STAT0013 Examination Paper 2020/2021 Page 1 STAT0013 – Stochastic Methods in Finance (2021) Section A A1 (a) Give the mathematical expression of the gamma and vega of a portfolio of derivatives. Explain briefly the meaning of these two quantities. [3] (b) A portfolio consists of stocks in the Asian markets. It is known that the portfolio has a negative delta. Suppose there is a small increase in the value of the stocks. Explain briefly the effect of this increase in the total value of the portfolio. [2] (c) Find the value of delta of a 9-month European call option on a stock with a strike price equal to the current stock price (t = 0). The interest rate is 8%. The volatility is = 0.10. [4] ANSWER: (a) (b) (c) We find d1 = (0.08 + (0.1)2 0.5)) 0.75 0.1 p0.75 0.736 and = N(0.736) 0.7704 A2 A 9-months forward contract on a non-dividend paying stock is entered into when the stock price is ￡40 and the risk-free interest rate is 5%. (a) What are the forward price and the value of the forward contract at the initial time [4] (b) Suppose that at the initial time the forward contract is being bought and sold in the market for ￡5, instead of being traded at the fair price. Outline a simple trading strategy to take advantage of the arbitrage opportunity, stating how much risk-free profit will be made per contract. [5] ANSWER: (a) F0 = S0erT = 40e0.05 0.75 = 41.528 and f0 = 0. TURN OVER STAT0013 Examination Paper 2020/2021 Page 2 (b) Consider a portfolio that it is short 1 unit of contract, long 1 stock, short ￡35 in bonds. At time t = 0 the value of the portfolio is 40 5 35 = ￡0. At expiry time, the value of the portfolio is ST (ST K) 35erT = F0 35erT = ￡5.191. A3 (a) Write down the Black-Scholes-Merton partial differential equation. State the boundary condition for the case of a derivative with payoff S2T + 2ST + 3. [2] (b) Consider a European put option and a European call option on the same underlying asset, with the same strike priceK = $5, and same expiry date T = 1. The put is traded at $3, the call at $10, and the underlying asset at $6. The interest rate is r = 0.05. i. Verify that put-call parity does not hold in this case. [3] ii. Describe a portfolio that exploits the arbitrage opportunity. State the risk-free profit that the portfolio will provide. [5] ANSWER: (a) (b) i. We have that St + Pt Ct = 6 + 3 10 = $1 < KerT = $4.756 ii. We construct the following portfolio: long 1 share, long 1 put, short 1 call, long 1 unit of risk-less bond. At present the portfolio has a value of 6 + 3 10 + 1 = $0. At expiry date the portfolio has a value ofK + er = 5 + 1.051 = $6.051. Thus, we have made a risk-free profit of $6.051. A4 Consider the Geometric Brownian Motion (GBM) process dSt = μStdt+ StdBt, S0 = 1. A stock price follows the above GBM, so that for the first two years, μ = 4 and = 2, and for the next two years, μ = 0 and = 2. Express the probability P [S4 < s ], for any s > 0, as a function of the cumulative distribution function, N(·), of the standard normal distribution. Hint: You can make use of the equivalent expression St = S0 exp((μ 22 )t+ Bt). [6] ANSWER: CONTINUED STAT0013 Examination Paper 2020/2021 Page 3 We have that: P[S4 < s ] = P[B4 < log s 2 ] = P[Z < log s 4 ] = N( log s 4 ) A5 Consider a European call option on a stock. The call option will expire in 6 months. The current stock price is $30, and the strike price of the call option is $20. At the expiration date, the stock price can either be $35 or it can be $25. The risk free interest rate is 4%. What is the value of the European call option today [6] ANSWER: We compute the risk-neutral probability p = erT d u d = e0.04 0.5 5/6 7/6 5/6 = 0.56 The value of the call option is f = erT (p fu + (1 p )fd) = e0.04 0.5 (0.56 15 + 0.44 5) = 10.388 TURN OVER STAT0013 Examination Paper 2020/2021 Page 4 Section B B1 Consider a given portfolio with delta equal to 2, 000 and vega equal to 60, 000. We plan to create a new portfolio that is both delta and vega neutral by adding to the given portfolio: i) units of the underlying stock, and ii) units of a traded option with delta equal to 0.5 and vega equal to 10. How many units of the underlying stock and the traded option will we need [5] ANSWER: We need to add a long position in 1, 000 units of the underlying sto
