Because the values of option contracts depend on a number of different variables in addition to the value of the underlying asset, they are complex to value. There are many pricing models in use, although all essentially incorporate the concepts of
rational pricing (i.e.
risk neutrality),
moneyness,
option time value, and
put–call parity. The valuation itself combines a model of the behavior (
"process") of the underlying price with a mathematical method which returns the premium as a function of the assumed behavior. The models range from the (prototypical)
Black–Scholes model for equities, to the
Heath–Jarrow–Morton framework for interest rates, to the
Heston model where volatility itself is considered
stochastic. See
Asset pricing for a listing of the various models here.
Basic decomposition In its most basic terms, the value of an option is commonly decomposed into two parts: • The first part is the
intrinsic value, which is defined as the difference between the market value of the
underlying, and the strike price of the given option • The second part is the
time value, which depends on a set of other factors which, through a multi-variable, non-linear interrelationship, reflect the
discounted expected value of that difference at expiration.
Valuation models As above, the value of the option is estimated using a variety of quantitative techniques, all based on the principle of
risk-neutral pricing and using
stochastic calculus in their solution. The most basic model is the
Black–Scholes model. More sophisticated models are used to model the
volatility smile. These models are implemented using a variety of numerical techniques. In general, standard option valuation models depend on the following factors: • The current market price of the underlying security • The
strike price of the option, particularly in relation to the current market price of the underlying (in the money vs. out of the money) • The cost of holding a position in the underlying security, including interest and dividends • The time to
expiration together with any restrictions on when exercise may occur • an estimate of the future
volatility of the underlying security's price over the life of the option More advanced models can require additional factors, such as an estimate of how volatility changes over time and for various underlying price levels, or the dynamics of stochastic interest rates. The following are some principal valuation techniques used in practice to evaluate option contracts.
Black–Scholes Following early work by
Louis Bachelier and later work by
Robert C. Merton,
Fischer Black and
Myron Scholes made a major breakthrough by deriving a differential equation that must be satisfied by the price of any derivative dependent on a non-dividend-paying stock. By employing the technique of constructing a risk-neutral portfolio that replicates the returns of holding an option, Black and Scholes produced a closed-form solution for a European option's theoretical price. At the same time, the model generates
hedge parameters necessary for effective risk management of option holdings. While the ideas behind the Black–Scholes model were ground-breaking and eventually led to Scholes and Merton receiving the
Swedish Central Bank's associated
Prize for Achievement in Economics (a.k.a., the
Nobel Prize in Economics), the application of the model in actual options trading is clumsy because of the assumptions of continuous trading, constant volatility, and a constant interest rate. Nevertheless, the Black–Scholes model is still one of the most important methods and foundations for the existing financial market in which the result is within the reasonable range.
Stochastic volatility models Since the
market crash of 1987, it has been observed that market
implied volatility for options of lower strike prices is typically higher than for higher strike prices, suggesting that volatility varies both for time and for the price level of the underlying security a so-called
volatility smile; and with a time dimension, a
volatility surface. The main approach here is to treat volatility as
stochastic, with the resultant
stochastic volatility models and the
Heston model as a prototype; An alternate, though related, approach is to apply a
local volatility model, where
volatility is treated as a
deterministic function of both the current asset level S_t and of time t . As such, a local volatility model is a generalisation of the
Black–Scholes model, where the volatility is a constant. The concept was developed when
Bruno Dupire and
Emanuel Derman and
Iraj Kani noted that there is a unique diffusion process consistent with the risk neutral densities derived from the market prices of European options. See
#Development for discussion.
Short-rate models For the valuation of
bond options,
swaptions (i.e. options on
swaps), and
interest rate cap and floors (effectively options on the interest rate) various
short-rate models have been developed (applicable, in fact, to
interest rate derivatives generally). The best known of these are
Black-Derman-Toy and
Hull–White. These models describe the future evolution of
interest rates by describing the future evolution of the short rate. The other major framework for interest rate modelling is the
Heath–Jarrow–Morton framework (HJM). The distinction is that HJM gives an analytical description of the
entire yield curve, rather than just the short rate. (The HJM framework incorporates the
Brace–Gatarek–Musiela model and
market models. And some of the short rate models can be straightforwardly expressed in the HJM framework.) For some purposes, e.g., valuation of
mortgage-backed securities, this can be a big simplification; regardless, the framework is often preferred for models of higher dimension. Note that for the simpler options here, i.e. those mentioned initially, the
Black model can instead be employed, with certain assumptions.
Model implementation Once a valuation model has been chosen, there are a number of different techniques used to implement the models.
Analytic techniques In some cases, one can take the
mathematical model and using analytical methods, develop
closed form solutions such as the
Black–Scholes model and the
Black model. The resulting solutions are readily computable, as are their
"Greeks". Although the
Roll–Geske–Whaley model applies to an American call with one dividend, for other cases of
American options, closed form solutions are not available; approximations here include
Barone-Adesi and Whaley,
Bjerksund and Stensland and others.
Binomial tree pricing model Closely following the derivation of Black and Scholes,
John Cox,
Stephen Ross and
Mark Rubinstein developed the original version of the
binomial options pricing model. It models the dynamics of the option's theoretical value for
discrete time intervals over the option's life. The model starts with a binomial tree of discrete future possible underlying stock prices. By constructing a riskless portfolio of an option and stock (as in the Black–Scholes model) a simple formula can be used to find the option price at each node in the tree. This value can approximate the theoretical value produced by Black–Scholes, to the desired degree of precision. However, the binomial model is considered more accurate than Black–Scholes because it is more flexible; e.g., discrete future dividend payments can be modeled correctly at the proper forward time steps, and
American options can be modeled as well as European ones. Binomial models are widely used by professional option traders. The
trinomial tree is a similar model, allowing for an up, down or stable path; although considered more accurate, particularly when fewer time-steps are modelled, it is less commonly used as its implementation is more complex. For a more general discussion, as well as for application to commodities, interest rates and hybrid instruments, see
Lattice model (finance).
Monte Carlo models For many classes of options, traditional valuation techniques are
intractable because of the complexity of the instrument. In these cases, a Monte Carlo approach may often be useful. Rather than attempt to solve the differential equations of motion that describe the option's value in relation to the underlying security's price, a Monte Carlo model uses
simulation to generate random price paths of the underlying asset, each of which results in a payoff for the option. The average of these payoffs can be discounted to yield an
expectation value for the option. Note though, that despite its flexibility, using simulation for
American styled options is somewhat more complex than for lattice based models.
Finite difference models The equations used to model the option are often expressed as
partial differential equations (see for example
Black–Scholes equation). Once expressed in this form, a
finite difference model can be derived, and the valuation obtained. A number of implementations of finite difference methods exist for option valuation, including:
explicit finite difference,
implicit finite difference and the
Crank–Nicolson method. A trinomial tree option pricing model can be shown to be a simplified application of the explicit finite difference method. Although the finite difference approach is mathematically sophisticated, it is particularly useful where changes are assumed over time in model inputs – for example dividend yield, risk-free rate, or volatility, or some combination of these – that are not
tractable in closed form.
Other models Other numerical implementations which have been used to value options include
finite element methods. ==Risks==