Financial Derivatives (Mainstream)
Due by 11:59 p.m. on Sunday, 11 September 2022
1. [12 marks] Single-period multi-state model. Consider a single-period market
model M = (B, S) on a finite sample space ? = {肋1, 肋2, 肋3}. We assume that the
money market account B equals B0 = 1 and B1 = 4 and the stock price S = (S0, S1)
satisfies S0 = 2.5 and S1 = (18, 10, 2). The real-world probability P is such that
P(肋i) = pi > 0 for i = 1, 2, 3.
(a) Find the class M of all martingale measures for the modelM. Is the market
modelM arbitrage-free? Is this market model complete?
(b) Find the replicating strategy (?00, ?10) for the contingent claim X = (5, 1,?3)
and compute the arbitrage price pi0(X) at time 0 through replication.
(c) Compute the arbitrage price pi0(X) using the risk-neutral valuation formula
with an arbitrary martingale measure Q from M.
(d) Show directly that the contingent claim Y = (Y (肋1), Y (肋2), Y (肋3)) = (10, 8,?2)
is not attainable, that is, no replicating strategy for Y exists inM.
(e) Find the range of arbitrage prices for Y using the class M of all martingale
measures for the modelM.
(f) Suppose that you have sold the claim Y for the price of 3 units of cash. Show
that you may find a portfolio (x, ?) with the initial wealth x = 3 such that
V1(x, ?) > Y , that is, V1(x, ?)(肋i) > Y (肋i) for i = 1, 2, 3.
2. [8 marks] Static hedging with options. Consider a parametrised family of
European contingent claims with the payoff X(L) at time T given by the following
expression
X(L) = min
(
2|K ? ST |+K ? ST , L
)
where a real number K > 0 is fixed and L is an arbitrary real number such that
L ≡ 0.
(a) Sketch the profile of the payoff X(L) as a function of the stock price ST and
find a decomposition of X(L) in terms of terminal payoffs of standard call and
put options with expiration date T . Notice that the decomposition of the payoff
X(L) may depend on values of K and L.
(b) Assume that call and put options are traded at time 0 at finite prices. For
each value of L ≡ 0, find a representation of the arbitrage price pi0(X(L)) of
the claim X(L) at time t = 0 in terms of prices of call and put options at time
0 using the decompositions from part (a).
(c) Consider a complete arbitrage-free market modelM = (B, S) defined on some
finite state space ?. Show that the arbitrage price of X(L) at time t = 0 is a
monotone function of the variable L ≡ 0 and find the limits limL↙3K pi0(X(L)),
limL↙﹢ pi0(X(L)) and limL↙0 pi0(X(L)) using the representations from part (b).
(d) For any L > 0, examine the sign of an arbitrage price of the claim X(L) in any
(not necessarily complete) arbitrage-free market modelM = (B, S) defined on
some finite state space · Justify your answer.