Partial Matches:
AVERAGE OPTION
A plain vanilla option pays out the difference between its predetermined strike price and the spot rate (or price) of the underlying at the time of expiry. The purchaser of an average option (average price, average strike, average hybrid, average ratio), on the other hand, will receive a pay-out which depends on the average value of the underlying. The average can be calculated in a number of ways (arithmetic or geometric, weighted or simple) from the spot rate on a predetermined series of dates (usually official fixing rates). An average rate (also known as average price) option is a cash-settled option with a predetermined (i.e., fixed) strike which is exercised at expiry against the average value of the underlying over the option’s life. In general, hedging with an average option is cheaper than using a portfolio of vanilla options, since the averaging process offsets high values with low ones and therefore lowers volatility and premium. Average rate options, also known as Asian options, are particularly popular in the currency and commodity markets.
In contrast, the strike for an average strike option is not fixed until the end of the averaging period which is typically much before the expiry. When the strike is set, the option is exercised against the prevailing spot rate. Unlike average price options, average strike options may be either cash or physically settled. In the case of an average hybrid option (also known as an average-in/average-out option), both the strike and settlement price of the option are determined using the average, where the strike averaging period typically precedes the settlement price averaging period. For the average ratio option, both the strike and settlement price of the option are determined using the average as in the hybrid case. The final payoff is determined by comparing the ratio of settlement price to strike and a fixed percent strike.
BARRIER OPTION
Barrier options, also known as knock-out, knock-in or trigger options, are path-dependent options which are either activated (knocked-in) or terminated (knocked-out) if a specified spot rate reaches a specified trigger level (or levels) between inception and expiry. Before termination knock-out options behave identically to standard European-style options, but carry lower initial premiums because they may be extinguished before reaching maturity. In contrast, knock-in options behave identically to European-style options only if they are activated/knocked-in and so also command a lower premium.
The standard barrier options have barrier levels that are monitored continually during the lifetime of the option. Single barrier options that have a barrier level above current spot are classified as up-and-out or up-and-in options. For single barriers below spot the usual terminology is down-and-out for the knock-out barrier option, and down-and-in for the knock-in barrier option.
An alternative terminology for single barrier options classifies barrier options where the barrier is out-of-the money with respect to the strike price as regular barrier options. In-the-money barrier options are further differentiated into reverse barrier options (for cases where the barrier may be breached as the underlying asset’s spot rate moves deeper in-the-money) and geared barrier options (examples where the barrier is in-the-money and lies between the strike and the underlying spot rate) A double barrier option has both an upper and lower barrier.
Many variations on the barrier theme are available. Barrier levels can be monitored continually, at discrete fixing times (discrete barrier options) or only at the final expiry date of the option (at-expiry barrier options). Barriers may be active only during distinct time intervals (window barrier options) or may change value at fixed points during the lifetime of the option (stepped barrier options). Barriers may need to be breached for a certain time before they are considered triggered (Parisian Barrier Options) or may allow for partial triggering depending upon how far beyond the trigger level the underlying asset is observed (Soft Barrier options). Barriers may reference a different underlying to that of the option itself – such barriers are known as outside barriers.
BARRIER RISK
The value and sensitivities (Greeks) of barrier options can be subject to large swings when the spot rate is at, or near, the trigger level. This is particularly true for reverse barrier options and geared barrier options, where the option has positive intrinsic value at the Barrier. The specific nature of these swings can make the management of such products riskier, hence barrier risk.
See also
stealth
BETA
1. The beta of an instrument is its standardized covariance with its class of instruments as a whole. Thus the beta of a stock is the extent to which that stock follows movements in the overall market. If a stock has a beta greater than one, it is more volatile than the market; if less than one, it is less volatile.
2. Beta trading is used by currency traders if they take the volatility risk of one currency in another. For example, rather than hedge a sterling/yen option with another sterling/yen option, a trader, either because of liquidity constraints or because of lower volatility, might hedge with euro/yen options. The beta risk indicates the likelihood of the two currencies’ volatilities diverging.
BILATERAL NETTING
Agreement between two counterparties whereby the value of all in-the-money contracts is offset by the value of all out-of-the money contracts, resulting in a single net exposure amount owed by one counterparty to the other. Bilateral netting can be multi-product and encompass portfolios of swaps, interest rate options, and forward foreign exchange.
BINARY OPTION
Unlike simple options, which have continuous pay-out profiles, that of a binary option is discontinuous and pays out a fixed amount if the underlying satisfies a predetermined trigger condition but nothing otherwise. Binary options are also known as digital or all-or-nothing options.
There are two major forms: at maturity and one-touch. At maturity binaries, also known as European binaries or at expiry binaries, pay out only if the spot trades above (or below) the trigger level at expiry. One-touch binary options, also known as American binaries, pay out if the spot rate trades through the trigger level at any time up to and including expiry. The pay-out of a one-touch binary may be due as soon as the trigger condition is satisfied or alternatively at expiry (one-touch immediate or one-touch deferred binaries). As with barrier options, variations on the theme include discrete binaries, stepped binaries, etc. Binary options are frequently combined with other instruments to create structured products, such as contingent premium options.
BINOMIAL TREE
Also called a binomial lattice. A discrete time model for describing the evolution of a random variable that is permitted to rise or fall with given probabilities. After the initial rise, two branches will each have two possible outcomes and so the process will continue. The process is usually specified so that an upward movement followed by a downward movement results in the same price, so that the branches recombine. If the branches do not recombine it is known as a bushy, or exploded, tree. The size of the movements and the probabilities are chosen so that the discrete binomial model tends to the normal distribution assumed in option models as the number of discrete steps is increased. Options can be evaluated by discounting the terminal pay-off back through the tree using the determined probabilities. Interest in binomial trees arises from their ability to deal with American-style features and to price interest rate options. For example, American-style options can readily be priced because the early exercise condition can be tested at each point in the tree.
BLACK-SCHOLES MODEL
The original closed-form solution to option pricing developed by Fischer Black and Myron Scholes in 1973. In its simplest form it offers a solution to pricing European-style options on assets with interim cash pay-outs over the life of the option. The model calculates the theoretical, or fair value for the option by constructing an instantaneously riskless hedge: that is, one whose performance is the mirror image of the option pay-out. The portfolio of option and hedge can then be assumed to earn the risk-free rate of return.
Central to the model is the assumption that market returns are normally distributed (i.e., have lognormal prices), that there are no transaction costs, that volatility and interest rates remain constant throughout the life of the option, and that the market follows a diffusion process. The model has five major inputs: the risk-free interest rate, the option’s strike price, the price of the underlying, the option’s maturity, and the volatility assumed. Since the first four are usually determined by the market, options traders tend to trade the implied volatility of the option.
BOND FUTURE
A futures contract on a bond. The underlying can be of two types. The first is a notional (theoretical) bond, for example the 10-year government bond futures contract on Marché à Terme International de France (Matif), which has a notional seven- to 10-year underlying government bond with a 10% coupon. The second type is a futures contract on a specific bond, for example, the futures contract on the Danish government bond 8% 2000, listed on the Guarantee Fund for Danish Options and Futures. The notional type is more common.
