Black-Scholes model

The term of Black-Scholes is used to indicate three very close concepts:

• the Modèle Black-Scholes is a mathematical model of the market for an action, in which the price of the action is a stochastic Processus
• the Black-Scholes PDE is an equation which must be solvée by the price of a derivative of an action
• the Equation Black-Scholes (commonly called Modèle Black-Scholes ) is the result obtained of Black-Scholes PDE applied to the options of the European type.

Robert C. Merton was first has to publish an article developing the mathematical aspect of a model of Pricing of option by quoting work of Fischer Black and Myron Scholes. Those, published in 1973, are based on the developments of theorists like Louis Bachelier or Paul Samuelson. The fundamental concept of Black and Scholes was to put in report/ratio the implicit price of the option and the price changes of the subjacent credit.

Merton and Scholes accepted in 1997 the “Nobel Prize of economy” for their work. Black, because of its death in 1995, was ineligible and was quoted like contributeur.

The model

The Black-Scholes model rests on a certain number of conditions

• the price of the subjacent credit S _T follows a Brownian Movement geometrical with a constant Volatilité $\backslash \; sigma$ and a constant derivative $\backslash \; driven$.

$dS\_t\; =\; \backslash \; driven\; S\_t\; \backslash ,\; dt\; +\; \backslash \; sigma\; S\_t\; \backslash ,\; dW\_t\; \backslash ,$

• It does not have there arbitration appropriatenesses
• the trading is carried out
• uninterrupted It is possible to carry out Short sales
• It does not have there costs of trading or taxes
• It is possible to borrow or to place at a Taux without risk
• All the subjacent ones are perfectly divisible ( e.g. one can buy 1/100e action)
• It does not have there a Dividendes

Points Ci above involve the following formula for the calculation of the price of option of a European type with:

• $\backslash \; mathcal\; \{\}\; S\_0$ the current value of the subjacent action
• $\backslash \; mathcal\; \{\}\; K$ the price of exercise fixed by the option
• $\backslash \; mathcal\; \{\}\; T$ the time which remains with the option before its expiry (expressed in years)
• $\backslash \; mathcal\; \{\}\; R$ the rate without risk
• $\backslash \; mathcal\; \{\}\; \backslash \; sigma$ the Volatilité of the price of the action

Thus for a cal one a:
$C\; (S,\; K,\; R,\; T,\; \backslash \; sigma)\; =\; S\; \backslash \; mathcal\; \{NR\}\; (d\_1)\; -\; K\; e^\; \{-\; rt\}\; \backslash \; mathcal\; \{NR\}\; (d\_2)$

By parity of the option one obtains for could:
$P\; (S,\; K,\; R,\; T,\; \backslash \; sigma)\; =\; -\; S\; \backslash \; mathcal\; \{NR\}\; (-\; d\_1)\; +\; K\; e^\; \{-\; rt\}\; \backslash \; mathcal\; \{NR\}\; (-\; d\_2)$

With:

• $\backslash \; mathcal\; \{NR\}$ the function of distribution of the normal Law centered reduced $\backslash \; mathcal\; \{NR\}\; \backslash \; left\; (0,1\; \backslash \; right)$, i.e. $\backslash \; mathcal\; \{NR\}\; (X)\; =\; \backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \{X\}\; \backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{2\; \backslash \; pi\}\}\; e^\; \{-\; \backslash \; frac\; \{1\}\; \{2\}\; u^2\}\; du$
• $d\_1\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; sigma\; \backslash \; sqrt\; \{T\}\}\; \backslash \; left\; \backslash \; ln\; \backslash \; left\; (\backslash \; frac\; \{S\}\; \{K\}\; \backslash \; right)\; +\; \backslash \; left\; (R\; +\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash \; sigma^2\; \backslash \; right)\; T\; \backslash \; right$
• $d\_2\; =\; d\_1\; -\; \backslash \; sigma\; \backslash \; sqrt\; \{T\}$

Extensions of the model

The model presented previously can easily be modified to support rates and nonconstant volatilities. The model can also be wide for the option European paying of the dividends

Continuous outputs

Outputs proportional

Greques

Formulas of derivation

Historical and economic importance

It was published in 1973, and constituted the prolongation of work completed by Paul Samuelson and Robert Merton. The French mathematician Louis Bachelier had inaugurated the study of the subject in 1900. The fundamental intuition of Black and Scholes was to put in report/ratio the implicit price of the option and the price changes of the subjacent credit. Their discovery had a considerable influence very quickly, and variations of their model are used in all the compartments of the financial markets. As of 1977, Oldrich Vasicek was inspired some to found the modern theory of the Interest rate.

