Option Terms Page

Binomial option pricing model
An option pricing model which uses binomial trees to model the price of the underlying.
This is the most common type of numerical model.
The key to the binomial or binomial lattice-based model is the binomial trial process.
This divides the time until option maturity into discrete intervals or steps and presumes that during each of these intervals the key parameter typically the price or yield of a security follows a binomial process moving from its initial value S, either up to value Su with probability p or down to value Sd with probability 1 minus p.
Representations of the resulting distribution resemble trees or lattices and so the series of values generated by the binomial trial process is known as a binomial tree.
(More complex versions exists: a trinomial tree would allow three possible movements, and a multinomial model more than that).
The binomial process is usually specified as being path-independent that is, a move up followed by a move down results in the same price as a move down followed by a move up so that the branches recombine.
Trees that do not incorporate this feature are said to be non-recombining, bushy or exploding.
They are much more computationally demanding.
By working backward through the lattice from expiration, at which time the value of the option is known, options can be evaluated by backward induction to discount the terminal payoff through the tree: the value of the option is that which avoids an arbitrage profit.
The advantage of binomial models is that they can deal with a range of different assets, options or market conditions.
So, a lattice-based model gives rise to an algorithm rather than a closed formula for determining the option value.
Such models are particularly useful for valuing American-style options.
The best-known is the Cox-Ross-Rubinstein model.
See backward induction, Cox-Ross-Rubinstein.

Black's Model
This, like Garman-Kohlhagen's model for foreign exchange, is a derivative of the original Black-Scholes model.
Originally this was a model for the pricing of options on futures but it has been extended in scope to include any situation in which it is necessary to value a European option on a variable which can be assumed to be lognormally distributed about a forward price and where interest rates are non-stochastic.
In particular it is frequently used to price interest rate caps and swaptions.

Black-Derman-Toy
A single-factor (in this case short-term interest rates) term structure option pricing model proposed by Fisher Black, Emanuel Derman and William Toy in 1990
which expanded on the Ho-Lee model by specifying a time-varying structure for volatility and incorporating it into a binomial tree of possible forward short rates.

Black-Karasinski
A single-factor model of the term-structure where the logarithm of the short-term interest rate is assumed to be a 'mean-reverting' process with time-varying coefficients.

Black-Scholes
Developed by Fischer Black and Myron Scholes in 1973, this is the classic modern option pricing model and the first general equilibrium solution for the valuation of options.
The model provides a no-arbitrage value for European-style call options on shares as a function of the forward price, the exercise price of the option, the risk-free interest rate and the variance of the stock price which is assumed to follow a lognormal distribution.
It does this by recognising that stocks and calls on them can be combined to construct a risk-free portfolio and that options on equities can therefore be valued using a dynamic hedging argument.
That is, the option writer can exactly offset his exposure to the underlying stock by continuously buying or selling it.
The model shows that, by combining the underlying stock and a money market instrument, a riskless hedge (the delta hedge) can always be formed that exactly replicates the payoff of the option to be hedged.
This means that a portfolio formed by the combination of the option and its riskless hedge must appreciate at the risk-free interest rate.
This riskless hedge method circumvents the difficulties of specifying investors' risk preference and allows the risk-free interest rate to be used in the valuation process rather than some other discount rate that reflects the appropriate risk level.
For any time period, the value of such a portfolio can be computed as its value at the end of the period discounted back one period at the risk-free rate.
Because the price of an option is a deterministic function of the price of the underlying asset at that time, given that the distribution of asset prices is known for each time period (and in this model it is assumed to be lognormal), then the initial value of the option can be deduced by working backwards in time.
For a European call option, the Black-Scholes pricing formula is:
C = SN(d1) - Ee-rT (N(d2) where d1 = ln (s/e) + (r + 0.5s2)T sT d2 = d1 - sT, N(d1) and N(d2) are the cumulative probability for a unit normal variable z
That is it is the probability N(d1) = ¥òd1 f(z)dz where f(z) is distributed normally with mean zero and standard deviation of one.
1n is the natural logarithm e is the exponential T is time to maturity s2 is the instantaneous variance of the stock price which is the measure of volatility of the stock.
The equation states that the value of the call is equal to the stock price, S, minus the discounted value of the exercise price, Ee-rT, each weighted by a probability.
The stock price is weighted by N(d1) which is also the hedge ratio. For each call written, the riskless hedge portfolio contains N(d1) shares of stock.
On the other hand, the discounted value of the exercise price is weighted by N(d2) which is the probability that the option will finish in the money.
The value of a European put, P, can be derived in a similar way as: P = Ee-rT N(-d2) - SN(-d1).
The model's great achievement is completeness: it provides a method for hedging options with the underlying asset, which allows for arbitrage pricing and hedging.
Its drawbacks are that it assumes no dividends, no taxes or transaction costs, constant short-term interest rates, no penalties for short sales, that volatility and interest rates are constant, that the market operates continuously and that stock price distribution is lognormal.
The generalizations of Black-Scholes address these problems, while extensions to it apply it in a modified form to options on futures (Black's Model), options on currencies (Garman and Kohlhagen's Model) and to exotic options.
The basic model has problems pricing short-dated options because volatility is not time-homogeneous and long-dated options because it fails to take into account mean reversion.
It systematically undervalues near-maturity options, deeply out-of-the-money options, options on low volatility stocks.
It systematically overvalues long-dated options, deeply in-the-money options and options on high volatility stocks.
All these problems are due to the model's assumption of the uniformity of variance across time. Other types of models address these problems.

Brownian bridge
A Brownian Bridge defined between two points in time t0 and t1 is an arithmetic Brownian motion conditioned to take specific values x0 and x1 at those points.
The distributional properties of the Brownian Bridge may sometimes be used in option pricing when time is approximated as being discrete
(e.g. in binomial trees or Monte Carlo techniques) to smooth out the effect of the discretization.

Brownian motion
Archetypal random motion.
Variants of this are used as the assumed path of securities prices in many financial models.

CEV- model
The CEV, or constant elasticity of variance, model was original proposed by Cox and Ross in 1976 to try to account for some empirical observations about equity price volatility.
The specification of volatility is given as sS-a so for a=0 we have the classical model of stock prices used within Black-Scholes.
With values of a greater than 0 the equity prices have high volatility when the equity price is low and low volatility when the equity price is high.

