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The Black model (sometimes known as the Black-76 model) is a variant the Black-Scholes option pricing model. It is widely used in the futures market and interest rate market for pricing bond options. It was first presented in a paper written by Fischer Black in 1976. The Black-Scholes model, often simply called Black-Scholes, is a model of the varying price over time of financial instruments, and in particular stocks. ...
In finance, a futures contract is a standardized contract, traded on a futures exchange, to buy or sell a certain underlying instrument at a certain date in the future, at a pre-set price. ...
For short-term mutual funds investing in money market securities, see Money fund: The money market is the financial market for short-term borrowing and lending, typically up to thirteen months. ...
In finance, a bond is a debt security, in which the issuer owes the holders a debt and is obliged to repay the principal and interest (the coupon). ...
Fischer Black (1938 - August 30, 1995) was an American economist, best known as one of the authors of the famous Black-Scholes equation. ...
The main problem with the Black model is that it does not easily deal with price correlation of multiple options. Each option is considered to be priced independently of other options and the linkages between the prices of different options is not easily incorporated into the model.
Black's model can be generalized into a class of models known as log-normal forward models.
The Black formula
The Black formula is similar to the Black-Scholes formula for valuating stock options except that the spot price of the underlying is replaced by the forward price. The Black-Scholes model, often simply called Black-Scholes, is a model of the varying price over time of financial instruments, and in particular stocks. ...
Main article: Option A stock option is a specific type of option that uses the stock itself as an underlying instrument to determine the options pay-off (and therefore its value). ...
The spot price of a commodity or a security or a currency is the price that is quoted for settlement (payment and delivery) of the transaction immediately. ...
The forward price is the agreed upon price of an asset in a forward contract. ...
The Black formula for a call option on an underlying struck at K, expiring T years in the future is
- c = e − rT(FN(d1) − KN(d2))
where - r is the risk-free interest rate
- F is the current forward price of the underlying for the option maturity
- σ is the volatility of the forward price.
- and N(.) is the standard cumulative Normal distribution function.
The put price is - p = e − rT(KN( − d2) − FN( − d1)).
Derivation and assumptions The derivation of the pricing formulas in the model follows that of the Black-Scholes model almost exactly. The assumption that the spot price follows a log-normal process is replaced by the assumption that the forward price follows such a process. From there the derivation is identical and so the final formula is the same except that the spot price is replaced by the forward - the forward price represents the expected future value discounted at the risk free rate. In finance, discounting is the process of finding the current value of an amount of cash at some future date, and along with compounding cash from the basis of time value of money calculations. ...
The risk-free interest rate is the interest rate that it is assumed can be obtained by investing in financial instruments with no risk. ...
See also Mathematical finance is the branch of applied mathematics concerned with the financial markets. ...
The Black-Scholes model, often simply called Black-Scholes, is a model of the varying price over time of financial instruments, and in particular stocks. ...
External links
References - Black, Fischer (1976). The pricing of commodity contracts, Journal of Financial Economics, 3, 167-179.
- Garman, Mark B. and Steven W. Kohlhagen (1983). Foreign currency option values, Journal of International Money and Finance, 2, 231-237.
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