15. More on Portfolio Choice Problem

Option pricing in Chapter 14 implicitly used martingale methodology. This section gives a more general treatment. Let stock \(S_t\) and bond \(B_t\) follow (15.1):

with \(r\) the risk-free rate and \(\{Z_t\}\) standard BM. A security maturing at \(T\) pays \(g(S_T)\) (\(g\) twice differentiable), with time-\(t\) value \(V(S_t,t)\).

15.1.2 Dynamic Replication

Form a portfolio \(\Pi_t\): long 1 security, short \(\theta_t\) shares (15.2):

Delta-neutral hedging \(\frac{\partial\Pi_t}{\partial S_t}=0\) gives (15.3):

By Itô, the random term of \(\Pi_t\) vanishes; no-arbitrage forces \(d\Pi_t=\Pi_t r\,dt\) (15.4):

Substituting (15.2), (15.3) into (15.4) gives the PDE the security satisfies (15.5):

Let the portfolio \(W_t=\theta_t S_t+B_t\) (15.6), with \(\theta_t\) from (15.2) and \(B_0=\Pi_0=V(S_0,0)-\theta_0 S_0\). Since \(B_t\) and \(\Pi_t\) are both risk-free bonds with the same initial value, no-arbitrage gives \(B_t=\Pi_t\), hence \(W_t=V(S_t,t)\) for all \(t\). So \(W_t\) is the wealth needed to replicate the security payoff \(g(S_T)\).

15.1.3 Feynman-Kac Theorem

The §14.1.3 discussion holds for a general payoff \(F(S_T)\) (not just a call). Writing \(F(S_T)=g(S_T)\), \(C_t(S_t)=V(S_t,t)\), Feynman-Kac gives (15.7):

By (14.12), (14.14) (zero dividend \(\delta=0\)), \(dZ_t^\star=dZ_t-\frac{r-\mu}{\sigma}dt\), with \(\{Z_t^\star\}\) standard BM under \(\mathbf Q\).

15.1.4 Combine the Results

An investor with initial wealth \(w_0\) can invest in stock \(S_t\) and bond \(B_t\) at any time, has CRRA utility and cares only about consumption at \(T\) (15.8):

subject to the dynamic budget constraint (15.9) and initial value (15.10):

By §15.1.2, \(W_t=V(S_t,t)\) and \(g(S_T)=V(S_T,T)\), so \(W_T=g(S_T)\) (15.11). Plugging into (15.7) at \(t=0\) gives the static budget constraint (15.12):

So the original problem (15.8)–(15.10) is rewritten as a static optimization over \(W_T\): \(\max_{W_T}\mathbb E_0[e^{-\rho T}\frac{W_T^{1-\gamma}}{1-\gamma}]\) s.t. \(w_0=\mathbb E_0^{\mathbf Q}[e^{-rT}W_T]\). (15.9) is the dynamic budget constraint, (15.12) the static one. This transformation is martingale methodology.

Drop intermediate consumption and dividends (the same problem as §6.3, with identical result). Stock/bond dynamics (15.13): \(\frac{dS_t}{S_t}=\mu(S_t)\,dt+\sigma(S_t)\,dZ_t\), \(\frac{dB_t}{B_t}=r\,dt\). The CRRA investor consumes only the terminal \(V(S_T,T)=g(S_T)\). Solve (15.14)–(15.16):

Step 1. Martingale methodology turns the dynamic constraint (15.15) into the static \(w_0=\mathbb E_0^{\mathbf Q}[e^{-rT}W_T]\), rewriting the problem as (15.17)–(15.18): \(\max_{W_T}\mathbb E_0[e^{-\rho T}\frac{W_T^{1-\gamma}}{1-\gamma}]\) s.t. \(w_0=\mathbb E_0^{\mathbf Q}[e^{-rT}W_T]\).

Step 2. Lagrangian (15.19):

Step 3. Convert \(\mathbb E_0^{\mathbf Q}\) from risk-neutral back to the physical measure. Define the state-price density \(\pi_t\) (the SDF, §3.2.1) (15.20): \(\frac{d\pi_t}{\pi_t}=\mu_\pi(S_t)\,dt-\sigma_\pi(S_t)\,dZ_t\), such that \(V(S_t,t)=\mathbb E_t[\frac{\pi_T}{\pi_t}g(S_T)]\) (15.21). Since \(M_t\equiv\mathbb E_t[\pi_T g(S_T)]=V(S_t,t)\pi_t\) is a martingale, \(\mathbb E_t[dM_t]=0\) (15.22); combining Itô with (15.5) and matching coefficients gives (15.28):

where \(\frac{\mu(S_t)-r}{\sigma(S_t)}\) is the market price of risk — the Sharpe ratio (risk premium per unit of standard deviation). Rewriting (15.21) with \(W_T=g(S_T)\) gives the physical-measure static constraint (15.30): \(w_0=\mathbb E_0[\pi_T W_T]\).