The theoretical price of a bond futures contract equals the spot price of the underlying bond plus its net cost of carry. The higher the cost of carry (measured by its implied repo rate) for the cash bond, the more value there is in holding a futures contract instead, which is simply another way of buying a cash bond but at a future date. If the financing costs of holding a bond are lower than the bond’s yield, the bond will have positive carry and the futures contract will trade at a discount. If the financing costs of the bond are higher than the bond’s yield, the futures will trade at a premium. Hence bond futures normally trade at a discount to the cash in an upward-sloping yield curve and at a premium in a downward-sloping one. The degree of premium or discount will decrease the closer the futures contract is to expiry.
When futures are based on a notional bond, the exchange has to specify which bonds are deliverable into the futures contract. These can change periodically as outstanding issues get shorter and reach maturity. Because bonds have differing coupons, the exchange has to specify a conversion factor that allows those bonds to be compared on an equal basis to the notional underlying. A bond’s conversion factor is the value of one unit of that bond were it to trade at a yield to maturity equal to that of the notional bond. Because of demand/supply conditions, some bonds will be cheaper to deliver than others. The cheapest to deliver will be those where the greatest profit can be made from holding the bond and delivering it into the futures contract (called cash-and-carry arbitrage). This is dependent on the supply of bonds in the market, and on a bond’s maturity and coupon.
BOND OPTION
An option offered on debt, usually government securities, although OTC options are available on corporate debt. The options can either be ex-change-traded, listed or OTC. Bond options have traditionally been standard European-style or American-style puts and calls. There is more interest in exotic structures such as yield curve options, inter-market spread options, and quanto options.
BOX
To buy/sell mispriced options and hedge the market risk using only options, unlike the conversion or the reversal, which use futures contracts. If a certain strike put is underpriced, the trader buys the put and sells a call at the same strike, creating a synthetic short futures position. To get rid of the market risk, he sells another put and buys another call, but at different strike prices.
CALENDAR SPREAD
A strategy that involves buying and selling options or futures with the same (strike) price but different maturities. Such a strategy is used in futures when one contract month is theoretically cheap and another is expensive. With options, the strategy is often used to play expected changes in the shape of the volatility term structure. For example, if one-month volatility is high and one-year volatility low, arbitrageurs might buy one-year straddles and sell short-term straddles, thereby selling short-term volatility and buying long-term volatility. If, all else being equal, short-term volatility declines relative to long-term volatility, the strategy makes money.
CAP
A contract whereby the seller agrees to pay to the purchaser, in return for an upfront premium or a series of annuity payments, the difference between a reference rate and an agreed strike rate when the reference exceeds the strike. Commonly, the reference rate is three- or six-month Libor. A cap is therefore a strip of interest rate guarantees that allows the purchaser to take advantage of a reduction in interest rates and to be protected if they rise. They are priced as the sum of the cost of the individual options, known as caplets.
See also
collar,
floor
CATASTROPHE OPTION
These options can be American-style or European-style, either paying out if a single specified catastrophe such as a hurricane or earthquake occurs, or alternatively, having a pay-out dependent on an index. For example, the index may represent the number of claims received by property insurance companies.
CHOOSER OPTION
A chooser option offers purchasers the choice, after a predetermined period, between a put and a call option. The pay-outs are similar to those of a straddle but chooser options are cheaper because purchasers must choose before expiry whether they want the put or the call. Also known as a hermaphrodite, or AC-DC option.
CLIQUET OPTION
Also known as a ratchet or reset option. A path-dependent+option that allows buyers to lock-in gains on the underlying security during chosen intervals over the life time of the option. Cliquet options were developed in France with the CAC 40 stock index as the underlying, although they are used in structured retail products elsewhere in Europe. The option’s strike price is effectively reset on predetermined dates. Gains, if any, are locked in. So if an index rises from 100 to 110 in year one, the buyer locks in 10 points and the strike price is reset at 110. If it falls to 97 in the next year the strike price is reset at that lower level, no further profits are locked in, but the accrued profit is kept.
See also
ladder option
COLLAR
The simultaneous purchase of an out-of-the-money call and sale of an out-of-the-money put (or cap and floor in the case of interest rate options). The premium from selling the put reduces the cost of purchasing the call. The amount saved depends on the strike rate of the two options. If the premium raised by the sale of the put exactly matches the cost of the call, the strategy is known as a zero cost collar. When used to hedge an outright position in the underlying, this locks the hedger into a range of values; this hedging strategy is known as a cylinder.
COMPOUND OPTION
An option on an option, permitting the purchaser to buy (or sell) an option on an underlying at a fixed price over a predetermined period. Usually sold on interest rate instruments (e.g., captions or floortions), or currencies. They are also used as components of more complex trades. Compound options are often bought to protect against increases in standard option prices during periods of high volatility. The upfront premium for a compound option is less than for a normal European-style option but if the option is exercised, the overall cost will be greater. Due to their greater flexibility the cost, if both options are exercised, is greater than a conventional option.
Compound options can also be constructed on options other than European style options (e.g., barrier options) or portfolios of options (e.g., compound on a cylinder). Indeed compound options on compound options, otherwise known as options">installment options are common (often as part of more complex structures). An installment option requires the holder to pay fixed amounts of premium (installment) at certain installment dates to benefit from the right of exercise of the underlying option. At any point that holder can elect to let the installment payments lapse and loses any right of exercise.
CONSTANT MATURITY TREASURY DERIVATIVE
Over-the-counter swaps and options which use longer-term, Treasury-based instruments for their floating rate reference than money market indexes, such as Libor. “Constant Maturity Treasury” (CMT) refers to the par yield that would be paid by a treasury bill, note or bond which matures in exactly one, two, three, five, seven, 10, 20 or 30 years. Since there may not be treasury issues in the market with exactly these maturities, the yield is interpolated from the yields on treasuries that are available. In the US, such rates have been calculated and published by the Federal Reserve Bank of New York and the US Treasury department on a daily basis every day for more than 30 years. The H.15 Report from the Federal Reserve Bank is often used as a source for CMT rates.
It is then possible for this interpolated yield to form the index rate for instruments such as floating rate notes, which pay interest linked to the CMT yield, options, which pay the difference between a strike price and the CMT yield, and swaps and swaptions, in which one of the cashflows exchanged is the CMT yield. Where necessary, the reference rate is reset at each settlement date. Typical uses of CMT derivatives as hedging tools include the purchase of CMT floors by mortgage servicing companies to protect the value of purchased mortgage servicing portfolios, and the purchase of CMT caps to protect investors with negatively convex mortgage-backed securities portfolios. It is possible to enter into derivatives in other currencies that are based, by analogy, on a “constant maturity interest rate swap” interpolated from the swap curve in the relevant currency. Such derivatives are known as constant maturity swap (CMS) derivatives. Unlike CMT derivatives, CMS derivatives incorporate the spread component of swaps.