Confidence interval
A range of values in which, with some specified probability, the value taken by a variable will lie.

Continuous variable
A variable such as time that can be subdivided into an infinite number of sub-units for measurement.
The unit of measurement can therefore be increased or decreased infinitesimally. See discrete variable.

Continuous probability distribution
A distribution where the variable can take any value within a specified range such as the gain or loss from a position in a financial asset over a specified interval of time.
Continuous probability distributions do not consider the probability of the variable taking on a specific value. See discrete probability distribution.

Control variates
This is a technique available to all numerical valuation techniques.
It consists of using the same technique to value not just the option whose value is required but also an option, the control, whose characteristics are close to the original option but whose value can be calculated in some more accurate manner (preferably analytically).
The difference between the control's values calculated analytically and numerically is an estimate of the error inherent in the numerical approach and is added on to the numerically calculated value of the original option.
Choosing an appropriate control can result in a greatly reduced variance of the estimate.

Convexity adjustment
The most common use of this term is to describe the adjustment that has to be made to the interest rate implied by the price of a Euro-deposit future to obtain the corresponding forward interest rate.
It has come to mean any adjustment to a price obtained under simplifying assumptions to account for real-world non-linearities,
for example adjusting the price of a barrier option to take account of non-uniform volatility over different deltas.

Correlation
A measure of the degree to which changes in two variables are related.
The standard measure of correlation is the correlation coefficient, a number between minus one and plus one that indicates the strength and direction of a linear relationship between two variables.
A correlation coefficient of minus one indicates that they are perfectly negatively correlated, zero that they are not correlated at all and one that they are perfectly correlated.
Correlation risk is the risk that two variables or instruments are unfavourably correlated.
Identifying and quantifying correlation risk has become a key element in pricing and hedging certain derivative instruments.
In some options such as spread options and cross-currency caps, the correlation between the underlying assets is a first-order effect as it directly affects the option price.
In quanto products, such as differential swaps, there is a second order or indirect effect, in that case between interest rates and exchange rates.
See second-order effect.

Covariance
A measure of how two random variables behave in relation to each other.
Matrices of covariances are used in several different financial models, the most famous of which is Sharpe's Capital Asset Pricing Model.
It differs from correlation in that it incorporates measurements of the magnitude of the variations as opposed to the correlation coefficient which is dimensionless.
The correlation coefficient between two random variables is equal to the covariance between them divided by the product of their standard deviations.

Cox-Ingersoll-Ross
A generalization of the Black-Scholes option pricing model incorporating the work of John Cox, Stephen Ross and Jonathan Ingersoll.
The model represents one of the two approaches followed by term structure option pricing models.
It models the expected returns from movements in the term structure in order to price them.
The second approach, followed by Ho-Lee, Heath-Jarrow-Morton, Black-Derman-Toy, and Hull-White
utilizes the volatilities of the various sectors of the term structure to derive a probability distribution for an arbitrage-free binomial, trinomial or multinomial lattice of the term structure.
These models all have one thing in common -
they allow for the whole-term structure to be stochastic instead of the price of a single underlying instrument or a single interest rate.
The whole-term structure is represented at each node of the lattice.
This methodology allows both long-term and short-term interest rate instruments to be priced with an internal consistency not possible if different models are used to price different instruments.

Cox-Ross-Rubinstein model
The classical binomial option pricing approach first proposed by Cox, Ross and Rubinstein in 1979.
It requires that u (the up-jump multiplier) is 1/d (the down-jump multiplier).

Crank-Nicholson technique
Technique for the numerical solution of partial differential equations, which is particularly useful for diffusion equations.
Since the equation satisfied by option prices is a diffusion equation it is frequently used in pricing by finite differences.

Discrete probability distribution
A distribution of the probabilities that a variable takes certain discrete values.

Discrete variable
A variable that can only take certain discrete values - such as whole numbers.

Differential equation
Equation where the values of variables are implicitly related to each other through their derivatives (or rates of change).
The analysis performed by Black and Scholes resulted in a differential equation where the derivatives of the value of an option with respect to time and the spot price of the underlying are related together.

Diffusion process
A continuous-time model of the behaviour of a random variable that uses geometric Brownian motion as its basic assumption.
In the Black-Scholes model, the price of the underlying follows a pure diffusion process - that is, it is assumed to move continuously from one point to another.
The consequence of this assumption is that the terminal distribution of share prices is lognormal.
Other models, particularly discrete-time models, use modifications of the process.

Equilibrium model
An equilibrium model of the yield curve makes assumptions about economic variables and economic behaviour and uses the requirement of equilibrium of the economic system to deduce the process followed by the yield curve.
From this process the processes followed by discount bonds and options can be deduced.
A fundamental obstacle to using this approach for pricing interest rate derivatives is that there is no guarantee that the initial term structure will be matched.

Fauré sequence
A particular type of quasi-random number sequence. See Halton sequence.

Finite difference methodology
An option pricing approach based on finding a numerical solution to the differential equation that the option valuation must satisfy.
It does this by converting the differential equation into a series of difference equations which are then solved iteratively.

Garch
Acronym for Generalized AutoRegressive Conditional Heteroscedasticity.
A variation of the pure Arch that generalizes the univariate Arch models into allowing the whole covariance matrix to change with time instead of just the variance.
Several other variations exist. See Arch.

Garman-Kohlhagen
The classic and commonly used extension of the Black-Scholes option pricing model to pricing currency options. Mark Garman and Steven Kohlhagen showed that much the same arguments apply to pricing currency options as apply to pricing stock options with adaptations to allow for the two interest rates and the fact that a currency can trade at a premium or discount forward depending on the interest rate differential. (The dividend yield is replaced by the foreign interest rate).

Gaussian distribution
See normal distribution.

Geometric Brownian motion
Describes the movements in a variable or asset price when the proportional change in its value in a short period of time is normally distributed. The proportional changes in two non-overlapping periods of time are uncorrelated, hence the alternative name for the process random walk. The term geometric refers to the fact that it is the proportional change in the asset price (not the absolute level) that is normally distributed. This means that the future value of a variable following geometric Brownian motion has a lognormal probability distribution and is always positive, unlike a variable following a Wiener process, whose value can become negative. This makes it mathematically useful and consequently it is the most common assumption for the movement of stock prices, stock indices, currencies and futures contracts. It is the assumption made for stock prices in the original Black-Scholes options pricing model.

