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Introduction to Asset Pricing

“A little knowledge of geometric series goes a long way” – Robert E. Lucas, Jr.
“Asset pricing is all about covariances” – Lars Peter Hansen


An asset is a claim on one or more future payoffs

The spot price of an asset depends primarily on

  • the anticipated dynamics for the stream of income accruing to the owners
  • attitudes to risk
  • rates of time preference

In this lecture we consider some standard pricing models and dividend stream specifications

We study how prices and dividend-price ratios respond in these different scenarios

We also look at creating and pricing derivative assets by repackaging income streams

Key tools for the lecture are

  • formulas for predicting future values of functions of a Markov state
  • a formula for predicting the discounted sum of future values of a Markov state

Pricing Models

In what follows let \(\{d_t\}_{t \geq 0}\) be a stream of dividends

  • A time-\(t\) cum-dividend asset is a claim to the stream \(d_t, d_{t+1}, \ldots\)
  • A time-\(t\) ex-dividend asset is a claim to the stream \(d_{t+1}, d_{t+2}, \ldots\)

Let’s look at some equations that we expect to hold for prices of assets under cum-dividend and ex-dividend contracts respectively

Risk Neutral Pricing

Our first scenario is risk-neutral pricing

Let \(\beta = 1/(1+\rho)\) be an intertemporal discount factor, where \(\rho\) is the rate at which agents discount the future

The basic risk-neutral asset pricing equation for pricing one unit of a cum-dividend asset is

(1)\[p_t = d_t + \beta {\mathbb E}_t [p_{t+1}]\]

This is a simple “cost equals expected benefit” relationship

Here \({\mathbb E}_t [y]\) denotes the best forecast of \(y\), conditioned on information available at time \(t\)

For an ex-dividend asset, the basic risk-neutral asset pricing equation is

(2)\[p_t = \beta {\mathbb E}_t [d_{t+1} + p_{t+1} ]\]

Pricing with Random Discount Factor

What happens if for some reason traders discount payouts differently depending on the state of the world?

Michael Harrison and David Kreps [HK79] and Lars Peter Hansen and Scott Richard [HR87] showed that in quite general settings the price of an ex-dividend asset obeys

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

for some stochastic discount factor \(m_{t+1}\)

The fixed discount factor \(\beta\) in (2) has been replaced by the random variable \(m_{t+1}\)

The way anticipated future payoffs are evaluated can now depend on various random outcomes

One example of this idea is that assets that tend to have good payoffs in bad states of the world might be regarded as more valuable

This is because they pay well when the funds are more urgently needed

We give examples of how the stochastic discount factor has been modeled below

Asset Pricing and Covariances

Recall that, from the definition of a conditional covariance \({\rm cov}_t (x_{t+1}, y_{t+1})\), we have

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

If we apply this definition to the asset pricing equation (3) we obtain

(5)\[p_t = {\mathbb E}_t m_{t+1} {\mathbb E}_t (d_{t+1} + p_{t+1}) + {\rm cov}_t (m_{t+1}, d_{t+1}+ p_{t+1})\]

It is useful to regard equation (5) as a generalization of equation (2)

  • In equation (2), the stochastic discount factor \(m_{t+1} = \beta\), a constant
  • In equation (2), the covariance term \({\rm cov}_t (m_{t+1}, d_{t+1}+ p_{t+1})\) is zero because \(m_{t+1} = \beta\)

Equation (5) asserts that the covariance of the stochastic discount factor with the one period payout \(d_{t+1} + p_{t+1}\) is an important determinant of the price \(p_t\)

We give examples of some models of stochastic discount factors that have been proposed later in this lecture and also in a later lecture

The Price-Dividend Ratio

Aside from prices, another quantity of interest is the price-dividend ratio \(v_t := p_t / d_t\)

Let’s write down some expressions that this ratio should satisfy

For the case of an ex-dividend contract, we can divide both sides of (3) by \(d_t\) to get

(6)\[v_t = {\mathbb E}_t \left[ m_{t+1} \frac{d_{t+1}}{d_t} (1 + v_{t+1}) \right]\]

For the cum-dividend case, the corresponding expression is

(7)\[v_t = 1 + {\mathbb E}_t \left[ m_{t+1} \frac{d_{t+1}}{d_t} v_{t+1} \right]\]

Below we’ll discuss the implications of these equations

Prices in the Risk Neutral Case

What can we say about price dynamics on the basis of the models described above?