CONTINGENT PREMIUM OPTION
An option for which the purchaser pays no premium unless the option is exercised. As a rule of thumb, the premium eventually paid is equal to the premium payable on a normal option divided by the option delta, hence the price increases dramatically for out-of-the-money options. Contingent options can usually be broken down into one or more binary options plus a conventional option. For example, a purchaser could synthesize a contingent call by buying a European-style call and selling enough European binary options with the same strike to pay for the premium on the call. If the options are not in-the-money at expiry, both the total premium paid and the total pay-out are zero. If they are in-the-money, the pay-out on the binary options is simply subtracted from the pay-out on the call. Further flexibility can be obtained by setting the strike for the digitals further out-of-the-money than the call.
See also
rebate,
mini-premium option
CONVERSION
1) A way of taking advantage of mispriced options by creating a synthetic short futures position and hedging market risk by buying a futures contract against it. Thus if a put is undervalued, a trader buys it, at the same time selling a fairly valued call and buying a futures contract. The same strategy can be applied if the call is mispriced. If the option is truly undervalued, the trader earns a riskless profit. The whole exercise relies on put-call+parity. 2) The act of converting a convertible bond into equity.
See also
box,
reversal
CONVEXITY
A bond’s convexity is the amount that its price sensitivity differs from that implied by the bond’s duration. Fixed-rate bonds and swaps have positive convexity: when rates rise the rate of change in their price is slower than suggested by their duration; when rates fall it is faster. Positive convexity is therefore a welcome attribute. The higher the bond’s duration, the more its convexity. Bonds or swaps with call options or embedded call options, e.g., collateralized mortgage obligations, have negative convexity: when rates rise their price fall is faster relative to the interest rate move. Convexity effectively describes the same attribute as gamma.
CORRELATION
Correlation is a measure of the degree to which changes in two variables are related. It is normally expressed as a coefficient between plus one, which means variables are perfectly correlated (in that they move in the same direction to the same degree) and minus one, which means they are perfectly negatively correlated (in that they move in opposite directions to the same degree). In financial markets correlation is important in three areas:
1. The model used for global asset allocation decisions, Sharpe’s capital asset pricing model (CAPM), has, as its linchpin, a covariance matrix that measures correlations between markets.
2. Correlation is also central to the pricing of some options, where two-factor or multi-factor models are used. For spread options, yield curve options and cross-currency caps, estimating the correlation between the underlying assets is of primary importance, the degree of correlation between them having a direct influence on the option price. For quantos such as guaranteed exchange rate options, or differential swaps, the correlation effect is the extent to which there is a relationship between movements in the underlying and movements in the ex-change rate, which has a secondary effect on the price of the option.
3. Correlation between markets is also used to offset an option position in one market against another with similar direction and volatility. Such a strategy might be used to reduce cost – to avoid hedging the positions separately, or because implied volatility in the second market is lower – or because hedging is difficult in the first market. Correlation can be estimated historically (like volatility) but tends to be unstable, and historic estimations may be poor predictors of future realized correlations.
CORRIDOR OPTION
The holder of a corridor option receives a coupon at the end of the lifetime of the corridor whose magnitude depends upon the behavior of a specified spot rate during the lifetime of the corridor. For each day on which the spot rate (typically an official fixing rate observation) remains within the chosen spot range (the accrual corridor) the holder accrues one day’s worth of coupon interest. At the end of the lifetime the accrued coupon is paid out. Its value is calculated according to the following formula:
A variation is the knockout corridor option. In this structure, the holder ceases to accrue coupon interest as soon as the spot rate leaves the range. Even if the spot rate subsequently re-enters the range, the holder does not continue to accrue coupon interest. At the end of the option’s lifetime, the accrued coupon is calculated according to the following formula:
If the accrual corridor is one-sided (the other side of the range being open-ended), it is known as a wall option. Typically, corridor options are imbedded in a structured note, sometimes called a range note, that pays a higher yield than the corresponding vanilla debt as long as the underlying rate remains sufficiently long within the accrual corridor. A similar option to the corridor option is the range binary, a binary option which pays a fixed coupon amount if the range is not breached but nothing if it is breached.
COX-INGERSOLL-ROSS MODEL
In its simplest form this is a lognormal one-factor model of the term structure of interest rates, which has the short rate of interest as its single source of uncertainty. The model allows for interest rate mean reversion and is also known as the square root model because of the assumptions made about the volatility of the short-term rate. The model provides closed-form solutions for prices of zero-coupon bonds, and put and call options on those bonds.
CREDIT DERIVATIVE
A bilateral financial contract which isolates credit risk from an underlying instrument and transfers that credit risk from one party to the contract (the Protection Buyer) to the other (the Protection Seller). There are two main categories of credit derivatives: the first consists of instruments such as credit default swaps in which contingent payments occur as a result of a credit event; the second, which includes credit spread options, seeks to isolate the credit spread component of an instrument’s market yield.
CREDIT OPTION
Put or call options on the price of either (a) a floating rate note, bond, or loan, or (b) an asset swap package, consisting of a credit-risky instrument with any payment characteristics and a corresponding derivative contract that exchanges the cashflows of that instrument for a floating rate cashflow stream, typically three- or six-month Libor plus a spread.
CURRENCY FORWARD
An agreement to exchange a specified amount of one currency for another at a future date at a certain rate. The exchange of currencies is priced so as to allow no risk-free arbitrage. In other words, pricing is not a market estimate of the spot rate at that date, but is made according to the two currencies’ respective interest rates. For example, assuming that Eurosterling interest rates are 10% and Eurodollar 5%, and the US dollar/sterling spot rate is 1.75, the forward rate should reflect the 5% interest rate advantage of depositing money in sterling. Thus the 12-month forward rate should be 1.6695.
Forwards are more appropriate than options if a company has a strong directional view of expected movements in exchange rates. But certainty is rare and hedging entirely with forwards may leave a company locked into unfavorable exchange rates. Unlike options, forwards do not enable companies to take advantage of favorable currency movements. The purchaser of a forward, unlike the purchaser of a future, carries the credit risk of the firm from which it makes the purchase. Since the contracts are not easily reassignable, it is difficult to reduce this risk.
DEFERRED PAY-OUT OPTION
A deferred pay-out option is a variation on American-style options similar to a shout option. The holder of the option may exercise it at any time, for the value taken by the underlying at that time, but the pay-out is delayed until the expiry date. This term is also applied to certain digital options whose pay-out is not paid when triggered, but deferred until the final maturity.
See also
option styles
DELTA
The delta of an option describes its premium’s sensitivity to changes in the price of the underlying. In other words, an option’s delta will be the amount of the underlying necessary to hedge changes in the option price for small movements in the underlying. The delta of an option changes with changes in the price of the underlying. An at-the-money option will have a delta of close to 50%. It falls for out-of-the-money options and increases for in-the-money options, but the change is non-linear: it changes much faster when the option is close-to-the-money. The rate of change of delta is an option’s gamma.
