Calculate dynamic replicating strategy for European call

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Reference no: EM132268053

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Q1. A stock price is currently £50. Over each of the next two 6-month periods, it is expected to go up by £10 or down by £10. The risk free interest rate is 20% per annum with semi-annual compounding.

(a) Compute the risk-neutral probabilities and use them to calculate the initial price of a 1-year European call with a strike price of £45. Briefly justify this pricing approach.

(b) Calculate the dynamic replicating strategy for this European call.

(c) Suppose instead that over each of the next two 6-month periods, the stock price can go up by £10, down by £10, or stay constant. How would you determine the value of the 1-year European call?

(d) Suppose 6 months have gone by, the risk-free interest rate is still 20% per annum with semi-annual compounding, and the price of the stock is £60. Over the next 6 months, the stock price can go up by £10, down by £10, or stay constant. A cash-or-nothing digital call option with 6-month maturity, strike price £62 and cash payment £30 trades at £19.09. What is the no-arbitrage price of a European call expiring in 6 months with strike price £55?

Q2. In this question, the instantaneous risk-free rate r follows an arbitrary stochastic process.

(a) Consider a forward contract, entered into at date 0, to buy a non-dividend paying underlying at date T.

i. Find the marked-to-market value V_t of this contract at time t < T in terms of the spot price at date t, S_t, and the forward price at date 0, F₀.

ii. Now suppose that S_t = S₀, i.e. the price of the underlying at date t is the same as its price at date 0. Find a condition on zero-coupon prices under which V_t = 0. Interpret.

(b) Let P(t, t') denote the time t price of a zero-coupon bond with maturity at time t' > t. All bonds have face value £1. Show that

P(t, T) = E_t^Q[e^{-t_∫S^r_udu}/P(T, S)],

for all t < T < S, where Q is the risk-neutral measure.

Reference no: EM132268053

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