Texts: a) The current stock price for XYZ Ltd is sh. 25. A European call option with an exercise price of sh. 28 will expire in 160 days. The yield on a 160-day Treasury bill is 5.18%. The standard deviation of annual returns on XYZ's stock is 21%. Compute the premium for a call option on this stock using the Black-Scholes model. (10 marks) c = SN(d1) - XN(d2)e^(-rT) You may use ln(s) + (r + δ^2)T d1 = (ln(s/X) + (r + δ^2/2)T) / (δ√T) d2 = d1 - δ√T b) Assume that the 3.75% US Treasury bond that matures on 15 August 2041 is priced to yield 5.14% for settlement on 15 October 2014. Coupons are paid semi-annually on 15 February and 15 August. The yield-to-maturity is stated on a street-convention semiannual bond basis. This settlement date is 61 days into a 184-day coupon period, using the actual/actual day-count convention. Compute the approximate modified duration and the approximate Macaulay duration for this Treasury bond assuming a 5 bp change in the yield-to-maturity. (10 marks)
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18%. Since the option expires in 160 days, we can use this yield as the risk-free rate (r). Show more…
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It is August 20th, and you have just entered a long position in a futures contract. The contract expires on December 20th and calls for the delivery of 500 tons of a commodity. Further, because this is a futures position, it requires the posting of 40% of the current futures price as the initial margin. The maintenance margin is 25% of the current futures price. Assume that the account is marked to market monthly. The following represent the contract delivery prices (in dollars per ton) that prevail on each settlement date: August 20th (initiation): $330.00 September 20th: 333.55 October 20th: 314.00 November 20th: 318.00 December 20th (delivery): 314.50 Calculate the equity value of your margin account on each settlement date, including any additional equity required to meet a margin call. Also compute the amount of cash that will be returned to you on December 20th. Calculate leverage multiplier for the contract. Calculate the cumulative total holding period returns and cumulative spot holding period returns for each month. If the investment pays no dividend and requires a storage cost of 3 percent per annum (of current value), calculate the current (i.e., August 20th) implied spot price for a ton of the commodity and November 20th implied price for the same ton. In your calculations, assume that an annual risk-free rate of 6 percent prevails over the entire contract life.
Akash M.
Adi S.
An asset that follows geometric Brownian motion will have a European call option with value c given by the risk-neutral expectation: c = e^{-rT}E^*left(max(Se^{(r-frac{1}{2}sigma^2)T+sigma W_T}-X,0) ight), where S is the current spot price of the asset, X is the strike price for the option, r is the annual continuously compounded interest rate, T is the time to expiry, sigma is the asset volatility, and W_T is a Wiener process. (a) Evaluate the integral above and show that this gives the Black-Scholes formula, c = SN(d_1) - Xe^{-rT}N(d_2), where N(z) indicates the (cumulative) standard normal probability distribution, and the parameters d_1 and d_2 are given by d_1 = frac{ln(S/X)+(r+frac{1}{2}sigma^2)T}{sigmasqrt{T}}, and d_2 = frac{ln(S/X)+(r-frac{1}{2}sigma^2)T}{sigmasqrt{T}}. (b) The same integral approach can be used to derive the expression for the Delta of the call option: Delta_{coll} = frac{partial c}{partial S} = N(d_1). i. What range of values can Delta have? ii. Explain why this range of values is to be expected, based on the pay-off behaviour of a call option with respect to asset price. iii. Determine an expression for the Gamma of the call option. (c) The vega indicates the dependence of the call option with respect to volatility: v_{call} = S_0sqrt{frac{T}{2pi}}e^{-frac{d_1^2}{2}}. i. How does the vega for a European call option compare to the vega of a European put option with the same strike price X and the same time to maturity T? Explain your answer. ii. Explain how and why options can be used to estimate the market perception of volatility.
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