Question

Binomial option pricing model

A stock currently trades for $41. In one month, the price will either be $47 or $34. The annual risk-free rate is 6%; assume daily interest compounding and 365 days per year. The value of a one-month call option with an exercise price of $39 is $______.

Answer 1

One step binomial option pricing model for call option price:

	Price after period 1		Value of call option [stock price - Strike price ($39)]
	$47	Move up (u=$47/$41=1.15)	$18
Current Price of stock($41)
	$34	Move down (d=$34/$41=0.83)	$0 (price is below the strike price so option will not exercise)

Probability of moving up, P = (e ^r*t – d)/ (u-d)

Where

Risk free rate, r = 6% per year

We have to calculate Effective annual rate (EAR)

Effective annual rate (EAR) = (1 + r/m) ^m – 1

Where,

Effective annual rate (EAR) =?

Where, nominal annual interest rate annual percentage rate (APR); r=6%

Number of compounding per year, m = 365 (daily compounding, where number of days in a year is 365)

Therefore

EAR= (1 + 6%/365) ^365 - 1

Or EAR= (1 + 0.06/365) ^365 -1 =0.0618 or 6.18%

Time period, t = 1 month or 1/12 year

Factor of moving up, u = 1.15

Factor of moving down, d = 0.83

Therefore,

P = (e^0.0618*1/12 – 0.83) / (1.15 -0.83)

P = 0.5548

And (1- P) = 1- 0.5548 = 0.4452 (probability of moving down)

Now the expected value of call option

= P * $18 + (1-P) * $ 0

= 0.5548 * $18 + 0.4452 *$0

= $ 9.9856

Therefore the value of the call option is $9.9856