Question

In addition to the five factors, dividends also affect the price of an option. The Black–Scholes Option Pricing Model with dividends is:

  

C=S×e−dt×N(d1)−E×e−Rt×N(d2)C=S×e−dt×N(d1)⁢−E×e−Rt×N(d2)

d1=[ln(S/E)+(R−d+σ2/2)×t](σ−t√)d1= [ln(S⁢  /E⁢ ) +(R⁢−d+σ2/2)×t ] (σ−t)

d2=d1−σ×t√d2=d1−σ×t

  

All of the variables are the same as the Black–Scholes model without dividends except for the variable d, which is the continuously compounded dividend yield on the stock.

The put–call parity condition is also altered when dividends are paid. The dividend-adjusted put–call parity formula is:

S×e−dt+P=E×e−Rt+CS×e−dt+P=E×e−Rt+C

where d is again the continuously compounded dividend yield.

  

A stock is currently priced at $84 per share, the standard deviation of its return is 60 percent per year, and the risk-free rate is 5 percent per year, compounded continuously. What is the price of a put option with a strike price of $80 and a maturity of six months if the stock has a dividend yield of 3 percent per year? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

  

Price of put option

$

Answer 1

The formula for European call & Put option using Black Scholes model is given by

Call = S0 x e^(-d x t) x N(d1) - K x e^(-rxT) x N(d2)

Put = K x e^(-r xT) x N(-d2)- (S0 x e^(-d x t) x N(-d1))

Where,

S0 = Price of the underlying, eg stock price.

K = Exercise price.

r = Risk free interest rate

t = Time to expiry

σ = Standard deviation of the underlying asset, eg stock.

d = Dividend yield

N(d1) =  standard normal cumulative distribution function using value of d1

N(d2) =  standard normal cumulative distribution function using the value of d2

The value of d1 & d2 is calculated using the below formula

d1 = (Ln(So/K) + ((r - d + σ ^2)/2) x t)) / ((σ^2) x t)^1/2)

d2 = d1 – σ x (t)^1/2

Here,

Where,

S0 = Price of the underlying, eg stock price = 84

K = Exercise price = 80

r = Risk free interest rate = 5%

t = Time to expiry = ^ months = 0.5 years

σ = Standard deviation of the underlying asset, eg stock. = 60%

d = Dividend yield = 3%

d1 = (Ln(84/80) + ((5% - 3% + (60%)^2)/2) x 0.5)) / ((60%^2) x 0.5)^1/2) = 0.3507

d2 = d1 – σ x (t)^1/2 = 0.3507 - 60% x 0.5^(1/2) = -0.07356

N(d1) = 0.6371 , N(d2) = 0.4707

Value of call = Call = S0 x e^(-d x t) x N(d1) - K x e^(-rxT) x N(d2)

= 84 x e ^(-0.03 x 0,.5) X 0.6371 - 80 x e^(-0.05 x 0.5) x 0.4707

Call = 52.72 - 36.72 = $15.99