# GenBS: Generalized Black Scholes model for pricing vanilla European... In jrvFinance: Basic Finance; NPV/IRR/Annuities/Bond-Pricing; Black Scholes

## Description

Compute values of call and put options as well as the Greeks - the sensitivities of the option price to various input arguments using the Generalized Black Scholes model. "Generalized" means that the asset can have a continuous dividend yield.

## Usage

 `1` ```GenBS(s, X, r, Sigma, t, div_yield = 0) ```

## Arguments

 `s` the spot price of the asset (the stock price for options on stocks) `X` the exercise or strike price of the option `r` the continuously compounded rate of interest in decimal (0.10 or 10e-2 for 10%) (use `equiv.rate` to convert to a continuously compounded rate) `Sigma` the volatility of the asset price in decimal (0.20 or 20e-2 for 20%) `t` the maturity of the option in years `div_yield` the continuously compounded dividend yield (0.05 or 5e-2 for 5%) (use `equiv.rate` to convert to a continuously compounded rate)

## Details

The Generalized Black Scholes formula for call options is
exp(-r * t) * (s * exp(g * t) * Nd1 - X * Nd2)
where
g = r - div\_yield
Nd1 = N(d1) and Nd2 = N(d2)
d1 = (log(s / X) + (g + Sigma^2/ 2) * t) / (Sigma * sqrt(t))
d2 = d1 - Sigma * sqrt(t)
N denotes the normal CDF (`pnorm`)
For put options, the formula is
exp(-r * t) * (-s * exp(g * t) * Nminusd1 + X * Nminusd2)
where
Nminusd1 = N(-d1) and Nminusd2 = N(-d2)

## Value

A list of the following elements

 `call` the value of a call option `put` the value of a put option `Greeks` a list of the following elements `Greeks\$callDelta` the delta of a call option - the sensitivity to the spot price of the asset `Greeks\$putDelta` the delta of a put option - the sensitivity to the spot price of the asset `Greeks\$callTheta` the theta of a call option - the time decay of the option value with passage of time. Note that time is measured in years. To find a daily theta divided by 365. `Greeks\$putTheta` the theta of a put option `Greeks\$Gamma` the gamma of a call or put option - the second derivative with respect to the spot price or the sensitivity of delta to the spot price `Greeks\$Vega` the vega of a call or put option - the sensitivity to the volatility `Greeks\$callRho` the rho of a call option - the sensitivity to the interest rate `Greeks\$putRho` the rho of a put option - the sensitivity to the interest rate `extra` a list of the following elements `extra\$d1` the d1 of the Generalized Black Scholes formula `extra\$d2` the d2 of the Generalized Black Scholes formula `extra\$Nd1` is `pnorm`(d1) `extra\$Nd2` is `pnorm`(d2) `extra\$Nminusd1` is `pnorm`(-d1) `extra\$Nminusd2` is `pnorm`(-d2) `extra\$callProb` the (risk neutral) probability that the call will be exercised = `Nd2` `extra\$putProb` the (risk neutral) probability that the put will be exercised = `Nminusd2`

jrvFinance documentation built on Nov. 5, 2021, 5:07 p.m.