Encyclopedia of Law and Economics

Living Edition
| Editors: Alain Marciano, Giovanni Battista Ramello

Option Prices Models

  • Peter-Jan EngelenEmail author
  • Danny Cassimon
Living reference work entry
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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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

As an option is a financial instrument which gives the holder the right, but not the obligation, to exercise, the typical payoff structure of an option is asymmetrical. The market value or premium of the option typically consists of the sum of the intrinsic value and the time value. It is indicated by the solid line in Fig. 1. The intrinsic value of an option is its value upon immediate exercise. The dash line in Fig. 1 indicates the intrinsic value and shows the typical hockey stick payoff profile of a call option. When the stock price 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.
Fig. 1

Value of a call option as a function of the stock price

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

This model assumes the stock price moves in every time period Δt 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.
Fig. 2

Two-period binomial trees

To develop a realistic tree of possible future stock prices, one needs to determine the size of one tree step and the upward and downward jumps of the stock prices. The time interval Δt 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.
Fig. 3

Possible price paths and distribution of stock prices at expiration

To determine the value of the upward and downward movement the standard binomial option model of Cox et al. (1979) fixes \( u={e}^{\sigma \sqrt{\Delta t}} \) and \( d=1/u \). Once both parameters are determined, one can develop the binomial tree and calculate the current option value by discounting at each node of the tree the upward and downward expected option values:
$$ C=\left\{\left[{C}_u\times p\right]+\left[{C}_d\times \left(1-p\right)\right]\right\}{e}^{-r} $$
(1)
where 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 right panel in Fig. 3 shows that increasing the number of tree steps to infinity produces a lognormal distribution of possible share prices at expiration. In that case, the binomial model converges to the continuous time model we discuss in the next section. In practice the binomial model produces a good approximation of the Black-Scholes model from 250 tree steps onwards (see Fig. 4).
Fig. 4

Convergence of the binomial model towards the Black-Scholes (BS) model

The Black-Scholes Model

Probably the best known model has been developed by Black and Scholes (1973). Its popularity is derived from its closed-form solution and fast and relatively simple computation. Its main disadvantage is due to the strict assumptions underlying the model (Hull 2011): (i) frictionless markets, implying no transaction costs or taxes, nor restrictions on short sales; (ii) continuous trading is possible; (iii) the risk-free (short term) interest rate is constant over the life of the option; (iv) the market is arbitrage-free; and (v) the time process of the underlying asset price is stochastic and exhibits a process assuming asset prices to be lognormally distributed and returns to be normally distributed. Obviously, any violation of some of these assumptions may result in a theoretical Black-Scholes option value which deviates from the price observed at the market. The option value C according to the Black-Scholes model can be calculated as:
$$ C=SN\left({d}_1\right)-X{e}^{-{r}_c\left(T-t\right)}N\left({d}_2\right) $$
(2)
$$ {d}_1=\frac{ \ln \left(\frac{S}{X}\right)+\left({r}_c+\frac{1}{2}{\sigma}^2\right)\left(T-t\right)}{\sigma \sqrt{T-t}} $$
(3)
$$ {d}_2=\frac{ \ln \left(\frac{S}{X}\right)+\left({r}_c-\frac{1}{2}{\sigma}^2\right)\left(T-t\right)}{\sigma \sqrt{T-t}}={d}_1-\sigma \sqrt{T-t}, $$
(4)
where 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).