Step 4. Rewrite the Lagrangian and take the state-by-state f.o.c. w.r.t. \(W_T\) (15.31):

\(\lambda\) is pinned by (15.30) (immaterial).

Step 5. Characterize wealth \(W_t\). Assume \(r\) and \(\sigma_\pi(S_t)=\sigma_\pi\) constant. By He (2019d) Prop 15.1 (\(dX_t=X_t(m\,dt+\sigma\,dB_t)\Rightarrow X_t=X_0 e^{[(m-\frac12\sigma^2)t+\sigma B_t]}\)) and (15.29), \(\pi_T\) (15.32), (15.33): \(\pi_T=\pi_t\exp\{[-r-\frac12\sigma_\pi^2](T-t)-\sigma_\pi(Z_T-Z_t)\}\). Substituting into (15.21), (15.31) gives (15.34): \(W_t=\pi_t^{-\frac1\gamma}e^{\frac{\rho T}{-\gamma}}\lambda^{\cdots}F(T-t)\), with \(F(T-t)\) depending only on \(T-t\).

Step 6. Find the Merton fraction \(\frac{\theta_t S_t}{W_t}\). Itô on (15.34) gives (15.35): \(\frac{dW_t}{W_t}=\frac1\gamma\sigma_\pi\,dZ_t+R\,dt\) (\(R\) the drift). The dynamic budget constraint (15.36): \(dW_t=\theta_t\sigma(S_t)S_t\,dZ_t+[\theta_t\mu(S_t)S_t+(W_t-\theta_t S_t)r]\,dt\). Matching \(dZ_t\) coefficients (15.37):

(15.37) is the wealth fraction in the risky asset, the Merton fraction, matching (6.36) from the §6.3 dynamic-programming approach in the simplified case — confirming the equivalence of martingale and dynamic-programming methods.

15.2.2 Generalization: Model with Dividends and Underlying State

Introduce a non-tradable state variable \(\delta\) (15.38): \(d\delta_t=\mu_\delta(\delta_t)\,dt+\sigma_\delta(\delta_t)\,dZ_t\). A stock \(S(\delta_t,t)\) pays dividend \(D_t\,dt\); a security \(V(\delta_t,t)\) pays \(h(\delta_t)\,dt\) with terminal \(V(\delta_T,T)=0\); risk-free rate \(r(\delta_t)\).

Short \(\theta_t\) shares, long 1 security, Delta-neutral \(\theta_t=\frac{\partial V/\partial\delta_t}{\partial S/\partial\delta_t}\). No-arbitrage \(\Pi_t r(\delta_t)\,dt=d\Pi_t+h(\delta_t)\,dt-\frac{\partial V/\partial\delta_t}{\partial S/\partial\delta_t}D_t\,dt\) (15.40); substituting Itô (15.39) and matching, defining \(-\mu_\delta^\star(\delta_t)\), gives the security PDE (15.41):

By Feynman-Kac, the solution to (15.41) is (15.42):

with \(dZ_t^{\mathbf Q}=dZ_t+v(\delta_t)\,dt\), \(d\delta_t=\mu_\delta^\star(\delta_t)\,dt+\sigma_\delta(\delta_t)\,dZ_t^{\mathbf Q}\). The investor's portfolio problem (with intermediate consumption) (15.45): \(\max_{\{\theta_t\}}\mathbb E_0[\int_0^T e^{-\rho t}u(C_t)\,dt]\) s.t. \(dW_t=\theta_t\,dS_t+(W_t-\theta_t S_t)r\,dt+h(\delta_t)\,dt-C_t\,dt\); static form (15.46): s.t. \(w_0=\mathbb E_0[\int_0^T\pi_t C_t\,dt]\). Defining the SDF \(\pi_t\) (15.48) and matching gives (15.56), (15.57):

Finally (CRRA, constant coefficients) the generalized Merton fraction: risky-asset wealth share \(=\frac{\mu_S(\delta_t)-r(\delta_t)}{\gamma\sigma_S(\delta_t)\sigma_\delta(\delta_t)}\).