Geske-Johnsonng
The Roll-Geske-Whaley model values call options on dividend-paying assets but is not applicable to American-style puts on such assets. Indeed there is no analytical solution. The Geske-Johnson model, an extension of the Roll-Geske-Whaley model, notes that there is a positive probability of early exercise of in-the-money puts which means that an American-style option can be viewed as an infinite sequence of options to exercise a European-style option. However, when the put is on an asset that pays dividends, the valuation procedure is simplified because it will not be optimal to exercise prematurely the option at any time near to but prior to an ex-dividend date. Because of its complexity, it uses trivariate normal density functions. Many market practitioners use binomial models instead.

Halton Sequence
A particular type of quasi-random number sequence. See Fauré sequence.

Heath-Jarrow-Morton
A multi-factor term structure option pricing model that uses all the information in the term structure and can handle multiple causes of term structure movement. This means that the returns on zero-coupon bonds of differing maturities are not assumed to be perfectly correlated (as is assumed, for example, by the Ho-Lee model). The most common form is a two-factor version, where the two factors are an underlying (in this case the entire term structure which is an input into the model in the same way that the current stock price is an input into Black-Scholes) and volatility - that is, a description of how the term structure fluctuates over time. This means that the model does not have to assume that all bond prices (in fact the model uses stochastic forward rates not zero coupon yields) are perfectly correlated. Instead, it assumes a random term structure of interest rates and is designed to be automatically consistent with both the observed term structure and the volatility functions input by the user. As a result of using a multi-factor model of the term structure, the model employs a multinomial instead of binomial model of term structure movement. The key difference between it and the spot rate models of Black-Derman-Toy, Vasicek, Hull-White and Cox-Ingersoll-Ross is that these models treat the spot interest rate as the underlying variable. Besides the current spot rate, these models include various parameters used to describe the possible future paths of the spot rate. Since the current term structure is not a direct input, these models try to fit the term structure by searching for parameter values which cause calculated zero coupon bond prices to match the market.

Heteroscedastic
In simple linear regression, an error term compensates for the fact that in modelling the relationship between two variables, one of which is assumed to be the major factor in the movements of the other, movements in one will in fact be imperfectly described by movements in the other because of factors not captured by the regression model. This error term is normally distributed with a mean of zero so that its effects cancel each other out. If the variance of the error terms is constant, the regression is said to be homoscedastic. If it is not, it is said to be heteroscedastic.

Ho-Lee
The first whole-term structure option pricing model, proposed by Thomas Ho and Sang-Bin Lee in 1986. Using a discrete-time binomial approach this single-factor model incorporates the whole term structure rather than just changes in a long or short interest rate. Given the term structure as known today, in the next time period the whole term structure can move up or down. However, the model makes a number of assumptions not borne out by empirical observations: it assumes that the returns of zero coupon bonds of different maturities (which it uses to represent the term structure at each node on the binomial lattice) move in a perfectly correlated manner and it requires that all interest rates both spot and forward have the same volatility. It also allows for negative interest rates as it does not incorporate mean reversion. Unlike Black-Scholes-type models, Ho-Lee establishes no explicit link between hedging and pricing.

Hull-White
A single factor model developed using a trinomial lattice. It is a yield-curve based model in the same mould as the Ho-Lee, Vasicek, Heath-Jarrow-Morton and Black-Derman-Toy models. A key feature of Hull-White is that it treats mean reversion as time-dependent.

Implied correlation
The correlation coefficient implied by the price of a two-factor option on the assumption that volatilities for the two assets involved are available. This concept is most relevant in the foreign exchange markets where there are liquid markets in, for instance, dollar-mark, dollar-yen and mark-yen options. The option prices quoted on these options give implied volatilities, the exchange rates and implied correlations between them.

Ito Process
A generalized Wiener process where the drift and dispersion parameters are a function of state and time.

Ito's Lemma
An equation giving the process followed by a function of an Ito process. This is also an Ito process whose coefficients are given in terms of the derivatives of the original process with respect to state and time. This lemma (proposition) is fundamental to Black-Scholes since an option price is a function of an underlying spot price whose behaviour is conventionally assumed to be an Ito process. Knowledge of the relationship between the dispersion parameters of the two processes given by the lemma enables a risk-neutral portfolio to constructed and hence a risk-neutral price for the option to be derived as a solution to a partial differential equation whose form is also determined by the lemma.

Jump diffusion process
The process proposed by Robert Merton whereby the price of the underlying neither simply jumps nor follows a pure diffusion process but moves by a combination of a jump followed by continuous diffusion. Option pricing models have been extended to incorporate these kinds of jump price dynamics with directional bias but there are still theoretical problems associated with jump diffusion models. For example, the underlying asset in a foreign exchange option is an exchange rate which can be denominated in either of two currencies. However, jump diffusion models do not give the same prices when compared in a common currency.

Jump process
A stochastic process for movements in the price of the underlying proposed by John Cox and Stephen Ross.
In it the price of the underlying does not follow the pure diffusion process assumed by the Black-Scholes model but rather jumps from one point to another in steps larger than traditional random processes would generate.
This idea was expanded in the Cox-Ross-Rubinstein binomial model.

Kurtosis
A measure of the extent to which probability is concentrated more around the mean and in the tails rather than in the mid-range relative to a normal distribution.
A normal distribution has a kurtosis coefficient of three.
Kurtosis of less than that indicates a distribution with a fat midrange on either side of the mean and a low peak - called platykurtic.
A kurtosis coefficient greater than that indicates a high peak, thin midrange and fat tails - called leptokurtic.
Empirically the latter is the phenomenon most frequently observed in financial prices and markets.
It results in implied volatilities of vanilla options which are higher for strikes above and below the forward than for options struck at the forward - the well-known volatility smile.
See volatility smile.

Least squares regression
One of a number of types of regression analysis that measure the relationship between variables.

Linear regression
Linear regression is a regression where the dependent variable is modelled as a linear combination of the dependent variables plus the error term.
The simplest case is simple linear regression where there is only one dependent variable.
Here the relationship is: y = a + bx + u, where a is a constant; b is the regression coefficient and u is the error or disturbance term.

