Mich 2010
Mathematical Finance: Short Solutions
3
15. Apply the Law of One Price to some suitable portfolios. The inequalities for the put option are (KerT S )+ P KerT . For the nal inequality compare portfolios containing put options and some cash. 16. A
Mathematical Finance
1. (a) (1 + 0.1/2)2 = 1.1025 so the eective rate is 0.1025. (b) (1 + 0.1/4)4 1 0.1038. (c) e0.1 1 0.1052. 2. (a) 6.93 years (b) 28.41 years. Time to reach D is t = log / log(1 + r ).
3. P (r ) = 0 is equivalent to a quadratic in r . S
Revision material on probability
Stock market prices are unpredictable and are modelled with random walks (or Brownian motion for continuous price models). The following material from Core A Probability will be assumed. Look in your course notes or a book
Revision for Mathematical Finance
1. For n > a, log(1 + a/n) = log(n + a) log n = n dx/x a(1/(n + a), 1/n) using the obvious bounds on the integrand and noting that the interval has signed length a. As na/(n + a) a as n it follows that n log(1 + a/n) a as
Revision for Mathematical Finance
1. Show that (1 + a/n)n ea as n , where a 0 is constant. Hint: Analyse the behaviour of (1 + a/x)x as x by taking log and applying Taylors theorem (treating the remainder with professional care!). Alternatively show direc
Q46
Stock price 100 1+r= 120 80 140 100 60 160 120 80 50 K = 90 Note! Top final price must be more than 140(1+r) 1.1
Risk-neutral p 0.750
0.800 0.700
0.850 0.750 0.533 Return from immediate exercise 0 0 10 (as we definitely wait at the first step). 0 0 30
Michlmas 2010
Mathematical Finance: Questions
6
40. Find the no-arbitrage price of a (K, T ) call option at time 0 on a stock that pays dividends dS (Ti ) at times 0 < T1 < T2 < T . 41. Use the put-call parity relation for options on a non-dividend paying
Michlmas 2010
Mathematical Finance: Questions
4
20. A random trial has three possible outcomes with odds o1 = 1, o2 = 2, o3 = 5. Is there a betting scheme based on which outcome occurs that results in a sure win? Suppose that bets on pairs of the outcomes
Mathematical Finance
1. What is the eective annual interest rate when the nominal interest rate of 10% is compounded (a) twice yearly (b) quarterly (c) continuously? 2. You deposit amount D in a bank which pays interest at nominal rate r . How many years
Pricing on binary trees rst steps
MF lec Suppose that a stock price changes in discrete time steps along a binary tree up to time T . 28/10 That is, for any price S at time t T 1 there are two possible time t + 1 prices which we denote by uS and dS , wher
Math Finance summary
Mich 10
10
5
Approximation methods
Stock brokers need to update prices for many millions of options as stock prices change. In practice they use approximation techniques of three basic types: calculations on nite binomial trees; simul
Math Finance summary
Mich 10
7
Asset price behaviour The cornerstone of a great many models in mathematical nance H.6 is the assumption that asset prices S (t), t 0 (at least for non-dividend paying stocks) are distributed as a geometric Brownian motion w
Brief Summary of Math Finance 1 Fundamentals
Stock markets around the world sell a great variety of dierent products. These include shares in companies, commodities, futures, currencies and options. We will consider a simplied market without commodities o
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A theorem of the alternative
The separating hyperplane theorem has a variety of applications. Amongst them is the very interesting result about existence of solutions to linear systems which we can use to determine conditions when arbitrages cannot exis
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Convexity
f x + (1 )y f (x) + (1 )f (y )
We dened convex functions f : I R in the lectures. These satisfy the condition for all x, y I (any real interval) and [0, 1]. The regularity properties of convex functions can demonstrated by studying the slopes
Mich 2010
Mathematical Finance: Short Solutions
4
29. We have K = 42, S (0) = 40, T = 1/3, = 0.15 and = 0.24 so 2 /2 = 0.1212. We have to calculate P(S (1/3) > K ). We know that W log(S (1/3)/S (0) N (0.0404, 0.0192) and consequently P(S (1/3) > K ) = = P