15.2.3 Discussion on Market Completeness

• The above always assumed a complete market. Here "complete" just means no fewer tradable instruments than sources of risk, so the riskless replicating portfolio can hedge each risk source with each instrument.
• If risks outnumber tradable instruments, the dynamic-replication step breaks down and the whole derivation fails.

Complete-market general equilibrium, exogenous aggregate dividend \(D_t=e^{\delta_t}\) (15.66): \(d\delta_t=\mu_\delta\,dt+\sigma_\delta\,dZ_t\). Stock price \(P_t\) is the claim to all future aggregate dividends; endogenous rate \(r_t\); SDF \(\pi_t\) (15.67): \(P_0=\mathbb E_0[\int_0^T\pi_t D_t\,dt]\). A continuum of agents \(i\in[0,1]\), CRRA \(\frac{C_{i,t}^{1-\gamma_i}}{1-\gamma_i}\), risk tolerance \(\rho_i=\frac1{\gamma_i}\in[\underline\rho,\bar\rho]\). Agent \(i\) holds initial wealth share \(\omega_i\): \(W_{i,0}=\omega_i P_0\), \(\int_0^1\omega_i\,di=1\), \(\omega_i\geq0\).

15.3.2 The Decentralized Problem

Agent \(i\) solves (15.68): \(\max_{\{C_{i,t}\}}\mathbb E_0[\int_0^T e^{-\phi t}\frac{C_{i,t}^{1-\gamma_i}}{1-\gamma_i}\,dt]\) s.t. \(W_{i,0}=\mathbb E_0[\int_0^T\pi_t C_{i,t}\,dt]\) (discount rate \(\phi\) common to all \(i\)), equivalently \(\omega_i P_0=\mathbb E_0[\int_0^T\pi_t C_{i,t}\,dt]\) (15.69). The Lagrangian f.o.c. gives (15.70):

\(g_t\) depends only on \(t\), not \(i\). From \(\pi_t\equiv e^{-\phi t-g_t}\) (15.71), \(\delta_t=\ln D_t\), rewrite (15.67) at \(t=0\) as (15.72). Substituting into (15.69) yields Equation 1 (15.73):

Equation 2. Define the cross-sectional density \(f_{\mathrm{CS}}(\omega_i,\rho_i\mid\delta_t)\) and cross-sectional expected consumption \(\mathbb E_{\mathrm{CS}}[C_{i,t}\mid\delta_t]=\int_i C_{i,t}f_{\mathrm{CS}}\,d\omega_i d\rho_i\). Market clearing (unit mass) \(\mathbb E_{\mathrm{CS}}[C_{i,t}\mid\delta_t]=D_t\) gives (15.74): \(\delta_t=\ln(\mathbb E_{\mathrm{CS}}[e^{\rho_i g_t}\omega_i\lambda_i\mid\delta_t])\).

Functional fixed point. By (15.74) (Equation 2), the implicit function theorem makes \(g_t\) a function \(g(\delta_t)\); but by (15.73) (Equation 1), \(\lambda_i\) depends on the whole function \(g(\delta_t)\), so (15.76): \(\delta_t=\ln(\mathbb E_{\mathrm{CS}}[\underbrace{e^{\rho_i g(\delta_t)}\omega_i}_{\text{Part A}}\underbrace{\lambda_i(g(\delta_t))}_{\text{Part B}}\mid\delta_t])\) is a functional fixed point, generally very hard to solve, but solvable by iteration.

15.3.3 Characterization of the Solution

Though no closed form, we can characterize derivatives of \(g(\delta_t)\). Differentiating the implicit \(g_t(\delta_t)\) in (15.74), and defining a new probability measure \(\star\) (density \(f_{\mathrm{CS}}^\star(\omega_i,\rho_i\mid\delta_t)=\frac{f_{\mathrm{CS}}e^{\rho_i g(\delta_t)}\omega_i\lambda_i}{\int f_{\mathrm{CS}}e^{\rho_i g(\delta_t)}\omega_i\lambda_i\,d\omega d\rho}\)), the first derivative (15.80):

Since \(\rho_i\in[\underline\rho,\bar\rho]\), \(g'(\delta_t)>0\) and \(g'(\delta_t)\in[\frac1{\bar\rho},\frac1{\underline\rho}]\). The second derivative (15.81): \(g''(\delta_t)=g'(\delta_t)\left(1-\frac{\mathbb E_{\mathrm{CS}}^{\star\star}[\rho_i\mid\delta_t]}{\mathbb E_{\mathrm{CS}}^\star[\rho_i\mid\delta_t]}\right)\), where \(\star\star\) weights higher \(\rho_i\) more, so \(\mathbb E_{\mathrm{CS}}^{\star\star}[\rho_i]>\mathbb E_{\mathrm{CS}}^\star[\rho_i]\), giving \(g''(\delta_t)<0\).