DERIVATIVE
A derivative instrument or product is one whose value changes with changes in one or more underlying market variables, such as equity or commodity prices, interest rates or foreign exchange rates. Basic derivatives include, forwards, futures, swaps, options, warrants and convertible bonds. In mathematical models of financial markets, derivatives are known as contingent claims.
DETERMINISTIC VOLATILITY
The family of options pricing models (including those of Dupire, Derman, Kani and Zou) that seek to incorporate the volatility skew and assume that the local volatility of the underlying stock is a deterministic function of time and the stock price itself.
DIGITAL OPTIONS
See
binary option
DISCRETE BARRIER OPTION
Barrier options where the trigger level is only active for part of the option’s lifetime. This includes barrier options where the trigger is only valid on certain fixing dates, as well as cases where the trigger is valid for sub-intervals of the option’s lifetime.
See also
barrier option
DISTRIBUTION
The probability distribution of a variable describes the probability of the variable attaining a certain value. Assumptions about the distribution of the underlying are crucial to option models because the distribution determines how likely it is that the option will be exercised. Many models assume the logarithm of the relative return has a normal distribution, which can be described by two parameters.
The first is the distribution’s mean; the second its standard deviation (equivalent, if annualized, to volatility). In practice, most empirically observed asset distributions depart from normality. This departure can be described in terms of the skew (how much it tilts to one side or the other) and kurtosis, which describes how fat or thin are the tails at either side. Most markets tend to have fat tails (to be leptokurtic) rather than thin tails (platykurtic). This pushes up the price of out-of-the-money options.
DUAL CURRENCY SWAP
Dual currency swaps are currency swaps that incorporate the foreign exchange options necessary to hedge the interest payments back into the principal currency for dual currency bonds.
EMBEDDED OPTION
An option, often an interest rate option, embedded in a debt instrument that affects its redemption. Examples include mortgage-backed securities and callable and putable bonds. Embedded options do not have to be interest rate options; some are linked to the price of an equity index (Nikkei 225 puts embedded in Nikkei-linked bonds) or a commodity (usually gold). Many so-called guaranteed products contain zero-coupon bonds and call options.
EXCHANGE OPTION
Depending on the context, this can either refer to an outperformance option, or, alternatively an option giving the purchaser the right to exchange one asset for another. The latter type of options are useful if there isn’t a cross-market, as with a barrel of oil priced in Euros. The purchaser of a Euro-oil exchange option would have the right to exchange a certain amount of Euros for a certain number of barrels of oil.
See also
integrated hedge
EXOTIC OPTION
Any option with a more complicated pay-out structure than a plain vanilla put or call option. The pay-out of a plain vanilla option is simply the difference between the strike price of the option and the spot price of the underlying at the time of exercise. For a European-style option, the exercise time is always the expiry date; other option styles offer greater flexibility.
There are a number of ways in which an option pay-out can differ from that of a plain vanilla. The pay-out could also be a function of:
• the difference between a strike and an average rate for the underlying (average options)
• the difference between prices for two different underlyings (difference options, exchange options), the same underlying at different times (high-low options)
• the correlation between two or more underlyings (outperformance options, outside barrier options)
• the difference between a strike and the spot rate at some time other than expiry (deferred pay-out options, shout options, lookback options, cliquet options, ladder options – see diagram
• a fixed amount (binary options)
Alternatively, or additionally, a pay-out may be conditional on certain trigger conditions being met. For example, barrier options are activated or nullified if a spot rate falls or rises through a predetermined trigger level. Multiple trigger conditions are possible (as the in case of corridor or mini-premium options).
EXPLODED TREE
A tree (binomial or trinomial) in which an up step followed by a down step gives a different outcome to a down step followed by an up step. Consequently, the number of nodes increases exponentially, compared with a recombining tree, in which the number increases quadratically. This makes their evaluation exceptionally computer-intensive. The advantage is that they can be used to price path-dependent options and they are important for modeling interest rate options. See binomial+tree for diagram.
FLEXIBLE OPTION
A flexible option (also known as a flexible exchange or flex option) is a customizable exchange-traded option which allows the buyer to customize contract terms such as expiry date and contract size in addition to the strike price. Flexible options with single stock, index, or even currency underlyings are traded on several major exchanges.
GAMMA
The rate of change in the delta of an option for a small change in the underlying. The rate of change is greatest when an option is at-the-money and decreases as the price of the underlying moves further away from the strike price in either direction – gamma is therefore -shaped. A long gamma position is one in which a trader is long options. For a position that is short gamma, the opposite holds. Gamma can be hedged by mirroring the options position. Alternatively, a trader may choose to adjust the position in the underlying continually in order to maintain delta neutrality.
See also
convexity
GARMAN-KOHLHAGEN MODEL
A model developed to price European-style options on spot foreign exchange rates. The model is based upon the Black-Scholes model with the addition of an extra interest rate factor for the foreign currency.
GARMAN-KOHLHAGEN MODEL
A model developed to price European-style options on spot foreign exchange rates. The model is based upon the Black-Scholes model with the addition of an extra interest rate factor for the foreign currency.
GOLD-LINKED NOTE
A note (or bond) with interest payments linked to the price of gold constructed by reducing the coupon (sometimes to zero) and buying (or selling) put or call options to gain exposure to an increasing (or decreasing) gold price.
See also
embedded option
GUARANTEED EXCHANGE RATE OPTION
An option (also known as a quanto option) on an asset in one currency denominated in a second currency. The exchange rate at which the purchaser converts the currency is fixed at the start. Such options are increasingly popular as investors want exposure to foreign assets without the foreign exchange risk. Most of the demand is for bond and stock index options. The extra cost of the option depends on the correlation between movements in the exchange rate and movements in the underlying. The higher (more positive) the correlation between the underlying and the exchange rate (expressed as the number of units of currency two per unit of currency one) the more expensive a call option will be and the cheaper a put option will be. Quanto options can, however, look cosmetically cheaper (or more expensive) depending on the forward interest rates in the two currencies. For example, buying a call on a US asset could be “cheaper” in euros if there is a wide interest rate differential between the euro and the dollar.
See also
joint option
HEDGE
To hedge is to reduce risk by making transactions that reduce exposure to market fluctuations; for example, an investor with a long equity position might compensate by buying put options to protect against a fall in equity prices. A hedge is also the term for the transactions made to effect this reduction.
HIGH-LOW OPTION
A combination of two lookback options. A high-low option pays the difference between the high and low of an underlying, such as a stock index. A speculative purchaser would be taking the view that the market would be more volatile than the implied volatilities of both lookback options incorporated in the structure.
HISTORICAL VOLATILITY
Historical volatility is a measure of the volatility of an underlying instrument over a past period. Historical volatility can be used as a guide to pricing options but isn’t necessarily a good indicator of future volatility. Volatility is normally expressed as the annualized standard deviation of the log relative return.
