Finance Mathmatics

Stochastic Calculus in Finance

Homework 1

Page 98 of the text, Problems 2, 4, 6, 8, 10, 12, 14

8)

(a) Determine Ω, F, and P for 3 tosses of a fair coin.

(b) Let X be the random variable that counts the total number of heads. What is the smallest σ-field G with respect to which X is measurable? What is the number of elements and atoms of this σ-field

(c) Find E[1𝐴 | G], where 𝐴 is the event that the first toss is a head.

9)

Consider the space of two coin tosses Ω2 = {HH , HT, TH, TT} and let stock prices be given by

S0 = 4, S1(H) = 8, S1(T) = 2, S2(HH) = 16,

S2(HT) = S2(TH) = 4, S2(TT) = 1.

Consider two probability measures given by

𝑃1 (HH) = 1/4, 𝑃1 (HT) = 1/4, 𝑃1 (TH) = 1/4, 𝑃1 (TT) = 1/4,

𝑃2 (HH) = 4/9, 𝑃2 (HT) = 2/9, 𝑃2 (TH) = 2/9, 𝑃2 (TT) = 1/9,

Define the random variable X = 1 if S2 = 4 otherwise 0.

(i) List all the sets in 𝜎(X). (ii) List all the sets in 𝜎( S1). (iii) Show that 𝜎(X) and 𝜎( S1) are independent under the probability

measure 𝑃1. (iv) Show that 𝜎(X) and 𝜎( S1) are not independent under the probability

measure 𝑃2. (v) Under 𝑃2, we have 𝑃2{ S1 = 8} = 2/3 and 𝑃2{ S1 = 2} = 1/3.

Explain intuitively why, if you are told that X = 1, you would want to revise your estimate of the distribution S1.

10) Consider a probability measure Ω with four elements, which we call a, b, c, and d. The σ-field F is the collection of all subsets of Ω.

We define a probability measure 𝑃 by specifying that

𝑃{a} = 1/6, 𝑃{b} = 1/3, 𝑃{c} = 1/4, 𝑃{d} = 1/4,

And as usual, the probability of other sets in F is the sum of the probabilities of the elements in the set. We next define two random variables X and Y by the formulas

X(a) = 1, X(b) = 1, X(c) = −1, X(d) = −1

Y(a) = 1, Y(b) = −1, Y(c) = 1, Y(d) = −1

We then define Z= X+ Y

(i) List all the sets in 𝜎(X). (ii) Determine E[Y|X], specify the values of this random variable for

a, b, c, and d. Verify that the partial-averaging property is satisfied. (iii) Determine E[Z|X]. Verify the partial-averaging property. (iv) Compute E[Z|X] - E[Y|X], Using the property of conditional

expectations, explain why you get X.

11) Let Y be a random variable on a probability space (Ω, F, 𝑃) and let G be a sub- 𝜎-algebra of F. Based on the information in G we can form the estimate

E[Y|G] of Y and define the error of estimation E𝑟𝑟 = Y − E[Y|G]. This is the random variable with expectation 0 and 𝑉a𝑟(E𝑟𝑟). Let X be some other G- measurable random variable which we regard as another estimation of Y. Show that

𝑉a𝑟(E𝑟𝑟) ≤ 𝑉a𝑟(Y− X). In other words, the estimate E[Y|G] minimizes the variance of the error among all estimates based on the information in G Hint: Let 𝜇 = E(Y− X). Compute the variance of Y− X as E[(Y− X− 𝜇)2 = E[((Y− E[Y|G]) + (E[Y|G] − X− 𝜇))2] then multiply it out.

12) Let X and Y be random variables on the probability space (Ω, F, 𝑃). Then Y = Y1 + Y2, where Y1 = E[Y|X] is 𝜎(X)-measurable and Y2 = Y − E[Y|X] . Show that Y2 and X are uncorrelated. More generally, show that Y2 is uncorrelated with every 𝜎(X)- measurable random variable.

13) Let (Ω, F, 𝑃) be Uniform measure on [0; 1) with the Borel 𝜎-algebra.

Suppose also that 𝑋(𝜔) = 𝜔3

4 and that G = 𝜎 ([0, 1/3), [1/3, 2/3), [2/3, 1)):

Explicitly find E[X|G].

14) Let 𝑋𝑛 denote the symmetric random walk, with 𝑍𝑛 = 𝑋𝑛 − 𝑋𝑛−1, (a). Calculate 𝑔(𝛼) = 𝐸[𝑒 𝛼𝑍𝑛 ]

(b). Show that 𝑀𝑛 = exp {𝛼𝑋𝑛 − 𝑛𝑙𝑜𝑔 𝑔(𝛼)} is a martingale.

15) Let 𝑀𝑛 be a martingale such that 𝑀0 = 0. Show that (a) 𝐸[𝑀𝑛 ] = 0

(b). 𝐶𝑜𝑣(𝑀𝑛+1, 𝑀𝑛 ) = 𝐸[𝑀𝑛 2]