MATH97009 MATH97101 An Introduction to Option Pricing 2021
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MATH97009 MATH97101
Mathematical Finance: An Introduction to Option Pricing
2021
Question 1 (Total: 20 marks) Consider the probability space Ω = {ω1,ω2,ω3}, with the probability P such that P({ω}) = 1/3 for every ω ∈ Ω. Define the random variables
ω |
1 |
2 |
3 |
S1 (ω) |
6 |
8 |
10 |
X1 (ω) |
−6 |
4 |
14 |
Y1 (ω) |
16 |
6 |
2 |
Consider the one-period trinomial model of the market (B,S) made of a bond B with initial price 1 (all prices in a fixed currency, say £), and interest rate r = 1, a stock whose initial price is S0 = 4, and whose final price is S1 . Answer the following questions about the two illiquid derivatives with payoffs X1,Y1 in the market (B,S).
(a) Is this model free of arbitrage?
(b) Is X1 replicable?
(c) Is Y1 replicable ?
(d) What is the set of arbitrage-free prices of X1? (e) What is the set of arbitrage-free prices of Y1?
(3 marks) (2 marks) (2 marks) (4 marks) (4 marks)
(f) Consider now a derivative with payoff Z1 which is not replicable in the (B,S) market. Now (5 marks) enlarge the (B,S) market, by assuming that Y is traded at the arbitrage-free price Y0 at time
0. Is the set of arbitrage-free prices of Z1 in the market (B,S,Y) a singleton? Explain your reasoning.
Question 2 (Total: 20 marks)
Suppose a British investor can:
1. deposit £ in a bank at the (domestic) interest rate r = 1/2
2. buy or sell $ (by paying/getting paid in £) at any time n with exchange rate Sn
3. deposit $ in a bank at the (foreign) interest rate f
The exchange rate Sn is defined as the number of units of £ needed to buy one unit of $ at time n, and is assumed to follow the one period binomial model with S0 = 2, u = 2,d = 1/2. Answer the following questions and justify carefully your answers.
(a) Explain why the formula (3 marks)
V := (V0 − hS0)(1 + r) + h(1 + f)S1
described the total wealth V1 := V (in £) at time 1 of the investor whose initial capital (in
£) is V0 = x, and at time 0 buys $ h ∈ R, and then deposits his £ and his $ in the banks. (b) Assume from now on that r = 1/2. For what values of f is the above model arbitrage-free? (4 marks) (c) From now on let f = 1. For what value of p˜ = Q(H) ∈ (0, 1) does the RNPF (Risk Neutral (4 marks)
Pricing Formula) V0 = EQ[V1/(1 + r)] provide the (only) arbitrage-free price of the payoff V1 ,
for any value of V1?
(d) Consider a forward contract on the exchange rate. In other words, consider the agreement (5 marks) which has no initial cost, and which states that its buyer will buy one $ at the price P0 (called the forward exchange rate) at time 1, where the constant P0 is determined by asking that the arbitrage-free price F0 of the forward contract is zero. What is the value of P0?
(e) What is the replicating strategy h of a call option (on the exchange rate) with strike price 2? (4 marks)
Question 3 (Total: 20 marks) In the framework of the N-period binomial model with constant parameters S0 = 6,u = ,d = ,r = 0, let S = (Sn)n(N)=0 be the stock price process,
Qn := S + (Si+1 − Si)2
its quadratic variation up to time n, and Yn := , where n = 0,...,N. Consider the option
denotes the process of coin tosses X which generates (Sn)n, and we take as filtration F the natural
(a) Use the risk-neutral pricing formula to express Vn in terms of Vn+1 . (b) Express Yn+1 as a function of Yn and Cn+1 := .
Sn independent of Fn under Q?
Is Y a Q-Markov process?
(3 marks) (5 marks) (2 marks) (4 marks)
(e) Work by backward induction to show that, for every n = 0,...,N , Vn admits the representation (6 marks) Vn = Svn(Yn), where vn : R → R, n = 0,...,N, are (deterministic) functions. Write explicitly vN and an explicit formula to express vn in terms of vn+1 for n = 0,...,N − 1.