The answer to this question depends on

  1. the process we specify for dividends
  2. the stochastic discount factor and how it correlates with dividends

For now let’s focus on the risk neutral case, where the stochastic discount factor is constant, and study how prices depend on the dividend process

Example 1: Constant dividends

The simplest case is risk neutral pricing in the face of a constant, non-random dividend stream \(d_t = d > 0\)

Removing the expectation from (1) and iterating forward gives

\[\begin{split}\begin{aligned} p_t & = d + \beta p_{t+1} \\ & = d + \beta (d + \beta p_{t+2}) \\ & \quad \vdots \\ & = d + \beta d + \beta^2 d + \cdots + \beta^{k-1} d + \beta^k p_{t+k} \end{aligned}\end{split}\]

Unless prices explode in the future, this sequence converges to

(8)\[\bar p := \frac{1}{1-\beta} d\]

This price is the equilibrium price in the constant dividend case

Indeed, simple algebra shows that setting \(p_t = \bar p\) for all \(t\) satisfies the equilibrium condition \(p_t = d + \beta p_{t+1}\)

The ex-dividend equilibrium price is \((1 -\beta)^{-1} \beta d\)

Example 2: Dividends with deterministic growth paths

Consider a growing, non-random dividend process \(d_{t+1} = g d_t\) where \(0 < g \beta < 1\)

While prices are not usually constant when dividends grow over time, the price dividend-ratio might be

If we guess this, substituting \(v_t = v\) into (7) as well as our other assumptions, we get \(v = 1 + \beta g v\)

Since \(\beta g < 1\), we have a unique positive solution for the cum-dividend case:

\[v = \frac{1}{1 - \beta g }\]

The cum-dividend price is then

(9)\[p_t = \frac{1}{1-\beta g} d_t\]

In view of (6), the ex-dividend formulas are

\[v = \frac{\beta g}{1 - \beta g } \quad \text{and} \quad p_t = \frac{\beta g}{1 - \beta g } d_t\]

If, in this example, we take \(g = 1+\kappa\) and let \(\rho := 1/\beta - 1\), then the ex-dividend price becomes

\[p_t = \frac{1 + \kappa}{ \rho - \kappa} d_t\]

This is called the Gordon formula

Example 3: Markov growth, risk neutral pricing

Next we consider a dividend process

(10)\[d_{t+1} = g_{t+1} d_t\]

The stochastic growth factor \(\{g_t\}\) is given by

\[g_t = g(X_t), \quad t = 1, 2, \ldots\]


  1. \(\{X_t\}\) is a finite Markov chain with state space \(S\) and transition probabilities
\[P(x, y) := \mathbb P \{ X_{t+1} = y \,|\, X_t = x \} \qquad (x, y \in S)\]
  1. \(g\) is a given function on \(S\) taking positive values

You can think of

  • \(S\) as \(n\) possible “states of the world” and \(X_t\) as the current state
  • \(g\) as a function that maps a given state \(X_t\) into a growth factor \(g_t = g(X_t)\) for the endowment
  • \(\ln g_t = \ln (d_{t+1} / d_t)\) is the growth rate of dividends

(For a refresher on notation and theory for finite Markov chains see this lecture)

The next figure shows a simulation, where

  • \(\{X_t\}\) evolves as a discretized AR1 process produced using Tauchen’s method
  • \(g_t = \exp(X_t)\), so that \(\ln g_t = X_t\) is the growth rate

The code can be found here


To obtain asset prices in this setting, let’s adapt our analysis from the case of deterministic growth

In that case we found that \(v\) is constant

This encourages us to guess that, in the current case, \(v_t\) is constant given the state \(X_t\)

In other words, we are looking for a fixed function \(v\) such that the price-dividend ratio satisfies \(v_t = v(X_t)\)

Staring with the cum-dividend case, we can substitute this guess into (7) to get

\[v(X_t) = 1 + \beta {\mathbb E}_t [ g(X_{t+1}) v(X_{t+1}) ]\]

