# Option Prices Models

**DOI:**https://doi.org/10.1007/978-1-4614-7883-6_356-1

## Abstract

Option prices models refer to the overall collection of quantitative techniques to value an option given the dynamics of the underlying asset. Option prices models can be classified into European-style versus American-style models, into closed-form versus numerical expressions and into discrete versus continuous time approaches.

## Keywords

Stock Price Call Option American Option Tree Step Expiration Date## Definition

Quantitative techniques applied to compute the theoretical value/market price of a financial option given the characteristics and properties of the underlying asset. They can be classified into European-style versus American-style models, closed-form versus numerical expressions, and discrete versus continuous time approaches.

## Introduction

Options give the holder the right to buy (in the case of a call option) or sell (in the case of a put option) the underlying asset (e.g., shares, stock indices, currencies) at an agreed price (strike price or exercise price) during a specific period (in the case of American options) or at a predetermined expiration date (in the case of European options). Holding an option has a particular value, determining the (market) price (the “premium”) at which this option can be bought and sold. While many financial options are actively traded on an exchange and these market prices are easily observable, financial actors also rely on option prices models to calculate the theoretical (market) value of an option. Option prices models refer to the overall collection of quantitative techniques to value an option given the dynamics of the underlying asset. Although a complete taxonomy of option prices models is outside of the confines of this contribution, they can be classified into European-style versus American-style models, into closed-form versus numerical expressions and into discrete versus continuous time approaches. In this contribution we mainly focus on stock options, while we provide references to option prices models for other underlying assets at the end.

## Basic Option Relationships

*S*is higher than the exercise price

*X*, the call option is said to be in the money as it takes a positive intrinsic value (

*S-X*). In case the stock price is lower than the exercise price, the call option will not be exercised as it is cheaper to buy a share directly at the stock market. The call option is said to be out of the money and its value amounts to zero. The call option is at the money when the stock price equals the exercise price. The intrinsic value of the option is its value at expiration date or when it is exercised earlier: the payoff to the holder of the option equals

*max(S-X,0)*. The option’s market price prior to expiration or early exercise is higher than the intrinsic value as one has to add the time value on top of it. The time value reflects the flexibility that the holder has in postponing the decision to exercise the option until the expiration date, at the latest. Figure 1 also indicates the upper and lower boundary conditions of the option value. The maximum value of the call option is the underlying stock price

*S*. An option can never be worth more than what it costs to acquire the stock directly. The minimum value of the call option is the intrinsic value, or

*max(S-X,0)*. Finally, the market value depends on five parameters: the stock price, the exercise price, the time to expiration, the volatility of the stock return, and the risk-free interest rate. With the exception of the exercise price, all parameters have a positive impact on the market value of the option. These are the five value drivers of any option and will be the input parameters of any option prices model. The next section discusses different modeling approaches to calculate the theoretical market price of an option.

## European Option Prices Models

European option prices models refer to quantitative techniques used to value options which can only be exercised at their expiration date and where the holder has no decision to make during the lifetime of the option. Such basic plain vanilla options are typically valued using a closed-form expression such as the continuous time Black-Scholes model. When exact formulas are not available, numerical expressions are used to put an approximate value on the option. The discrete time binomial option pricing model is an example of the latter. Both models have in common that option valuation is based on the idea of a replicating portfolio. By creating a portfolio of stocks and bonds with the same payoff as the option in every state of the world, the law of one price requires the current value of the replicating portfolio to be equal to the current option value. As one can easily observe the market prices of the assets in the replicating portfolio, one can thus also value the option. A detailed analysis of this concept is outside the confines of this contribution.

