Equity derivatives: models and methods

Last updated 10 November 2026

This page describes how GIAnalyzer and the Gamma Impact API value equity options and measure their risk: the options covered, the model and numerical method behind each, the market data each takes, and what the models do not do. It describes GIAnalyzer version 5.12.0; the Gamma Impact API covers the same instruments and models. It is provided for information and is not a contractual specification.

The function-level detail is in the function reference.

Inputs and conventions

• Dates: a valuation date, an expiry date and a delivery (settlement) date. Volatility applies over the time to expiry, in calendar days over 365; discounting applies to the delivery date, in the interest rate's day count.
• Interest rate: a flat rate or a zero-rate curve read at the delivery date, converted to continuous compounding.
• Dividends: a continuous dividend yield.
• Volatility: a single flat volatility, or, for vanilla options, a volatility surface by strike (as a proportion of spot) and time to expiry, interpolated between its points and held flat beyond them.

Instruments and models

European vanilla options

Three methods are available:

• Closed form: the Black-Scholes-Merton model with a continuous dividend yield (Black and Scholes, 1973; Merton, 1973).
• Binomial tree: a Cox-Ross-Rubinstein tree with the number of steps set by the user.
• Monte Carlo: simulation of the share price at expiry, with the number of paths and the random seed set by the user.

Implied volatility is found from a given premium by an iterative search.

American vanilla options

Two methods are available:

• Binomial tree: a Cox-Ross-Rubinstein tree with early exercise tested at every node.
• Approximation: the Bjerksund-Stensland (1993) closed-form approximation with a flat exercise boundary.

Digital options

European cash-or-nothing and asset-or-nothing options, in closed form (Reiner and Rubinstein, 1991).

One-touch and no-touch options

Continuously monitored touch options, paid at the hit or at expiry, with the closed-form formulas given by Wystup (2006).

Single-barrier options

Knock-in and knock-out options with an up or down barrier:

• Continuously monitored barriers: closed form (Merton, 1973; Reiner and Rubinstein, 1991), or a binomial tree in which the barrier is checked at every step.
• Barriers observed at expiry only: replicated with a vanilla and a digital option, or on a binomial tree.

Knock-in options are valued as the vanilla less the knock-out.

Double-barrier and KIKO options

Double-barrier options with continuously monitored barriers are valued with a series solution of the method-of-images type (Kunitomo and Ikeda, 1992). A KIKO option (one knock-in and one knock-out barrier) is valued as a single-barrier knock-out less a double-barrier knock-out.

Asian options

Discrete arithmetic average-rate options with a fixed strike, with a moment-matching lognormal approximation of the average (Levy, 1992; Haug, Haug and Margrabe, 2003). Fixings already observed are taken into account; fixings are assumed evenly spaced.

Executive and employee stock options

Executive stock options are valued on a binomial tree that reflects the features of employee options, in the spirit of Hull and White (2004):

• a vesting period, with forfeiture if the employee leaves before vesting;
• exit after vesting, with exercise at intrinsic value;
• voluntary early exercise when the share price reaches a set multiple of the strike;
• a strike that can change each year, an optional cap on the share price, and an optional barrier.

Liquidity risk of American put options

For a portfolio of American equity put options written, GIAnalyzer estimates the cash that may be paid out on early exercise: for each position, the strike amount weighted by the value of a one-touch option on the share price falling a set distance below the strike.

Risk measures

All sensitivities are computed by revaluing the option with a small change in one input (finite differences).

With a volatility surface, the volatility is held fixed when the share price is shifted.

What the models do not do

The equity models:

• take dividends as a continuous yield only: discrete cash dividends are not modelled;
• use no stochastic or local volatility: the volatility surface applies to vanilla options only, and exotic options use a flat volatility;
• treat closed-form barriers as continuously monitored, with no correction for discrete monitoring, and have no rebates;
• value Asian options with evenly spaced fixings, arithmetic average-rate only;
• provide no Greeks for executive stock options: the model returns the value only, and call options only.

References

• Bjerksund, P. and Stensland, G. (1993). Closed-form approximation of American options. Scandinavian Journal of Management 9, 87–99.
• Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy 81(3), 637–654.
• Cox, J. C., Ross, S. A. and Rubinstein, M. (1979). Option pricing: a simplified approach. Journal of Financial Economics 7(3), 229–263.
• Haug, E. G. (2007). The Complete Guide to Option Pricing Formulas, 2nd edition. McGraw-Hill.
• Haug, E. G., Haug, J. and Margrabe, W. (2003). Asian pyramid power. Wilmott Magazine.
• Hull, J. and White, A. (2004). How to value employee stock options. Financial Analysts Journal 60(1), 114–119.
• Kunitomo, N. and Ikeda, M. (1992). Pricing options with curved boundaries. Mathematical Finance 2(4), 275–298.
• Levy, E. (1992). Pricing European average rate currency options. Journal of International Money and Finance 11(5), 474–491.
• Merton, R. C. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science 4(1), 141–183.
• Reiner, E. and Rubinstein, M. (1991). Breaking down the barriers. Risk 4(8), 28–35; and Unscrambling the binary code. Risk 4(9), 75–83.
• Wystup, U. (2006). FX Options and Structured Products. Wiley.

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