If we condition on \(X_t = x\), this becomes

\[v(x) = 1 + \beta \sum_{y \in S} g(y) v(y) P(x, y)\]


(11)\[v(x) = 1 + \beta \sum_{y \in S} K(x, y) v(y) \quad \text{where} \quad K(x, y) := g(y) P(x, y)\]

Suppose that there are \(n\) possible states \(x_1, \ldots, x_n\)

We can then think of (11) as \(n\) stacked equations, one for each state, and write it in matrix form as

(12)\[v = {\mathbb 1} + \beta K v\]


  • \(v\) is understood to be the column vector \((v(x_1), \ldots, v(x_n))'\)
  • \(K\) is the matrix \((K(x_i, x_j))_{1 \leq i, j \leq n}\)
  • \({\mathbb 1}\) is a column vector of ones

When does (12) have a unique solution?

From the Neumann series lemma and Gelfand’s formula, this will be the case if \(\beta K\) has spectral radius strictly less than one

In other words, we require that the eigenvalues of \(K\) be strictly less than \(\beta^{-1}\) in modulus

The solution is then

\[v = (I -\beta K)^{-1} {\mathbb 1}\]

Similar reasoning in the ex-dividend case yields

(13)\[v = (I - \beta K)^{-1} \beta K{\mathbb 1}\]


Let’s calculate and plot the price-dividend ratio for the ex-dividend case at a set of parameters

As before, we’ll generate \(\{X_t\}\) as a discretized AR1 process and set \(g_t = \exp(X_t)\)

Here’s the code, including a test of the spectral radius condition

using QuantEcon
using Plots
using LaTeXStrings


n = 25  # size of state space
beta = 0.9
mc = tauchen(n, 0.96, 0.02)

K = mc.p .* exp.(mc.state_values)'

I = eye(n)
v = (I - beta * K) \  (beta * K * ones(n, 1))

    ylabel="price-dividend ratio",

Here’s the figure it produces


Why does the price-dividend ratio increase with the state?

The reason is that this Markov process is positively correlated, so high current states suggest high future states

Moreover, dividend growth is increasing in the state

Anticipation of high future dividend growth leads to a high price-dividend ratio

Asset Prices under Risk Aversion

Now let’s turn to the case where agents are risk averse

We’ll price several distinct assets, including

  • The price of an endowment stream
  • A consol (a type of bond issued by the UK government in the 19th century)
  • Call options on a consol

Pricing a Lucas tree

Let’s start with a version of the celebrated asset pricing model of Robert E. Lucas, Jr. [Luc78]

As in [Luc78], suppose that the stochastic discount factor takes the form

(14)\[m_{t+1} = \beta \frac{u'(c_{t+1})}{u'(c_t)}\]

where \(u\) is a concave utility function and \(c_t\) is time \(t\) consumption of a representative consumer

(A derivation of this expression is given in a later lecture)

Assume the existence of an endowment that follows (10)

The asset being priced is a claim on the endowment process

Following [Luc78], suppose further that in equilibrium, consumption is equal to the endowment, so that \(d_t = c_t\) for all \(t\)

For utility, we’ll assume the constant relative risk aversion (CRRA) specification

(15)\[u(c) = \frac{c^{1-\gamma}}{1 - \gamma} \ {\rm with} \ \gamma > 0\]

When \(\gamma =1\) we let \(u(c) = \ln c\)

Inserting the CRRA specification into (14) and using \(c_t = d_t\) gives

(16)\[m_{t+1} = \beta \left(\frac{c_{t+1}}{c_t}\right)^{-\gamma} = \beta g_{t+1}^{-\gamma}\]

Substituting this into (6) gives the ex-dividend price-dividend ratio formula

\[v(X_t) = \beta {\mathbb E}_t \left[ g(X_{t+1})^{1-\gamma} (1 + v(X_{t+1}) ) \right]\]

Conditioning on \(X_t = x\), we can write this as

\[v(x) = \beta \sum_{y \in S} g(y)^{1-\gamma} (1 + v(y) ) P(x, y)\]

If we let

\[J(x, y) := g(y)^{1-\gamma} P(x, y)\]

then we can rewrite in vector form as

\[v = \beta J ({\mathbb 1} + v )\]

