There are a number of terms used herein which whose meanings will be well known to those of working skill in the field of financial markets. However, for ease of reference, the following table contains definitions of terms:
Yield—The “yield” of an investment is a measure of its expected annualised return, (arising from income only or including capital appreciation or depreciation). In the case of most interest rate futures contracts, the “yield” which corresponds to the contract's price is 100% minus the price.
Options—The collective term for “call options” and “put options” (q.v.), and occasionally also for other “derivatives” (q.v.).
Call option or Call—A contract which, if “physically-settled,” gives one party, (the “option buyer”), the right, but not the obligation, to purchase from the other party, (the “option seller”), a certain quantity of an asset or other financial instrument at a fixed price, (the “strike price” or “strike”), at a specified time or times. If the call option is “cash-settled,” then, upon electing to exercise the option, the option buyer receives from the option seller a cash amount equal to the economic benefit that would arise if the option had been physically-settled, i.e. by reference to the excess of the underlying over the strike.
Put option or Put—A contract which, if “physically-settled,” gives one party, (the “option buyer”), the right, but not the obligation, to sell to the other party, (the “option seller”), a certain quantity of an asset or other financial instrument at a fixed price, (the “strike price” or “strike”), at a specified time or times. If the put option is “cash-settled,” then, upon electing to exercise the option, the option buyer receives from the option seller a cash amount equal to the economic benefit that would arise if the option had been physically-settled, i.e. by reference to the excess of the strike over the underlying.
Derivatives—Contracts (including options) whose economic performance are dependent on the evolution of the price, yield or level of an asset, basket, index, contract or other financial instrument, or on the evolution of some other economically significant variable (including weather and geological data).
Underlying—The asset, basket, index, contract, other financial instrument or other economically significant variable, (and, interchangeably, the price, yield or level thereof), to which a derivative contract's performance is linked.
Hedge—“To hedge” is to invest in an asset or other financial instrument or to enter into a financial contract, (in any instance, a “hedge”), so as to offset the risk associated with other assets, financial instruments and/or financial contracts.
Delta-hedge—“To delta-hedge” an option position is to manage dynamically the amount, (“the delta-hedge”), of a direct exposure to the underlying of that option so as to neutralise the option's instantaneous price sensitivity to small changes in the price, yield or level of the underlying.
Mark-to-market revaluation—The revaluation of a position based on its prevailing market price.
Bid price—A price proffered for the purchase of an asset or other financial instrument.
Offered price—A price proffered for the sale of an asset or other financial instrument.
Mid-market—The price midway between the best readily-available bid and offered prices.
Expiration—The last time, according to the terms of an option contract, that the option buyer can exercise his right.
Liquidity—The ease with which one can buy or sell an asset or other financial instrument quickly and in large volume without substantially affecting the asset or financial instrument's price, usually characterised by narrowly separated bid and offered prices.
Term structure—The observed dependence of a financial measure on term or maturity, (e.g. the dependence of yield on redemption date in the case of government bonds, or the dependence of implied volatility on the expiration date in the case of options, etc.)
Duration—A measure of the sensitivity to interest rates of the value of a portfolio of bonds or other interest-rate sensitive instruments. It is defined as that maturity of notional zero-coupon bond having the same monetary value as the portfolio which also would have the same sensitivity as the portfolio to a single small change in interest rates applied equally to all maturities. Duration therefore takes no account of non-uniform changes in the interest rate term structure.
Present-Value-of-a-Basis-Point—A measure of the sensitivity of a portfolio of bonds or other interest-rate sensitive instruments in monetary terms to a 0.01% change in interest rates applied to all maturities. Present-Value-of-a-Basis-Point therefore takes no account of non-uniform changes in the interest rate term structure.
Futures (and options) exchange—An organisation that brings together buyers and sellers of futures (and options) contracts by, for example, open outcry or electronic trading.
Strike price or Strike—See “Call option or Call” and “Put option or Put”.
