Asset Pricing, Lucas Trees, and Options

Undergraduate Computational Macro

Jesse Perla

jesse.perla@ubc.ca

University of British Columbia

Table of contents

• Overview
• Review of Preferences
• Consumption Based Asset Pricing
• Aggregate Endowment and Complete Markets
• Finite State Asset Pricing
• Risk Neutral Examples
• Risk Averse Examples
• Option Pricing

Overview

Motivation

• We have used asset pricing examples as practice in dynamic programming and EPDVs, but have not explored the economics of these models
• In the Permanent Income Model lectures we analyzed the role of intertemporal smoothing and risk-aversion in helping consumers smooth consumption.
• Here, rather than considering an exogenous interest rate we will consider where asset prices should come from in a general equilibrium model
  • We will follow a variation of Lucas (1978) and build connections to Harrison and Kreps (1979) and Hansen and Richard (1987)

Materials

• Adapted from QuantEcon lectures coauthored with John Stachurski and Thomas J. Sargent
  • Asset Pricing I: Finite State Models
  • Asset Pricing II: The Lucas Asset Pricing Model

using LinearAlgebra, Statistics
using Distributions, LaTeXStrings, QuantEcon
using Plots.PlotMeasures, NLsolve, Roots, Random, Plots
default(;legendfontsize=16, linewidth=2, tickfontsize=12,
         bottom_margin=15mm)

Review of Preferences

Period Utility

• Notation warning: will occasionally use derivatives, such as the utility \(u'(c)\) we mean derivative, but in other cases we will use write the problem recursively and reserve \(c'\) for the next period notation
  • Confusing at first, but you will see it used often in macroeconomics
• Consider utility which is strictly concave where:
  • \(u'(c) > 0\): More is better
  • \(u''(c) \leq 0\): (Weakly) Diminishing Marginal Utility
• Examples include
  • \(u(c) = \log(c)\) and \(u(c) = \frac{c^{1-\gamma}}{1-\gamma}\) for \(\gamma > 0\)
  • If \(u''(c) = 0\) then we have a linear utility function, \(u(c) \propto c\) and \(u'(c)\) is constant

Strictly Concave Utility

• Positive Marginal Utility of Consumption
• Diminishing Returns
• No (visible, at least) point of satiation

Marginal Utility

• \(u'(c) > 0\) but decreasing \(u''(c) < 0\)
• \(u'(c_1) = u'(c_2)\implies c_1 = c_2\)
• If \(u'(c_t) < u'(c_{t+1})\) then \(c_t > c_{t+1}\)
• The less they consume, the more valuable additional consumption in that period would be

Uncertainty

• What if the agent does not know \(\{c_t\}_{t=0}^\infty\) because it is random or uncertain?

• In that case, we can instead have the agent compare expected utility streams

  \[ \mathbb{E}_t\left[\sum_{j=0}^\infty \beta^j u(c_{t+j})\right] \]

  • Where \(\mathbb{E}_t[\cdot] \equiv \mathbb{E}[\cdot|I_t]\) with \(I_t\) the information set we make available at time \(t\) for forecasting in our model
  • This uses our model of expectation formation from the previous lecture

Risk Aversion vs. Inter-temporal Substitution

• If \(u(c)\) is strictly concave the agent:
  • Risk Averse: Prefers more deterministic consumption to those with a higher variance
  • Preferences for Consumption Smoothing: Will substitute between time periods rather than smoother consumption over time rather than large fluctuations
• One challenge in macroeconomics with these preferences is that the \(u(c)\) serves both purposes, which have different economic interpretations.
  • To disentangle, can use recursive preferences such as Epstein-Zin which decouple these two concepts

Smoothing Incentives

• Consider a simpler case where they live for two periods and don’t discount the future: \(V(c_1, c_2) \equiv u(c_1) + u(c_2)\)
• Consider two possible bundles: \(\{c_t, c_{t+1}\}\) and \(\{\bar{c}, \bar{c}\}\) where \(c_t + c_{t+1} = 2 \bar{c}\)
• If the agent is risk-neutral, we see that \(V(c_t, c_{t+1}) = V(\bar{c}, \bar{c})\)
• However, if the agent if risk-averse, then \[ V(c_t, c_{t+1}) < V(\bar{c}, \bar{c})\quad \text{unless }c_t = c_{t+1} = \bar{c} \]
  • They strictly prefer smoother consumption over time
  • i.e., would forgo consumption on average to gain smoother consumption