Lognormal distribution
A variable has a lognormal distribution if the natural logarithm of the variable is normally distributed.
A consequence of the classical assumption about the process followed by assets, that it is geometric Brownian motion with drift, is that the asset price at a particular point in time in the future has a lognormal distribution. Thus when valuing an option using the risk-neutral expectation approach, to obtain for instance the classical Black-Scholes option pricing formula, we would take the expectation with respect to that lognormal distribution.

Low-Discrepancy sequences
See quasi-random numbers.

Markov process
A class of stochastic processes or models which define a finite set of states. The essential property of the Markov process is that the future behaviour of the process (the progression of the set of states from one state to the next) is independent of past behaviour and determinable solely from the current state. Most option pricing models assume that movements in the price of the underlying or, in the case of interest rate options, the zero-coupon curve, are determined by a Markov process.

Mean
The sum of observation values divided by the number of observations. A statistical measure of central tendency.

Mean reversion
The statistical tendency of variables, most relevantly stock prices, interest rates, and volatility, to trend away from extremely high or low values and to gravitate towards a long-term average level. When the value of a mean-reverting variable reaches a very high level, it is more likely to go down than to go up. When it reaches a very low level, it is more likely to go up than to go down. Mean reversion is important in option pricing because it contradicts an assumption of many early models that the variance of the price of the underlying asset of an option is directly proportional to the option's term to expiration. This assumption implies that the statistical dispersion of asset prices will widen indefinitely further and further into the future. In interest-rate option pricing models it means that interest rates can become negative. (Interest rate models are further constrained in absolute terms: in a normal economy 100% rates are extremely unlikely.) The practical consequence for pricing is that the longer-dated an option, the more seriously it will be mispriced by models that ignore mean reversion. To account properly for mean reversion and hence estimate the volatility of an economic variable that demonstrates it, a more complicated underlying model than geometric Brownian motion is needed. Models such as Vasicek and Cox-Ingersoll-Roll incorporate mean reversion to account for the term structure of volatility. The Hull-White model goes further by proposing that mean reversion is time-dependent.

Monte-Carlo simulation
A generic technique involving the generation of random numbers to solve deterministic problems.
It is often used by numerical option pricing models as an alternative to the binomial process as a simulation of the underlying asset price.
Using computers, a Monte-Carlo simulation attempts to simulate the process that generates future movements in the price of the underlying.
Each simulation results in a terminal asset value and several thousand computer simulations give a distribution of terminal asset values
from which the expected asset value at option expiration can be extracted.
This method is used to value complex options, particularly path-dependent options for which there is no analytical solution.

Multi-factor model
An option pricing model in which there are two or more sources of randomness contributing to the option price.
The reasons for using multi-factor models can be split into two: either the option payout itself is a function of two variables
(e.g. a spread option on A or B, or an option on A that knocks out on B),
or the processes of the variables appearing in the option payout are defined in terms of many factors (e.g. stochastic volatility models).
See single-factor model.

Non-uniformity
In option pricing used to refer to the fact that volatility is expected to be higher on certain days than on others.

Normal distribution
The most widely occurring frequency distribution. The normal (or Gaussian) distribution is distinguished by its symmetrical bell shape and has the statistically desirable characteristics of being completely described by the mean and standard deviation of the distribution. The mean indicates the position of the centre of the bell, the standard deviation how spread out it is. If a variable is normally distributed, 68.27% of its values will fall within plus or minus one standard deviation of the mean; 95.45% will fall within plus or minus two standard deviations and 99.73% will fall within plus or minus three standard deviations from the mean.

Numerical model
An option pricing model which avoids the requirement to solve a stochastic differential equation by specifying a particular process for the underlying asset price and then using an iterative approach to solve the value of the option. Numerical models can be divided into three main classes: the binomial models, the finite difference models, and Monte Carlo simulations.

Poisson Process
A process useful for describing events which happen discretely but randomly in time, e.g. crashes, central bank rate hikes.
It is frequently used as a component of jump diffusion processes to describe the occurrence of the discrete jumps.

Pseudo-random numbers
Monte Carlo techniques for valuing derivatives require a supply of random numbers which are uniformly distributed and independent.
Implementation of these techniques is performed on computers which are deterministic machines and therefore not intrinsically good at creating randomness on demand.
There are deterministic algorithms which generate sequences of numbers which appear to have the appropriate properties of randomness.
However, because they are not truly random they are termed pseudo-random. See quasi-random numbers.

Probability density [function]
A function associated with any distribution which specifies mathematically the way in which probability is distributed over different possible values for that distribution. So, the density function of a normal distribution is the well-known bell-shape which implies that there is a high probability of being close to the mean and low probability of being a long way from it.

Quasi-random numbers
These sequences, like pseudo-random numbers, may be used as the random number generating process in Monte Carlo models. Successive elements of a sequence of pseudo-random numbers are designed to be independent whereas successive elements of a sequence of quasi-random numbers are not. In fact they are designed to have a certain structure which allows them to cover the probability space in as uniform a manner as possible and hence to improve the convergence of the estimation. See Fauré sequence, Halton sequence, Sobol sequence.

Rendleman and Barter
This is a single-factor model of the yield curve where the short rate is defined to follow Geometric Brownian Motion with drift. It is not in common use since it does not incorporate any of the mean reversion that is empirically observed. Risk-neutral pricing principle
Developed by John Cox and Stephen Ross, the theory that stock options may be valued as if the underlying stock's mean rate of growth is equal to the riskless rate. In particular, the value of a European option is the discounted present value of the payoff under the risk-adjusted probability distribution for the stock price at expiry.

Roll-Geske-Whaley
An extension to the Black-Scholes model incorporating the independent work of Richard Roll (1977), Robert Geske (1979) and Robert Whaley (1981) and providing a solution for the pricing of American-style call options on assets paying dividends. Behind the model is the observation that an American-option can be viewed as a portfolio of three options: a European option on the underlying; a European option to exercise the first option which will not be exercised until the instant before the ex-dividend date; and a compound option written on the first option (to incorporate the cost incurred by exercising the first two options of forfeiting the remaining life of the first option). The model can also be used to value calls on stock indices and American puts on stocks that do not pay dividends but cannot be used to value American puts on assets that pay dividends.

Second order effect
Error term which occurs due to non-linear effects in a model which has been approximated by a simpler linear model. Option gamma is a second order effect. See correlation risk.