HO-LEE MODEL
The first model that set out to model movements in the entire term structure of interest rates, not just the short rate, in a way that was consistent with the initially observed term structure. However, since the model only has a single random factor, it makes the simplifying assumption that the volatility structure remains constant along the yield curve. Heath-Jarrow-Morton later generalized this model, using a more general form of volatility and introducing continuous trading. In addition, Ho-Lee allows for the possibility of negative interest rates. The model was developed using a binomial tree, although closed-form solutions have now been found for discount bonds and discount bond options.
HULL-WHITE MODEL
An extension of the Vasicek model for interest rates, the main difference being that mean reversion is time-dependent. Both are one-factor models. The Hull-White model was developed using a trinomial lattice, although closed-form solutions for European-style options and bond prices are possible.
IMPACT FORWARD
A collared forward, such as one in which the purchaser buys a put and sells a call, both being out-of-the-money. The premiums on the two options balance out, so the strategy is zero cost. Upside and downside is limited to the gap between the strike prices.
See also
collar
IMPLIED DISTRIBUTION
The probability distribution of returns for an asset which is implied by options traded on that asset. The distribution is inferred by combining the variation of volatility with strike price (see volatility smile) and the assumptions made about the distribution in the option pricing model.
INTEREST RATE SWAP
An agreement to exchange net future cashflows. Interest rate swaps most commonly change the basis on which liabilities are paid on a specified principal. They are also used to transform the interest basis of assets. In its commonest form, the fixed-floating swap, one counterparty pays a fixed rate and the other pays a floating rate based on a reference rate, such as Libor. There is no exchange of principal – the interest rate payments are made on a notional amount. In floating-floating swaps the two counterparties pay a floating rate on a different index, such as three-month Libor versus six-month Libor.
Swaps usually extend out as far as 10 years, although 12–40 year maturities are available in some liquid currencies. However, the longer the maturity of the swap, the less liquid it becomes and credit risk increases. Credit enhancements such as mutual put options and collateral are used to ameliorate the credit risk of longer term swaps.
Interest rate swaps provide users with a way of hedging the effects of changing interest rates. For example, a company can convert floating-rate interest payments to fixed-rate payments if it thinks interest rates will rise (which would make its liabilities more expensive). Companies can also use interest rate swaps in conjunction with new debt issuance, raising money on, say, a fixed basis and swapping it into floating-rate debt. In an interest rate swap there is a fixed-rate payer (floating-rate receiver) and a fixed-rate receiver (floating-rate payer). If there is a preponderance of fixed-rate payers (for example, when companies want to lock into low rates), the swap spread (the yield spread over equivalent maturity government bonds) increases. If there is a preponderance of fixed-rate receivers, the swap spread declines.
Interest rate swaps were initially transacted back-to-back, with a bank acting as an intermediary. In return for putting the differing requirements together and for taking the credit risk on the interest rate payments, the bank took a fee – either fixed or a small proportion of the interest payments. Now banks run swap books and act as principals in transactions. As a result they have to hedge the swaps they put on their books. This can be done with another swap or with securities. Since US dollar and sterling interest rate swaps are priced as a spread over Treasuries, this is not too difficult, although movement in the swap spread does create basis risk.
Most dealers use short-term interest rate futures such as Eurodollar futures or Euroyen futures to hedge swap positions. Disparities between futures and cash bonds and notes can drive swap spreads up and down. Such hedges can be imperfect when swap payments differ from the contract’s maturity.
IN-THE-MONEY
Describes an option whose strike price is advantageous compared to the current forward market price of the underlying. The more an option is in-the-money, the higher its intrinsic value and the more expensive it becomes. As an option becomes more in-the-money, its delta increases and it behaves more like the underlying in profit and loss terms; hence deep in-the-money options will have a delta of close to one.
See also
at-the-money,
out-of-the-money
ITO’S LEMMA
A mathematical relationship that allows the stochastic process followed by a function of a variable to be deduced. Ito’s Lemma is fundamental to the derivation of a number of options pricing models.
KURTOSIS
A measure of how fast the tails or wings of a probability distribution approach zero, evaluated relative to a normal distribution. The tails are either fat-tailed (leptokurtic) or thin-tailed (platykurtic). Markets are generally leptokurtic. The fatter the tails, the greater the chance a variable will reach an extreme value, implying that models such as Black-Scholes – which assume perfect normal distribution – produce pricing biases for deep in- or out-of-the-money options.
LEVERAGE
The ability to control large amounts of an underlying variable for a small initial investment. Futures and options are regarded as leveraged products because the initial premium paid by the purchaser is generally much smaller than the nominal amount of the underlying. Leverage is usually measured as a quantity called lambda. Many structured notes are said to be leveraged because their coupon is governed by a multiple of a reference interest rate (such as Libor). It is also possible to deleverage a note by linking its coupon to a fraction of the reference rate.
LISTED OPTION
See warrant, exchange traded futures, options
LONGSTAFF-SCHWARTZ MODEL
A two-factor model of the term structure of interest rates. It produces a closed-form solution for the price of zero coupon bonds and a quasi-closed-form solution for options on zero coupon bonds. The model is developed in a Cox-Ingersoll-Ross framework with short interest rates and their volatility as the two sources of uncertainty in the equation.
LOOKBACK OPTIONS
Lookback options give the holder to right at expiry to exercise the option at the most favorable rate or price reached by the underlying over the life of the option. As with average options, the strike may be either fixed or floating. With an optimal rate lookback option, the strike is fixed at the outset and the option will pay out against the highest (for a call) or lowest spot rate (for a put) reached over the life of the option, irrespective of the spot rate at expiry. The option will usually be settled in cash. Since the option is likely to have a larger pay-out than the corresponding plain vanilla option, it commands a larger premium. The strike for an optimal strike lookback option, on the other hand, is not fixed until expiry, when it is set to be the highest (for a put) or lowest spot rate (for a call) over the option’s life and exercised for cash or physical against the spot rate prevailing at expiry.
See also
cliquet option,
ladder option,
look-forward options,
shout option
MONTE CARLO SIMULATION
A method of determining the value of a derivative by simulating the evolution of the underlying variable(s) many times over. The discounted average outcome of the simulation gives an approximation of the derivative’s value. This method may be used to value complex derivatives, particularly path-dependent options, for which closed-form solutions have not been or cannot be found. Monte Carlo simulation can also be used to estimate the value-at-risk (VAR) of a portfolio. In this case, a simulation of many correlated market movements is generated for the markets to which the portfolio is exposed, and the positions in the portfolio revalued repeatedly in accordance with the simulated scenarios. The result of this calculation will be a probability distribution of portfolio gains and losses from which the VAR can be determined. The principal difficulty with Monte Carlo VAR analysis is that it can be very computationally intensive.
MULTI-FACTOR MODEL
Any model in which there are two or more uncertain parameters in the option price (one-factor models incorporate only one cause of uncertainty: the future price). Multi-factor models are useful for two main reasons. Firstly, they permit more realistic modeling, particularly of interest rates, although they are very difficult to compute. Secondly, multi-factor options (for example, spread options) have several parameters, each with independent volatilities, and also the correlation between the underlyings must be dealt with separately.
