This module includes functions/components to value standard American and European options. Specific option models are also documented in this module.

Refer also to the Exotic Options 1 and Exotic Options 2 modules. These two modules cover many different exotic options. In addition also refer to Statistical Tools module that provide general Monte Carlo functionality for pricing and analyzing standard & custom built exotic options.

BlackScholes

 This module contains important information about and examples of how to use the Derivatives Expert functions/components that describe the Black & Scholes (1973) option pricing model, The Pricing of Options and Corporate Liabilities, Journal of Political Economy 81 (May-June 1973), pp. 637-54.

The Black & Scholes (1973) model was a breakthrough in option pricing because it specified the option price solely as a function of known variables. The model is applicable for valuing European Call and European Put options on non-dividend paying stock. The theoretical prices, implied volatilities, elasticities and the first and second order partial derivatives of the option pricing functions with respect to each of the arguments of the option pricing function are implemented in the Black Scholes module.

Black76

 This module contains important information about and examples of how to use the Derivatives Expert functions/components that describe the Black (1976) option pricing model, The Pricing of Commodity Contracts, Journal of Financial Economics, 3 (Jan. - Mar.), pp. 167-179.

The model is applicable for valuing European call and European put options on commodity futures. The exact nature of the underlying commodity varies and may be anything from a precious metal such as gold or silver to agricultural products. The theoretical prices, implied volatilities, elasticities and the first and second order partial derivatives of the option pricing functions with respect to each of the arguments of the option pricing function are implemented in the Black76 module.

CoxRossRubinstein

 This module contains important information about and examples of how to use the Derivatives Expert functions/components that describe the Cox, Ross & Rubinstein (1979) option pricing model, A Simplified Approach, Journal of Financial Economics 7, pp. 229-263.

The Cox, Ross & Rubinstein (1979) was another milestone in option pricing because it made it possible to price American and European options on dividend paying instruments in a rather simple manner using only non-arbitrage arguments.

The model is a discrete time model in contrast to the Black & Scholes model which is a continuous time model. Another important difference is that the Black & Scholes model is an analytical (exact) model while the Cox, Ross & Rubinstein (1979) model is an approximate model based on numerical methods.

The Cox, Ross & Rubinstein (1979) model also shows that the option price can be interpreted as the expected discounted future value in a risk-neutral world. Risk-neutral pricing methodologies have since been refined using martingale theory (from probability theory).

In Derivatives Expert five versions based on "A Simplified Approach" are implemented. They are named Binomial1 - Binomial5.

Binomial1 represents a model that converges to the Black & Scholes (1973) model when the number of trading periods equals a large positive integer. European call options can be priced with this model. It is included for illustration purposes only because it is normally better to use the Black & Scholes model directly for pricing purposes.

Binomial2 represents a non-dividend version of the general recursive model. American call and put options and European call options can be priced in Derivatives Expert. With no dividends a European call option and an American call option will have the same price but this is not necessarily the case with put options.

Binomial3 represents a constant dividend version of the general recursive model in non-calendar time. American call and put options can be priced in Derivatives Expert. With dividends, a European option and an American option will not have the same price.

Binomial4 represents a variable dividend version of the general recursive model in non-calendar time, and Binomial5 represents the same variable dividend version but in calendar time.

GarmanKohlhagen

 This module contains important information about and examples of how to use the Derivatives Expert functions/components that describe the Garman & Kohlhagen (1983) option pricing model, Foreign Currency Option Values, Journal of International Money and Finance, 2(3), pp. 231-238.

This model is applicable for valuing European call and European put options on foreign exchange. The theoretical prices, implied volatilities, elasticities and the first and second order partial derivatives of the option pricing functions with respect to each of the arguments of the option pricing function are implemented in the Garman Kohlhagen module.

ShastriTandon

  This module contains important information about and examples of how to use the Derivatives Expert functions/components that describe the Shastri & Tandon (1987) option pricing model: Valuation of American Options on Foreign Currency, Journal of Banking and Finance 11 (1987), pp. 245-269.

The model is applicable for valuing European call, American call, European put and American put options on foreign exchange. Using the analytical techniques developed by Geske and Johnson (1984), The American put valued analytically, Journal of Finance 39, Dec. pp. 1511-1524, these securities are priced as a sequence of compound options. A total of nine different functions from the Shastri and Tandon (1987) formulas are provided in the Shastri Tandon module.

The Garman & Kohlhagen (1983) model is a special case of this model.

BaroneAdesiWhaley

 This module contains important information about and examples of how to use the Derivatives Expert functions/components that describe the Barone-Adesi & Whaley (1987) option pricing model, Efficient Analytic Approximation of American Option Values, The Journal of Finance, Vol. XLII, No.2, June.

This model provides approximate (not fully analytical) functions for valuing American call, European call, American put and European put options on commodities and commodity futures contracts. The method is called the quadratic approximation method.

Quoting directly from the paper: "The exact nature of the underlying commodity varies, and may be anything from a precious metal such as gold or silver to a financial instrument such as a Treasury Bond, foreign currency, or a constant dividend paying stock".

The Black & Scholes (1973), Black (1976), Garman & Kohlhagen (1983) and Shastri & Tandon (1987) are all special cases of the Barone-Adesi & Whaley (1987) model, but the model is not accurate in all cases when the time to expiration of the option is more than one year. In this case e.g. the Binomial or a Monte Carlo model can be used instead.

Context

 Different specific option pricing models are implemented in the Derivatives Expert modules: Zhang, BlackScholes, Black76, CoxRossRubinstein, GarmanKohlhagen, ShastriTandon and BaroneAdesiWhaley.

These option pricing models are all well known and widely used by the financial community. The question is which option pricing model to use. The Choice of Model section gives a non-exhaustive list of intended uses of the option pricing models. Note that new studies may show other results than those found in the original articles.

Refer to Exotics, Exotic Options 1 and Exotic Options 2 for documentation and examples on how to price many different exotic option types. Also refer to Statistical Tools that provide general Monte Carlo functionality for pricing and analyzing options.

Contents

13.1 Introduction

Summary -- Context -- This Opens the Package

+ 13.2 Objects

CallOption -- PutOption -- EuropeanCallOption -- EuropeanPutOption
AmericanCallOption -- AmericanCallOption2 -- AmericanCallOption3
AmericanPutOption -- AmericanPutOption2 -- AmericanPutOption3

Black76 -- BlackScholes -- Binomial1 -- Binomial2
+Binomial3 -- +Binomial4 -- +Binomial5
GarmanKohlhagen -- ShastriTandon -- BaroneAdesiWhaley

13.3 Choice of Model

American and European Options (Examples)

Non-Dividend Paying Stock
Constant, Proportional Dividend Paying Stock
Variable, Non-Proportional Dividend Paying Stock
Foreign Exchange
Commodities
Commodity Futures
Stock Futures
Bond Futures

13.4 Cash Flow

Non-Calendar Time

CashFlow

Calendar Time

CashFlow

+ 13.5 Pricing

Non-Calendar Time

D1 -- D2 -- TheoreticalPrice -- ImplicitVolatility

Calendar Time

+Binomial5

13.6 Static Risk

Non-Calendar Time

Elasticity
All First and Second Order Partial Derivatives (Delta, Gamma, etc.)