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程序代写案例-MA50196

时间：2021-01-11

MA50196 Mock exam (based on January 2020 exam)

New formulations in blue. Note this mock exam has not undergone the usual peer-review for

normal exams and is for guidance only.

1. (a) Under what circumstances might it be beneficial for a company to acquire a call

option in a certain commodity to protect themselves against risk? How could the

company alternatively use a futures contract to hedge against the same risk? Briefly

describe how these two ways of hedging against such a risk differ from each other?

[Exam 2018/19 Q1 (a)] [5]

(b) Suppose the current price of an asset is 100 pounds, the interest rate is 1%

(compounded continuously). Suppose that a call option with strike 50 pounds

and maturity 1 year is available for C = 40 pounds. Without making any further

assumptions on the stock price, can you find an arbitrage strategy in this case?

[5]

(c) Give a lower bound for the price P of a European put option in a certain

foreign currency, in terms of the strike price K, the maturity T , the current

exchange rate S (pounds per unit of the foreign currency), the interest rate

r (compounded continuously) for the home currency and q the interest rate

(compounded contiuously) for the foreign currency.

Use this lower bound to show that if r = 0 and q > 0, then it is never worthwhile to

exercise an Amercian put in the foreign currency before the maturity date. [Exam

2018/19 Q2(c)] [5]

(d) Describe how to create a combined position in call and put options (with strike K

and maturity T ) plus cash that is equivalent to a long position in a forward contract

on the asset with delivery price K1, where K1 < K. [5]

Page 2 of 5 MA50196

2. Suppose the (Pounds Sterling) interest rate is r = 5% per annum compounded

continuously.

(a) Assuming no arbitrage opportunites exist, find the 1-year futures price of a futures

contract in a non-dividend-paying stock currently worth £100. Suppose the contract

is available with a futures price of £102, how could you achieve arbitrage? [5]

(b) Find the 2-year futures price for a futures contract in 1000 US dollars, if the current

exchange rate is 0.7 pounds per dollar and the interest rate in dollars is 8% per annum

compounded continuously. Assume the home currency is Pounds Sterling. [2]

(c) Find the futures price for a 1-year futures contract in a stock currently worth

£100, for which dividends of £5 are due in 3 months’ time and again in 9 months’

time. [4]

(d) Six months later, suppose the stock considered in part (c) is worth only £80. What

is the value, at that instant, of a short position in the futures contract considered

in part (c)? [5]

(e) Suppose a stock is currently worth £80 and will pay dividends in 3 months’ time

and in 9 months’ time. Each time, the size of the dividend will be 5% of the value

of the stock just before the dividend payment.

Find the 1-year futures price for this stock. Briefly explain how, if you have the

short position in this futures contract, you can arrange to exactly break even, i.e.,

to hedge against the uncertainty associated with this position. [4]

Page 3 of 5 MA50196

3. (a) Suppose that each year, a certain stock either goes up by 50% or goes down by a

factor of 1/3. Suppose it is currently worth £72, and that the risk free interest rate

is zero.

(i) What is the probability that the stock goes up rather than down, assuming

risk-neutrality? [2]

(ii) What is the price of a European call option with strike price £90 and maturity

in 3 years’ time? [4]

(iii) What is the value of the European call option mentioned in the previous part

after 1 year if the price goes up that year? [3]

(iv) What is the initial delta, and how much is initially invested in the stock in the

hedging strategy (replicating portfolio) for this option? [4]

(b) (i) Suppose that Government 1-year bonds for £100 with no coupon payments are

available for £95. What is the 1-year zero coupon yield rate? [2]

(ii) Suppose also that six-month bonds are available for £98. What is the six-

month zero-coupon yield rate? [2]

(iii) Finally, suppose also that 18-month bonds with face value £100 are available

for £100 with coupon payments £3 of at the end of every six-month period

(including the last six months). Find the implied 1.5-year zero-coupon yield

rate. [3]

Page 4 of 5 MA50196

4. (a) Let C denote the price at time 0 of a European call option for an asset which does

not bring in any dividends between times t = 0 and t = T years. Assuming the

asset price (currently S = S(0) = £100) follows a log-normal process with volatility

σ = 0.2, use the Black-Scholes formula

C = SΦ(x0 + σ

√

T )−Ke−rTΦ(x0), x0 = log(S/K) + (r − σ

2/2)T

σ

√

T

,

to compute the price of a 27-month call with strike price K = £100. Assume the

risk-free rate of interest is r = 6% per annum compounded continuously. [4]

(b) Using the formula for ∆call derived in lectures compute the ∆ for the call in part

(a). [3]

(c) Suppose that you hold 1000 call options as described in part (a). What position in

the underlying asset should you add to make your portfolio delta-neutral? [2]

(d) Using the formula for Γ of a call derived in lectures compute the Γ for this call

option. [3]

(e) Suppose that another call option is available with ∆ = 0.5 and Γ = 0.02. How many

units of the second call option and how many units of the underlying asset do you

need to add make your original portfolio of 1000 call options as described in part

(a) both gamma- and delta-neutral? [4]

(f) Explain using a suitable Taylor expansion, why it might be considered desirable to

make the ∆ and Γ of a portfolio close to zero. [4]

MDP Page 5 of 5 MA50196

New formulations in blue. Note this mock exam has not undergone the usual peer-review for

normal exams and is for guidance only.

