Question

Binomial option pricing model

A stock currently trades for $41. In one month, the price will either be $50 or $36. 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)]
	$50	Move up (u=$50/$41=1.22)	$18
Current Price of stock($41)
	$36	Move down (d=$36/$41=0.88)	$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 = 1.22

Factor of moving down, d = 0.88

Therefore,

P = (e^0.0618*1/12 – 0.88) / (1.22 -0.88)

= (1.0052 -0.88)/ (1.22 -0.88)

P = 0.3723

And (1- P) = 1- 0.3723 = 0.6277 (probability of moving down)

Now the expected value of call option

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

= 0.3723 * $18 + 0.6277 *$0

= $ 6.70

Therefore the value of the call option is $6.70