what is the annual continuously compounded interest rate?

QUESTION

Put-Call Parity
A put option with a maturity of five months sells for $6.33. A call with the same expiration sells for $9.30. If the exercise price is $75 and the stock is currently priced at $77.20, what is the annual continuously compounded interest rate?
Basic Concept: We find the discount factor by the put call parity formula and then find the rate of interest for 1 year by taking time period for discount factor as 5/12 (since option is for 5 months) The put call parity is given by the following formula : C- P = S D.K where C= Call price P = Put price S= Spot Price K= Strike price D= discount factor At equilibrium for no arbitrage to exist, the discount factor should be the risk-free interest rate . Explanation: We substitute the values given in the problem to the formula C-P = S D.K Thus we get, 9.3 6.33 = 77.2 75D 2.97 = 77.2 –

75D 75D = 74.23 D= 0.9897 The discount factor, D is equal to (e^-rt) where r = interest rate and t= 5/12 (5 months) Thus (0.9897=e^-rfrac512) (0.9897=frac1e^rfrac512) (e^rfrac512=1.0104) Taking log to the base e on both sides, (rfrac512 = ln (1.0104)) (rfrac512 = 0.0103) r = 0.0248 = 2.48% The annual continuously compounded interest rate is 2.48%

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