Mathematical Solutions for Finance

Steven E. Shreve

Stochastic Calculus for Finance I
Student’s Manual: Solutions to Selected
Exercises
December 14, 2004

Springer
Berlin Heidelberg NewYork
Hong Kong London
Milan Paris Tokyo

Contents

1

1

Probability Theory on Coin Toss Space . . . . . . . . . . . . . . . . . . . .

7

2.9 Solutions to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3

1

1.7 Solutions to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2

The Binomial No-Arbitrage Pricing Model . . . . . . . . . . . . . . . .

7

State Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.7 Solutions to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4

American Derivative Securities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.9 Solutions to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5

Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.8 Solutions to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6

Interest-Rate-Dependent Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.9 Solutions to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

1
The Binomial No-Arbitrage Pricing Model

1.7 Solutions to Selected Exercises
Exercise 1.2. Suppose in the situation of Example 1.1.1 that the option sells for 1.20 at time zero. Consider an agent who begins with wealth X0 = 0 and at time zero buys ∆0 shares of stock and Γ0 options. The numbers ∆0 and Γ0 can be either positive or negative or zero. This leaves the agent with a cash position of −4∆0 − 1.20Γ0 . If this is positive, it is invested in the money market; if it is negative, it represents money borrowed from the money market. At time one, the value of the agent’s portfolio