Consider the N period binomial model. Let 1 ≤m ≤N −1 and K > 0 be given. A chooser option is a contract sold at time zero that confers on its owner the right to receive either a call or a put at time m.

1 Theoretical Problems

1. This question presents an analogy of the Capital Asset Pricing Model
(CAPM) for the discrete models studied in class.

Consider a complete,arbitrage free single period model with payoff matrix A, price vector z, and risk-neutral probability vector q. Assume that there is a risk-free bond and the interest rate is r. Let p be the vector of real-world probabilities and let ζ = dQ
dP be the vector with entries ζi = qi pi

For any payoff vector Y , we use the notation EP[Y ] = pT Y , EQ[Y ] = qT Y . For
any portfolio x with nonzero price, define its return in scenario i to be:
Rxi = (Ax)i
zT x −1
(a) Using the notation cov(X,Y ) = ∑
i piXiYi −∑
i piXi ·∑
i piYi,
show that for any portfolio x
EP[Rx] −r = −cov(Rx,ζ) (1)

(b) Let x∗be the portfolio that replicates ζ and let R∗be its return and V ∗= zT x∗be its initial price. Show that for any portfolio x:
EP[Rx] −r = cov(Rx,R∗)
var(R∗) (EP[R∗] −r)
where the variance and covariance are calculated under the real- world probabilities.

2. (a) Consider the binomial model from the slides, with Sd < S0(1+r) < Su. Show that there exists a vector of probabilities ̃q satisfying
̃qu + ̃qd = 1, ̃qu, ̃qd > 0, and:
1
S0= ̃qu 1+r
Su + ̃qd 1+r
Sd (2)
Furthermore, show that the arbitrage-free price of any derivative security Y is:
S0
(
̃qu
Yu
Su
+ ̃qd
Yd
Sd
)
(3)
1

(b) We say that we are using an asset as “num ́eraire” when we express the value of a security in terms of the number of units of the asset that the security is worth, and then select a probability vector such that the current values of all securities in these units are equal to their expected future values (also measured in these units). What
is the usual num ́eraire in the binomial model (i.e. what did we use in the slides)? What did you use as the num ́eraire in part a)?

3. Give an example of an incomplete market, and an unattainable payoff in that market whose price bounds based solely on the absence of arbitrage are −∞< Price of Payoff < ∞.

4. Consider the N period binomial model. Let 1 ≤m ≤N −1 and K > 0 be given. A chooser option is a contract sold at time zero that confers on its owner the right to receive either a call or a put at time m.

The owner of the chooser may wait until time m before choosing. The call or put chosen expires at time N with strike price K. Show that the time-zero price of a chooser option is the sum of the time zero price of a put, expiring at time N and having strike price K and a call, expiring at time m and having strike price K (1+r)N−m

5. Consider the following payoff matrix and price vector.
A =


10 10 30
10 20 20
10 30 5

, z =


10
20
λ



(a) For what values of λ is there no arbitrage in the market?

(b) Is the market in part (a) complete? (You may suppose if you want that λ is such that there is no arbitrage.)

2 Computational Problem

1. Consider an arbitrary single period model defined by a payoff matrix A (including a column for a risk free bond defining the interest rate r) and a price vector z. Write functions for this model which determine:

(a) Whether there is arbitrage in the market (see hint below).

(b) If the market is complete.

(c) Whether a given payoff Y is attainable in the market. If so, return its price. If not, return the arbitrage free bounds on its price. Hint: One method for determining if there is arbitrage in the market is to solve the following linear program:
maxq,λ λ
AT q = (1 + r)z
qi ≥λ i = 1,…,M
where 1 is a vector of ones. The first constraint is the linear system defining risk neutral probabilities. If the optimal value of the optimiza- tion problem is strictly positive (or the problem is unbounded), then there is a solution to this linear system with all entries positive (i.e. a vector of risk-neutral probabilities), otherwise not. Then apply the first Fundamental Theorem of Asset Pricing

Consider the N period binomial model. Let 1 ≤m ≤N −1 and K > 0 be given. A chooser option is a contract sold at time zero that confers on its owner the right to receive either a call or a put at time m.