Optimal acquisition of a partially hedgeable house

Abstract

We characterize the optimal time to purchase a house by a risk-averse investor who has access to complete financial markets and whose objective is to maximize expected utility from wealth at some fixed horizon. The house purchase is financially attractive (due to high expected returns or tax deductions, for example), which provides an incentive to buy as soon as possible; however, its value is only partially correlated with financial markets and, therefore, its price risk cannot be perfectly hedged, which provides an incentive to delay purchase. Using martingale duality method, we fully characterize the optimal terminal wealth in the case of CARA utility, so that we can generate simple numerical solutions for different parameter values, and study the trade-off between the two conflicting incentives. We derive more explicit results when the price follows a Gaussian process. In particular, we identify conditions under which an optimal stopping time is deterministic.

Appendix

Lemma 7

Let $\tau$ be a stopping time adapted to $\digamma ^{1}$ and $l(s,t)$ be as in (3.4) for $\tau \le s<t\le T$. Moreover, let the processes $M(.)$ and $D(.)$ be given by Notation 1. Then

• (i) For all $p\ge 1$, we have $E\sup \limits _{0\le s<t\le T}\left| l(s,t)\right| ^{p}<\infty$.

• (ii) There exists a process $\phi \in L_{\digamma ^{1}}^{2}[\tau ,T]$ such that

  $$\begin{aligned} H(T)&= \frac{1}{a}\left\{ \int \limits _{\tau }^{T}(\phi -b)(t)dW^{1}(t)+\int \limits _{\tau }^{T}\left( \frac{1}{2}|\phi (u)|^{2}-c\right) (t)dt-\ln E_{\tau }[l(\tau ,T)]\right\} \\&-\int \limits _{\tau }^{T}I(u)du \text {, a.s.} \end{aligned}$$

• (iii) Similarly, there exists a process $\psi \in L_{\digamma ^{1}}^{2}[0,\tau ]$ such that $D(\tau )$ can be written as

  $$\begin{aligned} D(\tau )+\int \limits _{0}^{\tau }I(u)du= \frac{1}{a}\left\{ \int \limits _{0}^{\tau }(\psi -b)(t)dW^{1}+\int \limits _{0}^{\tau }\left( \frac{1}{2}\psi ^{2}-c\right) (t)dt-\ln E[M(\tau )]\right\} . \end{aligned}$$

Proof

(i) For $0\le s<t\le T$, since the process $b(t)$ is essentially bounded over $[0,T]$, the process $M(s,t)=e^{-\int \limits _{s}^{t}b(u)dW^{1}(u)-\frac{1}{2}\int _{s}^{t}b^{2}(u)du}$ is a martingale. Noting that $0\le a\le \gamma$, $H(.)\ge 0$, $c(.)\ge 0$ and $b(.)$ is bounded, we have $e^{-paH(t)-p\int _{s}^{t}c(u)du}\le 1$ a.s. and $e^{\int _{s}^{t} \frac{1}{2}b^{2}(u)du}<K$, for a constant K. Then, by writing the expression $l(s,t)$ as

$$\begin{aligned} l(s,t)=e^{-aH(t)-\int _{s}^{t}c(u)du}e^{\int _{s}^{t}(\frac{1}{ 2} b^{2}(u)-aI(u))du}M(s,t), \end{aligned}$$

we get $\left| l(s,t) \right| ^{p}\le Ke^{-pa\int _{s}^{t}I(u)du]}M^{p}(s,t)$. Then, by the Cauchy–Schwartz inequality,

$$\begin{aligned} E\sup \limits _{0\le s<t\le T}\left| l(s,t)\right| ^{p}\le K\sqrt{ Ee^{2ap\int \limits _{0}^{T}\left| I(u)\right| du}E\sup \limits _{0\le s<t\le T}M^{2p}(s,t)}. \end{aligned}$$

The rest of the proof follows by the exponential moment condition on $I(.)$ and the Burkholder–Davis–Gundy inequality.

