Introduction:
Options calculator provides user with the facility to calculate the price or volatility for equity options. Also it provides user with various risk measures (commonly referred to as Greek Letters). User can also view graphs of various Greek Letters. These graphs show user the variation of the Greek Letters with change in underline price.

Valuation Method:
There are two types of options currently being traded in the market, American and European. Since these two options are very different from each other, its not possible to use common valuation method. Hence separate valuation methods are used for pricing American and European options. The detailed methodology for pricing as well as calculating the Greek Letters is given below.

European options:
For valuation of European options, Black-Scholes method of valuing the options is used.

Various formulas used for pricing and calculating Greek Letters are given below:

Formula for pricing of call option,
c = S * N(d1) – X * exp(-rT) * N(d2)

Formula for pricing the put option,
p = X * exp(-r * T) * N(-d2) - S * N(-d1)

where, d1 = ( ln(S/X) + (r + v^2/2) * T ) / v * T^1/2
d2 = d1 - v * T1/2

Formula for calculating the Delta,
Delta (call option) = N(d1)
Delta (put option) = N(d1) - 1

Formula for calculating Gamma,
Gamma (for both put and call) = N’(d1) / S * v * T^1/2

Formula for calculating Theta,
Theta (for call option) = -S * N’(d1) * v / 2 * T^1/2 - r * X * exp( -r * T) * N(d2)
Theta (for put option) = -S * N’(d1) * v / 2 * T^1/2 + r * X * exp( -r * T) * N(-d2)

Formula for calculating Vega,
Vega (for both put and call) = S * N’(d1) * T^1/2

where
S = stock price
T = time left for expiry of option
X = strike price
r = risk free rate of interest
v = volatility of stock

For calculating the value of implied volatility, method of binomial convergence is used.

American options:
For valuation of American options, valuation method suggested by Roll, Geske and Whaley is used.

Formula for pricing American call option,
C = (S – D1 * exp(-r * t1 )) * N(b1) + (S - D1 * exp(-r * t1 )) * M(a1, -b1; - (t1/T)^1/2 ) – X * exp( -r * T) * M(a2, -b2; - (t1/T)^1/2 ) – (X - D1 ) * exp(-r * t1 )) * N(b2)

Where
a1 = ( ln((S - D1 * exp(-r * t1 )) / X) + (r + v^2 / 2) * T ) / v * T^1/2
a2 = a1 - v * T^1/2
b1 = ( ln((S - D1 * exp(-r * t1 )) / S*) + (r + v^2 / 2) * t1 ) / v * t1^1/2
b2 = b1 - v * t1^1/2

The function M ( a, b; r) is the cumulative probability in a standardized bi variate normal distribution.

The variable S* is the solution to
c(S*) = S* + D1 - X

where c(S*) is the Black-Scholes option price, D1 is the final dividend and t1 is the time of final dividend.

For the valuation of Greeks Letters (Delta, Gamma, Vega and Theta), the numerical method of calculation is used.

For calculating the value of implied volatility, method of binomial convergence is used.