Clearinghouse—The clearinghouse of a futures (and options) exchange acts as the seller to all buyers and the buyer to all sellers of futures (and options) contracts transacted. As participants may both buy and sell contracts on many occasions, this function of the clearinghouse eliminates the need for keeping track of the complex and long list of successive buyers and sellers of each contract. Additionally, participants are not exposed to default on the contracts by the other participants. Each participant holds an account with a “clearinghouse member” which margins the account of the participant. The clearinghouse margins the accounts of the clearinghouse members.
Margin—“Margin” is the amount of money that the clearinghouse or clearinghouse member requires as deposit in order to maintain a position. “Margining” is the practice, usually undertaken daily, of maintaining a minimum margin with a clearinghouse or clearinghouse member taking account of accrued profits and losses on participants' positions. The purpose of margining is to protect the clearinghouse and clearinghouse members from defaults.
Cash-settled—See “Call option or Call” and “Put option or Put”
At-the-money—An option is at-the-money when the price of the underlying and the strike are the same.
Discount factor—A multiplier used to convert a future cashflow to its present value, and is dependent on interest rates and the period until that cashflow occurs.
Premium—The cost of purchasing an option.
Basis risk—The residual risk arising from the use of a proxy hedge.
Rolling a contract—Closing a position in one futures or options contract and simultaneously establishing an identical position in the futures or options contract with a later expiration.
Contract month—The month in which a futures or options contract expires. Usually all contracts of the same type but with different expirations expire in different months, and so the contract month is used to distinguish them.
Proxy hedge—A hedge whose performance is not perfectly correlated with the performance of the asset, basket, index, contract, other financial instrument or other economically significant variable being hedged.
Straddle—An option position comprising a call option and a put option (q.v.) with identical underlyings, quantities, strikes and expirations.
Strangle—An option position comprising a call option and a put option (q.v.) with identical underlyings, quantities and expirations, but different strikes.
Physically-settled option—See “Call Option or Call” and “Put option or Put”.
Over-the-counter—An “over-the-counter” (OTC) market is an informal market that does not involve a futures exchange.
Open outcry—The method of trading futures and options contracts whereby brokers (and traders) congregate at the designated exchange premises and express their intentions to buy or sell by calling out and by the use of hand signals.
Electronic trading—The method of trading whereby brokers' and traders' orders are submitted to a computer system which identifies and executes matching trades.
In-the-money—An option is in-the-money if, in the case of a call option, the underlying exceeds the strike or, in the case of a put option, the strike exceeds the underlying.
Out-of-the-money—An option is out-of-the-money if, in the case of a call option, the strike exceeds the underlying or, in the case of a put option, the underlying exceeds the strike.
The pricing and hedging of options and certain other derivatives, (which will be collectively referred to as “options” herein for the sake of convenience), was given a strong mathematical foundation by the work of F. Black and M. Scholes. See, e.g., Black, F. and Scholes, M., (1973), “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, 81, 637. Black and Scholes made a number of idealised assumptions about markets and price behaviour. In particular, they assumed that the price, S, of the underlying asset of an option follows a geometric Brownian process governed by the stochastic differential equation:
      dS    S    =            μ      ⁢                          ⁢              ⅆ        t              +          σ      ⁢                          ⁢              ⅆ        z            where μ is a drift rate, σ is a diffusion constant known as the volatility, dt is an infinitesimal increment in time, and dz is the infinitesimal change in a variable, z, which follows a standard Wiener process, (where dz has expectation equal to zero and variance equal to dt).
From this model, Black and Scholes, and many others have been able to derive formulae and algorithms for valuing a wide variety of option types. Somewhat paradoxically, the option values do not depend on the drift rate, μ. The central problem for obtaining numerical values for option prices has thus been the determination of the volatility, σ, since this is not directly observable.
Although Black and Scholes' model assumes that the volatility of the underlying is constant, it is observed that volatilities in general are neither constant nor entirely predictable.
Future volatility can either be estimated from previously experienced volatility, (known as the “historical volatility”), or by deducing the volatility consistent with the prevailing observable prices of options, (known as the “implied volatility”). Because the latter is forward-looking, it is generally regarded as the better estimate.