References

  1. Amin KI, Jarrow RA (1991) Pricing foreign currency options under stochastic interest rates. J Int Money Financ 10(3):310–329CrossRefGoogle Scholar
  2. Barone-Adesi G, Whaley R (1987) Efficient analytic approximation of American option values. J Financ 42:301–320CrossRefGoogle Scholar
  3. Black F (1975) Facts and fantasy in the use of options. Financ Anal J 31(4):36–41, 61–72CrossRefGoogle Scholar
  4. Black F (1976) The pricing of commodity contracts. J Financ Econ 3(1):167–179CrossRefGoogle Scholar
  5. Black F, Scholes M (1973) The pricing of options and corporate liabilities. J Polit Econ 81:637–659CrossRefGoogle Scholar
  6. Brennan M, Schwartz E (1977) The valuation of American put options. J Financ 32:449–462CrossRefGoogle Scholar
  7. Brennan M, Schwartz E (1978) Finite difference methods and jump processes arising in the pricing of contingent claims: a synthesis. J Financ Quant Anal 13:461–474CrossRefGoogle Scholar
  8. Broadie M, Glasserman P (1997) Pricing American-style securities using simulation. J Econ Dyn Control 21:1323–1352CrossRefGoogle Scholar
  9. Cassimon D, Engelen PJ, Thomassen L, Van Wouwe M (2004) Valuing new drug applications using n-fold compound options. Res Policy 33:41–51CrossRefGoogle Scholar
  10. Cassimon D, Engelen PJ, Thomassen L, Van Wouwe M (2007) Closed-form valuation of American call-options with multiple dividends using n-fold compound option models. Financ Res Lett 4:33–48CrossRefGoogle Scholar
  11. Chance DM (1986) Empirical tests of the pricing of index call options. Adv Futur Options Res 1(Part A):141–166Google Scholar
  12. Cox JC, Ross SA, Rubinstein M (1979) Option pricing: a simplified approach. J Financ Econ 7(3):229–263CrossRefGoogle Scholar
  13. Crosbie P, Bohn J (2003) Modeling default risk, white paper. Moody’s KMV, USAGoogle Scholar
  14. Garman MB, Kohlhagen SW (1983) Foreign currency option values. J Int Money Financ 2(3):231–237CrossRefGoogle Scholar
  15. Geske R (1979a) A note on an analytic valuation formula for unprotected American call options on stocks with known dividends. J Financ Econ 7:375–380CrossRefGoogle Scholar
  16. Geske R (1979b) The valuation of compound options. J Financ Econ 7:63–81CrossRefGoogle Scholar
  17. Hilliard JE, Reis J (1998) Valuation of commodity futures and options under stochastic convenience yields, interest rates, and jump diffusions in the spot. J Financ Quant Anal 33(01):61–86CrossRefGoogle Scholar
  18. Ho T, Lee S (1986) Term structure movements and pricing interest rate contingent claims. J Financ 41(4):1011–1029CrossRefGoogle Scholar
  19. Kamrad B, Ritchken P (1991) Multinomial approximating models for options with k state variables. Manag Sci 37:1640–1652CrossRefGoogle Scholar
  20. Li Y, Engelen PJ, Kool C (2013) A barrier options approach to modeling project failure: the case of hydrogen fuel infrastructure, Tjalling C. Koopmans Research Institute Discussion Paper Series 13–01, 34pGoogle Scholar
  21. Longstaff F, Schwartz E (2001) Valuing American options by simulation: a simple least-squares approach. Rev Financ Stud 14:113–147CrossRefGoogle Scholar
  22. MacMillan L (1986) Analytic approximation for the American put option. Adv Futur Options Res 1:119–139Google Scholar
  23. Merton R (1973) Theory of rational option pricing. Bell J Econ Manag Sci 4(1):141–183Google Scholar
  24. Merton RC (1974) On the pricing of corporate debt: the risk structure of interest rates. J Financ 29(2):449–470Google Scholar
  25. Merton RC (1976) Option pricing when underlying stock returns are discontinuous. J Financ Econ 3(1):125–144CrossRefGoogle Scholar
  26. Moreno M, Navas J (2003) On the robustness of least-squares Monte Carlo (LSM) for pricing American derivatives. Rev Deriv Res 6:107–128CrossRefGoogle Scholar
  27. Ramaswamy K, Sundaresan SM (1985) The valuation of options on futures contracts. J Financ 40(5):1319–1340CrossRefGoogle Scholar
  28. Roll R (1977) An analytic formula for unprotected American call options on stocks with known dividends. J Financ Econ 5:251–258CrossRefGoogle Scholar
  29. Schaefer SM, Schwartz ES (1987) Time-dependent variance and the pricing of bond options. J Financ 42(5):1113–1128CrossRefGoogle Scholar
  30. Whaley R (1981) On the valuation of American call options on stocks with known dividends. J Financ Econ 9:207–211CrossRefGoogle Scholar

Recommended Readings

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

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Utrecht School of EconomicsUtrecht UniversityUtrechtThe Netherlands
  2. 2.Institute for Development Policy and ManagementUniversity of AntwerpenAntwerpenBelgium