ck and a short position in 6, 000 units of the option. B2 A stock price St follows the usual model dSt = μStdt+ StdBt with expected return μ = 0.16 and volatility = 0.35. The current price is ￡38. (a) What is the probability that a European call option on the stock with an exercise price of ￡40 and a maturity date in 6 months will be exercised [5] (b) Using the Black- Scholes formula, find the price of the call option if the risk-free interest rate is μ. Hint: You might have already calculated quantity d2 (required for finding the price of the call) in your answer in part (a). This can save you a lot of calculations. [4] (c) What is the probability that a European put option on the stock with the same exercise price and maturity date will be exercised Find the price of the put option, using put-call parity. [4] ANSWER: (a) We need to find P [ST > K ]. We obtain that: P [ST > K ] = 1N(0.00775) = 0.4969 (b) Using the Black-Scholes formula, we have that the price of this option is: S · N(d1)KerT · N(d2) = ￡4.25 (c) Using put-call parity, the price of the put option is C S +KerT = ￡3.18 CONTINUED STAT0013 Examination Paper 2020/2021 Page 5 B3 Consider a binomial model with T = 2 periods, S = 100, u = 1.6 and d = 0.6. The interest rate is r = 0.1. What is the price at time t = 0 for an American put option that has exercise priceK = 97 Hint: To avoid too many calculations, you are given that the risk-neutral probability of one-step up movement is p = 0.505, and that Su = 160, Sd = 60, Su2 = 256, Sd2 = 36, Sud = 96. [9] ANSWER: The value of the option is 16.7609. B4 (a) Assume in this question that all the derivatives have the same maturity date and the same underlying asset. Write explicitly the payoff function for a portfolio consisting of a short position in two European put options with exercise price ￡20 and a long position in three European put options with exercise price ￡25. [4] (b) Consider the following payoff function at maturity T and strike priceK: Payoff = min(ST K, 4). i. Draw the payoff diagram, that is the plot of payoff at date of maturity versus price of underlying asset. [4] ii. By using 1 unit of forward contract (in a long or short position) and 1 unit of call option (long or short position), create a portfolio with the payoff function given above. The strike prices of these two derivatives need not be equal toK. [4] ANSWER: (a) (b) i. The graph is as shown below. K K+4 4 -K ii. Portfolio: long 1 forward atK plus short 1 call atK + 4. TURN OVER STAT0013 Examination Paper 2020/2021 Page 6 B5 (a) Assume that Bt corresponds to a standard Brownian motion. i. Use Ito ’s formula to find the SDE satisfied by the process Yt such that Yt = sin(Bt) t 2 + 5. Indicate clearly the Ito ’s correction term, the drift term and the variance term in the SDE you have obtained. [4] ii. We know that Xt satisfies the SDE: dXt = dt+ 2 p XtdBt, X0 = 0. Solve this SDE to obtain an analytic expression for Xt. Hint: The solution is of the form Xt = Bpt , for an appropriate positive integer p 2. [4] (b) Assume that (Bt, t 0) and (Wt, t 0) are independent standard Brownian motions. i. Let Xt =Wt +Bt. Compute the covariance of Xt and Xs, for s < t. [3] ii. Show that the process Yt such that Yt = B2t t is a fair game process. [3] ANSWER: (a) i. We have that: dYt = cos(Bt) t 2 dBt + 2 sin(Bt) t dt 12 sin(Bt) t2dt ii. The solution is Xt = B2t . (b) i. Using the independence ofWs and Bt, we have cov(Xt, Xs) = 2s ii. For s < t we have E [Xt |Xs ] = Xs B6 Consider a portfolio that consists of: a long position on 8 shares of the underlying asset; a short position on 5 units of risk-less bond; a long position on 6 units of European puts on the above underlying. The risk free interest rate is r = 5%. The volatility is = 0.1. The current value of a share of the underlying is $10. The strike price for the call isK = $12 and the expiry date is T = 1 year. (a) Find the delta of this portfolio. [4] (b) Find the rho of a portfolio which is as the one described above, with the difference that it contains 0 units of European puts instead of 6 units of European puts. [3] CONTINUED STAT0013 Examination Paper 2020/2021 Page 7 ANSWER: (a) We have that the delta of the portfolio is equal to: = 2.6 (b) The rho of the portfolio is equal to P = 0. END OF PAPER