Merton and Scholes accepted in 1997 the “Nobel Prize of economy” for their work (Fisher Black, had unfortunately died to him in 1995).

Formulate of Black-Scholes

The formula of Black-Scholes makes it possible to calculate the theoretical value of a option starting from the five following data:

• $\backslash \; mathcal\; \{\}\; S\_0$ the current value of the subjacent action
• $\backslash \; mathcal\; \{\}\; T$ the time which remains with the option before its expiry (expressed in years)
• $\backslash \; mathcal\; \{\}\; K$ the price of exercise fixed by the option
• $\backslash \; mathcal\; \{\}\; R$ the Interest rate without risk
• $\backslash \; mathcal\; \{\}\; \backslash \; sigma$ the Volatilité of the price of the action

The theoretical price of a option to buy ( cal ), which gives the right but not the obligation to buy the credit S with the value K at the date T, is characterized by its pay off : $(\backslash \; mathcal\; \{S\}\; \_\; \{T\}\; -\; K)\; ^\; \{+\}\; =\; \backslash \; max\; (S\_\; \{T\}\; -\; K;\; 0)$

The price of the option is given by the hope under probability risks neutral final pay off updated $C\; =\; E\; \backslash \; times\; e^\; \{-\; rT\}$, is the formula of Black-Scholes:

$C\; (S,\; K,\; R,\; T,\; \backslash \; sigma)\; =\; S\; \backslash \; mathcal\; \{NR\}\; (d\_1)\; -\; K\; e^\; \{-\; rt\}\; \backslash \; mathcal\; \{NR\}\; (d\_2)$

In the same way, the theoretical price of a option to sell ( could ), of pay off $(K\; -\; \backslash \; mathcal\; \{S\}\; \_\; \{T\})\; ^\; \{+\}\; =\; \backslash \; max\; (K-S\_\; \{T\};\; 0)$ is given by:

$P\; (S,\; K,\; R,\; T,\; \backslash \; sigma)\; =\; -\; S\; \backslash \; mathcal\; \{NR\}\; (-\; d\_1)\; +\; K\; e^\; \{-\; rt\}\; \backslash \; mathcal\; \{NR\}\; (-\; d\_2)$

with

• $\backslash \; mathcal\; \{NR\}$ the function of distribution of the normal Law centered reduced $\backslash \; mathcal\; \{NR\}\; \backslash \; left\; (0,1\; \backslash \; right)$, i.e. $\backslash \; mathcal\; \{NR\}\; (X)\; =\; \backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \{X\}\; \backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{2\; \backslash \; pi\}\}\; e^\; \{-\; \backslash \; frac\; \{1\}\; \{2\}\; u^2\}\; du$
• $d\_1\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; sigma\; \backslash \; sqrt\; \{T\}\}\; \backslash \; left\; \backslash \; ln\; \backslash \; left\; (\backslash \; frac\; \{S\}\; \{K\}\; \backslash right)\; +\; \backslash \; left\; (R\; +\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash \; sigma^2\; \backslash \; right)\; T\; \backslash \; right$
• $d\_2\; =\; d\_1\; -\; \backslash \; sigma\; \backslash \; sqrt\; \{T\}$

The formula of Black-Scholes rests on the assumption that the outputs of the subjacent credit are Gaussian, or in an equivalent way that the value of the credit follows a geometrical Brownian diffusion.

The first four data are obvious, only volatility $\backslash \; mathcal\; \{\}\; \backslash \; sigma$ of the credit is difficult to evaluate. Two analysts will be able to have an opinion different on the value from $\backslash \; mathcal\; \{\}\; \backslash \; sigma$ to choose.

One can also apply the formula contrary. Being given the price of the option which is with dimensions in the markets, which value of $\backslash \; mathcal\; \{\}\; \backslash \; sigma$ must be selected so that formula B-S gives this price exactly. One thus obtains the “implicit Volatilité” which has a great practical and theoretical interest.

