Question

6.Determine put option price from the following data: Current stock price Rs. 1260, strike price Rs.1280, Time to expiration 3 months, Volatility 30%, Annual risk-free rate 11 12% Use Black-Scholes formula

Answer 1

Assumptions : This is a European Option. Volatility given is an annual rate. Annual risk-free rate is a continuously compounded rate.

Inputs :

Current Stock Price : S = Rs. 1260

Strike Price : X = Rs. 1280

Time to expiry : T = 3 months = 3 / 12 = 0.25 (in years)

Volatility : $\large \sigma$ = 30% or 0.30

Annual risk free rate : r = 0.12

We will first use Black-Scholes formula to find the value of the Call option, and then use the Put-Call Parity equation to find the Value of the Put option.

1) Black-Scholes formula :

Value of Call option : C = [ S x N(d₁) ] - [ X * e^{(-r x t)} x N(d₂) ]

(Here X * e^{(-r x t)} denotes the Present Value of the Strike Price)

d₁ = [ LN (S / X) + (r + $\large \sigma$² / 2) x t ] / $\large \sigma$ x $\small \sqrt{t}$

(LN means Natural Log) (We can calculate this using the "ln" function in Excel) (In excel type the formula : "=ln(1260/1280)"

d₁ = [LN (1260 / 1280) + (0.12 + 0.30² / 2) x 0.25] / 0.30 x √0.25

=  [-0.01575 + 0.04125] / 0.15

= 0.17

d₂ = d₁ - $\large \sigma$ x $\small \sqrt{t}$

= 0.17 - (0.30 x √0.25)

= 0.17 - 0.15

= 0.02

N(d) denotes the area under the standard normal distribution curve. We can get these values either from a Normal distribution table or by using the formula NORMSDIST in Excel : =NORMSDIST(value). I have calculated this using the excel formula.

N(d₁) = N(0.17) = 0.5675

N(d₂) = N(0.02) = 0.5080

Present Value of Strike Price = X * e^{(-r x t)} = 1280 x e^{-0.12 x 0.25} = 1280 x e^-0.03

(using the formula : "=EXP(-0.03)" in Excel we get  e^-0.03 = 0.970446)

= 1280 x 0.970446

= 1242.17

Value of Call option : C = [ 1260 x 0.5675 ] - [ 1242.17 x 0.5080 ]

= 715.05 - 631.02

= Rs. 84.03

2) Now, as per Put-Call Parity equation :

Value of Put = Value of Call + Present Value of Strike Price - Current Stock Price

P = C + PV (X) - S

= 84.03 + 1242.17 - 1260

= Rs. 66.20