MULTI-FACTOR OPTION
Any option whose pay-out is linked to the performance of more than one asset. Such options include outside barrier options, outperformance options, portfolio options, multiple strike options and spread options. Their value is usually strongly dependent on the correlation between the underlying assets. A multi-factor option is synonymous with a multi-colored rainbow option.
OPTION
A contract that gives the purchaser the right, but not the obligation, to buy or sell an underlying at a certain price (the exercise, or strike price) on or before an agreed date (the exercise period).
For this right, the purchaser pays a premium to the seller. The seller (writer) of an option has a duty to buy or sell at the strike price, should the purchaser exercise his right. With European-style options, purchasers may take delivery of the underlying only at the end of the option’s life. American-style options may be exercised, for immediate delivery, at any time over the life of the option. Holders of semi-American-style or Bermudan options may be exercised on specified dates – typically on a monthly or quarterly basis.
Options can be bought on commodities, stocks, stock indexes, interest rates, bonds, currencies, etc. The trading terminology, though, may change according to the product. In most cases, the right to buy the underlying is known as a call, and the right to sell, a put.
Options are traded on formal exchanges and in over-the-counter (OTC) markets. The exchanges, such as the Chicago Board Options Exchange, the London International Financial Futures and Options Exchange and the Philadelphia Stock Exchange, provide primarily standardized options; the OTC markets are able to provide tailored products to fit specific requirements. The choice between OTC and exchange-traded options will depend on the degree of tailoring required, the relative liquidity of both markets (this varies greatly according to the underlying) and credit concerns.
Such credit concerns increase with the option’s maturity, since the likelihood that a counterparty will default increases with the length of time that passes between the option being bought and being exercised. Several derivatives exchanges have tried to bridge the gap between OTC and exchange-traded options by introducing flexible options that can be customized by the purchaser.
Pricing models for simple or vanilla options have five major inputs: the option’s exercise or strike price; the time to expiration; the price of the underlying instrument; the risk-free interest rate on the underlying instrument, and the volatility of the underlying instrument (See also historical volatility, implied volatility).
European-style options are usually priced off a closed-form analytical model first published by Fischer Black and Myron Scholes in 1973, which has subsequently been modified to fit different underlyings (see Black-Scholes model).
At maturity, an option’s value will depend on the value of the right to buy or sell a product. If an option is purchased giving the right to buy gold at $375 an ounce and at expiration the price is $400, the option is worth $25.
The extent to which an option is in-the-money (how far the strike price is below/above the current forward market price) is called its intrinsic value. Where the strike price is less favorable than the market price, the option is said to be out-of-the-money, and where the two prices are the same it is at-the-money.
At any time before maturity, an option’s price will be a combination of its intrinsic value (which is always either greater than, or equal to, zero) and its time value. The latter includes the cost of carry and the probability that the price of the underlying will move into or remain in the money. Options can broadly be used in two ways – for speculation, or for insurance. Their usefulness, both from a buyer’s and a seller’s point of view, derives from their pay-outs.
In contrast to other types of hedge, options provide insurance against unfavorable moves in a product’s price and the opportunity to take advantage of favorable moves. Forwards and futures, for example, require buyers and sellers to lock into one rate. In return for assuming this risk, sellers of options receive a premium, effectively a risk-taking fee.
The pay-off of a purchased option means that the price risk of an option is limited to its premium – it is not as exposed to adverse movements as a position in the underlying.
For speculators selling (writing) options, this often means taking a naked option position and therefore being exposed to adverse movements in the underlying. Hedgers may sell options to garner premium to offset any expected slight downturn in a market. Since option premiums are only a fraction of the cost of the underlying product, it is possible to achieve a much greater exposure to price changes of the underlying compared with a similar investment directly in the product – this is called leverage.
OPTION COMBINATION STRATEGIES
Options may be combined so that their pay-outs produce a desired risk profile. Some combinations are primarily trading strategies, but option combinations can be useful in, for example, allowing investors to construct a strategy to take advantage of a particular view they have of the market. Other strategies allow purchasers to reduce their premiums by giving up some of the benefits they may have received from market movements.
See also
bear spread,
bull spread,
calendar spread,
call spread,
condor,
cylinder,
put spread,
ratio spread,
straddle,
strangle
OPTION STYLES
The purchaser of a European-style option has the right to exercise it on a predetermined expiry date. In contrast, the holder of an American-style option has the right to exercise it at any time during its lifetime, up to and including its expiry date. This flexibility means there is a greater probability of an American-style option being exercised than the corresponding European-style option with the same strike. Hence the early exercise feature of an American option adds value and makes it the more expensive of the two. Most exchange-traded options are American-style. Further variations on these styles also exist. A Bermudan option, so called because it falls between American- and European-style options, has more than one possible exercise date. For example, the holder of a Bermudan option with a two-year maturity might have the right to exercise it every quarter or half year during the life of the contract. Bermudans are also known as limited exercise or semi-American-style options. Another twist is the deferred pay-out option, a variation on American-style options in which the option can be exercised at any time during the option’s life, but the pay-out is delayed until the expiry date. With the similar shout option, the purchaser can lock in a profit at any time, but retains the right to profit from further favorable moves.
otc derivative
Over-the-counter derivatives are privately negotiated contracts that are traded directly between two parties, rather than on a centralized exchange. Some of the most common derivatives to be traded in the OTC market include swaps, forward rate agreements, and exotic options. The self-regulatory trade organization that oversees the over-the-counter derivatives market is the International Swaps and Derivatives Association (ISDA).
See also
OTC
PORTFOLIO INSURANCE
A strategy developed in the 1980s as a way of limiting losses on risky asset portfolios. Because put options were not widely available, the strategy synthetically reproduced the pay-out of a put option by a delta-hedging program. As long as markets move continuously, transaction costs are minimal and volatility is relatively stable, option returns can be easily replicated, although one can not predetermine a maximum cost.
The effectiveness of such a strategy was thrown into doubt with the crash of 1987. The unprecedented levels of volatility and the lack of liquidity made the strategy extremely difficult to implement. Its reputation suffered and it was widely blamed for exacerbating the severity of the collapse. Portfolio insurance has not entirely disappeared, though. Some fund managers still synthetically replicate option pay-outs rather than pay option premium, especially if they think volatility will fall. However, most such strategies now involve the covering of a certain amount of volatility risk by buying out-of-the-money options.
See also
asset allocation
PRINCIPAL-GUARANTEED PRODUCT
Any investment vehicle that allows investors to gain exposure to an asset while guaranteeing the return of their principal. Such products are normally constructed by buying a deep discount bond (often a zero-coupon bond) and using the rest of the money to buy embedded call or put options to gain exposure to a second asset, often a stock index.