Single-factor model
An option pricing model that incorporates only one uncertain parameter, the future price of the underlying. Such models make fixed assumptions about other variables such as the term structure of interest rates and volatility. Multi-factor models which can accept more than one parameter are better able to model interest rates and volatility and are necessary to price options on a number of underlying assets (such as spread assets) correctly. See multi-factor model.

Skew
In statistics skew is the asymmetry of a distribution around its mean.
Positive skew is an asymmetric tail extending toward positive values (right-hand side).
Negative skew is an asymmetry toward negative values (left-hand side).
In options skew is commonly used to refer to the volatility skew. See volatility skew.

Sobol sequences
A particular type of quasi-random number sequence. Stochastic volatility model
A multi-factor model where the volatility of the process followed by the underlying is itself a stochastic, usually mean-reverting, process.

Tail
The end (left or right hand section) of a probability distribution.
Also used by futures traders either for the change in the number of futures contracts needed to hedge a position because of variation margin flows or for the number of excess futures contracts in a basis trade.
Also used in the bond or note markets of a security with only a short time to maturity.

Tri-nominal tree
An extension of the binomial method of option pricing in which the variable being modelled (the price of the underlying) is allowed three possible outcomes instead of just two: move up, move down or stay the same.
This provides greater flexibility and is useful in pricing more complex products.

Variance
The statistical measure of how widely a variable is dispersed around the mean. Standard deviation is the square root of the variance.

Volatility smile
Refers to the influence of the out-of-moneyness of an option of a given maturity on its quoted implied volatility.
Generally the implied volatility of out-of-the-money options (that is options with low deltas) is greater than that of at-the-money options.
If the implied volatilities are plotted (Y-axis) versus the strike (X-axis), a curved line resembling a smile is obtained.
This is due to option sellers needing a premium for selling low delta (disaster insurance) options.
This phenomenon is not consistent with the basic Black-Scholes model which implies that asset volatility is constant.
If true, the implied volatility from European options of all strikes and maturities would be identical.
In fact, implied Black-Scholes' volatilities depend on the maturity and strike of the European option in question.
That is, the market may believe that extreme upward and downward movements are more likely than allowed by the Black-Scholes model.
In this case it is said that the implied market distribution is more leptokurtotic than that implied by Black-Scholes.
This can be seen when the implied volatility smile is convex in the strike price.
See kurtosis.

Volatility trading
Taking options positions that will profit not from moves in the price of the underlying but from changes in implied volatility.
Traders can take views on absolute levels of implied volatility by buying and selling combinations of options - classically delta-hedged straddles and strangles.
They can also trade future actual or realized volatility versus present implied volatility, profiting if future actual volatility is more or less than the implied volatility of the position when the trade is put on.
So if they believe that the volatility implied by an option is too low, then the option is cheap and they will buy it.
Buyers {sellers} of realized volatility against implied volatility profit when the underlying is more {less} volatile than the implied volatility predicted.
These trades are non-directional, that is they are hedged against absolute price moves in the underlying.
Traders can also trade between the different implied volatilities of options at different maturities or with different strikes (smile or skew trading).

Whole-term structure pricing model
An interest rate option pricing model that takes into account the relationships between spot rates at different points in the curve.
By using the information contained in the current term structure of interest rates and also the volatilities of each of the spot rates as inputs into binomial,
trinomial or multinomial trees which value the underlying debt instrument at each node,
the models provide the basis for a valuation of an option on that instrument.
The Ho-Lee and Heath-Jarrow-Morton models are of this type.
Such models are designed to enable the exposure on all interest-rate derivative products to be aggregated.
For example, the volatility exposure created by a long position in swaptions should be able to be offset by a short position in caps so that only the net volatility is hedged.

Wiener process
The description of movements in a variable when the change in its value in a short period of time is normally distributed and the changes in two non-overlapping periods of time are uncorrelated. Also known as arithmetic Brownian motion.

Adjustable Strike Options

Key concepts
Adjustable strike options are options whose strike is reset either automatically or by the holder, depending on the path/level of the underlying.
Depending on the terms of the reset mechanism they are also known as moving/floating strike options, indexed-strike options, periodic (reset) options, ratchet/ladder options and step-up/step-down options.
They are often combinations of vanilla and digital or barrier options and in the two pages covering those instruments a number of products are explained which, because they actually consist of a number of options packaged together, appear to have similar resettable strikes.
In this section we also include a small number of options whose unusual exercise conditions make them similar to adjustable strike options, namely fixed-strike lookbacks, lookforwards and shout options.
All these options share one common characteristic:
they enable the holder to create strike price conditions that more exactly suit their views especially their views on the dynamic path of spot not just its final resting place than conventional options.

Deferred strike price option
Also known as a forward start option, this is an option that allows the holder to set the strike price at a predetermined time or during a predetermined period after its trade date.
The strike price is usually expressed as a fixed ratio to or percentage of spot.
The option's premium is usually set on the trade date. These options allow the holder to lock in current levels of volatility in the expectation that volatilities will rise or fall without setting the delta of the option until the strike is set.
These are more commonly embedded in structured assets than used as naked options.
Example
An investor might want a three-year bond whose annual coupon captured the appreciation of a currency, say sterling against the dollar, in each year.
This would be constructed from a strip of two-one year forward start USD put/GBP call options plus a one year vanilla option.
The first could be struck at-the money spot with the two forward starting options setting at 100% of spot on the first business day of the year and an expiry on the last business day of the year.

Hi-lo option
An option which pays out the difference between the high and the low price or rate reached by the underlying over the term of the option.
Constructed from a combination of a lookback call and lookback put, the buyer is taking a view that the volatility of the underlying will be greater than the implied volatility of the component options.
Because the expected payout is high, the premium is high, and the option buyer is taking a large, long position in gamma and vega.

