MATH3075/3975
Financial Derivatives
School of Mathematics and Statistics
University of Sydney
Semester 2, 2020
Tutorial sheet 4
Background: Section 2.2 – Single-Period Market Models.
Exercise 1 Consider the elementary market modelM = (B, S) on a sample
space Ω = {ω1, ω2} with P(ω1) = p ∈ (0, 1). We assume that S0 > 0 and
0 < d < 1 + r < u.
(a) Find the probability measure P̂ such that EP̂(B̂T ) = B̂0 where the
process B̂ is defined by B̂t = Bt/St for t = 0, 1. Compute the Radon-
Nikodym density L of P̂ with respect to the martingale measure P̃ and
show directly that EP̃(L) = 1.
(b) Let X = g(ST ) be any contingent claim. Show that the price π0(X)
satisfies
π0(X) = S0 EP̂
(
X
ST
)
.
(c) Consider the put option with the payoff PT (K) = (K − ST )+ for some
K > 0. Show that the price P0(K) admits the following representation
P0(K) = K(1 + r)
−1 P̃(ST < K)− S0 P̂(ST < K).
Find an analogous representation for the price C0(K) of the call option
with strike K.
(d) Show that the extended model Me = (B, S, P (K)) is arbitrage-free,
in the sense of Definition 2.2.3 from the course notes. Here P (K) =
(P0(K), PT (K)) is the price process of the put option for some fixed
strike K > 0.
(e) Let a strike K such that S0d < K < S0u be fixed. Consider the modified market model N = (B,P (K)) where P (K) is now traded at time 0 at the price P0(K). Does the price of an arbitrary claim X computed in N = (B,P (K)) coincides with its arbitrage price computed in the original model M = (B, S)? In particular, find the arbitrage price at time 0 for the claim X = ST in the model N . 1 Exercise 2 Verify the equality (see Section 2.2) V̂t := Vt Bt = ( x− n∑ j=1 φjS j 0 ) + n∑ j=1 φjŜ j t (1) for t ∈ {0, 1} with B0 = 1 and B1 = 1 + r, and derive the equality Ĝ1(x, φ) = n∑ j=1 φj∆Ŝ j 1 = n∑ j=1 φj(Ŝ j 1 − Ŝ j 0) (2) where Ĝ1(x, φ) := V̂1(x, φ)− V̂0(x, φ). Exercise 3 Consider the market modelM = (B, S) with k = 3, n = 1, r = 1 9 , S0 = 5 and the random stock price S1 given by the table ω1 ω2 ω3 S1 60 9 40 9 30 9 Find the class M of all risk-neutral probability measures for this market model by making use of Definition 2.2.4. Exercise 4 We consider the market model M = (B, S1, S2) introduced in Example 2.2.1 in the course notes but with k = 4 and the stock prices in state ω4 given by S 1 1(ω4) = 20 9 and S21(ω4) = 120 9 . The interest rate equals r = 1 9 . Stock prices at time t = 0 are given by S10 = 5 and S 2 0 = 10, respectively, and stock prices at time t = 1 are given in the following table ω1 ω2 ω3 ω4 S11 60 9 60 9 40 9 20 9 S21 40 3 80 9 80 9 120 9 (a) Compute explicitly the random variables V1(x, φ), G1(x, φ), V̂1(x, φ) and Ĝ1(x, φ). (b) Does G1(x, φ) (or Ĝ1(x, φ)) depend on the initial endowment x? Exercise 5 (MATH3975) Consider again the market modelM = (B, S1, S2) introduced in Exercise 4. (a) Give an explicit representation for the linear space W ⊂ R4. (b) Find explicitly the linear space W⊥ ⊂ R4. (c) Is the market model M = (B, S1, S2) arbitrage free? (d) Find the class M of all risk-neutral probability measures for M using the equality M = W⊥ ∩ P+. Exercise 6 (MATH3975) Give a proof of Proposition 2.2.1. 2