Quiz 2 Solutions

Name:___________________________________________ No

Question

1

Which of the following statements is most accurate?

2

If we say that variance is mean reverting and follows it’s own stochastic differential equation governed by its is the mean volatility where  ξ is the volatility of the variance  V = mean volatility  α is the mean reversion rate Which of the following SDEs represents time varying volatility?

3

Which of the following statements is false?

ID:______________________________________

Choice 1 Time varying volatility cannot be incorporated into binomial models because of variable timesteps and resultant bushy trees, trinomial models must be used

Choice 2

Choice 3

Choice 4

Ans

Time varying volatility cannot easily be incorporated into binomial models because of variable timesteps and resultant bushy trees make cashflow and maturity dates difficult to incorporate. Trinomial models are better used

Time varying volatility cannot be incorporated into and lattice model (binomial, trinomial, etc).

Time varying volatility can be easily incorporated into any lattice model

2

α(V −V )dz + ξ V dt

α(V −V )dz + ξ V dt

α(V −V )dt + ξ V dz

dV =

dV =

ξ(V −V )dt + α V dz

3

With antithetic variance the mean of the jth normally distributed samples εj and -εj is zero.

With antithetic variance, more accurate estimates can be derived from M pairs of CT , j , CT , j

With antithetic variance, M pairs of CT , j , CT , j is computationally cheaper than 2M of CT , j

The perfect negative correlation of the normally distributed samples εj and -εj increased the standard error of the estimate

4

dV =

dV =

(

)

than from 2M of CT , j

(

)

4

What is the correct value for the array tree after the following code has been run? N=3; tree=zeros(N,N); counter = 0; for i=1:N for j=N:-1:N-(i-1) counter = counter+1; tree(j,i) = counter; end end

0 0 1

0 3 2

6 5 4

0 0 1

0 2 3

4 5 6

1 0 0

2 3 0

4 5 6

1 0 0

3 2 0

6 5 4

1

5

What is the best description of the graph below?

This represents the payoff of a written call on a lognormally distributed asset

This represents the probability distribution payoff of a written call on a lognormally distributed asset

This represents the probability distribution payoff of a written call on a normally distributed asset

This represents the payoff of a written call on a normally distributed asset

2

6

Observe the general 2-step binomial at the end of the paper. The initial stock price is €20, volatility is 20%, risk free rates are constant at 1 2 . Using an additive 3% and 2 binomial model with probabilities and

€18.03

€18.33

€18.63

€18.93

4

v= r− σ

1 1 v∆t + 4σ 2 ∆t − 3v 2 ∆t 2 2 2 , 3 1 ∆xd = v∆t − 4σ 2 ∆t − 3v 2 ∆t 2 2 2 ∆xu =

what is the stock price at node (2,0) given a time-step (∆t) of 1 week?

7

A European call option with a strike price of €19 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 probabilities, 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

€0

€2.14

€4.11

€4.28

2

€0.15

€0.61

€1.04

€2.77

3

22.2%

39.7%

61.4%

96.8%

4

10

In Figure 1 to the left, what kind of option is being valued?

Figure 1 - Code Snippet A “Cliquet Option”, the strike price is reset periodically to the spot level

• • • •

A “Bermudan Option”, the option is exercisable periodically

Answer all questions All questions carry equal 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

A “Digital Option” that pays a fixed amount or zero at maturity

A “Rainbow Option” that pays the best of two options

4

Figure 2 - General 2-step binomial