Question

According to Emanuel Derman, writing in the Journal of Derivatives, Winter, 2000, p. 64 he says “Good theories, like Black-Scholes-Merton, provide a theoretical laboratory in which you can explore the likely effect of possible causes. They give you a common language with which to quantify and communicate your feelings about value.”
Riding on that background and given the following information
Current Price of underlying asset k100
Strike price of underlying asset k80
One period risk free rate of return 10%
Stock price can either go up or down by 20%
Time period 6 months
Determine the call value
What is the time value for holding on to the option
What is the Put Option Price using the PUT-CALL Parity relationship

Answer 1

First we find the probability of upward movement of price. Let that be 'x'

Hence probability of price going down is '1-x'

Current price of the stock should be the present value of the weighted expected price of the stock

i.e 100 = PV (X *120 + (1-X) *80)

100 = 1/1.10 (120 X + 80 - 80X)

110 = 40X + 80

X = 0.75

Hence upward movement is 0.75 and downward movement is 0.25

Strike price is 80

Options gives a right but not an obligation to the holder to buy or sell an asset at a certain date at a certain rate

Payoffs

Call payoff = (Spot Price - Strike Price ) (Where S>X) or Zero (Where S<X)

Hence if the price moves upwards call will be valued (120-80) i.e 40 and if moves downwards call shall be of no value

Probability of upward movement is 40

Value of call today = PV 40 * 0.75 = 1/1.10 * 30 = 27.27

Value of call includes 2 values, intrinsic value and time value, If the call was to exercise today, we would receive 20 (100 - 80), hence 20 is the intrinsic value and balance of 7.27 is the time value.

Put Call Parity

Put-call parity states that a fiduciary call should be equal to protective put. Fiduciary call means buying a call and investing in the PV of the strike price which ensures we have cash to buy the asset. Protective put is a position where we buy a put and we also buy a stock so that it limits the downside of a stock.

i.e Call + PV (Strike Price) = Put + Stock

27.27 + PV (80) = Put + 100

27.27 + 1/1.10 * 80 = Put + 100

Put = Nil

Hence value of put as per put call parity is Nil.