Assuming that the spectral radius of \(J\) is strictly less than \(\beta^{-1}\), this equation has the unique solution

(17)\[v = (I - \beta J)^{-1} \beta J {\mathbb 1}\]

Here’s a plot of \(v\) as a function of the state for several values of \(\gamma\), with a positively correlated Markov process and \(g(x) = \exp(x)\)

The code with all details can be found here


Notice that \(v\) is decreasing in each case

This is because, with a positively correlated state process, higher states suggest higher future consumption growth

In the stochastic discount factor (16), higher growth decreases the discount factor, lowering the weight placed on future returns

Special cases

In the special case \(\gamma =1\), we have \(J = P\)

Recalling that \(P^i {\mathbb 1} = {\mathbb 1}\) for all \(i\) and applying Neumann’s geometric series lemma, we are led to

\[v = \beta(I-\beta P)^{-1} {\mathbb 1} = \beta \sum_{i=0}^{\infty} \beta^i P^i {\mathbb 1} = \beta \frac{1}{1 - \beta} {\mathbb 1}\]

Thus, with log preferences, the price-dividend ratio for a Lucas tree is constant

Alternatively, if \(\gamma = 0\), then \(J = K\) and we recover the risk neutral solution (13)

This is as expected, since \(\gamma = 0\) implies \(u(c) = c\) (and hence agents are risk neutral)

A Risk-Free Consol

Consider the same pure exchange representative agent economy

A risk-free consol promises to pay a constant amount \(\zeta> 0\) each period

Recycling notation, let \(p_t\) now be the price of an ex-coupon claim to the consol

An ex-coupon claim to the consol entitles the owner at the end of period \(t\) to

  • \(\zeta\) in period \(t+1\), plus
  • the right to sell the claim for \(p_{t+1}\) next period

The price satisfies (3) with \(d_t = \zeta\), or

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

We maintain the stochastic discount factor (16), so this becomes

(18)\[p_t = {\mathbb E}_t \left[ \beta g_{t+1}^{-\gamma} ( \zeta + p_{t+1} ) \right]\]

Guessing a solution of the form \(p_t = p(X_t)\) and conditioning on \(X_t = x\), we get

\[p(x) = \beta \sum_{y \in S} g(y)^{-\gamma} (\zeta + p(y)) P(x, y)\]

Letting \(M(x, y) = P(x, y) g(y)^{-\gamma}\) and rewriting in vector notation yields the solution

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

Pricing an Option to Purchase the Consol

Let’s now price options of varying maturity that give the right to purchase a consol at a price \(p_S\)

An infinite horizon call option

We want to price an infinite horizon option to purchase a consol at a price \(p_S\)

The option entitles the owner at the beginning of a period either to

  1. purchase the bond at price \(p_S\) now, or
  2. Not to exercise the option now but to retain the right to exercise it later

Thus, the owner either exercises the option now, or chooses not to exercise and wait until next period

This is termed an infinite-horizon call option with strike price \(p_S\)

The owner of the option is entitled to purchase the consol at the price \(p_S\) at the beginning of any period, after the coupon has been paid to the previous owner of the bond

The fundamentals of the economy are identical with the one above, including the stochastic discount factor and the process for consumption

Let \(w(X_t, p_S)\) be the value of the option when the time \(t\) growth state is known to be \(X_t\) but before the owner has decided whether or not to exercise the option at time \(t\) (i.e., today)

Recalling that \(p(X_t)\) is the value of the consol when the initial growth state is \(X_t\), the value of the option satisfies

\[ w(X_t, p_S) = \max \left\{ \beta \, {\mathbb E}_t \frac{u'(c_{t+1})}{u'(c_t)} w(X_{t+1}, p_S), \; p(X_t) - p_S \right\}\]

The first term on the right is the value of waiting, while the second is the value of exercising now

We can also write this as

(20)\[w(x, p_S) = \max \left\{ \beta \sum_{y \in S} P(x, y) g(y)^{-\gamma} w (y, p_S), \; p(x) - p_S \right\}\]

With \(M(x, y) = P(x, y) g(y)^{-\gamma}\) and \(w\) as the vector of values \((w(x_i), p_S)_{i = 1}^n\), we can express (20) as the nonlinear vector equation