### The Binomial Model

*t*either upward with factor

*u*or downward with factor

*d*. For instance, Fig. 2 illustrates how the stock price can move in a two-period binomial model. The left panel shows how the possible price path of the underlying stock price, while the right panel shows the corresponding option payoff tree. After two time periods there are three possible stock prices at maturity date and three corresponding option prices. As the option is at expiration, its value can easily be computed as it only takes intrinsic value: the stock price at each node minus the exercise price, or zero, the highest of both. With a process called backward induction one moves one step back in the option payoff tree to calculate the option value in earlier tree steps until one reaches the current moment.

*t*depends on the lifetime of the option (

*T-t*) and the number of tree steps

*n*. The left panel in Fig. 3 illustrates how one can expand the number of possible share prices at expiration by increasing the number of tree steps. For instance, if time to expiration

*T-t*is 1 year and one uses 250 tree steps

*n*, then the stock moves up or down every time interval \( \Delta t=\frac{T-t}{n}=\frac{1}{250} \) or approximately one step equals one trading day.

*C*is the current option price,

*C*

_{ u }is the expected option value in the upward world,

*C*

_{ d }the expected option value in the downward world,

*p*the risk-neutral probability that the stock moves upward with \( p=\frac{e^r-d}{u-d} \), and

*r*the risk-free interest rate.

### The Black-Scholes Model

*C*according to the Black-Scholes model can be calculated as:

*S*is the current stock price,

*X*the exercise price,

*T-t*the time to expiration (in years),

*σ*the annualized standard deviation of the stock return,

*r*

_{ c }the continuous risk-free interest rate, and

*N(d)*the cumulative normal probability density function.

## American Option Prices Models

American option prices models refer to quantitative techniques used to value options which can be exercised earlier than the expiration date or where the holder has to make other decisions during the lifetime of the option. The main difference between European and American option models is thus the incorporation of the early exercise feature. American call options on nondividend paying stocks are equal in value than an otherwise identical European call option. Early exercise of an American call options on a nondividend paying stock would render the holder the intrinsic option value but would lose the time value of the option. Put differently, the American call option on a nondividend paying stock is worth more alive than dead. Therefore, one can value an American call option on a nondividend paying stock with European option models. The story changes when the stock distributes one or more dividends during the lifetime of the option. In that case, early exercise of American call options on stocks is a valuable alternative and depends on the tradeoff between losing the time value of the call option and capturing the dividend(s). A closed-form solution for the valuation of an American call option with one single dividend payment is offered by Geske (1979a), Roll (1977) and Whaley (1981), while Cassimon et al. (2007) offer a similar solution for the case of multiple dividends. A popular approximation is offered by Black (1975). Other valuation models for American options include trinomial models (Kamrad and Ritchken 1991), finite difference models (Brennan and Schwartz 1977, 1978), Monte Carlo simulations (Broadie and Glasserman 1997; Longstaff and Schwartz 2001; Moreno and Navas 2003), and quadratic approximations (MacMillan 1986; Barone-Adesi and Whaley 1987). A detailed discussion of these models is beyond the confines of this contribution, but we refer the reader to the references for further details.

## Other Option Prices Models

Often more complicated option models are needed to handle the specific nature of the option features or the dynamics of the underlying asset. Such models include jump models (Merton 1976), compound option models (Geske 1979b; Cassimon et al. 2004), or barrier models (Li et al. 2013).

In the option literature one can also find option models for other financial assets, such as options on currency (Garman and Kohlhagen 1983; Amin and Jarrow 1991), options on futures (Black 1976; Ramaswamy and Sundaresan 1985), options on commodities (Hilliard and Reis 1998), options on (stock) indices (Merton 1973; Chance 1986), swaptions (Black 1976), options on interest rates (Ho and Lee 1986), and bond options (Schaefer and Schwartz 1987). Option models are also used in handling credit default risk (Merton 1974; Crosbie and Bohn 2003).

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## Recommended Readings

- Derivagem (2014) Option calculator at http://www-2.rotman.utoronto.ca/~hull/software/index.html
- Hull JC (2011) Options, futures, and other derivatives, 8th edn. Pearson, UpperSaddle RiverGoogle Scholar