Question 4 (Total: 20 marks) I bought at time 0 a derivative which gives me the right to purchase, at price c1 > 0 and time t1 > 0, a call option (on an underlying stock S) with expiration t2 > t1 and strike price K2 . Assume that the stock price S = (St)0≤t≤t2 follows the Black-Scholes model, and denote with c(x,,K) the price at time t of a call option on S with expiry T := t + > t and strike K > 0, if St = x (it can be proved that c does not depend on t, it only depends on ). Answer the following questions and justify carefully with either proofs or counterexamples.
(a) Prove that, for any value of (,K), the function x →7 c(x,,K) ∈ R is strictly increasing for (3 marks) x ∈ (0, ∞). Compute its limits as x ↓ 0 and as x ↑ ∞ .
(b) Write an equation whose solution is the value of b1 > 0 such that at time t1 if St1 > b1 I should (4 marks) exercise the derivative, if St1 < b1 I should not exercise it, and if St1 = b1 it is irrelevant what I
do. Prove that such equation admits one and only one solution. Warning: do not try to solve
the equation: it is transcendental, and so it does not have a closed-form analytic solution .
(c) Write a formula, which involves the price c1 and the pricing function c, for the value of the (3 marks) derivative at time t1 .
(d) Write down a formula for the function f = f(x,y) such that f(St1 ,St2 ) is the payoff of the (5 marks) option at time t2 .
(e) Consider a derivative G on S with payoff Gt2 = g(St1 ,St2 ) at time t2 , for some function (5 marks) g = g(x,y). Obtain an explicit formula for the price G0 at time 0 of the derivative G; this
formula must be of the form
(g ◦ h)(x,y) a(x,y) dxdy,
where h,a are functions which you have to determine; explain your reasoning.
Question 5 (Total: 20 marks) On a finite sample space Ω = {ωi}i=1,...,n endowed with some probability P s.t. P(ωi) > 0 for all i, consider a one-period arbitrage-free market model where the bank account has interest rate r = 0, and so we model it with the process B0 = B1 = 1, and there are two stocks S1,S2 . If we express the portfolio using the number h1,h2 ∈ R of shares of stocks S1 and S2 , and the amount of cash c ∈ R in the bank account (not of the initial capital x), then at time t = 0, 1 the wealth Vtc,h relative to (c,h) is given by
V = c + h · S0, V1c,h = c(1 + r) + h · S1 , (1)
where h · St denoted the usual dot product between h and St. For the non-replicable derivative with payoff X1 , we consider the problem of finding the smallest initial capital p of a portfolio (c,h) super-replicating X1 P a.s., i.e.
p := min{V : (c,h) ∈ R × R2 satisfies V1c,h(ωi) ≥ X1 (ωi) for all i}, (2)
and its dual linear program, i.e.
d := max{EQ(X1) : Q ∈ M}, where M := {Q probability on Ω : EQ(S − S) = 0, j = 1, 2} (3) is the set of martingale measures.
We denote by c∗ ,h∗ any trading strategy such that V0c∗ ,h∗ = p and V1c∗ ,h∗ ≥ X1 P a.s., and with Q∗
∗
(a) Is p an arbitrage-free price for X1? If it is not, find an arbitrage. (3 marks)
(b) Is it true that V1c∗ ,h∗ = X1 Q∗ a.s.? (3 marks)
(c) Is Q∗ equivalent to P? (2 marks)
(d) Assume from now on that it is not possible to borrow any money from the bank. Define what (2 marks) an arbitrage is in such a market, using formulas.
(e) Consider again the problem of finding the smallest initial capital p¯ of the portfolio (c,h) super- (4 marks) replicating X1 (but now without being able to borrow). Formulate this problem in a way analogous to (2), and formulate its dual problem in a way analogous to (3), and call the respective optimal values p¯ and d¯.
(f) Are p¯ and d¯ always equal? (3 marks)
(g) Is it true that p¯ ≥ p, for any choice of X1 and any choice of model (B,S1,S2)? What about (3 marks) p¯ > p?
2022-05-14