Smoothing and Concavity

• Recall \(\bar{c} \equiv (c_t + c_{t+1})/2\)
• 2 periods, \(\beta = 1\)
• Same “price” for \(c_t\) and \(c_{t+1}\)
• Two possible bundles:
  1. \(\{c_t, c_{t+1}\}\)
  2. \(\{\bar{c}, \bar{c}\}\)
• Later, \(\beta\) and prices will simply distort this exact tradeoff

Risk-Aversion Intuition

• Consider a utility \(u(c)\) and a lottery which is a random variable

  • \(C = \begin{cases} c_L & \text{with probability } \frac{1}{2} \\ c_H & \text{with probability } \frac{1}{2} \end{cases}\)
  • Let \((c_L + c_H)/2 = \bar{c}\)
  • We can form expected utility as \(\mathbb{E}[u(C)] = \frac{1}{2} u(c_L) +\frac{1}{2} u(c_H)\)

• Note if risk-neutral then \(\mathbb{E}[C] = \frac{1}{2} c_L + \frac{1}{2}c_H = \bar{c} = u(\bar{c})\)

• Then if an agent is risk-averse, \[ u(\mathbb{E}(C)) > \mathbb{E}[u(C)] \]

  • i.e., would forgo consumption on average to avoid the risk

Risk Aversion and Concavity

• Interpretation as fair, risk-neutral prices for lotteries
• Then compare choice between lotteries:
  1. \(\mathbb{E}[u(C)] \equiv \frac{1}{2} u(c_L) + \frac{1}{2} u(c_2)\)
  2. \(u(\mathbb{E}(C)) = u(\frac{1}{2} c_L + \frac{1}{2} c_H)\)
• The strict concavity of \(u(c)\) shows you are better off with the deterministic consumption

Consumption Based Asset Pricing

Why Study This Problem?

• Macro-finance and financial economics \(\neq\) pure finance. Different goals and questions, though sometimes common tools
• If you are interested in macro-finance, then this is the core theory of aggregate asset prices (“consumption-based asset pricing”)
• Even if you do not care about macro-finance or financial economics, macroeconomists need to understand asset prices because they are tightly connected to models of saving and investment
• Finally, if you have a model of asset pricing you can use it to invert consumer expectations of the economy from empirical asset prices
  • e.g., the yield cure (i.e., prices bonds of different maturity) can be used to infer the market’s expectations of future GDP growth

General Equilibrium for Asset Markets

• General Equilibrium (GE) refers to a model where all markets clear simultaneously. Supply equals demand, which determines the price
• The simplest models of asset pricing should have prices such as that of bonds, equities, insurance contracts, etc. determined by the same forces
• Agents might want to purchase assets in order to
  • Delivery in the future where they expect to want more consumption relative to today (i.e. \(u'(c_{t+j}) > u'(c_t)\) after discounting by \(\beta^j\), etc.)
  • Delivery in states of the world to hedge against bad outcomes. For example, if they think there is a 50% chance of a bad outcome, they might want to purchase an asset that pays off in that state to smooth consumption - even if it may decrease their average consumption today

Exchange Economies

• The simplest models to understand asset prices are when the “endowments” are exogenous (i.e., the amount of goods each agent cannot be changed by their behavior)
• Then, there may be gains from trade if different agents get their endowments in different states of the world or at different times.
  • e.g., the young may have more endowments relative to the retired
  • e.g., employed have endowments at different times than unemployed
• If agents are able to trade these exogenous endowments we call it a “pure exchange economy”

Representative Consumers

• Since we will be looking at prices emerging from supply and demand, it is important to be clear when agents are competitive vs. can exert market power
• We will assume that no individuals have large enough endowments relative to each other that they can unilaterally affect prices of traded assets
• It turns out that if we assume agents have identical preferences and there are complete markets for smoothing consumption, we can solve the model with a single representative agent to get the same (aggregate) results
  • The “endowments” of the representative agent are the sum of the endowments of all agents, i.e. the aggregate endowments
  • Using a representative agent is an aggregation result given particular assumptions on primitives, not an assumption itself

Supply of Goods

• In the simplest version, think of there being a “tree” which produces a random stream of fruit each period.

