CS代考 Prof. M. 659 – Problem 4 Fall 2022

Prof. M. 659 – Problem 4 Fall 2022

given: Sunday November 6: due: Thursday November 17.

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Subject: Inequalities on Option Prices and Two-period Option Pricing

Questions of the kind that follow will appear on the Midterm Exam: they require the use of

simple no-arbitrage arguments which the problems that follow will teach you to use. You will find

MM Lectures 9 and 10 useful. We also solve for a contingent market equilibrium of an economy

with log utility functions.

1. Consider an interval of time [0, T ] and the current date t ∈ [0, T ]: we think of [0, T ] as a relatively
small interval of time in the subset [0,∞) during which we assume there are no dividend payments
on the equity. For t ∈ [0, T ] let t denote the current time and let Qet denote the known current
price of equity: we let Q̃eT denote the random price of equity at date T . Let’s think of Ṽ

the random payoff of equity at date T .

The other securities traded are the riskless bond and options on the stock expiring at date T .

Let Qbt denote the price of the bond at date t and let V
T denote its payoff at date T with V

We assume the interest rate is constant so that Qbt is defined in terms of the interest rate r by the

relation Qbt = e

We consider put and call options with exercise price K, also called the strike price: a European

call option is a contract which gives the right to buy the stock at date T at the price K; a European

put option is a contract which gives the right to sell the stock at date T at the price K. An

American call option is a contract which gives the right to buy the stock at any date at or before

date T at the price K, while an American put option is a contract which gives the right to sell the

stock at any date at or before date T at the price K. Let Qct and Q
t denote the date t prices of the

European call and the put options, and let Qcat and Q
t denote the date t prices of the American

call and the put options, all with the common exercise price K.

(a) On a graph draw the payoffs of the call and put options as a function of the equity price

realized at date T . On the same graph draw the payoff of the equity as a function of its

date T price (i.e. the diagonal). It will be convenient to use this graph to give a geometric

interpretation to questions b, c, d.

(b) Prove the put-call parity relation for European options. Give a geometric interpretation of

this result using the figure in (a).

(c) Show that the following put-call inequality holds for American options

Qet −K ≤ Q

(d) Show that if K1 ≤ K2 then Qc1t ≥ Qc2t .

(e) Show that Qbt(K2 −K1) ≥ Qc1t −Qc2t .

(f) Show that the price of a call option is a convex function of its striking price, i.e. if K1 and

K2 are the striking prices of two different options and if a third option has a striking price

Kλ = λK1 + (1 − λ)K2 then Q

t + (1 − λ)Q

t with obvious notation. [Hint: find

a portfolio which gives a larger payoff than the option with striking price Kλ]. Explain the

intuition for the result.

(g) Consider the special case where T = 1 and there are only two periods, t = 0, 1. If there are

3 or more states of nature at date 1 then given only the equity and the bond, the markets

are incomplete and it is not possible to derive a unique price Qc0 for a call option. Prove the

following bounds for the price of the call option

, 0} ≤ Qc0 ≤ Q

(h) Note that the inequality in (g) may only imply rather weak bounds on the price of the call,

as the following example shows. Suppose the price of equity is 100 at date 0, 120 in state

1, 110 in state 2, 100 in state 3, let the interest rate be 7 % and suppose the strike price is

K = 107: find the interval in (g).

(i) Use the the more precise information implied by the equivalence of no-arbitrage and the

existence of state prices to find a more precise interval in which the price of the option must

lie. Show that the length of this interval is about 1/50 of the length of the interval given in

(h). Explain.

2. Option pricing with 2 states and 2 periods. Consider the model above in which there are S = 2

states of nature at date 1. We think of these as primitive shocks which affect the price of the stock

at date 1: the first is “up”, the second is “down” (good news and bad news). Let R = 1 + r denote

the return on the riskless bond with payoff V b = (1, 1) at date 1 and let u and d denote the returns

on equity in the “up” and “down” states respectively, with 0 < d < u. Let the date 0 price and the two prices at date 1 for the equity contract be given by Qe0 = Q, Q Since no dividends are paid on the equity, the payoff matrix at date 1 for the bond and the stock is given by V = [V b V e] = (a) Show that there are no arbitrage opportunities if and only if d < R < u. Interpret this condition. (b) Exhibit an arbitrage opportunity when (a) is violated. (c) In what follows assume that (a) holds. Find the vector of state prices π = (πu, πd) for the two states at date 1. (d) Find the price Qc at date 0 of the call option on the stock with exercise price K. Explain the idea underlying the method you are using: try to be as clear and thorough as you can! (e) Show that the formula in (d) can be written as , with µu > 0, µd > 0, and µu + µd = 1

How do you interpret this expression?

