Question

A stock price is currently AUD 60; the risk-free rate is 5% and the volatility is 30%. What is the value of a two-year American put option with a strike price of AUD 62

Answer 1

Answer:

Given:

Current market price = AUD 60

risk free rate = 5%

volatility = 30%

time to expiration = 2 years

Strike Price = AUD 62

With the given information, we can find out the value of american put option using Black and Scholes model.

The formula of Black and Scholes model is as under:

First we need to find out the value of call option

c = Pa N(d1) - Pe N(d2) e-rt

c = 60 x 0.6443 - 62 x 0.4761 x e -0.05 x 2

c = 38.658 - 62 x 0.4761 x 0.9048

c = 38.658 - 26.708

c = AUD 11.95

Hence,value of call option is AUD 11.95.

Value of put option

p = c - Pa + Pe x e-rt

p = 11.95 - 60 + 62 x e-0.05x2

p = 11.95 - 60 + 56.10

p = AUD 8.05

Hence,value of put option is AUD 8.05.

where,

c = value of call option

pa = current market price = 60

pe = exercise price = 62

r = risk free rate of interest = 5%

t = time to expiration = 2 yrs

e = exponential constant

Workings

Now first we will find out the value of d1

d1 = ln (Pa/Pe) + (r + 0.5 s2) t / s x square root of t

d1 = ln (60/62) + (0.05 + 0.5 x 0.3 x 0.3) x 2 / 0.3 x square root of 2

d1 = -0.03279 + 0.19 / 0.424

d1 = 0.3708 or d1 = 0.37

d2 = d1 - s x square root of t

d2 = 0.37 - 0.3 x square root of 2

d2 = -0.054 or d2 = -0.06

The next step is to find out the N(d1) and N(d2).

N(d1) = 0.1443 + 0.5 = 0.6443 (by looking at the standard normal distribution table)

Here we have added 0.50 to 0.1443 since the value of d1 is positive.

N(d2) = 0.0239 - 0.5 = 0.4761 (by looking at the standard normal distribution table)

Here we have subtracted 0.50 from 0.239 since the value of d2 is negative.