[SOLVED] finance Assume the following single period model

Assume the following single period model

City University of Hong Kong

Department of Economics and Finance

Course EF5210Assignment #4

Question 1 (due April 25, 2018)

With an accumulator contract, investor agrees to buy a certain amount (M) of the underlying asset at a

fixed price (F) over a period of time at settlement dates {1, 2, , n}. There is also a knock out price

(L F) that terminates the contract. The risk-neutral valuation of the accumulator contract is given by

f0 M ( eri(Si F)
m
i 1| S0 )

where m is the last settlement date prior to the termination of the contract. There is no need to settle the

contract in case when it is being terminated at settlement date.

Use Monte-Carlo simulation to price an accumulator contract with the following parameters :

Current price of the underlying assetS0 $100,

Annualized risk-free interest rate r 5%,

Annualized volatility of the underlying asset 20%,

Number of settlement dates in the accumulator contract n 4,

Settlement dates in year of the accumulator contract { 1 0.25, 2 0.5 , 2 0.75, 2 1.00 },

Knock out price of the contract L $120,

Agreed purchase price of the contract F $96,

Agreed purchase unit of the contract M 100,

Number of samples in the simulation Nsample 100000, and

Number of simulated steps between two settlement dates N 100.

Use antithetic variate method to reduce the standard error in the simulation.

Note : You can use any programming language of your own preference. Please submit your numerical

results together with the source code.

( 40 points )

Question 2 (due May 2, 2018)

Suppose the risk-neutral process for the short rate is given by

drt dt (t)dzt

where is a constant factor and (t) is a function of time.

(a) Show that the forward price Pt(T) of a zero-coupon bond with unit dollar par value and maturity at T is

given by

Pt(T) ($1) A(t, T) e
B(t,T) rt

whereB(t,T) (T t)

A(t,T) e
(T t)2 2(s)(T s)2ds

T

t

(15 points)

(b) Consider the option with maturity at T written on a zero-coupon bond with unit dollar par value and

maturity at a later time T
*
. The maturity payoff of this option is given by

fT P(T
*)PT(T

*),where T

Use Jamshidian formula to determine the current price of this option.

(15 points)