  • We are using “fruit” instead of dollars because it is important to consider that this is a physical good, not just a nominal value

• The random sequence of consumption goods (fruit) is \(\{d_t\}_{t=0}^{\infty}\)

• Let the process determining the fruit be Markov, where for some \(w_{t+1}\) iid

  \[ d_{t+1} = h(d_t, w_{t+1}) \]

  • Since Markov, could also write \(d' = h(d, w)\) for IID \(w\)

• Assume the “fruit” is not storable

Preferences

• At time \(t\) the consumer has preferences \[ \mathbb{E}_t\left[\sum_{j=0}^\infty \beta^j u(c_{t+j})\right] \]
  • For now, assume that \(u(\cdot)\) is strictly concave, but we will consider cases where it is not in the limit (e.g., \(\lim_{\gamma \to 0} \frac{c^{1-\gamma}}{1-\gamma} = c\))
• We will solve a competitive equilibrium were the consumer buys and sells claims to the fruit of the tree (i.e., assets) to smooth consumption

Prices and Claims

• Let \(p_t\) = price of a claim to the fruit of the tree at time \(t\) giving the right to
  • Claim a unit share of the fruit that falls at time \(t\)
  • Sell that claim in time \(t\) or \(t+1\), where the (equilibrium) price will be forecast at \(p_{t+1}\) given time \(t\) information
• If \(d_t\) is varying this is “equity” rather than a bond, because there is no guarantee of how many pieces of fruit will fall at that time
• Let the state variable of the firm be \(\pi_t\) which is the number of claims to the fruit of the tree they own at time \(t\)

Budget Constraint

• Normalize the price of fruit to \(1\) at each time period, so \(p_t\) is in real terms
  • Think of this as spot markets for the fruit which we use as a price level
• The consumer has \(\pi_t\) claims to the tree, which delivers \(\pi_t d_t\) pieces of fruit
  • They can sell the fruit for \(\pi_t d_t \times 1\)
  • They can sell the claim itself for \(p_t \pi_t\)
• They may want to:
  • Purchase \((c_t - \pi_t d_t)\) additional fruit at price \(1\)
  • Change the number of future claims by purchasing (or selling) \((\pi_{t+1} - \pi_t)\) claims at price \(p_t\)
• Putting together, the budget constraint is: \(c_t + p_t \pi_{t+1} = \pi_t (d_t + p_t)\)

Consumers Problem

• The agent is a price taker at \(p_t\) (i.e., this is a competitive equilibrium)

• State: \(\pi_t\) and \(d_t\) (and information sets for \(d_{t+j}\) and \(p_{t+j}\) forecasts)

• Taking prices as given, the consumer solves

  \[ \begin{aligned} \max_{\{c_{t+j}, \pi_{t+j+1}\}_{j=0}^{\infty}} &\mathbb{E}_t\left[\sum_{j=0}^\infty \beta^j u(c_{t+j})\right]\\ \text{s.t. }& c_{t+j} + p_{t+j} \pi_{t+j+1} = \pi_{t+j} (d_{t+j} + p_{t+j}),\text{ for all }j\geq 0\\ \end{aligned} \]

  • The first order conditions for this problem will yield a demand function claims to the the fruit tree and the fruit itself

• If \(d_t\) is Markov, we can write this problem recursively as a Bellman equation

Dynamic Programming

• Let the Markov price be \(p(d)\), then the Bellman equation for the consumer is

  \[ \begin{aligned} V(\pi, d) &= \max_{c, \pi'}\left[u(c) + \beta \mathbb{E}[V(\pi', d') | d]\right]\\ \text{s.t. }& c + \pi'p(d) = \pi (d + p(d)) \end{aligned} \]

  • They forecast \(d'\) and \(p(d')\) based on their information set

• Substituting the budget constraint into the Bellman equation

  \[ V(\pi, d) = \max_{\pi'}\big[u(\underbrace{\pi(d + p(d)) - \pi'p(d)}_{c(\pi, \pi', d)}) + \beta \mathbb{E}[V(\pi', d') | d]\big] \]

Euler Equation

• Take the \(\partial_{\pi'}\) of the Bellman equation

  \[ 0 = -p(d) u'(\pi(d + p(d)) - \pi'p(d)) + \beta \mathbb{E}[\partial_{\pi}V(\pi', d') | d] \]

