Generalized Black-Scholes-Merton on Variance Form [Loxx]

4 oct. 2022

Generalized Black-Scholes-Merton on Variance Form [Loxx] is an adaptation of the Black-Scholes-Merton Option Pricing Model including Numerical Greeks. The following information is an excerpt from Espen Gaarder Haug's book "Option Pricing Formulas". This version is to price Options using variance instead of volatility.

Black- Scholes- Merton on Variance Form
In some circumstances, it is useful to rewrite the BSM formula using variance as input instead of volatility, V = v^2:

c = S * e^((b - r) * T) * N(d1) - X * e^(-r * T) * N(d2)

p = X * e^(-r * T) * N(-d2) - S * e^((b - r) * T) * N(-d1)

where

d1 = (log(S / X) + (b + V^2 / 2) * T) / (V * T)^0.5

d2 = d1 - (V * T)^0.5

BSM on variance form clearly gives the same price as when written on volatility form. The variance form is used indirectly in terms of its partial derivatives in some stochastic variance models, as well as for hedging of variance swaps. The BSM on variance form moreover admits an interesting symmetry between put and call options as discussed by Adamchuk and Haug (2005) at wilmott.com .

c(S, X, T, r, b, V) = -c(-S, -X, -T, -r, -b, -V)

and

p(S, X, T, r, b, V) = -p(-S, -X, -T, -r, -b, -V)

It is possible to find several similar symmetries if we introduce imaginary numbers.

b = r ... gives the Black and Scholes (1973) stock option model.
b = r — q ... gives the Merton (1973) stock option model with continuous dividend yield q.
b = 0 ... gives the Black (1976) futures option model.
b = 0 and r = 0 ... gives the Asay (1982) margined futures option model.
b = r — rf ... gives the Garman and Kohlhagen (1983) currency option model.

Inputs
S = Stock price.
X = Strike price of option.
T = Time to expiration in years.
r = Risk-free rate
cc = Cost of Carry
V = Variance of the underlying asset price
cnd (x) = The cumulative normal distribution function
nd(x) = The standard normal density function
convertingToCCRate(r, cmp ) = Rate compounder

Numerical Greeks or Greeks by Finite Difference
Analytical Greeks are the standard approach to estimating Delta, Gamma etc... That is what we typically use when we can derive from closed form solutions. Normally, these are well-defined and available in text books. Previously, we relied on closed form solutions for the call or put formulae differentiated with respect to the Black Scholes parameters. When Greeks formulae are difficult to develop or tease out, we can alternatively employ numerical Greeks - sometimes referred to finite difference approximations. A key advantage of numerical Greeks relates to their estimation independent of deriving mathematical Greeks. This could be important when we examine American options where there may not technically exist an exact closed form solution that is straightforward to work with. (via VinegarHill FinanceLabs)

Things to know

• Only works on the daily timeframe and for the current source price.
  
• You can adjust the text size to fit the screen
  

4 oct. 2022

Notes de version

Corrected static T input.