SDF. From \(g(\delta_t)=-\phi t-\ln\pi_t\), \(\pi_t=e^{-\phi t}e^{-g(\delta_t)}\) (15.85); Itô gives (15.86):

So the SDF's loading on \(dZ_t\) (absolute value \(g'(\delta_t)\sigma_\delta\), the market Sharpe ratio) decreases as \(\delta_t\) rises (since \(g'\) falls). Interest rate: as the SDF drift is always \(-r_t\,dt\) (15.87), matching gives \(r_t=\phi+g'(\delta_t)\mu_\delta+\frac12 g''(\delta_t)\sigma_\delta^2-\frac12(g'(\delta_t))^2\sigma_\delta^2\).

Comparing agents \(i,j\). From (15.73), (15.70), \(\frac{C_{i,t}}{C_{j,t}}=e^{(\rho_i-\rho_j)g(\delta_t)}\frac{\omega_i\lambda_i}{\omega_j\lambda_j}\), so (15.88): \(\frac{\partial}{\partial\delta_t}\frac{C_{i,t}}{C_{j,t}}=(\rho_i-\rho_j)g'(\delta_t)e^{(\rho_i-\rho_j)g(\delta_t)}\frac{\omega_i\lambda_i}{\omega_j\lambda_j}\). If \(\rho_i>\rho_j\) (\(i\) more risk-tolerant), \(i\)'s consumption share rises as aggregate consumption rises — the risk-tolerant bear risk when aggregate consumption is low (low consumption) and enjoy the upside when high, as intuition suggests.

Price \(P_t\). By (15.66), \(\delta_{t+\tau}=\delta_t+\mu_\delta\tau+\sigma_\delta\sqrt\tau x_t\) (15.89), \(x_t\sim\mathcal N(0,1)\). By (15.67), (15.85), \(P_t=e^{g(\delta_t)}\frac1\phi\mathbb E_t^{x_t}[\int_0^{T-t}\phi e^{-\phi\tau}e^{\delta_{t+\tau}-g(\delta_{t+\tau})}\,d\tau]\) (15.90), where \(f^\star(\tau)=\phi e^{-\phi\tau}\) is an exponential density truncated at \(T-t\), renormalized as (15.91), giving the numerically simulable form (15.92).

Homogeneous special case. If all agents are homogeneous \(\rho_i=\rho=\frac1\gamma\), (15.70) becomes \(e^{-\phi t}D_t^{-\gamma}=\xi\pi_t\), \(\pi_t=\frac1\xi e^{-\phi t-\gamma\delta_t}\) (15.93). Substituting into (15.67) gives the price-dividend ratio (15.95):

When \(\gamma>1\), \(\frac{P_t}{D_t}\) decreases in \(\mu_\delta\): the intertemporal elasticity of substitution is $<1$, so the agent cares more about the present; as prospects improve (\(\mu_\delta\) rises) they sell to consume now, depressing today's price.

15.4.1 Setup

Exogenous aggregate dividend \(D_t=e^{\delta_t}\) (15.96): \(d\delta_t=\mu_\delta\,dt+\sigma_\delta\,dZ_t\), aggregate consumption \(C_t=D_t\). The single risky asset is the stock \(P_t\) (claim to all future dividends); endogenous rate \(r_t\); SDF \(\pi_t\) (15.97): \(P_t=\mathbb E_t[\int_t^T\frac{\pi_\tau}{\pi_t}D_\tau\,d\tau]\). A continuum of infinitely-lived agents \(i\in[0,1]\) whose utility depends on both own consumption \(C_{i,t}\) and others' \(C_{j,t}\) (\(j\neq i\)):

The heterogeneous coefficient \(\eta_i\) takes two values: Regular Jones \(\eta_i=\eta_J<1\), mass \(1-\lambda\); Beating the Jones \(\eta_i=\eta_B>1\), mass \(\lambda\). Initial wealth share \(\omega_{i,0}\): \(W_{i,0}=\omega_{i,0}P_0\), \(\int_0^1\omega_{i,0}\,di=1\).