1. (a) Under what circumstances might it be beneficial for a company to acquire a call

option in a certain commodity to protect themselves against risk? How could the

company alternatively use a futures contract to hedge against the same risk? Briefly

describe how these two ways of hedging against such a risk differ from each other?

[Exam 2018/19 Q1 (a)] [5]

(b) Suppose the current price of an asset is 100 pounds, the interest rate is 1%

(compounded continuously). Suppose that a call option with strike 50 pounds

and maturity 1 year is available for C = 40 pounds. Without making any further

assumptions on the stock price, can you find an arbitrage strategy in this case?

[5]

(c) Give a lower bound for the price P of a European put option in a certain

foreign currency, in terms of the strike price K, the maturity T , the current

exchange rate S (pounds per unit of the foreign currency), the interest rate

r (compounded continuously) for the home currency and q the interest rate

(compounded contiuously) for the foreign currency.

Use this lower bound to show that if r = 0 and q > 0, then it is never worthwhile to

exercise an Amercian put in the foreign currency before the maturity date. [Exam

2018/19 Q2(c)] [5]

(d) Describe how to create a combined position in call and put options (with strike K

and maturity T ) plus cash that is equivalent to a long position in a forward contract

on the asset with delivery price K1, where K1 < K. [5]

Page 2 of 5 MA50196

2. Suppose the (Pounds Sterling) interest rate is r = 5% per annum compounded

continuously.

(a) Assuming no arbitrage opportunites exist, find the 1-year futures price of a futures

contract in a non-dividend-paying stock currently worth £100. Suppose the contract

is available with a futures price of £102, how could you achieve arbitrage? [5]

(b) Find the 2-year futures price for a futures contract in 1000 US dollars, if the current

exchange rate is 0.7 pounds per dollar and the interest rate in dollars is 8% per annum

compounded continuously. Assume the home currency is Pounds Sterling. [2]

(c) Find the futures price for a 1-year futures contract in a stock currently worth

£100, for which dividends of £5 are due in 3 months’ time and again in 9 months’

time. [4]

(d) Six months later, suppose the stock considered in part (c) is worth only £80. What

is the value, at that instant, of a short position in the futures contract considered

in part (c)? [5]

(e) Suppose a stock is currently worth £80 and will pay dividends in 3 months’ time

and in 9 months’ time. Each time, the size of the dividend will be 5% of the value

of the stock just before the dividend payment.

Find the 1-year futures price for this stock. Briefly explain how, if you have the

short position in this futures contract, you can arrange to exactly break even, i.e.,

to hedge against the uncertainty associated with this position. [4]

Page 3 of 5 MA50196

3. (a) Suppose that each year, a certain stock either goes up by 50% or goes down by a

factor of 1/3. Suppose it is currently worth £72, and that the risk free interest rate

is zero.

(i) What is the probability that the stock goes up rather than down, assuming

risk-neutrality? [2]

(ii) What is the price of a European call option with strike price £90 and maturity

in 3 years’ time? [4]

(iii) What is the value of the European call option mentioned in the previous part

after 1 year if the price goes up that year? [3]

(iv) What is the initial delta, and how much is initially invested in the stock in the

hedging strategy (replicating portfolio) for this option? [4]

(b) (i) Suppose that Government 1-year bonds for £100 with no coupon payments are

available for £95. What is the 1-year zero coupon yield rate? [2]

(ii) Suppose also that six-month bonds are available for £98. What is the six-

month zero-coupon yield rate? [2]

(iii) Finally, suppose also that 18-month bonds with face value £100 are available

for £100 with coupon payments £3 of at the end of every six-month period

(including the last six months). Find the implied 1.5-year zero-coupon yield

rate. [3]

Page 4 of 5 MA50196

4. (a) Let C denote the price at time 0 of a European call option for an asset which does

not bring in any dividends between times t = 0 and t = T years. Assuming the

asset price (currently S = S(0) = £100) follows a log-normal process with volatility

σ = 0.2, use the Black-Scholes formula

C = SΦ(x0 + σ

√

T )−Ke−rTΦ(x0), x0 = log(S/K) + (r − σ

2/2)T

σ

√

T

,

to compute the price of a 27-month call with strike price K = £100. Assume the

risk-free rate of interest is r = 6% per annum compounded continuously. [4]

(b) Using the formula for ∆call derived in lectures compute the ∆ for the call in part

(a). [3]

(c) Suppose that you hold 1000 call options as described in part (a). What position in

the underlying asset should you add to make your portfolio delta-neutral? [2]

(d) Using the formula for Γ of a call derived in lectures compute the Γ for this call

option. [3]

(e) Suppose that another call option is available with ∆ = 0.5 and Γ = 0.02. How many

units of the second call option and how many units of the underlying asset do you

need to add make your original portfolio of 1000 call options as described in part

(a) both gamma- and delta-neutral? [4]

(f) Explain using a suitable Taylor expansion, why it might be considered desirable to

make the ∆ and Γ of a portfolio close to zero. [4]

MDP Page 5 of 5 MA50196