Lemma of Itô (abstract)

Either $x$ a variable according to a process of Itô, $dx\; =\; has\; \backslash \; left\; (X,\; T\; \backslash \; right)\; dt\; +\; B\; \backslash \; left\; (X,\; T\; \backslash \; right)\; dz$ and $G$ a function d'$x$ and $t$. One makes a limited development of $G$ in discrete time and one obtains:

$\backslash \; delta\; G\; =\; \backslash \; frac\; \{\backslash \; partial\; G\}\; \{\backslash \; partial\; X\}\; \backslash \; delta\; X\; +\; \backslash \; frac\; \{\backslash \; partial\; G\}\; \{\backslash \; partial\; T\}\; \backslash \; delta\; T\; +\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash \; frac\; \{\backslash \; partial\; ^2\; G\}\; \{\backslash \; partial\; x^2\}\; \backslash \; delta\; x^2\; +\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash \; frac\; \{\backslash \; partial\; ^2\; G\}\; \{\backslash \; partial\; t^2\}\; \backslash \; delta\; t^2\; +\; \backslash \; frac\; \{\backslash \; partial^2\; G\}\; \{\backslash \; partial\; X\; \backslash \; partial\; T\}\; \backslash \; delta\; X\; \backslash \; delta\; T\; +\; \backslash \; ldots$

In discrete time, we know that $\backslash \; delta\; Z\; =\; \backslash \; epsilon\; \backslash \; sqrt\; \{\backslash \; delta\; T\}$ One thus has: $\backslash \; delta\; X\; =\; has\; \backslash \; left\; (X,\; T\; \backslash \; right)\; \backslash \; delta\; T\; +\; B\; \backslash \; left\; (X,\; T\; \backslash \; right)\; \backslash \; epsilon\; \backslash \; sqrt\; \{\backslash \; delta\; T\}$ It is known that the variance of $\backslash \; delta\; Z\; =\; 1$ and that its average is null. One thus finds:

$E\; \backslash \; left\; \backslash \; epsilon^2\; \backslash \; right\; -\; E\; \backslash \; left\; \backslash \; epsilon\; \backslash \; right\; =\; VAr\; \backslash \; left\; \backslash \; epsilon\; \backslash \; right$

$\backslash \; Leftrightarrow\; E\; \backslash \; left\; \backslash \; epsilon^2\; \backslash \; right\; -\; 0\; =\; 1$

$\backslash \; Leftrightarrow\; E\; \backslash \; left\; \backslash \; epsilon^2\; \backslash \; right\; =\; 1$

$\backslash \; Rightarrow\; \backslash \; delta\; X\; =\; has\; \backslash \; left\; (X,\; T\; \backslash \; right)\; \backslash \; delta\; T\; +\; B\; \backslash \; left\; (X,\; T\; \backslash \; right)\; \backslash \; sqrt\; \{\backslash \; delta\; T\}$

One replaces now $\backslash \; delta\; x$ in the equation $\backslash \; delta\; G\; =\; \backslash \; ldots$ and one keeps only the terms of order 1, one will thus neglect the terms of a higher nature.

$\backslash \; delta\; G\; =\; \backslash \; frac\; \{\backslash \; partial\; G\}\; \{\backslash \; partial\; X\}\; \backslash \; left\; (\backslash \; left\; (X,\; T\; \backslash \; right)\; has\; \backslash \; delta\; T\; +\; B\; \backslash \; left\; (X,\; T\; \backslash \; right)\; \backslash \; sqrt\; \{\backslash \; delta\; T\}\; \backslash \; right)\; +\; \backslash \; frac\; \{\backslash \; partial\; G\}\; \{\backslash \; partial\; T\}\; \backslash \; delta\; T\; +\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash \; frac\; \{\backslash \; partial\; ^2\; G\}\; \{\backslash \; partial\; x^2\}\; \backslash \; left\; (has\; \backslash \; left\; (X,\; T\; \backslash \; right)\; \backslash \; delta\; T\; +\; B\; \backslash \; left\; (X,\; T\; \backslash \; right)\; \backslash sqrt\; \{\backslash \; delta\; T\}\; \backslash \; right)\; ^2\; +\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash \; frac\; \{\backslash \; partial\; ^2\; G\}\; \{\backslash \; partial\; t^2\}\; \backslash \; delta\; t^2\; +\; \backslash \; frac\; \{\backslash \; partial^2\; G\}\; \{\backslash \; partial\; X\; \backslash \; partial\; T\}\; \backslash \; left\; (\backslash \; left\; (X,\; T\; \backslash \; right)\; has\; \backslash \; delta\; T\; +\; B\; \backslash \; left\; (X,\; T\; \backslash \; right)\; \backslash \; sqrt\; \{\backslash \; delta\; T\}\; \backslash \; right)\; \backslash \; delta\; T\; +\; \backslash \; ldots$

