Black model

Last updated

The Black model (sometimes known as the Black-76 model) is a variant of the Black–Scholes option pricing model. Its primary applications are for pricing options on future contracts, bond options, Interest rate cap and floors, and swaptions. It was first presented in a paper written by Fischer Black in 1976.

In finance, a bond option is an option to buy or sell a bond at a certain price on or before the option expiry date. These instruments are typically traded OTC.

An interest rate cap is a type of interest rate derivative in which the buyer receives payments at the end of each period in which the interest rate exceeds the agreed strike price. An example of a cap would be an agreement to receive a payment for each month the LIBOR rate exceeds 2.5%.

A swaption is an option granting its owner the right but not the obligation to enter into an underlying swap. Although options can be traded on a variety of swaps, the term "swaption" typically refers to options on interest rate swaps.

Contents

Black's model can be generalized into a class of models known as log-normal forward models, also referred to as LIBOR market model.

The LIBOR market model, also known as the BGM Model is a financial model of interest rates. It is used for pricing interest rate derivatives, especially exotic derivatives like Bermudan swaptions, ratchet caps and floors, target redemption notes, autocaps, zero coupon swaptions, constant maturity swaps and spread options, among many others. The quantities that are modeled, rather than the short rate or instantaneous forward rates are a set of forward rates, which have the advantage of being directly observable in the market, and whose volatilities are naturally linked to traded contracts. Each forward rate is modeled by a lognormal process under its forward measure, i.e. a Black model leading to a Black formula for interest rate caps. This formula is the market standard to quote cap prices in terms of implied volatilities, hence the term "market model". The LIBOR market model may be interpreted as a collection of forward LIBOR dynamics for different forward rates with spanning tenors and maturities, each forward rate being consistent with a Black interest rate caplet formula for its canonical maturity. One can write the different rates dynamics under a common pricing measure, for example the forward measure for a preferred single maturity, and in this case forward rates will not be lognormal under the unique measure in general, leading to the need for numerical methods such as monte carlo simulation or approximations like the frozen drift assumption.

The Black formula

The Black formula is similar to the Black–Scholes formula for valuing stock options except that the spot price of the underlying is replaced by a discounted futures price F.

Suppose there is constant risk-free interest rate r and the futures price F(t) of a particular underlying is log-normal with constant volatility σ. Then the Black formula states the price for a European call option of maturity T on a futures contract with strike price K and delivery date T' (with ) is

The risk-free interest rate is the rate of return of a hypothetical investment with no risk of financial loss, over a given period of time.

Futures contract standardized legal agreement to buy or sell something (usually a commodity or financial instrument) at a predetermined price (“forward price”) at a specified time (“delivery date”) in the future

In finance, a futures contract is a standardized forward contract, a legal agreement to buy or sell something at a predetermined price at a specified time in the future, between parties not known to each other. The asset transacted is usually a commodity or financial instrument. The predetermined price the parties agree to buy and sell the asset for is known as the forward price. The specified time in the future—which is when delivery and payment occur—is known as the delivery date. Because it is a function of an underlying asset, a futures contract is a derivative product.

The corresponding put price is

where

and N(.) is the cumulative normal distribution function.

Note that T' doesn't appear in the formulae even though it could be greater than T. This is because futures contracts are marked to market and so the payoff is realized when the option is exercised. If we consider an option on a forward contract expiring at time T' > T, the payoff doesn't occur until T' . Thus the discount factor is replaced by since one must take into account the time value of money. The difference in the two cases is clear from the derivation below.

Forward contract non-standardized contract between two parties to buy or sell an asset at a specified future time at a price agreed upon today

In finance, a forward contract or simply a forward is a non-standardized contract between two parties to buy or to sell an asset at a specified future time at a price agreed upon today, making it a type of derivative instrument. The party agreeing to buy the underlying asset in the future assumes a long position, and the party agreeing to sell the asset in the future assumes a short position. The price agreed upon is called the delivery price, which is equal to the forward price at the time the contract is entered into.

Time value of money value of current money with interest after time

The time value of money is the greater benefit of receiving money now rather than an identical sum later. It is founded on time preference.

Derivation and assumptions

The Black formula is easily derived from the use of Margrabe's formula, which in turn is a simple, but clever, application of the Black–Scholes formula.

The payoff of the call option on the futures contract is max (0, F(T) - K). We can consider this an exchange (Margrabe) option by considering the first asset to be and the second asset to be the riskless bond paying off $1 at time T. Then the call option is exercised at time T when the first asset is worth more than K riskless bonds. The assumptions of Margrabe's formula are satisfied with these assets.

The only remaining thing to check is that the first asset is indeed an asset. This can be seen by considering a portfolio formed at time 0 by going long a forward contract with delivery date T and short F(0) riskless bonds (note that under the deterministic interest rate, the forward and futures prices are equal so there is no ambiguity here). Then at any time t you can unwind your obligation for the forward contract by shorting another forward with the same delivery date to get the difference in forward prices, but discounted to present value: . Liquidating the F(0) riskless bonds, each of which is worth , results in a net payoff of .

See also

Discussion

Online tools

Related Research Articles

The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return. The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. It is widely used, although often with adjustments and corrections, by options market participants.

Geometric Brownian motion

A geometric Brownian motion (GBM) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion with drift. It is an important example of stochastic processes satisfying a stochastic differential equation (SDE); in particular, it is used in mathematical finance to model stock prices in the Black–Scholes model.

In financial mathematics, put–call parity defines a relationship between the price of a European call option and European put option, both with the identical strike price and expiry, namely that a portfolio of a long call option and a short put option is equivalent to a single forward contract at this strike price and expiry. This is because if the price at expiry is above the strike price, the call will be exercised, while if it is below, the put will be exercised, and thus in either case one unit of the asset will be purchased for the strike price, exactly as in a forward contract.