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

To solve (21), form the operator \(T\) mapping vector \(w\) into vector \(Tw\) via

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

Start at some initial \(w\) and iterate to convergence with \(T\)

Here’s a plot of \(w\) compared to the consol price when \(P_S = 40\)


The code with all details can be found here

In large states the value of the option is close to zero

This is despite the fact the Markov chain is irreducible and low states — where the consol prices is high — will eventually be visited

The reason is that \(\beta=0.9\), so the future is discounted relatively rapidly

Risk Free Rates

Let’s look at risk free interest rates over different periods

The one-period risk-free interest rate

As before, the stochastic discount factor is \(m_{t+1} = \beta g_{t+1}^{-\gamma}\)

It follows that the reciprocal \(R_t^{-1}\) of the gross risk-free interest rate \(R_t\) in state \(x\) is

\[{\mathbb E}_t m_{t+1} = \beta \sum_{y \in S} P(x, y) g(y)^{-\gamma}\]

We can write this as

\[m_1 = \beta M {\mathbb 1}\]

where the \(i\)-th element of \(m_1\) is the reciprocal of the one-period gross risk-free interest rate in state \(x_i\)

Other terms

Let \(m_j\) be an \(n \times 1\) vector whose \(i\) th component is the reciprocal of the \(j\) -period gross risk-free interest rate in state \(x_i\)

Then \(m_1 = \beta M\), and \(m_{j+1} = M m_j\) for \(j \geq 1\)


The file provides some functions for computing prices of the Lucas tree, consol and call option described above

Its contents are as follows


@authors: Spencer Lyon, Tom Sargent, John Stachurski


using QuantEcon

# A default Markov chain for the state process
rho = 0.9
sigma = 0.02
n = 25
default_mc = tauchen(n, rho, sigma)

mutable struct AssetPriceModel{TF<:AbstractFloat, TI<:Integer}
    beta :: TF    # Discount factor
    gamma :: TF   # Coefficient of risk aversion
    mc :: MarkovChain  # State process
    n :: TI           # Number of states
    g :: Function      # Function mapping states into growth rates

function AssetPriceModel(;beta::AbstractFloat=0.96, gamma::AbstractFloat=2.0,
                          mc::MarkovChain=default_mc, g::Function=exp)
    n = size(mc.p)[1]
    return AssetPriceModel(beta, gamma, mc, n, g)

Stability test for a given matrix Q.
function test_stability(ap::AssetPriceModel, Q::Matrix)
    sr = maximum(abs, eigvals(Q))
    if sr >= 1 / ap.beta
        msg = "Spectral radius condition failed with radius = $sr"

Computes the price-dividend ratio of the Lucas tree.

function tree_price(ap::AssetPriceModel)
    # == Simplify names, set up matrices  == #
    beta, gamma, P, y = ap.beta, ap.gamma,,
    y = reshape(y, 1, ap.n)
    J = P .* ap.g.(y).^(1 - gamma)

    # == Make sure that a unique solution exists == #
    test_stability(ap, J)

    # == Compute v == #
    I = eye(ap.n)
    Ones = ones(ap.n)
    v = (I - beta * J) \ (beta * J * Ones)

    return v

Computes price of a consol bond with payoff zeta

function consol_price(ap::AssetPriceModel, zeta::AbstractFloat)
    # == Simplify names, set up matrices  == #
    beta, gamma, P, y = ap.beta, ap.gamma,,
    y = reshape(y, 1, ap.n)
    M = P .* ap.g.(y).^(- gamma)

    # == Make sure that a unique solution exists == #
    test_stability(ap, M)

    # == Compute price == #
    I = eye(ap.n)
    Ones = ones(ap.n)
    p = (I - beta * M) \ ( beta * zeta * M * Ones)

    return p

Computes price of a perpetual call option on a consol bond.

function call_option(ap::AssetPriceModel, zeta::AbstractFloat, p_s::AbstractFloat, epsilon=1e-7)

    # == Simplify names, set up matrices  == #
    beta, gamma, P, y = ap.beta, ap.gamma,,
    y = reshape(y, 1, ap.n)
    M = P .* ap.g.(y).^(- gamma)