• Next the envelope theorem tells us how the value function changes with respect to the state variable \(\pi\) \[ \partial_{\pi}V(\pi, d) = u'(c)(d + p(d)) \]

• Use \(c = \pi(d + p(d)) - \pi'p(d)\), and \(d' = h(d, w)\) for \(\mathbb{E}[\cdot]\)

  \[ p(d) = \mathbb{E}\left[\beta \frac{u'(c')}{u'(c)}(d' + p(d')) \bigg| d\right] \]

Consumption in Equilibrium

• This is the celebrated consumption-based asset pricing equation

  \[ p(d) = \mathbb{E}\bigg[\underbrace{\beta \frac{u'(c')}{u'(c)}}_{m(c,c')}(d' + p(d')) \bigg| d\bigg] \]

  • Includes properties specific to the asset (e.g., \(p(d)\) and \(d\))
  • Includes consumers’ preferences and process for consumption. Collect into \(m(c,c')\) the stochastic discount factor(SDF)

• If the consumer’s consumption is tightly connected to the fruit of this particular asset, then there may be a correlation between \(c\) and the \(d\) and hence between \(m(c,c')\) and \(d' + p(d')\)

Sequential Notation

• In that case, lets directly use the \(m_{t+1}\) has a stochastic process

• It could have any correlation with a particular \(d_{t+1}\) process

  • In fact, maybe being negatively correlated is a good thing for smoothing risks?

• In that notation, the asset pricing equation is

  \[ p_t = \mathbb{E}_t\left[m_{t+1} (d_{t+1} + p_{t+1})\right] \]

  • However, this is just notation and we can switch for convenience

• Note that the first payoff of the “dividend” occurs at \(t+1\). This is called ex-dividend pricing

Reminder: Permanent Income Model

• In the permanent income model, the consumer could purchase a 1-period riskless asset which paid \(1\) with certainty.
  • Extending so the price of the risk-free asset might change as \(R_t\)
• The Euler Equation \[ \begin{aligned} u'(c_t) &= \beta R_t\, \mathbb{E}_t[u'(c_{t+1})]\\ p_t^{RF} \equiv \frac{1}{R_t} &= \mathbb{E}_t\left[\beta \frac{u'(c_{t+1})}{u'(c_t)}\right] \end{aligned} \]
  • Converts gross interest rate \(R_t\) to a price on 1 period asset \(p_t^{RF}\)

Connecting to the Asset Pricing Formula

• Back to our current setup. Since the risk-free asset has no future claims, \(p^{RF}_{t+1} = 0\) and since it is risk-free the \(d_{t+1} = 1\) \[ \begin{aligned} p^{RF}_t &= \mathbb{E}_t\left[m_{t+1} (d_{t+1} + p^{RF}_{t+1})\right]\\ p^{RF}_t &= \mathbb{E}_t\left[m_{t+1}(1 + 0)\right]\\ &= \mathbb{E}_t\left[\beta \frac{u'(c_{t+1})}{u'(c_t)}\right] = \frac{1}{R_t} \end{aligned} \]
  • Previously: Given an \(R_t\), find \(c_t, c_{t+1}\)
  • Now: Given the \(c_t, c_{t+1}\), could we find the \(R_t\) that would reconcile the asset pricing equation with consumer’s optimality?

Aggregate Endowment and Complete Markets

Example: Claim to the Aggregate Endowment

• Consider if the tree is the full output of the economy
  • Interpretation: a claim to real GDP per capita
• In that case, the
  • demand is determined by the asset pricing equation
  • supply is inelastic (since it is an endowment)
• Market clearing requires that \(c = d\) for all states
• Substitute into the equation to get the price of a claim to the aggregate endowment (e.g., a perfectly diversified equity index)

Asset Pricing Equation

• We can now write down the equation determining the price of a claim to the aggregate endowment \[ p(d) = \mathbb{E}\bigg[\beta \frac{u'(d')}{u'(d)}(d' + p(d')) \bigg| d\bigg] \]
  • Where the process \(d' = h(d, w)\) defines the conditional expectations
• This \(p(d)\) is now a recursive equation which we can solve for all \(d\)

Interpretation of the SDF for \(c = d\)

