Question

Consider a European option on a non-dividend-paying stock when the stock price is $30, the exercise price is $29, continuously compounded risk-free interest rate is 5%, volatility is 25% per annum, and time to maturity is 4 months (assume 4 months equals 120 days).

- Find the value of a call option at strike price of $29 using the Black-Scholes option pricing model. Show all steps using Excel.

- Find the value of a put option at strike price of $29 using the Black-Scholes option pricing model. Show all steps Using Excel.

- Show that put-call parity holds using the put and the call at strike price of $29. (you need to find the value of both sides of the put-call parity for this.) Using Excel.

Answer 1

Black and scholes model of option valuation ,

S = Current value of the underlying asset

K = Strike price of the option

t = Life to expiration of the option

r = Riskless interest rate corresponding to the life of the option

?2 = Variance in the ln (value) of the underlying asset

Given data in question below,

Input Data
Stock Price now (S)		30
Exercise Price of Option (K)		29
Number of periods to Exercise in years (t)		=120/360= 0.33
Compounded Risk-Free Interest Rate (rf)		5.00%
Standard Deviation (annualized s)		25.00%

Output data , Answer for A and B

Output Data
Present Value of Exercise Price (PV(K))		28.5207
s*t^.5		0.1443
d1		0.4225
d2		0.2782
Delta N(d1) Normal Cumulative Density Function		0.6637
N(d2)*PV(K)		17.3851
Puting all the value in above given Black Schole valuation model
Value of Call		2.5251
Value of Put		1.0458

Answer C)

Put-call parity defines the relationship between the price of European put options and European call options of the same asset class at a price.

The equation expressing put-call parity is:

C +Ke^-rt = P + S

where, C = price of the European call option, K = The strike price , P = value of the European put and S = spot price of the asset

Using the above formula ,

value of call , C = P + S - Ke^-rt

C = 1.0458 + 30 - 29 * e^{-0.05 * 0.33}

C= 2.521

Similarly for Value of Put , P = C +Ke^-rt - S

P = 2.52 + 29 * e^{-0.05 * 0.33} - 30 = 1.045.