(ii) By (the multiplicative form of) the Itô’s Representation Theorem and part (i), there is a (unique) progressively measurable process $\phi$ such that

$$\begin{aligned} l(\tau ,T)=E_{\tau }[l(\tau ,T)]\exp \left( -\int \limits _{\tau }^{T}\phi (u)dW^{1}(u)-\frac{1}{2} \int \limits _{\tau }^{T}|\phi (u)|^{2}du\right) . \end{aligned}$$

Then, by taking the logarithm of both sides and using Eq. (3.4), we get

$$\begin{aligned} \ln E_{\tau }[l(\tau ,T)]&= \ln l(\tau ,T)+\int \limits _{\tau }^{T}\phi (u)dW^{1}(u)+ \frac{1}{2}\int \limits _{\tau }^{T}|\phi (u)|^{2}du\\&= -a[H(T)+\int \limits _{\tau }^{T}I(u)du]+\int \limits _{\tau }^{T}(\phi -b)(u)dW^{1}(u)\\&+\int \limits _{\tau }^{T}\left( \frac{1}{2}|\phi |^{2}-c\right) (u)du. \end{aligned}$$

After rearranging the integrals and solving for $H$, the result follows. (iii) It is similar to the part (ii) after observing that the process $M(\tau )$ has a representation

$$\begin{aligned} M(\tau )=E[M(\tau )]\exp \left( -\int \limits _{0}^{\tau }\psi (u)dW^{1}(u)-\frac{1}{2} \int \limits _{0}^{\tau }|\psi (u)|^{2}du\right) , \end{aligned}$$

for some $\psi \in L_{\digamma ^1 }^{2}[0,\tau ]. \square$

Lemma 8

Let $\pi ^{*}$ be as in (3.6) for a given $\tau \in [0,T)$. Then we have

$$\begin{aligned} X^{\pi ^{*}}(T)+H(T)=X(\tau )-\frac{1}{\gamma (1-\rho ^{2})}\ln E_{\tau }[l(\tau ,T)]- \frac{1}{\gamma }\ln \tilde{Z}(\tau ,T),\;\text {a.s.} \end{aligned}$$

(6.1)

Proof

Let the parameters $a$, $b(\cdot )$ and $c(\cdot )$ be as before. By Eq. (2.2) and Lemma 7 (ii), $X^{\pi ^{*}}(T)+H(T)$ can be written as

$$\begin{aligned} X(\tau )&- \frac{\ln E_{\tau }[l(\tau ,T)]}{a}+\int \limits _{\tau }^{T}\left[ \frac{\phi -b}{a }+\pi ^{*}\rho \sigma \quad \sigma \pi ^{*}\sqrt{1-\rho ^{2}} \right] dW\\&+\int \limits _{\tau }^{T}\left( \frac{1}{2a}\phi ^{2}-\frac{c}{a}+\mu \pi ^{*}\right) dt \end{aligned}$$

which is equivalent to the right hand side of (6.1) if and only if

$$\begin{aligned}&\int \limits _{\tau }^{T}\left[ \frac{\phi -b}{a}+\pi ^{*}\rho \sigma \quad \sigma \pi ^{*}\sqrt{1-\rho ^{2}}\right] dW+\int \limits _{\tau }^{T}\left( \frac{\phi ^{2}-2c}{2a}+\mu \pi ^{*}\right) dt\nonumber \\&=\frac{1}{\gamma } \left\{ \int \limits _{\tau }^{T}\sigma ^{Z}dW+\int \limits _{\tau }^{T}\frac{|\sigma ^{Z}|^{2}}{2}du\right\} . \end{aligned}$$

(6.2)

Writing $\phi =\frac{\mu -\gamma (1-\rho ^{2})\sigma ^{2}\pi ^{*}}{\rho \sigma }$ from (3.6), using (3.7) and comparing the drift and diffusion terms in both sides of the Eq. (6.2), the result follows.$\square$

Lemma 9

Let $0\le \tau <T$ be given and $X(\tau )=x$ be fixed. Then, the followings hold:

• (i) The process $\{\tilde{Z}(\tau ,t)$, $\tau \le t\le T\}$ is a martingale under $P$.

