Black–Scholes
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The term Black–Scholes refers to three closely related concepts:
The Black–Scholes model is a mathematical model of the market for an equity, in which the equity's price is a stochastic process.
The Black–Scholes PDE is a partial differential equation which (in the model) must be satisfied by the price of a derivative on the equity.
The Black–Scholes formula is the result obtained by solving the Black-Scholes PDE for European put and call options.
Robert C. Merton was the first to publish a paper expanding the mathematical understanding of the options pricing model and coined the term "Black-Scholes" options pricing model, by enhancing work that was published by Fischer Black and Myron Scholes. The paper was first published in 1973. The foundation for their research relied on work developed by scholars such as Louis Bachelier, A. James Boness, Sheen T. Kassouf, Edward O. Thorp, and Paul Samuelson. The fundamental insight of Black-Scholes is that the option is implicitly priced if the stock is traded.
Merton and Scholes received the 1997 Nobel Prize in Economics for this and related work. Though ineligible for the prize because of his death in 1995, Black was mentioned as a contributor by the Swedish academy.
Contents
1 The model
2 The Equation
3 Extensions of the model
3.1 Instruments paying continuous yield dividends
3.2 Instruments paying discrete proportional dividends
4 Black–Scholes in practice
4.1 The volatility smile
4.2 Valuing bond options
4.3 Interest rate curve
4.4 Short stock rate
5 Formula derivation
5.1 Elementary derivation
5.2 PDE based derivation
5.2.1 The Black–Scholes PDE
5.2.2 Other derivations of the PDE
5.2.3 Solution of the Black–Scholes PDE
6 Remarks on notation
7 See also
8 Notes
9 References
9.1 Primary references
9.2 Historical and sociological aspects
10 External links
10.1 Discussion of the model
10.2 Derivation and solution
10.3 Revisiting the model
10.4 Computer implementations
10.5 Historical
The model
The Black-Scholes model of the market for an equity makes the following assumptions:
The price of the underlying instrument St follows a geometric Brownian motion defined by
where Wt is a Wiener process with constant drift μ and volatility σ.
It is possible to short sell the underlying stock.
There are no arbitrage opportunities.
Trading in the stock is continuous.
There are no transaction costs or taxes.
All securities are perfectly divisible (i.e. it is possible to buy any fraction of a share).
It is possible to borrow and lend cash at a constant risk-free interest rate.
The stock does not pay a dividend (see below for extensions to handle dividend payments).
The Equation
In the model, there is a unique price for any derivative of the stock. In particular, a European call option, which gives the right to buy one share at price K after T years, has the price C given by:
where
Where S is the current price of the stock, r is the continuously-compounded risk-free interest rate, and σ is the constant stock's volatility.
Here Φ is the standard normal cumulative distribution function, .
Interpretation: Φ(d1) and Φ(d2) are the probabilities of the option expiring in-the-money under the equivalent exponential martingale probability measure (numéraire = stock) and the equivalent martingale probability measure (numéraire = risk free asset), respectively. The equivalent martingale probability measure is also called the risk neutral probability measure. Note that both of these are "probabilities" in a measure theoretic sense, and neither of these is the true probability of expiring in-the-money under the real probability measure.
Intuitively, Φ(d1) represents the non-discounted delta hedge ratio in the underlying asset; and Φ(d2) gives the unit price of the binary call.
The price of a put option may be computed from this by put-call parity and simplifies to
The Greeks under the Black–Scholes model are calculated below:
WhatCallsPutsdeltagammavegathetarho
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