The implied volatility at which options trade for many underlyings, (or the mid-market implied volatility between bid and offered prices), usually varies from maturity to maturity so that there is a discernible term structure of implied volatility. Financial institutions frequently model this by assuming that the volatility of the underlying varies through time in an entirely deterministic way, (i.e. they assume σ=σ(t)), which we refer to as the “time-dependent Black-Scholes framework”.
Because of the enormous growth in the volume of options traded by financial institutions, (as well as by end users), financial institutions have become very sensitised to their exposure to changes in both (i) future realised volatility—which affects the cost or benefit of delta-hedging their option positions—and (ii) implied volatility—largely driven by supply and demand factors and which affects the mark-to-market revaluation of their option positions.
The sensitivity of the value of an option position, X, to small changes in the overall level of implied volatility is usually referred to as the “vega,” κ, of the option position, defined as:
  κ  =            ∂      X              ∂      σ      
However, financial institutions with portfolios containing many option positions are not just exposed to uniform changes in volatility, but also are exposed to non-uniform changes in the implied volatility term structure. Just as the Duration or the Present-Value-of-a-Basis-Point of a bond portfolio is no longer considered a sufficiently accurate measure of interest rate exposure, the overall vega of an option portfolio is no longer considered a sufficiently accurate measure of volatility exposure.
In the case of interest rates, financial institutions are able to hedge their exposure to changes in the term structure, (at least, in respect of the major currencies), very efficiently by trading the highly liquid interest rate futures and bond futures on the futures exchanges. By selecting the appropriate contract or contracts, exposure to interest rates at a specific maturity can be hedged.
In the case of volatility, however, the ability of financial institutions to hedge their exposure to changes in the term structure on the futures exchanges is much more restricted. Whilst regular exchange-traded call and put options may be used to hedge volatility exposure, they suffer from a number of disadvantages:
1) The liquidity of exchange-traded options, (except at short maturities), is typically relatively low. This can be attributed to the fact that, for each maturity, many different option strikes are available, which has the effect of fragmenting liquidity.                2) The price, (and hence the profitability), of regular call and put options are sensitive to a number of other variables besides volatility, not the least of which is the level of the underlying.        3) The vega of regular call and put options is not constant and is itself dependent on a number of other variables, particularly the level of the underlying. This means that, unless the terms of the calls or puts traded match quite closely the options being hedged, the net exposure to volatility may behave quite unpredictably over time.        
Points 2 and 3 are also important considerations for financial institutions or speculators who wish to take positions that express a pure volatility view. If the volatility exposure of a position is contaminated by a number of other exposures, then the profitability of that position will be contaminated by the profitability (or otherwise) relating to those other exposures.
It would be desirable to provide a contract which enables speculators and financial institutions to take a view on volatility uncontaminated by other exposures. It also would be desirable to provide a contract which enables financial institutions to hedge volatility exposure uncontaminated by other exposures.
Aside from being exposed to changes in the term structure of volatility, market professionals are also exposed to sudden jumps in the underlying. Sudden jumps in the underlying can cause a material mismatch between the performance of the delta-hedge and the performance of the option portfolio being hedged. This risk is usually measured by the “gamma,” Γ, of the portfolio, which is the local sensitivity of the “delta,” (the theoretical quantity of delta-hedge required to hedge the portfolio), to changes in the underlying. The delta is:
  Δ  =            ∂      X              ∂      S      and therefore the gamma is:
  Γ  =                    ∂        Δ                    ∂        S              =                            ∂          2                ⁢        X                    ∂                  S          2                    
It would be desirable if the contract that is provided also enables financial institutions to hedge their gamma exposures.
Other relevant background is provided in F. Black, “The Pricing of Commodity Contracts,” (1976), Journal of Financial Economics, 3, 167; and D. R. Cox and H. D. Miller, “The Theory of Stochastic Processes,” (1965), London: Chapman & Hall.
USPTO Backgrounds
USPTO Backgrounds