• (ii) The process $\{\tilde{W}(t)=W(t)+\int \limits _{\tau }^{t} \sigma ^{Z}(u)du: \tau \le t\le T\}$ is a Brownian motion under $\tilde{P}$.

• (iii) $X^{\pi }(t)-\int _{\tau }^{t}$ $I(u)du,$ with $\tau \le t\le T,$ is a martingale under $\tilde{P}$ for any admissible portfolio process $\pi$. In particular, $\tilde{E}_{\tau }^{\tau ,x}[X^{\pi }(t)-\int \limits _{\tau }^{t}$ $I(u)du]=x$, for any $\tau \le t\le T$.

Proof

(i)–(ii) The diffusion vector $\sigma ^{Z}=[\sigma ^{Z1}\,\sigma ^{Z2}]$ is a square integrable process as a linear function of $\pi ^*$ (3.7). Hence the proofs follow from the dynamics of $\tilde{ Z}$ and Girsanov theorem.

(iii) By Itô’s rule,

$$\begin{aligned} d(X\tilde{Z})&= \tilde{Z}[\pi \rho \sigma -\sigma ^{Z1}X\ \ \ \pi \sigma \sqrt{1-\rho ^{2}}-\sigma ^{Z2}X]dW+\tilde{Z}\pi [\mu -\rho \sigma \sigma ^{Z1}\\&-\sigma \sqrt{ 1-\rho ^{2}}\sigma ^{Z2}]dt+\tilde{Z}Idt \end{aligned}$$

where the drift term $[\mu -\rho \sigma \sigma ^{Z1}-\sqrt{1-\rho ^{2}}\sigma \sigma ^{Z2}]$ vanishes by (3.7). Then, it is easy to see that

$$\begin{aligned} X(t)\tilde{Z}(\tau ,t)-\int \limits _{\tau }^{t}\tilde{Z}(\tau ,u)I(u)du&= x+\int \limits _{\tau }^{t} \tilde{Z}(\tau ,u)[\pi \rho \sigma \\&-\sigma ^{Z1}X\quad \pi \sigma \sqrt{ 1-\rho ^{2}}-\sigma ^{Z2}X](u)dW(u) \end{aligned}$$

is a square integrable martingale with

$$\begin{aligned} \tilde{E}_{\tau }^{\tau ,x}[X^{\pi }(t)-\int \limits _{\tau }^{t}I(u)du]=E_{\tau }^{\tau ,x}\left[ X(t)\tilde{Z}(\tau ,t)-\int \limits _{\tau }^{t}\tilde{Z}(\tau ,u)I(u)du\right] =x, \quad \tau \le t\le T. \end{aligned}$$

$\square$

Corollary 6

Assume that all the model parameters are deterministic. Then the process $\phi$ in the representation of $\pi ^{*}(t)=\frac{\mu -\rho \sigma \phi }{\gamma (1-\rho ^{2})\sigma ^{2}}(t)$ is given by

$$\begin{aligned} \phi (t)=e^{aH(t)}\left\{ a\sigma ^{H}(t)E_{t}[l(t,T)H(T)]+aE_{t}\left[ l(t,T)\int \limits _{t}^{T}{\mathcal {D}} _{t}I(u)du\right] +b(t)E_{t}[l(t,T)]\right\} . \end{aligned}$$

(6.3)

where ${\mathcal {D}}$ is the Malliavin derivative operator with respect to $W^{1}$. Moreover, if the income rate process $I$ is also deterministic, then

$$\begin{aligned} \phi (t)=e^{aH(t)}\{a\sigma ^{H}(t)E_{t}[l(t,T)H(T)]+b(t)E_{t}[l(t,T)]\}. \end{aligned}$$

(6.4)

Proof

The random variable $l(\tau ,T)$ can be written as

$$\begin{aligned} l(\tau ,T)=E_{\tau }[l(\tau ,T)]-\int \limits _{\tau }^{T}l(\tau ,u)\phi (u)dW^{1}(u) \end{aligned}$$

(6.5)

where, by the Clark–Ocone formula,

$$\begin{aligned} -l(\tau ,t)\phi (t)=E_{t}[{\mathcal {D}}_{t}l(\tau ,T)], \end{aligned}$$

