MATH0031 Exam 2022
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MATH0031
Answer all questions .
NOTE: In the questions which follow the current price of an asset (or similar instrument) will often be denoted either by St or simply by S with the time subscript suppressed. Reference may be made to the following definitions:
where K denotes the exercise price, r the riskless rate, σ the volatility and t is the
time to expiry. The Black-Scholes formula for pricing a European call is C = SΦ(d1 ) - Ke尸rtΦ(d2 ).
1.
(a) Bitcoin is traded as both a spot and a future denominated in US dollars. The current spot price is BTCUSD = $30, 000. Assume there are no divi- dends or interest payments or other costs associated with holding Bitcoin.
(i) Write down a formula for the fair value of a one month future on BTCUSD given that one month USD interest is 2% continuously com- pounded. Hence calculate the fair value to the nearest USD.
(ii) Suppose you have bought an amount X of Bitcoin. A client agrees to borrow your Bitcoin position for one month and pay interest of 10% continuously compounded. What happens to your fair value for the one month BTCUSD future?
(b) European options on a share S can be replicated using a replicating port- folio H = (u, v) where u is the number of riskless assets and v the number of shares in the portfolio H . Consider the following model for S:
|
S(0, ω) |
S(1, ω) |
1 |
4 |
8 |
2 |
4 |
2 |
where we assume interest rates are zero.
(i) Construct the replicating portfolios HC for a European call option and HP for a European put option both on S with strike K = 3.
(ii) Explain why in general the replicating portfolio of a European call op- tion has a positive v whereas the replicating portfolio of a European put option has negative v .
(iii) A straddle is a portfolio H0 = (C, P) consisting of one long European call option and one long European put option both with the same strike K . Find the strike 2 < K0 < 8 where the replicating portfolio for H0 consists purely of riskless assets. Hence or otherwise value the straddle struck at K0 . (20 marks)
2. (a) Suppose Bitcoin is trading at $30, 000. The binomial model for the future Bitcoin price over the next two years is given by:
|
S(0, ω) |
S(1y, ω) |
S(2y, ω) |
1 |
30 |
60 |
120 |
2 |
30 |
60 |
30 |
3 |
30 |
15 |
30 |
4 |
30 |
15 |
7.5 |
where the numbers in the table represent thousands of dollars. Assume the USD interest rate is zero and Bitcoin does not earn interest.
(i) Show that the risk-neutral probability of Bitcoin falling to $7 , 500 is four times greater than the risk-neutral probability of Bitcoin reaching $120 , 000.
(ii) I decide to buy Bitcoin for $30, 000 today but as I am concerned about the price falling I also buy a two-year put option struck at $21 , 000. Calculate the value of the put option today and hence deduce the maximum gain and loss for my portfolio.
(b) S(t) is a 1-period model for share prices with values (pS(0), S(0)/p) at time 1. Prove that in the absence of interest rates it is not possible to have a risk-neutral measure Q = ( , ) in this model for any value of p > 1.
(c) Suppose you are given an envelope containing $A. You have the choice of keeping it or exchanging for an envelope containing either $2A or $A/2 with equal probability.
(i) Show that the expected profit from exchanging the envelope is strictly posi- tive.
(ii) Using the result in part (b), show that there must exist an arbitrage oppor- tunity. State any theorems you use to justify your conclusion. (20 marks)
3. St (ω) denotes the share price at time t for a given path ω . The share S satisfies the following filtration (Pt ), where T = 2 years, the interest rate r = 0 and the prices are in USD. Note that these share prices assume no dividend will be paid at any time.
ω |
S(0) |
S(1y) |
S(2y) |
1 |
4 |
8 |
16 |
2 |
4 |
8 |
4 |
3 |
4 |
2 |
4 |
4 |
4 |
2 |
1 |
(i) A single dividend of 40 cents per share is announced to be paid after 18 months. Calculate the premium of both the European and American call options with strike K = 3 and expiration in 2 years. Explain why the American call option has a greater premium than the European call option.
(ii) Typically the size of the dividend payment is not announced in advance. Consider the American call option struck at 3 and let D be the unknown dividend amount in USD. Show that for at least one path early exercise is always optimal for any dividend payment $0 < D < $1 and value the American in terms of D .
(iii) Show that the American call option is greater in value than the European call option by D/3 USD. Give an intuitive reason why this is the case.
(iv) Describe what effect a positive USD interest rate will have on the American option value. (20 marks)
4. (a) Let the process (B(t))t—0 be a standard Brownian motion and suppose c > 0 is a constant. Let τ be a stopping time such that B(τ ) = c and B(u) < c for all u < τ .
(i) Explain what stopping time means. (ii) Define
Show that Z(u) is also a Brownian motion for u > τ .
(iii) Draw a chart with an example path for both B(t) and the corresponding path for Z(t). Both paths must include the stopping time τ .
(b) Let W (t) be Brownian motion. Using Ito’s lemma evaluate
T
(4W3 (t) - 12tW (t))dW (t)
0
(c) St follows a stochastic process given by
dSt = (a - bSt )dt + σdWt
where Wt is a Brownian motion and a, b, σ are all positive.
The process Yt is given by Yt = exp(bt)St . Calculate dYt . (20 marks)
5. Use the Black-Scholes formulae at the start of this paper.
(a) Describe Put-Call Parity and calculate the Black-Scholes formula for a Eu- ropean put option from a European call option struck at K with expiration T.
(b) (i) Show that d2(2) = d1(2) - 2 log(Sert /K). Hence, or otherwise, show that the delta of a European call option is
∂C
= Φ(d1 ).
(ii) Explain how an option trader can use delta-hedging to manage their risk. (iii) What is the delta of a European put option?
(iv) Prove that the Black-Scholes formula for a European put option converges to the payoff equation as we approach expiration.
(c) An at-the-money forward option has a strike given by K = exp(rT) * S(0)
where the risk-free interest rate is r . Using the Black-Scholes formula for a Eu- ropean call option, calculate the value of an at-the-money-forward European call option. (20 marks)
2023-07-13