15.4.2 The Decentralized Problem

Agent \(i\) solves (15.98): \(\max_{\{C_{i,t}\}}\mathbb E_0[\int_0^T\frac{e^{-\phi t}}{1-\gamma}(\frac{C_{i,t}}{C_t^{\eta_i}})^{1-\gamma}\,dt]\) s.t. \(\omega_{i,0}P_0=\mathbb E_0[\int_0^T\pi_t C_{i,t}\,dt]\) (15.99) (each \(i\) is infinitesimal and cannot affect \(C_t\)). The f.o.c. gives (15.100):

\(g_t\) depends only on \(t\). From \(\pi_t=e^{-\phi t-g_t}\) (15.101), (15.102) and market clearing \(\mathbb E_{\mathrm{CS}}[C_{i,t}\mid\delta_t]=D_t=e^{\delta_t}\), two fixed-point equations arise: Equation 1 (from clearing, pins \(g_t\)) and Equation 2 (15.103): \(\omega_{i,0}\psi_i=e^{-\frac1\gamma\ln\xi_i}\), \(\psi_i\equiv\frac{P_0}{\mathbb E_0[\int_0^T e^{-\phi t-(1-\frac1\gamma)g_t-\eta_i\frac{1-\gamma}\gamma\delta_t}\,dt]}\). As in Veronesi, \(g(\delta_t)\) is a functional fixed point (15.104), solved by iteration.

15.4.3 Characterization of the Solution

Again characterize derivatives: first derivative \(g'(\delta_t)>0\) (15.105) (since \(\omega_{i,0}\psi_i=e^{-\frac1\gamma\ln\xi_i}>0\) and the exponentials are positive); second derivative \(g''(\delta_t)\) (same form as 15.81, involving \(\eta_i\)). SDF (15.106)–(15.107):

Matching the drift gives the interest rate \(r_t=\phi+g'(\delta_t)\mu_\delta+\frac12 g''(\delta_t)\sigma_\delta^2-\frac12(g'(\delta_t))^2\sigma_\delta^2\) and \(\sigma_\pi=g'(\delta_t)\sigma_\delta\) (15.108).

Comparing \(J\) (Regular) and \(B\) (Beating). From (15.103), (15.100), \(\frac{C_{B,t}}{C_{J,t}}=\frac{\omega_{B,0}\psi_B}{\omega_{J,0}\psi_J}e^{(\eta_J-\eta_B)\frac{1-\gamma}\gamma\delta_t}\), so (15.110): \(\frac{\partial}{\partial\delta_t}\frac{C_{B,t}}{C_{J,t}}=(\eta_J-\eta_B)\frac{1-\gamma}\gamma\frac{\omega_{B,0}\psi_B}{\omega_{J,0}\psi_J}e^{(\eta_J-\eta_B)\frac{1-\gamma}\gamma\delta_t}>0\) (when \(\gamma>1\), since \(\eta_J-\eta_B<0\)). So for \(\gamma>1\), Beating-the-Joneses agents consume relatively more as aggregate consumption rises: more eager to "beat the average," they consume heavily in good times; the gap shrinks as the aggregate dividend falls.

Wealth. By (15.111)–(15.114), \(\frac{dW_{i,t}}{W_{i,t}}=(\frac1\gamma g'(\delta_t)-\eta_i\frac{1-\gamma}\gamma)\sigma_\delta\,dZ_t+\frac{R_i}{W_{i,t}}dt\) (15.114); for \(\gamma>1\) Beating-the-Joneses agents load more on risk.

Portfolio allocation. Let \(dP_t=P_t\mu_P\,dt+P_t\sigma_P\,dZ_t\) (15.116); matching \(dZ_t\) in the dynamic budget constraint gives the risky-asset wealth share (15.118): \(\frac{\theta_{i,t}P_t}{W_{i,t}}=(\frac1\gamma g'(\delta_t)-\eta_i\frac{1-\gamma}\gamma)\frac{\sigma_\delta}{\sigma_P}\). For \(\gamma>1\), Beating-the-Joneses agents hold a larger risky exposure and less risk-free saving — eager to outdo others, they invest more in the risky asset and save less risk-free.

Consumption process. Itô on (15.100): \(\frac{dC_{i,t}}{C_{i,t}}=\sigma_{C,i}\,dZ_t+(\cdots)dt\), \(\sigma_{C,i}\equiv(\frac1\gamma g'(\delta_t)-\eta_i\frac{1-\gamma}\gamma)\sigma_\delta\). For \(\gamma>1\) and \(\eta_B>\eta_J\), \(\sigma_{C,B}>\sigma_{C,J}\) (reversed for \(\gamma<1\)): under high risk aversion the Beating-the-Joneses consumption bears more risk.

Price \(P_t\). As in Veronesi (truncated exponential \(\tau\) + normal \(\delta_{t+\tau}\)), the numerically simulable (15.122) and price-dividend ratio follow.