Finance -compounding interest rate
Two bhp European call options are available for $1.378
Subject: Business / Finance
Question
1. Two bhp European call options are available for $1.378 and $1.205, with strike prices $40.50 and $42.50, respectively. Both calls will expire in three months. The continuously compounding interest rate is r = 4.5% pa and bhp shares are currently trading at $39.540.
(a) Why do these two calls have di?erent premiums?
(b) Calculate the implied volatility for both calls to three signi?cant ?gures.
2. A European call written on shares has strike price $20. Consider a two-step binomial model. Using crr notation, the underlying share prices are calculated using S = 22, u = 1.1 and d = 1/u. The variable returns are R(0,0) = 1.05, R(1,1) = 1.03 and R(1,0) = 1.09.
(a) Calculate the risk neutral probabilities ?(0,0), ?(1,1) and ?(1,0) (if you wish, you may present these results in a binomial pricing tree).
(b) Construct a two-step binomial pricing tree for the European call.
(c) A European put has the same strike price, underlying asset and time to expiry as the European call. Construct a two-step binomial pricing tree for the European put.
(d) Put-call parity still holds when interest rates are variable,
C(t) ? P(t) ? S(t) + pvt(K) = 0,
but now the present value pvt(K) is di?erent at di?erent nodes. The present values for K in the two-step binomial model are given in the following binomial pricing tree.
(The graph is in the file)
Show that put-call parity holds (to ?ve decimal places) for your values of the put and the call at every node of the two-step binomial model.
(e) (not assessed) Given that ?(0,0) = 1, calculate all state prices at the call’s expiry, ?(2,2), ?(2,1) and ?(2.0). Use the state prices ?(2,j) with j = 0,1,2 to calculate the premium of the European call and con?rm that it agrees with the premium obtained from the binomial pricing tree in part (c).
3. An American and European put both have strike price $30, the same underlying asset and the same time to expiry. Consider a three-step binomial model. The underlying asset price is, in crr notation, S = 29, u = 1.2 and d = 1/u. The return over each time step is R = 1.1. (a) Construct a three-step binomial pricing tree for the European put and ?nd the premium.
(b) Construct a three-step binomial pricing tree for the American put and ?nd the premium.
(c) Now say the European put is a down-and-out barrier option with barrier at the asset price 21. Construct a three-step binomial pricing tree for this barrier Euro- pean put and ?nd the premium.
(d) Now say the American put is a down-and-out barrier option with barrier at the asset price 21. Construct a three-step binomial pricing tree for this barrier Amer- ican put and ?nd the premium.
(e) Discuss the four premium values calculated in parts (a)–(d). In particular, discuss why some premiums are higher than others and why some are equal.
