# Black model

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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 ${\displaystyle T'\geq T}$) 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.

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.

${\displaystyle c=e^{-rT}[FN(d_{1})-KN(d_{2})]}$

The corresponding put price is

${\displaystyle p=e^{-rT}[KN(-d_{2})-FN(-d_{1})]}$

where

${\displaystyle d_{1}={\frac {\ln(F/K)+(\sigma ^{2}/2)T}{\sigma {\sqrt {T}}}}}$
${\displaystyle d_{2}={\frac {\ln(F/K)-(\sigma ^{2}/2)T}{\sigma {\sqrt {T}}}}=d_{1}-\sigma {\sqrt {T}},}$

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 ${\displaystyle e^{-rT}}$ is replaced by ${\displaystyle e^{-rT'}}$ since one must take into account the time value of money. The difference in the two cases is clear from the derivation below.

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.

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 ${\displaystyle e^{-r(T-t)}F(t)}$ 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: ${\displaystyle e^{-r(T-t)}[F(t)-F(0)]}$. Liquidating the F(0) riskless bonds, each of which is worth ${\displaystyle e^{-r(T-t)}}$, results in a net payoff of ${\displaystyle e^{-r(T-t)}F(t)}$.