Quiz 1 Solutions

Name:___________________________________________ No 1

2

3

Question The Geometric Brownian Motion (GBM) process for a non-dividend paying asset is governed by which of the following SDE's? Assume μ and σ are constants, dS represents a change in asset price over a small time interval dt and dz is a random component A stock price increased from €47.00 to €47.80. An in-of-the-money put option on the stock moved from €2.25 to €1.95. What is the gearing of this option to two significant decimals? 3 19.384 2 18.9113 1 18.45 0 18 0

0 0 0 18.5 0 0 18 0 17.6 17.13 16.71 1

2

3

The excel sheet above shows a 3-step binomial process for a stock. What is the missing number at row 1, col 2?

ID:______________________________________

Choice 1 dS = µSdt + σSdt

Choice 2 dS = µSdt + σSdz

Choice 3 dS = µSdz + σSdt

Choice 4 dC = Sdt + σdz

2

Ans

-0.38

-2.67

12.7%

-7.83

4

17.56

17.50

17.46

17.44

1

4

uS Cu = max(uS-K,0)

∆=

(u − d ) S Cu − C d

∆=

S − 12 (uS + dS ) (u − d )

∆=

Cu − C d (T − t ) S

∆=

Cu − Cd (u − d ) S

4

S C

dS Cd = max(dS-K,0)

Δt t=0

t=T

Given a 1-step binomial as depicted here. At t=0 we have a portfolio consisting of Δ shares with a value S and short one call option value C. Δ is given by which equation? 5

x+Δxu pu x

pd x+Δxd

Δt The diagram above shows a 1-step Additive Binomial Process. If the process is described by

dx = vdt + σdz where v = r − 12 σ 2

what is the mean E[Δx] over Δt?

v∆t

x

∆x

r

1

6

Observe the general 2-step binomial at the end of the paper. The initial stock price is €18, volatility is 25%, risk free rates are constant at 4% and

€16.55

€16.68

€16.79

€16.81

3

€1.29

€0.38

€0

-€0.33

1

€1.25

€0.68

€0.65

€0

4

€0

€0.33

€0.68

€1.29

2

22%

32%

42%

52%

4

v = r − 12 σ 2 . Using an additive

binomial model with equal jump sizes

∆x = σ 2 ∆t + v 2 ∆t 2 , what is the stock price at node (2,0) given a time-step (∆t) of 1 week? 7

An at-the-money European call option matures in two weeks. Given the information from Question 6, what is the value of the option at Node(2,2)?

8

Using data from Q6 and Q 7 and again using an additive binomial model with equal jump sizes and

pu =

1 1 v∆t + , what is the value 2 2 ∆x

of the call option at node (1,0)? Using data from Q6 Q7 and Q8 and again using an additive binomial model with equal jump sizes and, what is the value of the call option at node (0,0)? Estimate delta, the rate of change of the option price relative to the change in stock price

9

10

• • • •

Answer all questions All questions carry equal marks, except Q.9 which carries 2 marks Answer 1, 2, 3 or 4 on the right-hand side Ensure you have entered your name and Student ID on the first page

Figure 1 - General 2-step binomial