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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.
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 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 corresponding put price is
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.
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 .
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 some adjustments, by options market participants.
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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.
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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 with the underlying at its current price it is said to be out of the money, and if the current underlying 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.
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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.
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