In finance, the binomial options pricing model (BOPM) provides a generalizable numerical method for the valuation of options. Essentially, the model uses a "discrete-time" model of the varying price over time of the underlying financial instrument, addressing cases where the closed-form Black–Scholes formula is wanting.

In finance, moneyness is the relative position of the current price of an underlying asset with respect to the strike price of a derivative, most commonly a call option or a put option. Moneyness is firstly a three-fold classification: if the derivative would have positive intrinsic value if it were to expire today, it is said to be in the money; if it would be worthless if expiring at the current price it is said to be out of the money, and if the current price and strike price are equal, it is said to be at the money. There are two slightly different definitions, according to whether one uses the current price (spot) or future price (forward), specified as "at the money spot" or "at the money forward", etc.

Lookback options, in the terminology of finance, are a type of exotic option with path dependency, among many other kind of options. The payoff depends on the optimal underlying asset's price occurring over the life of the option. The option allows the holder to "look back" over time to determine the payoff. There exist two kinds of lookback options: with floating strike and with fixed strike.

A binary option is a financial exotic option in which the payoff is either some fixed monetary amount or nothing at all. The two main types of binary options are the cash-or-nothing binary option and the asset-or-nothing binary option. The former pays some fixed amount of cash if the option expires in-the-money while the latter pays the value of the underlying security. They are also called all-or-nothing options, digital options, and fixed return options (FROs).

Rational pricing is the assumption in financial economics that asset prices will reflect the arbitrage-free price of the asset as any deviation from this price will be "arbitraged away". This assumption is useful in pricing fixed income securities, particularly bonds, and is fundamental to the pricing of derivative instruments.

In financial mathematics, the Hull–White model is a model of future interest rates. In its most generic formulation, it belongs to the class of no-arbitrage models that are able to fit today's term structure of interest rates. It is relatively straightforward to translate the mathematical description of the evolution of future interest rates onto a tree or lattice and so interest rate derivatives such as bermudan swaptions can be valued in the model.

In finance, a foreign exchange option is a derivative financial instrument that gives the right but not the obligation to exchange money denominated in one currency into another currency at a pre-agreed exchange rate on a specified date. See Foreign exchange derivative.

A variance swap is an over-the-counter financial derivative that allows one to speculate on or hedge risks associated with the magnitude of movement, i.e. volatility, of some underlying product, like an exchange rate, interest rate, or stock index.

In mathematical finance, the Black–Derman–Toy model (BDT) is a popular short rate model used in the pricing of bond options, swaptions and other interest rate derivatives; see Lattice model (finance) #Interest rate derivatives. It is a one-factor model; that is, a single stochastic factor—the short rate—determines the future evolution of all interest rates. It was the first model to combine the mean-reverting behaviour of the short rate with the lognormal distribution, and is still widely used.

In statistics, stochastic volatility models are those in which the variance of a stochastic process is itself randomly distributed. They are used in the field of mathematical finance to evaluate derivative securities, such as options. The name derives from the models' treatment of the underlying security's volatility as a random process, governed by state variables such as the price level of the underlying security, the tendency of volatility to revert to some long-run mean value, and the variance of the volatility process itself, among others.

In finance, a spread option is a type of option where the payoff is based on the difference in price between two underlying assets. For example, the two assets could be crude oil and heating oil; trading such an option might be of interest to oil refineries, whose profits are a function of the difference between these two prices. Spread options are generally traded over the counter, rather than on exchange.

Option (finance) right to to buy or sell a certain thing at a later date at an agreed price

In finance, an option is a contract which gives the buyer the right, but not the obligation, to buy or sell an underlying asset or instrument at a specified strike price prior to or on a specified date, depending on the form of the option. The strike price may be set by reference to the spot price of the underlying security or commodity on the day an option is taken out, or it may be fixed at a discount or at a premium. The seller has the corresponding obligation to fulfill the transaction – to sell or buy – if the buyer (owner) "exercises" the option. An option that conveys to the owner the right to buy at a specific price is referred to as a call; an option that conveys the right of the owner to sell at a specific price is referred to as a put. Both are commonly traded, but the call option is more frequently discussed.

In finance, the Heston model, named after Steven Heston, is a mathematical model describing the evolution of the volatility of an underlying asset. It is a stochastic volatility model: such a model assumes that the volatility of the asset is not constant, nor even deterministic, but follows a random process.

Black–Scholes equation

In mathematical finance, the Black–Scholes equation is a partial differential equation (PDE) governing the price evolution of a European call or European put under the Black–Scholes model. Broadly speaking, the term may refer to a similar PDE that can be derived for a variety of options, or more generally, derivatives.

In mathematical finance, Margrabe's formula is an option pricing formula applicable to an option to exchange one risky asset for another risky asset at maturity. It was derived by William Margrabe in 1978. Margrabe's paper has been cited by over 1500 subsequent articles.

The Datar–Mathews method is a method for real options valuation. The method provides an easy way to determine the real option value of a project simply by using the average of positive outcomes for the project. The method can be understood as an extension of the net present value (NPV) multi-scenario Monte Carlo model with an adjustment for risk aversion and economic decision-making. The method uses information that arises naturally in a standard discounted cash flow (DCF), or NPV, project financial valuation. It was created in 2000 by Professor Vinay Datar, Seattle University, and Scott H. Mathews, Technical Fellow, The Boeing Company.

In finance, Black's approximation is an approximate method for computing the value of an American call option on a stock paying a single dividend. It was described by Fischer Black in 1975.

References