One rearranges and one obtains:

$\backslash \; Rightarrow\; \backslash \; delta\; G\; =\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; G\}\; \{\backslash \; partial\; X\}\; has\; \backslash \; left\; (X,\; T\; \backslash \; right)\; +\; \backslash \; frac\; \{\backslash \; partial\; G\}\; \{\backslash \; partial\; T\}\; +\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash \; frac\; \{\backslash \; partial^2\; G\}\; \{\backslash \; partial\; x^2\}\; b^2\; \backslash \; left\; (X,\; T\; \backslash \; right)\; \backslash \; right)\; dt\; +\; \backslash \; frac\; \{\backslash \; partial\; G\}\; \{\backslash \; partial\; X\}\; b^2\; \backslash \; left\; (X,\; T\; \backslash \; right)\; dz$

Partial derivative equation of Black-Scholes

In discrete time, there is $\backslash \; delta\; S\; =\; \backslash \; driven\; S\; \backslash \; delta\; T\; +\; \backslash \; sigma\; S\; \backslash \; delta\; z$ and is $f$ dépndante of $S$ and $t$.

According to the lemma of Itô, one a:

$\backslash \; delta\; F\; =\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; S\}\; \backslash \; driven\; S\; +\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; T\}\; +\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash \; frac\; \{\backslash \; partial^2\; F\}\; \{\backslash \; partial\; S^2\}\; \backslash \; sigma^2\; S^2\; \backslash \; right)\; \backslash \; delta\; T\; +\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; S\}\; \backslash \; sigma\; S\; \backslash \; delta\; z$

One leaves the principle which one has a wallet provided with an action and one of his derivative products. One can thus neglect the process of Wiener. One buys $\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; S\}$ actions.

There is thus a wallet $\backslash \; pi\; =\; -\; F\; +\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; S\}\; S$

$\backslash \; Rightarrow\; \backslash \; delta\; \backslash \; pi\; =\; -\; \backslash \; delta\; F\; +\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; S\}\; \backslash \; S$ delta

One substitutes $\backslash \; delta\; f$ and $\backslash \; S$ delta in the preceding equation and one obtains:

$\backslash \; delta\; \backslash \; pi\; =\; \backslash \; left\; (-\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; T\}\; -\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash \; frac\; \{\backslash \; partial^2\; F\}\; \{\backslash \; partial\; S^2\}\; \backslash \; sigma^2\; S^2\; \backslash \; right)\; \backslash \; delta\; t$

We are in the case of a placement without risk with a possibility of short sale. One from of deduced:

$\backslash \; delta\; \backslash \; pi\; =\; R\; \backslash \; pi\; \backslash \; delta\; t$

One replaces $\backslash \; delta\; \backslash \; Pi$ and $\backslash \; Pi$ and while reorganizing, one arrives at the desired equation:

$\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; T\}\; +\; R\; S\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; S\}\; +\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash \; sigma^2\; S^2\; \backslash \; frac\; \{\backslash \; partial^2\; F\}\; \{\backslash \; partial\; S^2\}\; =\; R\; f$

Solution of the equation

Model of Black-Scholes

The formula of Black-Scholes can be shown rigorously if a certain number of conditions are established. One speaks then about model of Black-Scholes, or one says that one is in the case Black-Scholes. The financial markets correspond rather well to this model, but not exactly of course and, in particular, contrary to the central assumption of the model, time is not there continuous. There is thus a certain difference between this model and the reality, which can become important when the markets are agitated with frequent discontinuities of course.

The conditions of the model are the following ones:

1. the price of subjacent follows a Brownian Movement geometrical;
2. volatility is known in advance and is constant;
3. it is possible to constantly buy and sell the subjacent one and without expenses;
4. the short sales are authorized (where one borrows a certain quantity of subjacent to sell it);
5. it does not have there a Dividende;
6. interest rate is known in advance and is constant;
7. the exercise of the option can be done only at the expiration date, not before (option with European exercise, known as European option).

For an option known as American, the exercise of the option can be carried out before the expiration date.

The model of Black and Scholes in practice

The fundamental thesis of the model of Black and Scholes was that the price of the option to buy is indicated implicitly if the subjacent one is exchanged on the markets.