Indexed-strike option
Also known as a periodic reset option, this is an option whose initial strike price moves up or down according to a preset schedule or depending on the path of a reference asset or index.
The size, timing and direction of the reset mechanism can depend on almost any contingency required by the buyer/seller.
It may rely simply on pre-set trigger points being hit by the underlying at any time during the life of the option; it may require that the underlying move a certain amount relative to the last fix within a given sub-period of the option's life
(sometimes called momentum or gap options); or it can be automatic with the option's strike price resetting at a pre-agreed spread above or below the reference index or at a series of pre-agreed absolute levels for each successive period without the underlying having had to hit any predefined level (sometimes known as a moving strike option).
Many of these products are combinations of vanilla and exotic options.
The holder of a momentum cap is long a conventional cap and short a series of digital caps.
The benefits to him versus a vanilla cap will depend on the value of the sold options.
In a positive yield curve environment, the steeper the curve the higher the chance that the trigger Libor rise will be breached and the higher the probability that the strike will be raised and the higher their value.
Also known as step-up/-down options.
Example
Momentum options and gap options illustrate the subtleties available with resettable strike options.
They enable the holder to hedge against or benefit from dramatic movements in the price of the underlying.
An option on Libor struck at 7.50% would pay out if Libor rises by more than, say, 75 bp in the next three months.
It therefore has two triggers, the gap trigger (75bp) and the speed trigger (one month).
Regardless of whether Libor did rise by 75bp in the first three-month period, the strike would then be reset to current Libor at the beginning of the next payment period.
This structure is usually altered so that the strike price ratchets up by a predetermined amount.
So a borrower with a three-year US dollar loan based on three-month Libor could buy a three-year momentum cap with a 7.50% strike and a 75bp trigger amount.
If In any three-month period three-month Libor rises by more than 75bp, then the cap strike is reset 25bp higher with a maximum cap rate of 8.50%.

Ladder option
An option whose strike resets automatically when the underlying hits predetermined levels ('rungs'). When the strike is reset the intrinsic value of the option is automatically locked in regardless of whether the underlying subsequently moves disadvantageously.
Ladder options are strips of capped/exploding options with the cap level of one option set equal to the strike level of the next and each cap level a rung in the ladder.
Every time a rung/cap level is reached that option is exercised for its intrinsic value locking in that gain and a new option is triggered with a strike equal to the previous cap level and a new cap level higher (call)/lower (put) than the previous one.)
As these options are sometimes known as cliquet options (because cliqueter is French for to knock and the automatic exercise became known as the cliquet clause) cliquet option can also be used as a name for ladder option. These options are more expensive than vanilla options, particularly if the put asset has a lower interest rate than the call asset.
Example
A ladder call on the EUR/USD rate with a strike of 1.0500 might have a rung every 1 cent up to a maximum of 1.0800 and have a payout of the greater of (i) the closing spot less the original strike and (ii) the highest rung reached less the strike.
The more frequent the rungs, the more expensive it is. Other ladder options have only a minimum settlement level. Once the underlying has risen by, say, 10%, that gain is locked in regardless of the future path of the underlying price.
If it subsequently rises above 10%, the investor still participates, but he also has a floor at 110.
In exchange for this downside protection the maximum return is generally capped.

Lookback option
An option that allows the buyer, at maturity, to choose the most advantageous exercise conditions that have occurred over the life of the option, or in the case of a partial lookback option, during a pre-set sub-period (usually between one and three months) of the life of the option after which it becomes a standard European- or American-style option. A lookback period limited to the first part of the option's life will help improve the timing of any market entry; one limited to the last part of the option's life will help with market exit timing. Lookbacks come in two varieties.
The lookback strike option/floating-strike lookback, instead of having a specified strike price, allows the buyer at expiration to look back over the life of the option and set as the strike the most favourable price that has occurred during that time.
A lookback call {put} allows the buyer to choose the lowest {highest} price that has occurred over the life of the option.
These strikes are then compared with the spot price at expiration to determine the option's payoff.
The lookback spot option/fixed-strike lookback has a strike set at the outset but then at maturity allows the buyer to look back over the life of the option and choose the most favourable exercise point to maximize profit between strike and exercise.
Lookbacks, like conventional options, are most profitable to the buyer (net of premium) if the realized volatility of the underlying price is higher than the implied volatility.
If a buyer believes that there will be a sharp move in price but is not sure when and for how long the price will move, lookbacks are attractive. Because they allow the buyer to choose the best exercise conditions with perfect hindsight, lookbacks command much higher premiums than conventional options. Also known as hindsight options and lookforward options. A fixed-strike lookback struck at-the-money spot is sometimes called a lookforward option.
This gives the buyer the difference between the asset price at the beginning of the period covered by the option and its high (call) or low (put) over that period.

Ratchet option
A type of indexed strike option whose strike price resets favourably if the underlying moves out-of-the-money relative to the initial strike and hits certain trigger or ratchet levels but which does not reset in the other direction if the underlying subsequently moves in the other direction.
A ratchet call option is a call struck at the ratchet option's strike price, plus a series of bought knock-in put options each struck at a ratchet level and a series of sold knock-out puts with strikes staggered one rung behind the purchased options.
Confusingly, like ladder options, ratchet options are also sometimes known as cliquet options because vilbréquin à cliquet is French for ratchet brace.

Roll-up option
An option whose strike price is favourably reset at the same time as the option itself is converted into knock-out option if the price of the underlying asset trades through a predetermined trigger point, usually struck at a point where the underlying has moved significantly against the original option.
So, a roll-up put with an original at-the-money strike of 80 might be converted into an out-of-the-money knock-out (up-and-out) put with a strike of 100 and a knock-out level of 110 if the underlying traded to 100.
The holder has a new, more favourable put strike, but if the underlying continues to rise (i.e. in his favour as long as his put is hedging an existing position) then the put is knocked-out (at a point where he does not need protection).
The roll-up put outperforms the standard put if the roll-up trigger is reached but the trigger is not.
If the roll up trigger is not reached, then the roll-up put and vanilla put behave the same.
Only if the roll-up trigger and the trigger are reached does the roll-up underperform the vanilla instrument.
The trigger price for the up-and-out put is set in advance and is above the roll-up strike.

