# Glossary

• ### Value-at-risk

Formally, the probabilistic bound of market losses over a given period of time (known as the holding period) expressed in terms of a specified degree of certainty (the confidence interval). Put more simply, the value-at-risk (VaR) is the worst-case loss expected over the holding period within the probability set out by the confidence interval. Larger losses are possible, but with a low probability. For instance, a portfolio whose VaR is \$20 million over a one-day holding period, with a 95% confidence interval, would have only a 5% chance of suffering an overnight loss greater than \$20 million. Calculation of VaR entails modelling the possible market moves over the holding period, incorporating correlations among market factors, calculating the impact of such potential market moves on portfolio positions, and combining the results to examine risk at different levels of aggregation. The three main approaches to this analysis are historical simulation, the analytical approach using a correlation matrix or empirical (Monte Carlo) simulation. Major trading houses expend considerable energies on their VaR methodologies and have lobbied regulators to recognise their efforts, with some success.

• ### Vanilla option

Also known as a plain vanilla option. A vanilla option is a standard call or put option in its most basic form.

• ### Vanna

The vega of an option is not constant. Vega changes as spot changes and as volatility changes. The vanna of an option measures the change in vega for a change in the underlying spot. As spot moves deeper out-of-the-money for a vanilla option the vega is lower. If spot and volatility movements are positively correlated the holder of an option with positive vanna will be expected to profit from this correlation.

• ### Variable notional option/swap

An option or swap where the notional value is linked to the underlying asset price or rate. Usually changes in the notional will be directly proportional to changes in the underlying price; ie, they both decrease or increase together. Such derivatives have two main uses. In an equity swap, the fixed-rate receiver can opt to receive the return of either a fixed number of stocks, or the number of stocks that could be purchased for a fixed sum. The former case amounts to a variable notional amount for the swap. An example using an option is the case of a firm that sells more exports as exchange rates decline and its products therefore become cheaper abroad. Since it now has greater foreign currency revenue to hedge, it would purchase a variable notional currency option for this purpose.

• ### Variance gamma model

A jump model that better captures the characteristics of the volatility smile for shorter-dated options than stochastic volatility models.

• ### Variance swap

The cash payout of a variance swap is equal to notional multiplied by the difference between the realised variance of the underlying index over the life of the swap and the strike variance.

• ### 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.

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 A measure of the variability (but not the direction) of the price of the underlying instrument. It is defined as the annualised standard deviation of the natural log of the ratio of two successive prices. Historical volatility is a measure of the standard deviation of the underlying instrument over a past period. Implied volatility is the volatility implied in the price of an option. All things being equal, higher volatility will lead to higher vanilla option prices. In traditional Black-Scholes models, volatility is assumed to be constant over the life of an option. Since traders mainly trade volatility, this is clearly unrealistic. New techniques have been developed to cope with volatility's variability. The best known are stochastic volatility, Arch and Garch.

• ### Volatility skew

The difference in implied volatility between out-of-the-money puts and calls. In most equity option markets out-of-the money calls have lower implied volatility than out-of-the-money puts. This is mostly ascribed to the greater supply of volatility above, rather than below, the money since fund managers are happy to write calls and not so happy to write puts. Volatility skews can be very pronounced in the currency markets although whether puts or calls are favoured depends on market sentiment and demand and supply.

• ### 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 swap

The cash payout of a volatility swap is equal to notional multiplied by the difference between the realised volatility of the underlying index over the life of the swap and the strike volatility.

• ### 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.