Mathematical Finance: An Introduction to Option Pricing
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Assessed CW, due 21/11 at 1pm BST
Option Pricing
This document contains 2 questions.
1. [default,O3e]
Consider the probability space Ω = fω1 , ω2 , ω3 g, with the probability P such that P(fωg) = 1/3 for every ω 2 Ω . Deine the random variables
ω |
ω 1 |
ω2 |
ω3 |
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 ixed currency, say £), and interest rate r = 1, a stock whose initial price is S0 = 4, and whose inal price is S1 . In the questions (a-e) we consider the market model (B, S), and the random variables X1 and Y1 represent the payofs of two illiquid derivatives (with S as underlying); in question (f) Y1 represents instead the value at time 1 of a traded asset, which is part of the market model (B, S, Y).
(a) Is the market model (B, S) free of arbitrage?
A. No B. Yes
(b) Is X1 replicable?
A. No B. Yes
(c) Is Y1 replicable?
A. No B. Yes
(d) What is the set of arbitrage-free prices of X1?
A. (2,4) B. [2,4] C. f4g D. f2g E. None of the above
(e) What is the set of arbitrage-free prices of Y1?
A. (3,9/2) B. (6,9) C. f3g D. f6g E. None of the above
(f) Now enlarge the (B, S) market, by assuming that Y is traded at the arbitrage-free price Y0 at time 0. If an illiquid derivative has a payof Z1 which is not replicable using only bonds and stocks (i.e. Z is not replicable in the (B, S) market), does Z1 have a unique arbitrage-free price in the market (B, S, Y)?
A. Not enough info to answer (it depends on Z1) B. No C. Yes
2. [default,O21b]
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 deined as the number of units of £ needed to buy one unit of $ at time n 2 f0, 1g, and is assumed to follow the one-period binomial model: we assume that S0 = 2, and S1 takes the values 4 and 1.
(a) The formula
V1(x;h) := (V0 - hS0)(1 + r) + h(1 + f)S1
describes the total wealth V1 := V1(x;h) (in £) at time 1 of the investor whose initial capital (in £) is V0 = x, and at time 0 buys $h 2 R, and then deposits his £ and his $ in the banks. What is the value (in £) of the $ held by the investor?
A. h(1 + f)(S1 - S0)
B. h(1 + f)S1 - hS0(1 + r)
C. h(1 + f)S1
D. None of the above
(b) Assume from now on that r = 1/2. For what values of f is the above model arbitrage-free? A. f 2 (- , 2) B. f 2 (0, 2) C. f 2 [- , 2] D. f 2 [0, 2] E. None of the above
(c) From now on let f = 1. For what value of ˜(p) = Q(H) 2 (0, 1) does the RNPF (Risk Neutral Pricing Formula)
V0 = EQ [V1/(1 + r)] provide the (only) arbitrage-free price of the payof V1, for any value of V1?
Hint: of course the value of˜(p) depends on f. To fnd it, proceed as done in the lecture titled ‘Formulas for the binomial model’ .
A. ˜(p) = B. ˜(p) = C. ˜(p) = D. None of the above
(d) Consider a forward contract on the exchange rate. In other words, consider the agreement which has no initial cost, and which states that its buyer will buy one s 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?
Hint: frst determine what is the value F1 of the forward contract at time 1 .
A. P0 = S0
B. P0 = (1 + r)S0
C. P0 = (1 + r)(1 + f)S0
D. P0 = S0
E. None of the above
(e) What is the replicating strategy h of a call option (on the exchange rate) with strike price 2? A. h = 2/3 B. h = 1 C. h = 1/3 D. None of the above
2023-12-22