Shout option
Confusingly there are two completely different types of option that are called shout options.
(i) A path dependent option that combines the features of lookback, ratchet and ladder options.
A shout option allows the purchaser to lock in a minimum payout (the intrinsic value of the option at the time of the 'shout') while retaining the right to benefit from further upside.
So-called because when the option holder thinks the market has reached a high (call) or low (put), he 'shouts' and locks in that level as the minimum and, with a one-shout option, still holds a European option with the original strike price for the remaining life of the option.
If the market finishes higher (call) lower (put) than the shout level, the holder benefits further i.e. the payout of a shout option is the greater of the intrinsic value locked in by the shout and the intrinsic value at maturity.
This type of shout option is similar to a ladder option in which profits are locked in when the underlying rises/falls sufficiently to hit a pre-determined 'rung' level, but in the shout option the rungs are not set in advance.
This makes the shout option more expensive than the ladder option, the more so when multiple shouts are allowed (multiple shout options are very expensive).
As with a ladder option, the more shouts that are allowed, the more like a lookback the shout option becomes.
The ability to lock in gains before expiry makes the shout more expensive than a standard European option, and the fact that even after a shout,
the option holder effectively has another option struck at the shout level, makes it more expensive than an American-style option.
Example
A corporate treasurer might be bullish on EUR/USD rates but also expects the cross to be very volatile.
He is worried that using a vanilla option will mean that he misses out on temporary highs.
A shout call solves the problem. If the EUR/USD rate rises above the strike price, but ends up below the shout level
(wherever the treasurer eventually chooses that to be), the treasurer receives a profit of the shout level less the strike level.
If the exchange rate closes above the shout level, the investor will receive that additional profit as well.
The payout is therefore the maximum of (shout strike) and (close strike).
The second type of shout option is a call or put option that gives the buyer or seller the right once and only once during a pre-specified period
to 'shout' the option and reset the strike to the then prevailing spot rate (or some percentage thereof)).
These shout options are therefore, in a sense, halfway between a vanilla and a lookback option.
They are more expensive to buy and generate less premium when sold.

Surge options
An option whose strike price is reset on a daily basis to a fixed spread above or below a moving average.
This hedges against the risk of rapid price changes rather than absolute price trends over longer periods.
Commonest in the commodity markets, a put surge option on the price of crude oil could work like this whenever the spot oil price falls below the 45-day moving average less two cents, the option is in-the-money.
The settlement amount is determined by the difference between the spot price and the strike price multiplied by the number of barrels to be priced each day.
A call would move into the money if the spot price moved above the moving average plus a fixed spread.

Asian Options

Key concepts
Asian or average rate options are options whose payout or strike price is based on an average of the price of the underlying over the life of the option.
The averaging process can begin at any point during the option period (for example a one-year option whose payout depends on the average underlying price in the final month).
The sampling process frequency and interval of underlying price observations can also be tailored.
The number and timing of price (strike) observations is determined in advance and may start at the beginning or near the end of the life of the option.
Observations may also be weighted in favour of prices (strikes) observed on specific dates.
Unlike a straight American- or European-style option, an average option can be settled more than once over its life.
So for example, the holder of a one-year average option can choose to settle the option monthly versus the average price of the underlying the previous month.
Average options are nearly always cheaper than conventional options because the averaging process smoothes out the underlying price movements thereby reducing volatility and hence the premium of the option.

average price/rate option (ARO)
Unlike a conventional option, which is settled by comparing the strike with the spot rate at expiration, an average rate option's payout is the difference, if positive, between the predetermined, fixed strike price and the average of spot rates observed over the option's life. This hedges against the average prevailing spot over the life of the trade. It also removes the reliance of the option's expiration value to the underlying cash price on a particular day. Typically the volatility of an average rate option is about 58% of the volatility of a conventional option and so is cheaper. AROs are cash-settled, not deliverable, so when hedging an underlying exposure, cash flows need to be converted in the underlying market on the relevant fixing dates.
This ensures that the hedge instrument effectively offsets the aggregate FX rate of the cashflow conversions.
There are three main varieties:

Barrier or Trigger Options

Key concepts
Barrier or trigger options are conventional options except that they are cancelled or activated or, more generally, changed in a pre-determined way when the underlying trades at predetermined barrier/trigger levels.
So a knock-in option pays nothing at expiry unless at some point in its life the underlying reaches a pre-set barrier and brings the option to life as a standard call or put.
A knock-out option is a conventional option until the price of the underlying reaches a pre-set barrier price, in which case it is extinguished and ceases to exist.
Barrier options have a strike price and a barrier price and the barrier can be above or below the strike price.
In all variations the barrier can be made to be active for either part or all of the option's life.
Because of the importance of the barrier event in determining the value of the option, users must ascertain at the outset of the transaction the definition of a barrier event.
For example, is it to be based on quoted rates or transactions; how is the issue of crosses to be dealt with in illiquid currency pairings; when can barrier events occur (outside normal trading, only on hourly fixes, at the end of day fix, only on certain dates and so on. Where barrier events can only occur at certain times (and under certain circumstances) there is said to be barrier discontinuity. This makes the options more difficult to price and value). The concept can be applied to every type of option and some option combinations- caps, floors, collars, digitals, swaptions and in any asset class.
The two basic classes of barrier options are the standard (or out-of-the-money) barrier options and the reverse (or in-the-money) barrier options.
These options should not be confused with capped {floored} calls {puts} which are also sometimes known as trigger options.

Balloon option
An option whose notional principal increases if a preset trigger level is breached.
For example, an equity investor might believe that the FTSE-100 will rise from 4900 to 5000 and then, if it breaches this resistance level, rise strongly again.
He could buy a 4900 call with a trigger of 5000 and a multiple of two, meaning that if the index stays below 5000 the option behaves like a vanilla call but if it rises above 5000 then the option's notional principal doubles.
The balloon option's premium is more expensive on the original notional principal than a vanilla option because it is a combination of two options a vanilla call struck at 4900 and a knock-in call struck at 4900 with a knock in at 5000. However, if the trigger is reached, the premium on the ballooned premium is cheaper.
The greater the ballooning the higher the premium; the further the trigger level is relative to the strike, the cheaper the premium.

