$$

where we use (B.1) and (B.3), and

$${V}_\mathrm{ OBPI}(t,\lambda)= \hat{V}_\mathrm{ OBPI} ( {K_t}, {\lambda\hat{X}_t} ) ={\lambda\hat{X}_t} \biggl\{U_\mathrm{ OBPI} \biggl( P_t \frac{K_0}{\lambda} \biggr)+1 \biggr\}.$$

Note here that U _OBPI is convex, that $U_{\mathrm{ OBPI}} \in C^{1}({\mathbb{R}}_{>0})\cap C^{2} ({\mathbb{R}}_{>0} \setminus\{ \frac{1+\kappa}{\kappa}\} )$, and that ∂ _x U _OBPI(x−)=∂ _x U _OBPI(x+) at $x=\frac{1+\kappa}{\kappa}$. So, we apply the Itô–Meyer–Tanaka formula and the occupation times formula to deduce

$$d {U}_\mathrm{ OBPI} (P_t) =U'_\mathrm{ OBPI}(P_t)dP_t +\frac{1}{2} U''_\mathrm{ OBPI}(P_t)1_{ \{ P_t < \frac{1+\kappa}{\kappa} \}} d\langle P \rangle_t.$$

Hence, the integrand of the martingale part of dV _OBPI(t,λ) is computed as

From this, the expression for $\hat{\pi}^{\mathrm{ OBPI}}$ follows, letting $\lambda:=\hat{\lambda}_{\mathrm{ OBPI}}(x)$.

Similarly, recalling $\hat{V}_{\mathrm{ DFP}} \in C^{2}({\mathbb{R}}_{>0}^{2})$, we use Itô’s formula to deduce

The expression for $\hat{\pi}^{\mathrm{ DFP}}$ follows from this, letting $\lambda:=\hat{\lambda}_{\mathrm{ DFP}}(x)$.

(3) The expression of G follows from the expression of $\hat{V}_{\mathrm{ OBPI}}$, and the expression of $\hat{\pi}^{\mathrm{OBPI}\mbox{-}\mathrm{G}}$ follows by using a similar calculation as in the proof of (2).

(4) Recall that for i∈{OBPI,DFP}, $\hat{\lambda}_{i}(x)$, respectively, satisfies

$$\hat{V}_{i} \bigl(K_0, \hat{\lambda}_i(x)\bigr)=x$$

for any x≥K ₀. Thus,

$$\hat{V}_\mathrm{ OBPI} \biggl(K_0, \frac{\kappa}{1+\kappa}\hat{\lambda}_\mathrm{ DFP}(x) \biggr) =\frac{\kappa}{1+\kappa} \hat{V}_\mathrm{ DFP}\bigl( K_0, \hat{\lambda}_\mathrm{ DFP}(x) \bigr) =\frac{\kappa}{1+\kappa}x$$

follows for any x≥K ₀, which implies

$$\hat{\lambda}_\mathrm{ OBPI} \biggl( \frac{\kappa}{1+\kappa} x \biggr)=\frac{\kappa}{1+\kappa}\hat{\lambda}_\mathrm{ DFP}(x)$$

for any x≥K ₀. Similarly, we see that