Question

Derivatives Markets

3. You own a 6-month European put option on a XYZ Company's stock with a strike price of $100. The spot price of the stock is $102, it has a return volatility of 30% and currently pays no dividends. The continuously compounded risk free rate of interest is 3%. Using Black-Scholes you calculate the value of your put option to be $6.84. Now you have just learned the company is considering paying a one-time dividend in 3 months in the amount of $10. Their decision will be announced in the next few hours. Assuming the company decides in favor of the dividend, using Black-Scholes formula what is the change in value that you expect for your put option?

Answer 1

Ans.3

As per Black Scholes Model with dividend

we have

C= (S-D.e-xt) N(d1)- X e-rT N (d2)

Where

C= Price of the call option=?

S= Price of underlying stock= $102

X= Option Exercise price= $100

r= Risk Free rate of return=3%

σ= Standard Deviation=0.30

D= Dividend per stock= $10

t= 90 days i.e., 90/360=1/4 years

T= 180 days i.e.. 180/360= 1/2 year

N()= Area under normal curve

d1= (Log((S-D*e-rt)/X)+(r+σ2/2)T)/σ T1/2

d2= d1- σ T1/2

Substituting the above given values for finding out N(d1) and N(d2)

we get as follows

d1=( Log((102-10*e-0.03*0.25)/100)+(0.03+(0.3)^2/2)0.5)0.3*0.5^1/2

d1= (Log((102-10*.9925)/100)+0.0375)/0.212

= (log(92.075/100)+0.0375)0.212

= (-0.03586+0.0375)/0.212

=0.00164/.212

=0.007735

Now

d2= d1-σ T1/2

d2= 0.007735-0.212

d2= -0.2043

so, C= (S-D*e-xt)* N(d1)- X e-rT N(d2)

C= 92.075* N(0.007735)-100*1.0151* N(-0.2043)

C= 92.075*0.503085-101.51*.41906

C=46.32-42.53

C=3.79

Now we can use formula of put call parity to find the value of put:-

S+P=C+X*e-rT

102+P=3.79+100*1.0151

P= 105.3-102

P= $3.30

Value of put option is decreases from $6.84 to $3.30