• The “fruit” process (e.g., GDP) effects asset prices through two channels
• First consider how \(d' > d\) affects \(m(d,d')\)
  • Due to market clearing, more endowment tomorrow relative to today means that the ratio of marginal utilities will be higher
  • Hence the asset prices will be need to rise to make the consumer indifferent between consuming today and tomorrow (after discounting)
  • Higher asset prices deter borrowing, which ensures that markets can clear given the fixed endowment today
  • Otherwise, the consumer would want to borrow against the future (i.e., Permament Income model)

Interpretation of the Dividend and Price Forecasts

• Next, the \(d' + p(d')\) term is more mechanical in \[ p(d) = \mathbb{E}\bigg[\beta \frac{u'(d')}{u'(d)}(d' + p(d')) \bigg| d\bigg] \]

  • If \(d'\) is higher (in expectation) then the \(p(d)\) will be higher since it is a claim to the future endowment
  • In addition, if there is an any persistence in \(d\) then a higher \(d\) today will lead to the probability of a higher \(d'\) tomorrow, which will also raise the price of the claim to the endowment

• Suggests crucial to understand how \(m'\) and \(d'\) are correlated

Assets under Complete Markets

• Consider a case with complete market where the consumer can purchase financial assets to help smooth consumption against all possible idiosyncratic and aggregate states of the world
  • In particular, if there income/endowment fluctuates over time, they would trade with people who have the opposite fluctuations
  • If the income fluctuates idiosyncratically, trade with people in the opposite states
• Consider more broadly than just financial assets
  • e.g., insurance contracts, implicit contracts with family, government social insurance, etc.
• Can’t smooth fluctuations to aggregate endowment (e.g., GDP)

Complete Markets and Aggregate Endowment

• In a world with complete markets and identical preferences, you can show that all idiosyncratic preferences will be hedged against, and any individual asset cannot affect the aggregate.
• \(m(c,c')\) is the right way to discount for claims to the aggregate endowment, which can have its own stochastic process
• But more importantly, given the perfect diversification, the consumer should use that same \(m(c,c')\) for all assets!
  • Otherwise, there would be arbitrage opportunities

Conditional Covariances

• For any random variables \(x_{t+1}\) and \(y_{t+1}\)

• The definition of the conditional covariance \({\rm cov}_t (x_{t+1}, y_{t+1})\) is

  \[ {\mathbb E}_t (x_{t+1} y_{t+1}) \equiv {\rm cov}_t (x_{t+1}, y_{t+1}) + {\mathbb E}_t x_{t+1} {\mathbb E}_t y_{t+1} \]

• The key to understanding the price of an asset with payoff process \(d_{t+1}\) will be its covariance with the SDF

Covariances and Asset Prices

• Apply this decomposition to the asset pricing equation

  \[ \begin{aligned} p_t &= \mathbb{E}_t\left[m_{t+1} (d_{t+1} + p_{t+1})\right]\\ &= {\mathbb E}_t m_{t+1} {\mathbb E}_t (d_{t+1} + p_{t+1}) + {\rm cov}_t (m_{t+1}, d_{t+1}+ p_{t+1}) \end{aligned} \]

• Recall: \(m_{t+1}\) measures value of consumption in different states

• For example, if consumption in a state is lower relative to today means \(u'(c_{t+1})/u'(c_t)\) is higher and \(m_{t+1}\) is higher

  • Then, if \(d_{t+1}\) has a positive covariance with \(m_{t+1}\), (i.e., it pays more in states where the SDF is higher) the price of the asset will be higher
  • Asset hedges against bad states

Risk-Free Asset and SDF

• Risk-free asset is a claim to one unit of consumption tomorrow with certainty

• The SDF \(m_{t+1}\) is a random variable which says how much you value payoff tomorrow in various states of the world

• Given the complete markets in the economy we see that

  \[ \frac{1}{R^{RF}_t} = \mathbb{E}_t\left[\beta \frac{u'(c_{t+1})}{u'(c_t)}\right] = \mathbb{E}_t\left[m_{t+1}\right] \]

• Powerful tool: given asset prices such as the interest rate, and a functional form of \(m_{t+t}\) you can infer the market expectations of \(c_{t+1}/c_t\)

Finite State Asset Pricing

Finite State Markov Processes

• Using our tools from above, lets consider that the \(m_t\) and \(d_t\) follow a finite state Markov process (i.e., a Markov Chain)
• The processes will have variance degrees of covariance
  • The extreme example is if \(d_t = c_t\) as in the previous example, then the \(m_t\) will be perfectly correlated with \(d_t\)
  • A perfect hedge against GDP would be have a perfect negative correlation
• Let the underlying random variable which generates the random states of both \(m_t\) and \(d_t\) processes be \(X_t\)