See also
guaranteed return on investment
PUT-CALL PARITY
The relationship between a European-style put option and a European-style call option on the same underlying with the same exercise price and maturity. Put-call parity states that the pay-off profile of a portfolio containing an asset plus a put option is identical to that of a portfolio containing a call option of the same strike on that same asset (with the rest of the money earning the risk-free rate of return). In practice, a put option on, say, a stock index, can be constructed by shorting the stock and buying a call option. The relationship means that traders are able to arbitrage mispriced options.
See also
box,
conversion,
reversal
QUANTO PRODUCT
An asset or liability denominated in a currency other than that in which it is usually traded, typically equity index futures, equity index options, bond options and interest rate swaps (differential swaps). One example is the Chicago Mercantile Exchange’s Nikkei 225 stock index contract, which uses the nominal price of the yen-denominated index applied to a US dollar notional principal. Quanto products can be hedged with an offsetting position in a local currency product. Variable asset and foreign exchange exposures will arise with changes in the foreign exchange rate and in the underlying, so the structures must be continually dynamically hedged in a similar fashion to option products.
See also
guaranteed exchange rate option
RAINBOW OPTION
The term “rainbow option” is synonymous with “multi-factor option”. The underlying factors are referred to as colors in the context of rainbow options. Hence a two-factor option (such as a spread option) would be a two-color rainbow option.
RATIO SPREAD
A ratio spread involves buying different amounts of similar options with differing strike prices. The purchase of an in-the-money option is financed by the selling of more out-of-the-money options. Conversely, the out-of-the-money options are financed by selling less of an in-the-money option.
REBATE
Barrier options often have a rebate associated with the trigger level(s). A rebate is an amount paid to the holder of the derivative if the instrument is knocked out or is never activated during its lifetime as partial recompense for their initial investment. One example is the rebate range binary.
RELATIVE PERFORMANCE RISK
The risk that a fund manager’s choice of investments will fail to match the performance of the benchmark against which the fund is measured, prompting fund redemptions. A similar risk is run by corporate treasury risk managers who are measured against benchmark hedge levels. One way to address this type of risk is with outperformance options. Relative performance risk is also used to refer to the risk that an individual asset will underperform relative to its asset class. For equities, this may be measured by a stock’s beta, its standardized covariance with respect to the relevant equity index.
See also
specific risk
REPLICATION
To replicate the pay-out of an option by buying or selling other instruments. Creating a synthetic option in this way is always possible in a complete market. In the case of dynamic replication this involves dynamically buying or selling the underlying (or normally, because of cheaper transaction costs, futures) in proportion to an option’s delta. In the case of static replication the option (usually an exotic option) is hedged with a basket of standard options whose composition does not change with time – e.g., an at-expiry digital option can be replicated with a call spread.
REVERSAL
To take advantage of mispriced options by creating a synthetic long futures position and hedging it by selling futures contracts against it. A trader may buy an undervalued call, at the same time selling a fairly valued put and buying a futures contract. The same strategy could be applied if the put was undervalued. The ability to undertake this riskless arbitrage relies on put-call parity.
See also
box,
conversion
RISK REVERSAL
1) See cylinder
2) The term “risk reversal” is also used, by currency option traders, to denote the difference in implied volatility between out-of-the-money call and put options which both have a delta of 25%. The level of the risk reversal is often used as a sentiment indicator in currency markets as it indicates the relative demand for calls versus puts.
SKEW
A skewed distribution is one which is asymmetric. Skew is a measure of this asymmetry. A perfectly symmetrical distribution has zero skew, whereas a distribution with positive (negative) skew is one where outliers above (below) the mean are more probable. An example of an asymmetric distribution in the financial markets is the distribution implied by the presence of a volatility skew between out-of-the-money call and put options.
SPREAD OPTION
The underlying for a spread option is the price differential between two assets (a difference option) or the same asset at different times or places.
An example of a financial difference option is the credit spread option, the underlying for which is the spread between two debt issues which derives from the relative credit rating of the issuers. Another is the cross-currency cap, where the underlying is the spread between interest rates in two different currencies. A calendar spread, a pair of options with the same strike price but different maturities, pays out the price difference for a single asset on two different dates. Spread options, including calendar spreads, are particularly popular in the commodity markets. Variations include:
• Location spreads, based on the price of the same commodity at two different locations. These can be used to hedge the basis risk incurred when taking delivery of a commodity at one location but required at another.
• Processing spreads, known as crack spreads in the crude oil market and frac spreads in the natural gas market. These are based on the price differential between a feedstock (e.g., crude oil or natural gas) and the products that can be obtained by refining or fractionating it (e.g., heating oil or propane).
• Quality spreads, based on the differential between different grades of the same commodity, such as “sweet” and “sour” crudes or heating oils of varying sulfur content.
STATIC REPLICATION
Static replication is a method of hedging an options position with a position in standard options whose composition does not change through time. The method attempts to replicate the pay-out of the instrument in a more manageable fashion than dynamic replication, where a position in the underlying or futures contracts must be dynamically adjusted if it is to remain effective.
Because it uses options to hedge options, a static replication portfolio is a better hedge for gamma and volatility, as well as delta, than dynamic replication. Static replication can be used for hedging a position in exotic options with vanilla options, or for replicating a long-term option with short-term options. In practice, however, it is not always possible to hedge using static replication. The number of different options and notional amounts required can quickly become unmanageable.
See also
synthetic asset,
replication,
delta-hedging
STEALTH
Stealth measures the percentage difference between the strike and the trigger level for a barrier option. Stealth is a particularly important measure for reverse and geared barrier options, where it measures the percentage intrinsic value at the trigger level of the option.
See also
barrier risk
STRANGLE
1) As with a straddle, the sale or purchase of a put option and a call option on the same instrument, with the same expiry, but at strike prices that are out-of-the-money. The strangle costs less than the straddle because both options are out-of-the-money, but profits are only generated if the underlying moves dramatically, and the break-even is worse than for a straddle. Sellers of strangles make money in the range between the two strike prices, but lose if the price moves outside the break-even range (the strike prices plus the premium received).
2) The term strangle is also used, by currency option traders, to denote the average difference in implied volatility between out-of-the-money call and put options with a 25% delta and the implied volatility of at-the-money forward options.
STRUCTURED NOTE
Structured notes are over-the-counter products, which bundle several disparate elements to create a single product, generally by embedding options in a debt instrument such as a medium-term note. They are often view-oriented and are generally tailored to be attractive to investors with highly focused risk/reward appetites and opinions on the market. For example, a structured note might embed equity or currency options or forwards in a debt issue in an effort to enhance the yield of a normal debt holding. Heavily promoted in the early 1990s, structured notes fell out of favor somewhat in 1993–94 as a sequence of surprise market moves and widely publicized losses pointed to the difficulty of pricing and trading such instruments, as well as the cost of taking the incorrect market view. During this time, the comparatively undeveloped secondary market for structured notes allowed sophisticated relative value players to buy “broken” structured notes on an asset swapped basis much more cheaply than vanilla assets from the same issuers.
