Put-Call Parity of a Hedged Portfolio This problem is to be understood in terms

Question

Put-Call Parity of a Hedged Portfolio

This problem is to be understood in terms of a single-period binomial process. Let the initial value of the underlying equal $100. The underlying can either go up by 50% (u=1.50) or down by 20% (d=0.80). The risk-free rate is equal to 6%. A hedge portfolio consisting of a long position in the underlying and a short position in a European call option to purchase the underlying at the end of one period at an exercise price of $100 was created.

In this problem, use put-call parity to price a European put to sell the underlying at the end of one period for an exercise price of $100.

Explanation / Answer

Stock Prices:
SU = 100 x 1.5 = 150

SD = 100 x 0.8 = 80

So, Call valueU = Max [SU - K, 0] = Max [(150 - 100), 0] = 50

  Call valueD = Max [SD - K, 0] = Max [(80 - 100), 0] = 0

Risk Neutral Probability P* = (er - d) / (u - d) = (e0.06 - 0.8) / (1.25 - 0.8) = (1.06184 - 0.8) / 0.45 = 0.5819, So (1 - P*) = 0.4181

So Call Price at t=0:

C = e -r x [P* x CU + (1 - P*) x CD] = e -0.06 x [0.5819 x 50 + 0.4181 x 0] = 27.4

Using Put-Call Parity:

Put + Underlying Asset = Call + Strike Price

Or, Put = Call + Strike Price - Underlying Asset = 27.4 + 100 - 100 = 27.4

Put = Call because Strike Price = Underlier Price

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