The use of the model and the Black-Scholes formula is very widespread on the financial markets, so much so that certain quotations rather give each other in level volatility than in absolute price. Indeed, the other parameters of the model (lasted at the limit, price of exercise, interest rate without risk and price of subjacent) are easily observable on the markets.

However, the model of Black and Scholes do not make it possible to model the real-world precisely. The experiment shows that actually volatility depends on the price of exercise and maturity.

In practice, the surface of volatility (implicit volatility according to the price of exercise and maturity) is not punt. Often, for a given maturity, implicit volatility compared to the price of exercise has a form to smile (called the spalling hammer of volatility ): with the currency, implicit volatility is lowest and more one moves away from the currency, plus it is high. It is noted in addition that the spalling hammer is often not symmetrical on the stock market: higher of with dimensions the than of with dimensions the cal could. That is due to the fact that the actors of market are more sensitive to the risk of fall than to the risk of rise of the action.

For a price of exercise given, the difference between implicit volatility observed and that with the currency are called the skew .

The surface of volatility of subjacent also evolves/moves in time. The actors of the market revalue it unceasingly, modifying their anticipation of the probability, for each price of exercise and maturity, that an option does not finish in the currency.

Extensions of the formula

The formula of price of option above is employed for the evaluation of European options on the actions not paying a Dividende S. the Black-Scholes model can be easily wide with the options on instruments paying of the dividends. For the options on indices (such as FTSE or CAC 40) where each company entering its calculation can pay a dividend one or twice a year, it is reasonable to suppose that the dividends are paid without interruption.

The payment of the dividends during one period of time $\backslash \; left\; T,\; t+\; \backslash \; delta\; T\; \backslash \; right$ is then noted:

$Q\; \backslash ,\; S\_t\; \backslash ,\; dt$

for a Q constant. Under this formulation the arbitration-free price according to the Black-Scholes model can be shown as being:

$C\; (S,\; T)\; =\; e^\; \{-\; qT\}\; S\_0\; NR\; (d\_1)\; -\; e^\; \{-\; rT\}\; kN\; (d\_2)\; \backslash ,$

$P\; (S,\; T)\; =e^\; \{-\; rT\}\; kN\; (-\; d\_2)\; -\; e^\; \{-\; qT\}\; S\_0\; NR\; (-\; d\_1)\; \backslash ,$

where now:

$F\; =\; e^\; \{(r-q)\; T\}\; S\_0\; \backslash ,$

is the price modified of before which occurs under the terms D ₁ and D ₂. This formula is generally known like Black-Scholes-Merton .

Exactly the same formula is employed to evaluate options on rates of foreign currencies, except that now Q takes the role of interest rate foreign without-risk and S immediate foreign exchange rate. It is the model of Garman-Kohlhagen (1983).

It is also possible to extend the Black-Scholes framework to the options on instruments paying of the discrete dividends. It is useful when the option is based on simple actions.

A typical model must suppose that a proportion $\backslash \; delta$ price (course) of actions paid like dividend at the predetermined dates $T\_1,\; T\_2\dots$.

The price of the actions is then modelled like: $S\_t\; =\; S\_0\; (1\; \backslash \; delta)\; ^\; \{N\; (T)\}e^\; \{\backslash \; W\_t\; sigma\; +\; \backslash \; driven\; T\}$

where N ( T ) is the number of dividends which were paid at time T .

The price of an option to buy on such actions is still:

$C\; (S\_0,\; K,\; R,\; T,\; Q,\; \backslash \; sigma)\; =\; FN\; (d\_1)\; -\; Ke^\; \{-\; rT\}\; NR\; (d\_2)\; \backslash ,$

$P\; (S\_0,\; K,\; R,\; T,\; Q,\; \backslash \; sigma)\; =\; Ke^\; \{-\; rT\}\; NR\; (-\; d\_2)\; -\; FN\; (-\; d\_1)\; \backslash ,$

where now:

$F\; =\; S\_0\; (1\; \backslash \; delta)\; ^\; \{N\; (T)\}e^\; \{rT\}\; \backslash ,$

is the price in advance of the actions paying of the dividend.

It is more difficult to evaluate American options, and a choice of the models is (for example) Whaley (binomial Modèle of options).

See too

• Mathematical financial
• Evaluation of option
• binomial Model, other models usually used to evaluate options
• Greek Lettres in financial mathematics, functions derived resulting from the formula from Black and Scholes
• Option (finance), Call, Could, Straddle, Warrant

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