(6.6)

When the parameters are deterministic, ${\mathcal {D}}_{t}l(\tau ,T)$ can be computed as

$$\begin{aligned} {\mathcal {D}} _{t}l(\tau ,T)&= l(\tau ,T)\left\{ -a\sigma ^{H}(t)H(T)-a{\mathcal {D}} _{t}\int \limits _{\tau }^{T}I(u)du-b(t)\right\} \\&= -l(\tau ,T)\left\{ a\sigma ^{H}(t)H(T)+a\int \limits _{t}^{T}{\mathcal {D}} _{t}I(u)du+b(t)\right\} . \end{aligned}$$

Noting that $\frac{l(\tau ,T)}{l(\tau ,t)}=e^{aH(t)}l(t,T)$, the result follows from (6.6). When $I(t)$ is deterministic, the Malliavin derivatives vanish and we get (6.4)$\square$

Remark 7

The conditional expectations on the right-hand side of (6.3) or (6.4) cannot be solved explicitly. However using Euler discretizations, they can be solved numerically through Monte Carlo simulations, thanks to the conditional Gaussian distribution of the time-discretized processes $l(t_{i},t_{i+1})$ and $l(t_{i},t_{i+1})H(t_{i+1})$ over a partition $\{ \tau =t_{0}<t_{1}<\cdots <t_{n}=T\}$ of $[\tau ,T]$.⁶

Corollary 7

With the deterministic model parameters, the process $\psi$ in the representation of $\pi _{*}(t)= \frac{\mu -\rho \sigma \psi }{\gamma (1-\rho ^{2})\sigma ^{2}}(t)$ is given by $\psi (t)=-E_{t}[D_{t}M(\tau ]/M(t)$, where

$$\begin{aligned} E_{t}[{\mathcal {D}}_{t}M(\tau ]=E_{t}\left[ M(\tau )(a\delta \sigma ^{H}(t)H(\tau )+\frac{{\mathcal {D}}_{t}E_{\tau }[l(\tau ,T)]}{E_{\tau }[l(\tau ,T)]}-a\int \limits _{t}^{\tau }{\mathcal {D}}_{t}I(u)du-b(t))\right] . \end{aligned}$$

Proof

Again, by the Clark–Ocone formula, $\psi (t)=-E_{t}[{\mathcal {D}} _{t}M(\tau )]/M(t)$ and

$$\begin{aligned} -E_{t}[{\mathcal {D}}_{t}M(\tau )]&= aE_{t}\left[ M(\tau )({\mathcal {D}}_{t}D(\tau )+{\mathcal { D}}_{t}\int \limits _{0}^{ \tau }I(u)du+b(t))\right] \nonumber \\&= aE_{t}\left[ M(\tau )(-\delta \sigma ^{H}(t)H(\tau )- \frac{{\mathcal {D}}_{t}E_{\tau }[l(\tau ,T)]}{aE_{\tau }[l(\tau ,T)]} +\int \limits _{t}^{\tau }{\mathcal {D}} _{t}I(u)du+b(t))\right] .\nonumber \\ \end{aligned}$$

(6.7)

$\square$

Lemma 10

Let $\pi _{*}$ be the optimal portfolio before $\tau$ as in Corollary 7, and $X^{\pi }(0) = x_{0}$ . Then,

• (i) $X^{\pi _{*}}(\tau _{-})+D(\tau )=x_{0}- \frac{1}{ \gamma (1-\rho ^{2})}\ln E[M(\tau )]-\frac{1}{\gamma }\ln \tilde{Z} (\tau )$

• (ii) $-\gamma (\tilde{E}\int \limits _{0}^{\tau }I(u)du+\tilde{E}[D(\tau )])-\tilde{E}[\ln \tilde{Z}(\tau )]=\frac{1}{1-\rho ^{2}} \ln E[M(\tau )]$

Proof

Using Lemma 7 (iii) and the notation of Definition 1, it is similar to that of Lemma 8.$\square$

