Question

10. Use DerivaGem to complete this problem where you have an option on a non-dividend paying stock when the stock price is $30, the exercise price is $29, the risk-free interest rate is 5% per annum, the volatility is 25% per annum, and the time to maturity is four months: a. What is the price of the option if it is a European call? b. What is the price of the option if it is an American call? c. What is the price of the option if it is a European put? d. Verify that put-call parity holds.

Answer 1

A

Pricing a European Call Option Formula

Price Call = P0N(d1) – Xe-rtN(d2)

Where,

d1 = [ln(P0/X) + (r+v2/2)t]/v √t and d2 = d1 – v √t

P0= Price of the underlying security

X= Strike price

N= standard normal cumulative distribution function

r = risk-free rate

v= volatility

t= time until expiry

Applying the formula option price is 2.5251

B

American options:

For valuation of American options, valuation method suggested by Roll, Geske and Whaley is used.

Formula for pricing American call option,

C = (S – D1 * exp(-r * t1 )) * N(b1) + (S - D1 * exp(-r * t1 )) * M(a1, -b1; - (t1/T)^1/2 ) – X * exp( -r * T) * M(a2, -b2; - (t1/T)^1/2 ) – (X - D1 ) * exp(-r * t1 )) * N(b2)

Where

a1 = ( ln((S - D1 * exp(-r * t1 )) / X) + (r + v^2 / 2) * T ) / v * T^1/2

a2 = a1 - v * T^1/2

b1 = ( ln((S - D1 * exp(-r * t1 )) / S*) + (r + v^2 / 2) * t1 ) / v * t1^1/2

b2 = b1 - v * t1^1/2

Applying the formula option price is 2.5363

C

Pricing a European Put Option Formula

Price Put = Xe-rt *(1-N(d2)) – P0*(1-N(d1))

Where d1 and d2 can be calculated in the same way as in the pricing of call option explained above.

Applying the formula option price is 1.0458