(f) Here’s another way of pricing the call. Find the portfolio (∆, B) of the stock and the bond

which replicates the date 1 payoff of the option. Use this to deduce the price Qc: check that

it coincides with what you found in (d).

(g) Here’s yet another way of pricing the call ! Show that there is a portfolio which consists of

buying ∆ units of stock and going short one unit on the call option which generates a riskless

income stream at date 1: use this hedge to price the call. Give the intuition underlying the

3. Suppose the equity price is Qe0 = 40 and is expected to go up by 10% or down by 10% for each of

the next two three-month periods. Suppose the interest rate is known to be 12% per annum with

continuous compounding for each period. Find the value of a six-month European put option with

strike price K = 42, and the value of a six-month American put option with the same strike price.

4. In question 5 below we will calculate a contingent market equilibrium: to this end we begin by

deriving a preliminary result on demand functions of agents with log utility functions vi. There is

a simple piece of gymnastics that all of you should know and that only needs to be done once: for

ever after life is simple. Consider a one period economy E(|RL, u, ω) with L goods and with spot
markets for these goods. Suppose that an agent has a so-called utility function for

bundles of the L goods i.e. v(x1, . . . , xL) = γ1 log(x1) + . . . + γL log(xL). Suppose also that the

agent has a vector of initial endowments of the L goods, ω = (ω1, . . . , ωL) ∈ |RL+ and faces spot

prices for the goods p = (p1, . . . , pL)� 0. Show that the agent’s demand function (i.e. the solution
of his utility maximizing problem over his budget set—it should be clear what his budget set is in

this context) is given by f(p, pω) = (f1(p, pω), . . . , fL(p, pω)) with

f`(p, pω) =

, ` = 1, . . . , L

Give a simple interpretation of this result and explain the proportion of his income that the agent

spends on each good.

5. Now lets calculate the contingent market equilibrium of a stochastic economy in which agents

have log preferences. Consider a one-good two period-economy E(|RS+1, u, ω) in which agents have
log Bernouilli utility functions

ui(xi) = log(αi + x

ρs log(αi + x
s), αi ∈ |R, i = 1, . . . , I

where δ is the common discount factor of the agents, and 0 < δ ≤ 1. Suppose agents can buy and sell on contingent markets: let π = (1, π1, . . . , πS) denote the vector of present-value prices. Define ρ̃ = (ρ̃0, ρ̃1, . . . , ρ̃S) by ρ̃0 = 1, ρ̃s = δρs, s = 1, . . . , S, and ∆ = s=0 ρ̃s. Let 1̃ = (1, 1, . . . , 1) ∈ |RS+1. In this problem we will be a bit sloppy about the non-negativity constraints for consumption, and assume that the admissible consumption for agent i in any state is the set of ξ ∈ |R such that αi + ξ > 0.

(a) Using the change of variable Xis = αi + x
s, s = 0, . . . , S and question (4), show that agent

i’s demand function is given by

π(ωi + αi1̃)

− αi, s = 0, 1, . . . , S, i = 1, . . . , I

Interpret.

(b) Derive the aggregate excess demand function Z : |RS+1++ → |R
S+1 defined by

(f i(π, πωi)− ωi)

(c) Let S = 3 (three states at date 1). Explain why we only need to solve the equations

Zs(π, ω) = 0, for s = 1, 2, 3, to find the equilibrium present-value prices. [Hint: use the

fact that every agent satisfies his budget equation: add them and then notice that this im-

plies that the aggregate excess demands across the states are not independent. This is a very

important property to understand.]

(d) Find the equilibrium prices for the case (c) and show that they can be expressed as simple

functions of the aggregate output w =

i, the aggregate coefficient α =
i=1 αi, the

discount factor δ, and the vector of probabilities ρ.

(e) Show that the equilibrium consumption of agent i is of the form

x̄i = b̄iw + āi1̃ where b̄i =
π̄(ωi + αi1̃)

π̄(w + α1̃)

Find āi. Clearly
i b̄i = 1. Check that

i āi = 0.

(f) Let T i(ξ) denote the risk tolerance of agent i defined by

where Ai(ξ) is the risk aversion of the agent (defined in Problem Set#1). Show that for the

log Bernouilli functions

T i(ξ) = αi + ξ

What does this imply about the way agents can differ in their risk tolerance, for this class of

utility functions? Show that if i 6= j are two agents with ωi = ωj and αi < αj , then b̄i < b̄j and āi > āj in (e). Interpret.

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