Growth Rates of “Dividends”

• Given that the growth rates of payoffs (and its correlation to the SDF) will be essential, define the growth rate of the endowments (e.g. dividends) as \[ d_{t+1} = G_{t+1} d_t \]

  • Assume for simplicity that the growth rates are themselves IID

• Since the underlying random variable is \(X_t\) we can write this as \[ G_{t+1} = G(X_{t+1}) \]

• Similarly, the SDF is IID and may be correlated with \(G_t\) through \(X_t\) \[ m_{t+1} \equiv m(X_{t+1}) \]

Finite States

• Consider if \(X_t \in \{x_1, \ldots x_N\}\) a Markov Chain where

  \[ P_{ij} \equiv \mathbb P ( X_{t+1} = x_j \,|\, X_t = x_i ),\quad \text{ for }i=1,\ldots N, j=1,\ldots N \]

• Baseline growth factor: \(G(x_i) = \exp(x_i)\), with \(x_i > 0\) for all \(i=1,\ldots N\), and hence \(\log G(x_i) = x_i\)

• Baseline process for \(X_t\): discretized AR(1) process using Tauchen’s Method

  • e.g. \(X_{t+1} = \rho X_t + \sigma w_{t+1}\) where the mean of the stationary distribution is \(X_{\infty} = 0\) and hence \(G(X_{\infty}) = 1\). No growth on average
  • Correlation \(\rho\) helpful for interpretation

Price to Dividend Ratio

• Let the price to dividend ratio be \(v_t \equiv p_t/d_t\)
• Divide the pricing equation by \(d_t\) \[ \begin{aligned} p_t &= \mathbb{E}_t\left[m_{t+1} (d_{t+1} + p_{t+1})\right]\\ \frac{p_t}{d_t} &= \mathbb{E}_t\left[m_{t+1} \frac{d_{t+1}}{d_t}\left(1 + \frac{p_{t+1}}{d_{t+1}}\right)\right]\\ v_t &= \mathbb{E}_t\left[m_{t+1}G_{t+1}(1 + v_{t+1})\right]\\ v(X_t) &= \mathbb{E}\left[m(X_{t+1})G(X_{t+1})(1 + v(X_{t+1}))\big| X_t\right] \end{aligned} \]
• This lets us describe the price-to-dividend ratio which is scaleless. Similarly, as \(m_{t+1}\) is typically a ratio of marginal utilities, it is also scaleless

Price to Dividend Ratio with Markov Chain

• Price to dividend called Price to Earnings (P/E) ratio in equity markets
• Continuing with this example, given the Markov Chain \[ \begin{aligned} v(X_t) &= \mathbb{E}\left[m(X_{t+1})G(X_{t+1})(1 + v(X_{t+1}))\big| X_t\right]\\ v_i &= \sum_{j=1}^N m(X_j) G(x_j) (1 + v_j) P_{ij} \end{aligned} \]
  • We can stack these equations for all \(i=1,\ldots N\) into a vector \(v\)
  • Then solve for the \(v\) vector - which is a linear equation for any \(G(\cdot)\) and \(m(\cdot)\)

Risk Neutral Examples

Risk-Neutral Asset Pricing

• If risk-neutral, then \(m_{t+1} = \beta\) for all \(X_t\)

• Given the finite number of states, we can find a vector \(v_t = v(X_t)\)

• Define the matrix \(K\) where \(K_{ij} \equiv G(x_j) P_{ij}\) and

  \[ \begin{aligned} v_i &= \beta \sum_{j = 1}^N K_{ij} (1 + v_j)\quad \text{for }i=1,\ldots N\\ v &= \beta K (\mathbb 1 + v)\\ v &= (I - \beta K)^{-1} \beta K{\mathbb 1} \end{aligned} \]

  • Assuming the \(\max\{|\text{ eigenvalue of } A|\} < 1/\beta\) as in LSS examples

