Question

Find the fair value of an European call option and an American put option using the incoherent and coherent binomial option tree if the underlying asset pays dividend of 4 PLN in one and half month. The initial stock price is 60 PLN, the strike price of 58 PLN is expiring at the end of the third month, the continuously compounded risk-free interest rate is 10% per annum, and the stock volatility is 20%.

Answer 1

The values can be determined by using the black scholes method. Formula for which is:

C = SP e^-dt N(d₁) - ST e^-rt N(d₂)

P = ST e^-rt N(-d₂) - SP e^-dt N(-d₁)

d₁ = ( ln(SP/ST) + (r - d + (σ²/2)) t ) / σ √t

d₂ = ( ln(SP/ST) + (r - d - (σ²/2)) t ) / σ √t = d₁ - σ √t

Where:

C is the value of the call option,

P is the value of the put option,

N (.) is the cumulative standard normal distribution function,

SP is the current stock price (spot price),

ST is the strike price (exercise price),

e is the exponential constant (2.7182818),

ln is the natural logarithm,

r is the current risk-free interest rate (as a decimal),

t is the time to expiration in years,

σ is the annualized volatility of the stock (as a decimal),

d is the dividend yield (as a decimal).

Accordingly, the value of European call option for :

Spot Price (SP)	60
Strike Price (ST)	58
Time to Expiration (t)	90
Volatility (v)  %	20
Risk-Free Interest Rate (r)  %	10
Dividend Yield (d)  %	6.67	(4/60)*100

The fair value of European call option is 3.71.

Using the same formula for American put option, the fair value of American put option is 1.31.