exam-2001

QUEEN’S UNIVERSITY BELFAST

Graphical calculators are NOT allowed.

210AMA307

LEVEL 3 EXAMINATION

Applied Mathematics AMA307

Financial Mathematics

Monday 28 May 2001 2.30 pm – 5.30 pm

Examiners

Professor E. A. G. Armourand the internal examiners

Answer FOUR questions

All questions carry equal marks

Write on both sides of the answer paper

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1. A supermarket chain anticipates that it needs 100,000 bushels of corn in the nearfuture and wants to agree on a price now for delivery of the corn in 3 months time.Assume that the spot price for corn is $207 per 100 bushels, the risk-free sterlinginterest rate is 8% per annum, the risk-free dollar interest rate is 4% per annum andthe spot exchange rate $/£ is 1.50. Suppose that you are a corn trader. Obtainsolutions to (a)-(d) below, showing your working in each case.

(a) Construct an arbitrage argument to determine the arbitrage-free forward pricein dollars per 100 bushels that you should charge the supermarket chain.

(b) Suppose that the supermarket chain wants to pay in pounds instead of dollars.Derive a new arbitrage argument to determine the forward price of corn inpounds per 100 bushels, remembering that you need to buy the corn usingdollars.

(c) Assume the storage cost for corn up to 200,000 bushels is $1,000 per month.This cost must be paid at the beginning of each month. Taking this intoaccount, derive a new arbitrage-free forward price (in $) for corn in dollars per100 bushels. What would be the forward price if the supermarket chain onlywanted 50,000 bushels?

(d) In the case of (c), if the forward price is higher than the arbitrage-free pricehow would you form a portfolio to get a riskless profit?

/continued. . .

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2. Suppose the current time is only one period δt prior to an option expiration. Thestrike price of the option is X. We know that the underlying share price can go upto Su or down to Sd while the spot price of the share is S. The risk-free interestrate for the period δt is r.

(a) Suppose you form a portfolio of ∆ numbers of shares and cash amount B inrisk-free bonds. Find ∆ and B to replicate the long position of the call withyour portfolio. Do you have to borrow cash or lend cash?

(b) Derive an expression for the risk-neutral probability and justify the probabilityyou found.

(c) When the variables are X = $21; S = $20; r = 12%; δt = 3 months; u =1.1; d = 0.9, what is the European put option price calculated using thebinomial model? Show your work to answer the question.

(d) For the same variables as in (c), determine the range of an American putoption price which has the same conditions.

/continued. . .

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3. Let S denote the price of a non-dividend-paying asset which undergoes a stochasticprocess described by

dS

S= µdt + σdZ

where the drift µ and the volatility σ are assumed constant and dZ is the standardWiener process.

(a) Using Ito’s lemma, write the stochastic differential equation for a derivativec(S).

(b) Construct a risk-free portfolio by shorting one derivative and longing under-lying assets.

(c) Using an arbitrage-free argument, derive the Black-Scholes equation.

(d) Describe the delta hedging portfolio and outline the difficulties of delta hedg-ing.

/continued. . .

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4. The following is the contract specification for the corn futures

Initial Margin: $ 473 per contractMaintenance Margin: $ 350 per contractContract Size: 5,000 bushels

Let us consider the following transaction.

Day 1 A customer longs two corn futures contract.

Current futures price: $ 2.07 per bushelClosing futures price of day 1: $2.05 per bushel

Day 2 The price of the corn futures falls to $2.04 per bushel.

(a) What is the excess margin/margin call on the day 2? Show your work toanswer the question.

(b) Suppose the price of the corn futures rises to $2.08 per bushel on the day 3.Assuming the position is still open, what is the excess margin/margin call onthe day 3? Show your work to answer the question.

(c) Prove that the forward price for a contract with a certain delivery date is thesame as the futures price for a contract with the same delivery date when therisk-free interest rate is constant and the same for all maturities.

/continued. . .

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5. The Black equation for futures derivatives f is

−∂f

∂τ+

1

2

∂2f

∂F 2σ2F 2 = rf.

where r is the risk-free interest rate, τ is the time to mature, F is the underlyingfutures and σ is its volatility. In answering the questions, recall that the cumulativenormal probability distribution is

N(x) =1

√

2π

∫ x

−∞

e−t2/2dt

(a) Denoting y = ln F and w = erτf , show that the Black equation becomes

∂w

∂τ−

σ2

2

(

∂2w

∂y2−

∂w

∂y

)

= 0.

(b) The fundamental solution of the above partial differential equation is

φ(y, τ) =1

σ√

2πτe−rτ exp

−

(

y −σ2τ)

2

2σ2τ

.

Using the fundamental solution of the Black equation, find the price for aEuropean call option on the futures. Show your work.

(c) Using the fundamental solution of the Black equation, find the price for aEuropean put option on the futures. Show your work.

(d) Derive the put-call parity for the European futures options.

/continued. . .

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6. (a) At time t = 0, a company sells a forward rate agreement (FRA) to lend forthe period of time T to T ∗ (T < T ∗) at a preset interest rate, RK .

i. The value of the agreement at the initial time t = 0 is zero if the presetrate is fairly agreed. In this case, show that the preset interest rate, RK , inthe FRA should be the same as the forward rate at the time the contractis initiated.

ii. Define at time t

rt (r∗t ): the spot rate applying for T − t (T ∗− t) years

ft : the forward rate for the period of time between T and T ∗.

Calculate the value of the FRA at time t.

iii. Show that we can value the FRA at time t by calculating the presentvalue of cash flows on the assumption that the current forward rate, ft, isrealised.

(b) To study the forward rate curve (plotted against the maturity), we need theinstantaneous forward rate. If r is the spot rate of interest applying for T

years and r∗ is the spot rate for T ∗ years where T ∗ > T , the forward interestrate for the period of time between T and T ∗ is

f =r∗T ∗

− rT

T ∗− T

.

Find the instantaneous forward rate. Discuss the forward rate curve plottedagainst the maturity with the reference of the bond-yield curve.

/continued. . .

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7. We would like to derive the governing equation for the bond price using the arbitragepricing approach. The bond price is assumed to follow the following stochasticdifferential equation

dB

B= µB(r, t)dt + σB(r, t)dZ

where µB(r, t) and σB(r, t) are respectively the drift rate and the variance of thestochastic process and dZ is the standard Wiener process. We also assume that thebond price depends only on the spot interest rate r, current time t and maturitytime T and the spot rate r(t) follows a continuous Markovian stochastic processdescribed by

dr = u(r, t)dt + w(r, t)dZ

where u(r, t) and w(r, t)2 are the instantaneous drift and variance of the processfor r(t).

(a) With use of Ito’s lemma, prove that the drift rate µB(r, t) and the varianceσB(r, t) can be written as

µB(r, t) =1

B

(

∂B

∂t+ u

∂B

∂r+

1

2w2

∂2B

∂r2

)

and

σB(r, t) =1

Bw

∂B

∂r.

(b) To hedge the risk, take a portfolio which consists of a bond of dollar value V1

with maturity T1 and another bond of dollar value V2 with maturity T2. Withhelp of the portfolio find the governing equation for the bond price.

(c) Consider two securities which depend on the spot interest rate. Suppose se-curity A has an expected return of 5% per annum and a volatility of 10%per annum, while security B has a volatility of 20% per annum. Suppose theriskless interest rate is 7% per annum. Find the market price of interest raterisk and the expected return per annum of security B.

exam-2001

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