Risk-Neutral Simulation

n = 25
mc = tauchen(n, 0.96, 0.02)
sim_length = 80
X_0_ind = 12
X_t = simulate(mc, sim_length; init = X_0_ind)
G_t = exp.(X_t)
d_0 = 1
d_t = d_0 * cumprod(G_t) 

series = [X_t G_t d_t log.(d_t)]
labels = [L"X_t" L"G_t" L"d_t" L"\log(d_t)"]
plot(series; layout = 4, labels)

Risk-Neutral Simulation

Price-Dividend Ratios for Risk-Neutral Assets

beta = 0.9
K = mc.p .* exp.(mc.state_values)'
v = (I - beta * K) \ (beta * K * ones(n, 1))

plot(mc.state_values, v; xlabel = L"X_t",
     label = L"v_t \equiv \frac{p_t}{d_t}",
     size = (600, 400))

Interpretation

• Remember that \(m_{t+1} = \beta\), so this is not driven by the SDF or the correlation between the SDF and the dividend process
• Why does the price-dividend ratio increase with the state?
  • The Markov process is positively correlated, so high current states suggest high future states
  • Moreover, dividend growth is increasing in the state, which is persistent
• Hence, high future dividend growth leads to a high price-dividend ratio

Risk Averse Examples

Pricing with CRRA and Lucas Tree SDF

• Utility: \(u(c) = \frac{c^{1-\gamma}-1}{1 - \gamma} \ {\rm with} \ \gamma > 0\)

  • Then \(u'(c) = c^{-\gamma}\), nesting \(\log\) utility if \(\gamma = 1\)

• With complete market, \(d_t = c_t\) and the SDF is

  \[ m_{t+1} = \beta \frac{u'(c_{t+1})}{u'(c_t)} = \beta \left(\frac{c_{t+1}}{c_t}\right)^{-\gamma} = \beta G_{t+1}^{-\gamma} \]

Price-Dividend Ratio for CRRA

• Substitute this into the formula for the price-to-dividend ratio

  \[ \begin{aligned} v(X_t) &= \beta {\mathbb E}_t \left[ G(X_{t+1})^{-\gamma}G(X_{t+1}) (1 + v(X_{t+1}) ) \right]\\ v_i &= \beta \sum_{j = 1}^N G(x_j)^{1-\gamma} (1 + v_j) P_{ij} \end{aligned} \]

• Rearranging as a fixed point with \(J_{ij} \equiv G(x_j)^{1-\gamma} P_{ij}\) \[ \begin{aligned} v &= \beta J ({\mathbb 1} + v )\\ v &= (I - \beta J)^{-1} \beta J {\mathbb 1} \end{aligned} \]

Implementation

function asset_pricing_model(; beta = 0.96, gamma = 2.0, G = exp,
                         mc = tauchen(25, 0.9, 0.02))
    G_x = G.(mc.state_values)
    return (; beta, gamma, mc, G, G_x)
end
# price/dividend ratio of the Lucas tree
function tree_price(ap)
    (; beta, mc, gamma, G) = ap
    P = mc.p
    y = mc.state_values'
    J = P .* G.(y) .^ (1 - gamma)
    @assert maximum(abs, eigvals(J)) < 1 / beta # check stability
    v = (I - beta * J) \ sum(beta * J, dims = 2)
    return v
end

Price-Dividend for Various Risk-Aversion Parameters

gammas = [1.2, 1.4, 1.6, 1.8, 2.0]
p = plot(; xlabel = L"X_t", size=(600,400))

for gamma in gammas
    ap = asset_pricing_model(; gamma)
    states = ap.mc.state_values
    plot!(states, tree_price(ap);
         label = L"\gamma = %$gamma")
end
p

Interpretation

• Keep in mind that this is with perfectly correlated \(m_{t+1}\) and \(d_{t+1}\)
• Notice that \(v\) is decreasing in each case, in contrast to the risk-neutral case
• This is because, with a positively correlated state process, higher states suggest higher future consumption growth.
• In the stochastic discount factor, higher growth decreases the discount factor, lowering the weight placed on future returns
• Special cases:
  • If \(\gamma = 1\) then the \(v\) is constant, as the forces exactly cancel
  • If \(\gamma = 0\) then the \(v\) nests the risk-neutral case