    # == Make sure that a unique console price exists == #
    test_stability(ap, M)

    # == Compute option price == #
    p = consol_price(ap, zeta)
    w = zeros(ap.n, 1)
    error = epsilon + 1
    while (error > epsilon)
        # == Maximize across columns == #
        w_new = max.(beta * M * w, p - p_s)
        # == Find maximal difference of each component and update == #
        error = maximum(abs, w-w_new)
        w = w_new

    return w

Exercise 1 asks you to make use of this code


Exercise 1

Consider the following primitives

n = 5
P = 0.0125 .* ones(n, n)
P .+= diagm(0.95 .- 0.0125 .* ones(5))
s = [1.05, 1.025, 1.0, 0.975, 0.95]
gamm = 2.0
bet = 0.94
zet = 1.0

Let \(g\) be defined by \(g(x) = x\) (that is, \(g\) is the identity map)

Compute the price of the Lucas tree

Do the same for

  • the price of the risk-free consol when \(\zeta = 1\)
  • the call option on the consol when \(\zeta = 1\) and \(p_S = 150.0\)

Exercise 2

Let’s consider finite horizon call options, which are more common than the infinite horizon variety

Finite horizon options obey functional equations closely related to (20)

A \(k\) period option expires after \(k\) periods

If we view today as date zero, a \(k\) period option gives the owner the right to exercise the option to purchase the risk-free consol at the strike price \(p_S\) at dates \(0, 1, \ldots , k-1\)

The option expires at time \(k\)

Thus, for \(k=1, 2, \ldots\), let \(w(x, k)\) be the value of a \(k\)-period option

It obeys

\[w(x, k) = \max \left\{ \beta \sum_{y \in S} P(x, y) g(y)^{-\gamma} w (y, k-1), \; p(x) - p_S \right\}\]

where \(w(x, 0) = 0\) for all \(x\)

We can express the preceding as the sequence of nonlinear vector equations

\[w_k = \max \{ \beta M w_{k-1}, \; p - p_S {\mathbb 1} \} \quad k =1, 2, \ldots \quad \text{with } w_0 = 0\]

Write a function that computes \(w_k\) for any given \(k\)

Compute the value of the option with k = 5 and k=25 using parameter values as in Exercise 1

Is one higher than the other? Can you give intuition?


Exercise 1

n = 5
P = 0.0125 .* ones(n, n)
P = P .+ diagm(0.95 - 0.0125 .* ones(5))
s = [0.95, 0.975, 1.0, 1.025, 1.05]  # state values
mc = MarkovChain(P, s)

gamma = 2.0
beta = 0.94
zeta = 1.0
p_s = 150.0

Next we’ll create an instance of AssetPriceModel to feed into the functions.

ap = AssetPriceModel(beta=beta, mc=mc, gamma=gamma, g=x -> x)
v = tree_price(ap)
println("Lucas Tree Prices: $v\n")
Lucas Tree Prices: [29.474,21.9357,17.5714,14.7252,12.7222]
v_consol = consol_price(ap, 1.0)
println("Consol Bond Prices: $(v_consol)\n")
Consol Bond Prices: [753.871,242.551,148.676,109.251,87.5686]
w = call_option(ap, zeta, p_s)

Exercise 2

Here’s a suitable funciton:

function finite_horizon_call_option(ap::AssetPriceModel,

    # == Simplify names, set up matrices  == #
    beta, gamma, P, y = ap.beta, ap.gamma,,
    y = y'
    M = P .* ap.g.(y).^(- gamma)

    # == Make sure that a unique console price exists == #
    test_stability(ap, M)

    # == Compute option price == #
    p = consol_price(ap, zeta)
    w = zeros(ap.n, 1)
    for i in 1:k
        # == Maximize across columns == #
        w = max.(beta * M * w, p - p_s)

    return w
lines = []
labels = []
for k in [5, 25]
    w = finite_horizon_call_option(ap, zeta, p_s, k)
    push!(lines, w)
    push!(labels, "k = $k")
plot(lines, labels=reshape(labels,1,length(labels)))

Not surprisingly, the option has greater value with larger \(k\). This is because the owner has a longer time horizon over which he or she may exercise the option.