Double barrier option
A general term for any barrier option incorporating two knock-out or knock-in levels, one either side of spot.
These are commonest in the FX markets where users may have strong views on both a support and a resistance level.
The illustrated of a knock-out rebate option on this page is an example of a double barrier option.

in-the-money/Reverse barrier option
A barrier option whose barrier is in-the-money relative to the strike.
So, the barrier level would be above the strike for a call (up-and-in/out calls) and below it for a put (down-and-in/out puts).
These are priced and behave very differently from standard barrier options since they have intrinsic value when they are knocked-in or out, making knock-ins relatively more expensive and knock-outs relatively cheaper for a given proximity to the strike.
So, unlike standard options, which become more valuable as volatility increases, in-the-money knock-outs become cheaper.
That is, they have negative vega: the probability of knock-out increases with increasing volatility, reducing the chance that the option will pay out and making them cheaper.
In-the-money barrier options can be used both to hedge an underlying position (as in example 1 below) and to take outright speculative positions (as in example 2).
Example 1
In-the-money barrier option could be used by a USD-based exporter wishing to reduce the cost of protection against Euro weakness.
The company could purchase a EUR put/USD call with an in-the-money knock-out struck at 1.0200 with the knock-out set at 1.0000.
This option could be up to half as cheap again as the out-of-the-money knock-out, but carries the risk that if EUR/USD does trade down to the knock-out level,
then the corporation has not only lost its hedge, but also has to re-hedge at much worse levels than with the standard knock-out.
The hedger has to have a stronger view on exchange rate movements than with either vanilla options or standard knock-outs.
Example 2
An investor believes that over the next six months the dollar will strengthen against the yen by around 10% and by at the very most 12%.
If dollar yen is trading at 125 he could buy a reverse knock-out USD call/JPY put struck at the money forward at, say, 121.50, with a trigger at 140.50, 0.5 yen above his predicted high of 140 and another knock-out at 119.
Then if the dollar does strengthen, but trades either below 119 or above 140.5 over the life of the option, the call will disappear.
If the dollar strengthens, but never reaches 140.50 or 119 over the life of the option, the call will behave like an ordinary call and the investor will exercise the call and make the same profit as the ordinary call.
If the dollar does not close above the call strike, the option will expire worthless like an ordinary option.

Knock-in cylinder/collar
Barrier options are often used to modify simpler option spreads.
The knock-in collar is an alternative to selling an out-of-the-money call {put} to finance the purchase of an out-of-the-money put/(call) to create a standard collar.
Instead the holder sells a knock-in call {put} instead of a vanilla option.
This allows the holder of the position to participate fully in the upside {downside} of any moves in the underlying, until the trigger level is breached.
Example
A USD-based exporter has EUR receivables due in six months.
The current spot EUR/USD rate is 1.0310 and the six-month forward points are 130 so the six-month forward outright is 1.0440.
The company wants protection against Euro weakening and favours options over selling Euros forward because of the high potential opportunity cost of locking in an outright forward rate.
However it does not wish to incur any upfront premium at all or does not wish to be long volatility.
In this case they could execute a zero premium standard risk-reversal/collar, selling a call at 1.0550 to finance the purchase of put protection at 1.0340.
This risk-reversal can be as narrow or as wide as the corporation wishes.
Alternatively, instead of selling a vanilla EUR call the hedger could sell a knock-in call struck at 1.0500 with a 1.0900 trigger level.
Although the strikes of the call and put would be less advantageous than a regular collar, the corporation would have full protection and only give up its upside if the knock-in were triggered.

Knock-out cylinder/collar
Similar in concept to the knock-in collar, this is the substitution of the short call of a simple zero cost collar with a knock-out call.
In the example above, the corporation would buy a 1.0240 EUR put and sell a 1.0500 EUR call with a knock-out trigger at 1.0000.
If the knock-out were triggered, then upside is no longer limited.
Knock-out trigger option
A capped call {floored put} which incorporate an out-of-the-money knock-out level.
If the underlying trades through this, the whole option is cancelled.

One factor barrier option
A barrier option whose barrier event and payout are based on the same underlying.
Also known as inside barrier options.
In a two-factor/outside barrier option the barrier event and the payout are based on two different underlying assets.
So, the payout might be a function of a foreign exchange rate but the barrier event may be the breach of a level in the price of a commodity.

Out-of-the-money/Standard barrier option
A barrier option whose barrier is set out-of-the-money relative to the strike.
So the barrier level would be below the strike at the start of the option contract for a call (down-and-in/out calls) and above it for a put (up-and-in/out puts).
These options cost less than standard options because the price of a vanilla option takes the entire probability distribution of possible prices
for the underlying into account, while the knock-out feature removes many of those possible values.
The exact premium reduction is determined by how likely the barrier event is to occur.
The more likely the option is to be knocked out or the less likely it is to be knocked-in, the greater the premium reduction, and vice versa.
The likelihood of the barrier event depends on how near the spot/forward level the extinguishing trigger level is, on the maturity of the option and on volatility. Example A USD-based exporter finds a vanilla EUR put/USD call too expensive.
A knock-out EUR put/USD call struck at 1.0340 with a barrier level at 1.0600 costs less (and so has a lower break-even),
gives the a guaranteed minimum of 1.0340 USD for every EUR of receivables and only disappears when the underlying is moving in the hedger's favour,
giving it the flexibility to examine other hedging strategies.
The main risk is that if the option is knocked-out, the corporation is exposed to any subsequent dollar weakness and
so may have to put on a second forward or option-based hedge that may cost more than the original vanilla put option.

Rebate option
An option that pays a fixed amount if it would otherwise have expired worthless due to some barrier event.
So a knock-in rebate option pays a pre-set fixed amount if it has never been knocked-in (even if the option then expires out-of-the-money)
and a knock-out rebate option pays the option holder a pre-set fixed amount if it is knocked out.
The commonest structure is the knock-out rebate option a call or put with an in-the-money knock-out level and rebate almost always set equal to the initial premium paid.
This is usually just called a rebate option.
A second, out-of-the-money knock-out level can be incorporated which, if traded, cancels the entire option.
This structure is sometimes, confusingly, called a knock-out rebate option.

Sloping/moving/jumping barrier option
A barrier option whose knock-in/knock-out level changes during the life of the option, for example to match moving technical levels in the underlying.
This change may be either linear (i.e. sloping) or move in discrete steps (i.e. jumps).
For example, our USD-based exporter could set the knock-out level for the first three months at 1.0500 and then 1.0600 for the last three months.

Step-up {down} barrier option
A barrier option whose barrier increases {decreases} over time.

Switchback option
The simultaneous purchase of both a capped call {floored put} and a knock-in put {or call}.
The trigger levels of the knock-in barrier options typically equal the cap/floor strike prices.
If the underlying hits the trigger levels, the capped option is automatically locked-in and the knock-ins activated.
The holder would typically set the strikes at a point he believed to be around a peak {trough} in the underlying.
The position benefits from that level being reached and then switches back from call to put (or vice versa) as the underlying itself switches back,
retreating {rising} from its peak {trough}.