MATH0031
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:
(x)+ = max{x, 0},
1 Zx −t2 Φ(x) = √ exp(
)dt, φ(x) = √2π exp( 2 ),
2π−∞ 2 1 −x2
log(S/K) + (r + 21 σ2)t d1= √ ,
σt
log(S/K) + (r − 21 σ2)t d2= √ ,
σt
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−rtΦ(d2).
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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) ω1 4 8 ω2 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)
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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) ω1 30 60 120 ω2 30 60 30 ω3 30 15 30 ω4 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 = (21, 12) 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)
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3. St(ω) denotes the share price at time t for a given path ω. The share S satisfies the following filtration (Pt)Tt=0, 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) ω1 4 8 16
ω2 4 8 4
ω3 4 2 4
ω4 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)
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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
(B(u), for 0 ≤ u < τ, 2c−B(u), foru>τ.
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
Z(u) =
path for Z(t). Both paths must include the stopping time τ.
(b) Let W(t) be Brownian motion. Using Ito’s lemma evaluate
ZT
(4W 3 (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.
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(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 = d21 − 2 log(Sert/K). Hence, or otherwise, show that the
delta of a European call option is
∂C = Φ(d1).
∂S
(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)
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