Consider a six-month European call option on a non-dividend-paying stock. The stock price is $30, the strike price is $29, and the continuously compounded risk-free interest rate is 6% per annum. The volatility of the stock is 20% per annum.

1) Value this option using the Black-Scholes formula. Illustrate each step in your

calculation.

2) Please use a one-step binomial tree to value this option.

3) Please use a two-step binomial tree to value this option.

4) Compare the results from 2) to 3) with what you get using the Black-Scholes-

Merton formula.

ANSWER 1: Value of call option using Black-Scholes Model = $2.7315

ANSWER 2: Value of call option using One-Step Binomial Model = $3.085

ANSWER 3: Value of call option using Two-Step Binomial Model = $2.245

ANSWER 4: Difference between BSM and one-step binomial model = $0.3535 ; Difference between BSM and two-step binomial model = $0.4865

Step-by-step explanation

ANSWER 1: BLACK-SCHOLES MODEL

Call option price = [Stock price * N(d1)] - [N(d2) * Strike price * Exponential value^{-Risk-free rate * Time}]

where,

N = Cumulative standard normal distribution

d1 = [LN(Spot price / Strike price) + (r + (standard deviation² / 2)) * Time] / [Standard deviation * Time^0.50]

d2 = d1 - [Standard deviation * Time^0.50]

 

Note: LN = Natural Log

 

Stock price = $30.00

Strike price = $29.00

Risk-free rate = 6.00%

Standard deviation = 20%

Time = 0.50 years

 

d1 = [LN (30.00 / 29.00) + (6.00% + (20%² / 2)) * 0.50] / [20% * 0.50^0.50]

= 0.522563

d2 = 0.522563 - [20% * 0.50^0.50] = 0.381142

 

Cumulative standard normal distribution value of d1 = 0.699361

Cumulative standard normal distribution value of d2 = 0.648451

Note: NORM.S.DIST function can be used to calculate cumulative standard normal distribution values.

N(d1) = 0.699361

N(d2) = 0.648451

 

Call option price = [$30 * 0.699361] - [0.648451 * $29.00 * 2.718^-0.06*0.50]

= $2.7315

 

ANSWER 2: ONE-STEP BINOMIAL MODEL

 

Current price = $30

Strike price = $29

Risk-free rate = 6.00%

Volatility = 20%

Time period = 0.50 years

Number of steps = 1.00

Delta T = Time period / Number of steps = 0.50 / 1 = 0.50

 

u = Exponential value^{Standard deviation} * ^{√Delta T}

= 2.718^0.20*√0.50= 1.1519

d = 1 / u = 1 / 1.1519 = 0.8681

 

Spot price if it moves up (Su) = $30 * 1.1519 = $34.557

Spot price if it moves down (Sd) = $26.043

 

Probability of moving upside (p) = Exponential value^{Risk-free rate * Delta T} - d / [u - d]

= 2.718^0.06*0.50 - 0.8681 / [1.1519 - 0.8681] = 0.5721

Probability of moving downside (q) = 1 - 0.5721 = 0.4279

 

Call option price if it moves up (Cu) = Maximum of (Spot price - Strike price, 0.00)

= MAX($34.557 - $29, 0.00) = $5.557

Call option price if it moves down (Cd) = Max ($26.043 - $29.00, 0.00) = 0.00

 

Current value of call option = Exponential value^{-Risk-free rate * Delta T} * [(p * Cu) + (q * Cd)]

= 2.718^-0.06*0.50 * [(0.5721 * 5.557) + (0.4279 * 0.00)]

= $3.085

 

ANSWER 3: TWO-STEP BINOMIAL MODEL:

Current price = $30

Strike price = $29

Risk-free rate = 6.00%

Volatility = 20%

Time period = 0.50 years

Number of steps = 2.00

Delta T = Time period / Number of steps = 0.50 / 2 = 0.25

u = Exponential value^{Standard deviation} * ^{√Delta T}

= 2.718^0.20*√0.25= 1.105

d = 1 / u = 1 / 1.105 = 0.905

Probability of going up (p) = Exponential value^{Risk-free rate * Delta T} - d / [u - d]

= 2.718^{0.06 * 0.25} - 0.905 / [1.105 - 0.905] = 0.5506

Probability of going down (q) = 1 - 0.5506 = 0.4494

Binomial tree:

At node D, E, and F, the value of call option premium will be:

Maximum of (Spot price - Strike price, 0.00)

Value at Node D (Vd) = MAX(36.63 - 29, 0.00) = 7.63

Value at Node E (Ve) = MAX(30 - 30, 0.00) = 0.00

Value at Node F (Vf) = MAX(24.57 - 30, 0.00) = 0.00

Value at Node B (Vb) = Exponential value^{-Risk-free rate * Delta T} * [(p * Vd) + (q * Ve)]

= 2.718^{-0.06 * 0.25} * [(0.5506 * 7.63) + (0.4494 * 0.00)]

= $4.1385

Value at Node C (Vc) = Exponential value^{-Risk-free rate * Delta T} * [(p * Ve) + (q * Vf)]

= 2.718^{-0.06 * 0.25} * [(0.5506 * 0.00) + (0.4494 * 0.00)]

= $0.00

Value at Node A (current value) = Exponential value^{-Risk-free rate * Delta T} * [(p * Vb) + (q * Vc)]

2.718^-0.06*0.25 * [(0.5506 * 4.1385) + (0.4494 * 0.00)]

= $2.245

 ANSWER 4:

Difference between the call option value by Black-Scholes Model and call option value by one-step binomial model = $3.085 - $2.7315 = $0.3535

Difference between the call option value by Black-Scholes Model and call option value by two-step binomial model = $2.7315 - $2.245 = $0.4865

There is a difference because Binomial Model is based upon the probabilities of moving up and moving down. Moreover, under binomial model, expiration period can be divided into multiple time periods in which probabilities of going up and going down can be applied.

Please see the attached file for the complete solution