A Risk-Free Consol

• A risk-free consol pay a constant amount, a fixed coupon each period forever
• Asset has
  • \(\zeta\) in period \(t+1\) (i.e., \(d_{t+1} = \zeta\))
  • the right to sell the claim for \(p_{t+1}\) next period
  \[ \begin{aligned} p_t &= \mathbb{E}_t \left[ m_{t+1} ( \zeta + p_{t+1} ) \right]\\ p_t &= {\mathbb E}_t \left[ \beta G_{t+1}^{-\gamma} ( \zeta + p_{t+1} ) \right]\\ p_i &= \beta \sum_{j = 1}^N G(X_j)^{-\gamma} (\zeta + p_j) P_{ij} \end{aligned} \]

Linear System

• Letting \(M_{ij} \equiv P_{ij} G(X_j)^{-\gamma}\) and rewriting in vector notation yields the solution

  \[ p = (I - \beta M)^{-1} \beta M \zeta {\mathbb 1} \]

Implementation

function consol_price(ap, zeta)
    (; beta, gamma, mc, G) = ap
    P = mc.p
    y = mc.state_values'
    M = P .* G.(y) .^ (-gamma)
    @assert maximum(abs, eigvals(M)) < 1 / beta

    # Compute price
    return (I - beta * M) \ sum(beta * zeta * M, dims = 2)
end

Consol Price

ap = asset_pricing_model(; beta = 0.9)
zeta = 1.0
strike_price = 40.0

x = ap.mc.state_values
p = consol_price(ap, zeta)
plot(mc.state_values, p; xlabel = L"X_t",
     label = L"p_t",
     size = (600, 400))

Option Pricing

Pricing an Option to Purchase the Consol

• An option is a contract that gives the owner the right, but not the obligation, to buy or sell an asset at a specified price
• Many problems in macro are isomorphic to option-pricing problems
  • e.g.firm entry/exit decisions
• Consider an option to purchase a consol at a price \(p_S\)
  • This will never expire (infinite horizon, or “perpetual” option)
  • The “call” option gives the owner the right to buy the asset
  • The price \(p_S\) is called the strike price
• Let the dynamics of the console be driven by the SDF \(m_{t+1}\) and the growth process \(G_{t+1}\)

Exercising an Option

• Let \(w(X_t, p_S)\) be the value of the option given known \(X_t\) but before the owner has decided whether or not to exercise the option
  • Discounts with the SDF \(m(X_{t+1})\)
• \(p(X_t)\) remains the price of the consol itself
• Bellman equation is \[ w(X_t, p_S) = \max \left\{ {\mathbb E}_t\left[m(X_{t+1}) w(X_{t+1}, p_S)\right], \; p(X_t) - p_S \right\} \]
  • Left term is value of waiting, right is exercising now.

Option Pricing with Finite State Markov Process

• Using our SDF process

  \[ w(x_i, p_S) = \max \left\{ \beta \sum_{j = 1}^N P_{ij} G(X_j)^{-\gamma} w (x_j, p_S), \; p(x_i) - p_S \right\} \]

• If we define \(M_{ij}\equiv P_{ij} G(X_j)^{-\gamma}\) and stack prices then

  \[ w = \max \{ \beta M w, \; p - p_S {\mathbb 1} \} \]

Fixed Point

• To solve this problem, define an operator \(T\) mapping vector \(w\) into vector \(T(w)\) via

  \[ T(w) = \max \{ \beta M w,\; p - p_S {\mathbb 1} \} \]

  • To solve this, we can find the fixed point of \(T(w) = w\)
  • Also a linear complementarity problem in this case

Implementation

# price of perpetual call on consol bond
function call_option(ap, zeta, p_s)
    (; beta, gamma, mc, G) = ap
    P = mc.p
    y = mc.state_values'
    M = P .* G.(y) .^ (-gamma)
    @assert maximum(abs, eigvals(M)) < 1 / beta
    p = consol_price(ap, zeta)

    # Operator for fixed point, using consol prices
    T(w) = max.(beta * M * w, p .- p_s)
    sol = fixedpoint(T, zeros(length(y), 1))
    converged(sol) || error("Failed to converge in $(sol.iterations) iter")
    return sol.zero
end

Example

ap = asset_pricing_model(; beta = 0.9)
zeta = 1.0
strike_price = 40.0

x = ap.mc.state_values
p = consol_price(ap, zeta)
w = call_option(ap, zeta, strike_price)

plot(x, p; xlabel = L"X_t", size=(600,400),
     label = L"p(X_t)")
plot!(x, w; label = L"w(X_t, p_S)")