Question

A) You try to price the options on XYZ Corp. The current stock price of XYZ is $100/share. The risk-free rate is 5%. You project the stock price of XYZ will either be $90 or $120 in a year. Assume you can borrow or lend money at the risk-free rate. Use risk-neutral approach to price the 1-year call option on XYZ Corp with the strike price of $105. (Attention: using a wrong approach will cost you all the credits)

B) You try to price the options on XYZ Corp. The current stock price of XYZ is $100/share. The risk-free rate is 5%. You project the stock price of XYZ will either be $90 or $120 in a year. Assume you can borrow or lend money at the risk-free rate. Use risk-free portfolio approach to price the 1-year put option on XYZ Corp with the strike price of $105. (Attention: using a wrong approach will cost you all the credits)

C) Use put-call parity to verify the call and put option prices you just calculated in A and B

Answer 1

First, we need to compute the probability of price going up [ P(u) ] and probability of price going down [ P(d) ] -

wher, V_s = current stock price = $100, r = continuously compounded risk free rate = 5% or 0.05, t = time to maturity in years = 1, LP = Lower price = $90, UP = Upper price = $120

P(d) = 1 - P(u) = 1 - 0.504333333 = 0.495666667

Value of option is computed as follows -

where, V_O = Value of option, V_UP = Intrinsic Value of option when price goes up, V_LP = Intrinsic Value of option when price goes down

A) Call option

Intrinsic value of a call option when price goes up = Market price - Strike price = $120 - $105 = $15

Intrinsic value of a call option when price goes down = 0 (as we do not exercise the option when price goes down since it will result in a loss)

or, Value of call option = $7.20

B) Value of Put option

Intrinsic value of a put option when price goes up = $0 (as we do not exercise the option when price goes up since it will result in a loss)

Intrinsic value of a put option when price goes down = Strike price - Market price = $105 - $90 = $15

or, Value of put option = $7.08

C) As per put-call parity -

E = (S + P - C) x (1 + r)^t

where, E = Exercise price or strike price, S = current market price of stock, P = Value of put option, C = value of call option, r = risk free rate, t = time to maturity in years

E = ($100 + $7.08 - $7.20) x (1 + 0.05)¹ = $104.874 or $105

which is equal to the strike price on the options.