STRUCTURED YIELD INVESTMENTS
Any security (normally a structured note) whose yield is conditional on certain trigger conditions being met. Such a security is normally constructed by embedding path-dependent options (such as binary options) in a vanilla debt issue. The investor’s return on the note will then vary according to the pay-out of the options.
SWAPTION
An option to enter an interest rate swap. A payer swaption gives the purchaser the right to pay fixed, a receiver swaption gives the purchaser the right to receive fixed (pay floating).
Apart from those in the sterling market, many swaptions are capital-market driven. Good-quality borrowers are able to issue putable or callable bonds and use the swaptions market to reduce their financing costs. In the case of callable bonds, the issuer effectively buys an option from the investor in return for a slightly higher coupon, so that it may benefit if rates decline. Because many of these embedded options have traditionally been underpriced, good-quality borrowers have been able to monetize this anomaly by selling an equivalent swaption (a receiver swaption) to a bank at market rates.
The profit from this arbitrage lowers funding costs. If the swaption is exercised against the issuer, it calls the bonds (although the issuer would almost certainly have called the issue given the reduction in rates). In the case of putable bonds, the borrower sells a swaption to the swaption market. The premium gained lowers the funding cost at the expense of leaving the borrower unsure of the maturity of the debt.
SYNTHETIC ASSET
A synthetic asset is a combination of long and short positions in financial instruments which has the same risk/reward profile as another instrument. For example, it is possible to replicate the pay-out and exposure of a short futures position by going short European-style call options and long European puts with identical strikes and expiries. Synthetic index options can be generated either through positions in the underlying and futures contracts, or with a basket of vanilla options.
See also
replication
THETA
This measures the effect on an option’s price of a one-day decrease in the time to expiration. The more the market and strike prices diverge, the less effect theta has on a vanilla option’s price. Theta is also non-linear for vanilla options, meaning that its value decreases faster as the option is closer to maturity. Positive gamma is generally associated with negative theta and vice versa.
TRIGGER CONDITION
Path-dependent derivatives such as barrier options and binary options have pay-outs which depend in some way on a market variable satisfying a specific condition during the derivative’s life. If this “trigger condition” is met, the derivative may pay out immediately (early exercise) or at some other specified time (such as expiry). Alternatively, the option may only become effective (be knocked-in) or be de-activated (knocked out) when the trigger condition is met (see barrier options).
The most common condition is that the spot rate or price of the underlying must breach a specified level, meaning that it must trade through the barrier, either from above or below. Many other trigger conditions are possible, however. Some examples include:
• the spot rate must breach the trigger, and remain above/below it for a specified time (see Parisian options);
• the spot trades at the trigger level at a specified time (e.g., expiry) or at any time during the option’s life;
• the spot trades within or breaks out of a range (for example, range binaries);
• there is more than one trigger level, with the pay-out conditional upon or increasing with the number of triggers activated and possibly the order in which they are activated (for example, a mini-premium option);
• some combination of these.
VARIANCE GAMMA MODEL
A jump model that better captures the characteristics of the volatility smile for shorter-dated options than stochastic volatility models.
VASICEK MODEL
An interest rate model that incorporates mean reversion and a constant volatility for the short interest rate. It is a one-factor model from which discount bond prices and options on those bonds can be deduced. All have closed-form solutions.
VEGA
Measures the change in an option’s price caused by changes in volatility. Vega is at its highest when an option is at-the-money. It decreases the more the market and strike prices diverge. Options closer to expiration have a lower vega than those with more time to run. Positions with positive vega will generally have positive gamma. To be long vega (to have a positive vega) is achieved by purchasing either put or call options. Positions that are long vega benefit from increases in implied volatility but also from actual volatility if the option is being delta hedged. They will also lose from reductions in volatility. Spread options can be an exception: a reduction in the volatility of one of the assets may actually increase the price of the option because the correlation between the two assets decreases. Vega is sometimes known as kappa or tau.
See also
gamma
VERTICAL SPREAD
Any option strategy that relies on the difference in premium between two options on the same underlying with the same maturity, but different strike prices. Thus put spreads and call spreads would both be vertical spreads.
VOLATILITY SMILE
A graph of the implied volatility of an option versus its strike (for a given tenor) typically describes a smile-shaped curve – hence the term “volatility smile”. This can be attributed to the belief that the underlying distribution is leptokurtic, since this tends to increase the value of out-of-the-money options.
VOLATILITY TERM STRUCTURE
The term structure of volatility is the curve depicting the differing implied volatilities of options with differing maturities. Such a curve arises partly because implied volatility in short options changes much faster than for longer options. However, the volatility term structure also arises because of assumed mean reversion of volatility. The effect of changes in volatility on the option price is less the shorter the option. Most market-makers take advantage of differing volatilities to hedge their books or to trade perceived anomalies in volatility. Such strategies have to be weighted because of the differing vega effects.
VOLATILITY TRADING
A strategy based on a view that future volatility in the underlying will be more or less than the implied volatility in the option price. Option market-makers are volatility traders. The most common way to buy/sell volatility is to buy/sell options, hedging the directional risk with the underlying. Volatility buyers make money if the underlying is more volatile than the implied volatility predicted. Sellers of volatility benefit if the opposite holds. Other methods of buying/ selling volatility are to buy/sell combinations of options, the most usual being to buy/sell straddles or strangles. Other strategies take advantage of the difference between implied volatilities of differing maturity options, not between implied and actual volatility. For example, if implied volatility in short-term options is high and in longer options low, a trader can sell short-term options and buy longer ones.
VOMMA
The vega of an option is not constant. Vega changes as spot changes and as volatility changes. The vomma of an option is defined as the change in vega for a change in volatility. Vomma measure the convexity of an option price with respect to volatility. Vega is to vomma (volatility gamma) as delta is a to gamma for spot movements. Holders of options with a high vomma benefit from volatility of volatility.
See
vega
WARRANT
(1) A certificate giving the purchaser the right, but not the obligation, to purchase a specified amount of an asset at a certain price over a specified period of time. Warrants differ from options only in that they are usually listed. Underlying assets include equity, debt, currencies and commodities.
(2) The document of title to metal stored in a London Metal Exchange-registered warehouse. The warrant is a bearer instrument and states the brand of metal, its weight, the number of pieces and the rent payable. Warrants tend to be stored and transferred electronically in the LME electronic system known as “SWORD.”
See also
equity warrant
WEATHER DERIVATIVE
Typically swaps and vanilla options such as calls, puts, caps, floors and collars with payoffs linked to temperature, precipitation, humidity or wind speed. Most instruments are linked to heating degree days or cooling degree days. These two indexes measure the deviation of the average of a day’s high and low temperature from a baseline reference temperature.
YIELD CURVE OPTION
An option that allows investors to take a view on the shape of a yield curve without taking a view on a bond market’s direction. It is normally structured as the yield of a longer maturity bond minus the yield of a shorter one. A call would therefore appreciate in value as a curve flattened. A put would decrease in value. Such options were developed in the US in 1991 in response to a steepening yield curve.