Lemma 11

Let $H(t)$ satisfy (5.1) with $H(0)=h$, $l(s,t)$ be as in (3.4), and introduce

$$\begin{aligned} \alpha (s,t)&= a\{H(s)+(t-s)(\sigma ^{I}W^{1}(s)+\mu ^{I}+h\mu ^{H}\}+c(t-s) \\ \beta (s,t)&= ah\sigma ^{H}+b+a\sigma ^{I}(t-s), \end{aligned}$$

for    $0\le s\le t\le T$. Then

• (i) The expression $V^{\tau ,x}=\sup \limits _{\pi \in {\mathcal {U}}(\tau ,T)}E_{\tau }^{\tau ,x}[-e^{-\gamma Y(T)}],$ of the problem (3.1) can be written as

  $$\begin{aligned} V^{\tau ,x}=-e^{-\gamma x-\frac{1}{1-\rho ^{2}}(\alpha (\tau ,T)-\frac{1}{2} \int _{\tau }^{T}\beta ^{2}(u,T)du)} \end{aligned}$$

  (6.8)

  for given $(\tau ,x)$, with $0\le \tau <T$ and $x=X(\tau )>0$.

• (ii) Let $\delta (t)$ be a deterministic function on $[0,T]$. Then, for $0\le s\le t\le T$, the process $M(\cdot )$ in Notation 1 can be written as $M(t)=M(s)e^{\xi (s,t)}$ where

  $$\begin{aligned} \xi (s,t)&= -a[(1-\delta (s))(H(t)-H(s))-H(t)(\delta (t)-\delta (s))+\int \limits _{s}^{t}I(u)du \nonumber \\&-(h\mu ^{H}+\mu ^{I})(t-s)]-[a\sigma ^{I}(T-t)+b](W^{1}(t) -W^{1}(s))\nonumber \\&-\frac{1}{2} \int \limits _{s}^{t}\beta ^{2}(u,T)du, \end{aligned}$$

  (6.9)

  with $\xi (s,s)=0$ and the constants $a,b$ and $c$ are as in (3.3): $a=\gamma (1-\rho ^{2}),b=\frac{\mu \rho }{\sigma }$ and $c=\frac{ \mu ^{2} }{2\sigma ^{2}}$.

Proof

(i) Noting that $\int \limits _{s}^{t}W^{1}(u)du=\int \limits _{s}^{t}(t-u)dW^{1}(u)+W^{1}(s)(t-s)$, we have $\int \limits _{s}^{t}I(u)du=\int \limits _{s}^{t}\mu ^{I}du+\sigma ^{I}(\int \limits _{s}^{t}(t-u)dW^{1}(u)+W^{1}(s)(t-s))$. Then the exponent term of $l(s,t)$ in (3.4) with constant parameters reduces to $-\alpha (s,t)-\int \limits _{s}^{t}\beta (u,t)dW^{1}(u),$ after rearranging the terms involved. Since this expression has a conditional Gaussian distribution (given $\digamma _{s}$) with conditional mean $-\alpha (s,t)$ and variance $\int \limits _{s}^{t}\beta ^{2}(u,t)du$, we get $E_{s}[l(s,t)]=e^{-\alpha (s,t)+\frac{1}{2}\int \limits _{s}^{t}\beta ^{2}(u,t)du}$. Then the result follows directly from Theorem 1.

(ii) Note that $M(t)$ can be written as

$$\begin{aligned} M(t)=e^{a(\delta H(t)-\int \limits _{0}^{t}I(u)du)-\int \limits _{0}^{t}b(u)dW^{1}(u)-\int \limits _{0}^{t}c(u)du}E_{t}[l(t,T)] \end{aligned}$$

(6.10)

where $I(t)=\mu ^{I}+\sigma ^{I}W^{1}(t)$ and $E_{t}[l(t,T)]=e^{-\alpha (t,T)+ \frac{1}{2}\int \limits _{t}^{T}\beta ^{2}(u,T)du}$. Then by comparing $M(t)$ with $M(s)$, for $s \le t$ using (6.10), and simplifying the terms in the exponent of the ratio $M(t)/M(s)$, the result follows.$\square$