What are Black-Scholes Model and Binomial Option Pricing Method? How to use those model for pricing the options?

The Black- Scholes Model is introduced by Fischer Black, Myron Scholes in 1973 in a paper entitled “The Pricing of Options and Corporate Liabilities". They introduce the formula to calculate fair price of call and put option which are affected by several factors like premium amount, market risk etc. It is used for pricing of European options and not for American option (exercise anytime before expiration date). 

Formula:

V_c = S*N(d₁)-X/e^rt N(d₂)

V_p = V_c +X /e ^rt –S

D₁= [In(S/X) +(r+ (v²/2) t]/ v √ t

 D₂= d1-v√t

Where,

 S = stock price

X = strike price

t = time remaining until expiration, expressed in years

 r = current continuously compounded risk-free interest rate

 v = annual volatility of stock price or standard deviation of the short-term returns over one year)

 In = natural logarithm

 N(x) = standard normal cumulative distribution function

 e = the exponential function (2.718281)

Assumptions:

·         The option exercise only on expiration date.

·         The risk free rate (investment is free from market risk) is constant.

·         The volatility of underlying asset is constant.

·         There is no tax and transaction cost.

·         There is no dividend on share.

Example: Find out the value of European call and put option price with the help of black schools model if the spot price is Rs.70 and strike price is Rs.65. Volatility of stock is 25% and risk free rate is 3%. The period of expiration is 7 months.

Solution: Let’s calculate d₁,

(d₁) = [In(S/X) +(r+ (v²/2) t]/ v √ t

= [In (70/65) + (0.03+ (0.25²/2) 0.5833]/ 0.25 √ 0.583

= [0.07410+ (0.03+ 0.03125)0.5833]/ 0.1909

= [0.0741+0.0357]/ 0.1909

= 0.5752

(d₂) = d₁-v√t

= 0.3843

Then interpolating to find 0.5752 and 0.3843,

N(d₁) = (0.5752 – 0.57) / (0.59 – 0.57)*(0.7224-0.7157)

= (0.0052/0.02)*0.0067

= 0.001742

N(d₁) = 0.7157 + 0.001742

= 0.7174

N(d₂) = (0.3843 – 0.38) / (0.40 – 0.38)*(0.6554-0.6480)

= (0.0043/0.02)*0.0074

= 0.001591

N(d₂) = 0.6480 + 0.001591

= 0.6496

Now we find out the call option,

V_c = S*N(d₁)-X/e^rt N(d₂)

= 70*0.7174 -65/2.71828^0.03*0.5833 * 0.6496

= 50.218 – 41.492

= 8.726

V_p = V_c +X /e ^rt –S

= 8.726 + 63.88 - 70

= 2.596

Binomial Option Pricing Model:  It is developed in 1979 by Rubinstein, Cox and Ross. In this model tree like diagram is constructed where different nodes shows underlying asset price at given time. 

Assumptions:

·         It assumes there is no tax and transaction cost.

·         It considers that the underlying assets price is either increases or decreases before expiration date.

·         There is constant risk free rate.

·         There is no dividend paid on share.

·         There is no arbitrage system.

Formula:

S₀ = Current stock price

Probability of up state = p

 Probability of down state = 1-p

The diagram is look like this:

              

Let’s assume the strike price of call option = X then if the option holder exercise the option the diagram is shown as

                          

Probability of up state:

P = e^{rt / n} – D / U – D

U = e^v√∆t

D = e^-v√∆t

Where,

r = risk free rate

P = probability of up state

U = up state factor

D = down state factor

Payoff of call and put option

C_n = max(S_n-X,0)

P_n = max( X-S_n,0)

Value of American Call and put option

P_n = max( X-S_n, e^-r∆t(pV_u+(1-p)V_d))

C_n = max( S_n-X, e^-r∆t (pV_u+(1-p)V_d))

Value of European call and put option

V_n= e^-v∆t (pV_u+(1-p)V_d))

V_u= exercise value of upper state

V_d= exercise value of down state

Example: Find out the American call option price if the spot price (current price) is Rs.100 and strike price is Rs.80.The risk free rate is 4% and the time period is 3 months. The standard deviation is 25%.

Solution: 

∆t = years of maturity/number of step = 0.8333

U = e^v√∆t

U = e^{0.25*√0.8333} = 1.0748

D = 1/u = 0.9303

P = e^{rt / n} – D / U – D

= 0.07968/0.1445

= 0.5515

1-p = 0.4485

First we have to find out the D, E and F node value then B and C and with the help of it A node is calculated

D node value:

C_n = max (S_n-X, e^-r∆t (pV_G+ (1-p)V_H))

= (18.191, 0.9967(25.535*0.5515+11.347*0.4485)

= (18.191, 19.10)

If we immediate exercise the option then the value is Rs.18.191 and if holding the option then Rs.19.10.

E node value:

= 6.2578*0.9967

=Rs.6.237

F node value = 0

Similarly future value of node B and C are Rs.12.229 and Rs.2.742

With the help of B and C node we can calculate node A future value that is Rs.7.469