Quiz 3 Solutions

Name:___________________________________________ No

Question

1

Which of the following statements is true?

2

Ho & Lee 1986 were the first to develop a model consistent with the initial yield curve. Supposing we have a series of options on discount bonds where the price of these options is denoted by marketi where i=1,…,M One way to calibrate our model is to minimise which of the following with respect to σ? What is the correct value for the array tree after the following code has been run?

3

N=3; tree=zeros(N,N); counter = 0; for i=N:N for j=N:-1:N-(i-1) counter = counter+1; tree(j,i) = counter; end end

ID:______________________________________

Choice 1 The CIR (1985) model has the advantage of non-negative rates and allowing for less variability at times of high interest rates and greater variability when rates are low.

Choice 2 The CIR (1985) model has the advantage of negative rates and allowing for less variability at times of high interest rates and greater variability when rates are low.

Choice 3 The CIR (1985) model has the advantage of non-negative rates and allowing for greater variability at times of high interest rates and less variability when rates are low.

Choice 4 The CIR (1985) model has the advantage of negative rates and allowing for greater variability at times of high interest rates and less variability when rates are low.

Ans 3

  marketi - modeli ( σ ) M  marketi - modeli ( σ ) M  marketi - modeli ( σ ) M  modeli ( σ ) − market Or i         ∑ ∑ ∑ ∑ σ market i 3  marketi modeli ( σ ) i =1  i =1  i =1  i =1  4

M

0 0 0

0 0 0

3 2 1

0 0 1

0 2 3

4 5 6

1 0 0

2 3 0

4 5 6

1 0 0

0 3 0

0 0 6

1

4

Let P(i+1) represent the price of a pure discount bond that matures at time (i+1)∆t. The following values are given: P(0) = 1, P(1) = 0.9753 and P(2) is 0.9512. Given the following information:

P ( i +1) = ∑Qi , j j

5 6

[

1

1 + U ( i )eσj

∆t

A montecarlo 2-factor model has a stock process driven by the SDE

dS = Sµdt +σSdz S

And a volatility driven by dV = α V −V dt + ξ V dz V Where dzs and dzv have a correlation ρ

(

0.000

0.1000

0.1013

4

0.086

0.096

0.106

0.116

2

It contains an option delta

It contains an option gamma

It contains an option vega

It contains an option rho

1

Decrease

Increase

No change

Not possible to say

2

]∆t

Q0,0 = 1, Q1,-1 = Q1,1 = 0.495, ∆t = 0.25 and σ = 10%, what is the value of U(0)? Using the same details defined in Q5, what is the value of U(1) to 3 decimal places? What is the most likely purpose of the variable “qVar”?

for j=1:noSimulations stockPrices(j,2) = stockPrices(j,1)*exp(mudt+randn*sigdt); if (stockPrices(j,2)<strike) payOff(j,1) = exp(mudt+randn*sigdt); else payOff(j,1) = 0; end stdDevCalc(j,1) = payOff(j,1)^2; end qVar = -exp(-intRate*T)*mean(payOff(:,1))

7

1.000

)

All other variables being equal, if V is increased, what will happen the price of a vanilla put option?

8

Given the details in Q7, all other variables being equal, if ξ is increased, what will happen the price of a vanilla call option? A hedged option can closely resemble the payoff on the option as given by the following equation

9

Decrease

Increase

No change

Not possible to say

2

Cash flows from rebalancing the hedge

The strategy cash flows

The difference between implied and actual delta

The hedging error

4

C t0 e r (T −t0 ) = CT −  N −1 ∂C ti  S ti +1 − E S ti e r (T −ti +1 )  + η ∑  i =0 ∂S 

(

[ ])

What does η represent? The floating strike Asian option has a payoff based on the difference between the underlying at expiration , and the average of the underlying prior to expiration. What formula best describes the payoff on an asian put option where AT is the average of n samples of the asset?

10

• • • •

PT = max( AT − S T ,0) PT = max( S T − AT ,0) PT = min( AT − S T ,0) PT = min( S T − AT ,0)1

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