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Volatility-Based Margining

Copyright 2008-2014 FX BridgeTechnologies Corp.
U.S. Provisional Patent Application - Serial No.: 61/038,171
Title: VOLATILITY-BASED MARGINING

As everyone knows, modeling options pricing is the science, entering the correct parameters is the art. FX Bridge employs a modified Black-and-Scholes model for options to accommodate the domestic and foreign interest rate differentials associated with foreign exchange options. This model is known as the Garman-Kolhagen model, or the Biger and Hull Model - both having been published near the same time.

The formula was developed by Fischer Black and Myron Scholes and published in 1973. They built on earlier research by Edward O. Thorp, Paul Samuelson, and Robert C. Merton. The fundamental insight of Black and Scholes is that the option is implicitly priced if the underlying asset is traded.

Merton and Scholes received the 1997 Nobel Prize in Economics for this and related work; Black was ineligible, having died in 1995.
The key assumptions of the Black–Scholes model are seen in this derivative equation:

The price of the underlying instrument St follows a geometric Brownian motion with µ (the percentage drift) and s ('the percentage volatility') are constants and Wt is a Wiener process or Brownian motion:

It is possible to short sell the underlying asset.

There are no arbitrage opportunities.

Trading in the underlying asset is continuous.

There are no transaction costs or taxes.

All assets are perfectly divisible (e.g. it is possible to buy 1/100th of an underlying asset).

It is possible to borrow and lend cash at a constant risk-free interest rate.

The above lead to the following formula for the price of a call and put options with a strike price K on a underlying asset currently trading at price S, i.e., the right to buy the underlying asset at the exchange rate of K after T years. The constant interest rate is r, and the constant underlying asset volatility is s.

S = Current underlying asset price
K = Strike price
T = current time-to-expiration (in years)
r = risk-less return (annualized)
q = underlying asset volatility (annualized)
N() = is the standard normal cumulative distribution function

where:

or:

(so that d + = d1 and d - = d2)

The Garman-Kohlhagen Model (also called "Biger and Hull" Model) is a currency option evaluation model developed in 1983. It is based on the Black-Scholes option model, but includes separate terms for foreign and domestic interest rates. Biger and Hull published essentially the same model at about the same time. The Garman Kohlhagen model is suitable for evaluating European style options on spot foreign exchange. This model has expanded the Black-Scholes assumption that borrowing and lending is performed at the same rate. In the real foreign exchange markets the risk free rate is not identical in both countries. Any interest rate differential between the two currencies will impact the value of the currency option. The Garman-Kohlhagen model treats the risk free foreign interest rate as a continuous dividend yield being paid on the foreign currency. Since an option holder does not receive any cash-flows paid from the underlying instrument, the present value of the continuous cash-flow is subtracted from the price of the underlying instrument. This results in a lower call price and a higher put price.

The Garman-Kolhagen model is slightly different from the Black-Scholes model:

T = time of expiration of option
S = current price of one unit of foreign currency
K = strike price of option
rd = continuously compounded domestic risk-free rate of interest for maturity T
rf = continuously compounded foreign risk-free rate of interest for maturity T
C = value of European call option to buy one unit of foreign currency
P = value of European put option to buy one unit of foreign currency
q = volatility of the underlying asset formulae

where

Fortunately, the FX Bridge Technologies engineers and, more importantly, years of real-world usage, have taken care of the math. All you need to do it set a few simple parameters for volatility and you are done. To learn more about our proprietary price modeling facility that makes pricing a snap, contact us today.

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