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From the parabolic 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 given the risk of the security and its expected return (instead replacing the ...
In mathematical finance, the Black–Scholes equation, also called the Black–Scholes–Merton equation, is a partial differential equation (PDE) governing the price evolution of derivatives under the Black–Scholes model. [1]
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 ) {\displaystyle \max(0,F(T)-K)} .
As in the Black–Scholes model for stock options and the Black model for certain interest rate options, the value of a European option on an FX rate is typically calculated by assuming that the rate follows a log-normal process. [3] The earliest currency options pricing model was published by Biger and Hull, (Financial Management, spring 1983).
The Black–Scholes formula (hereinafter, "BS Formula") provides an explicit equation for the value of a call option on a non-dividend paying stock. In case the stock pays one or more discrete dividend(s) no closed formula is known, but several approximations can be used, or else the Black–Scholes PDE will have to be solved numerically.
Itô's lemma can be used to derive the Black–Scholes equation for an option. [2] Suppose a stock price follows a geometric Brownian motion given by the stochastic differential equation dS = S(σdB + μ dt). Then, if the value of an option at time t is f(t, S t), Itô's lemma gives
The Greeks in the Black–Scholes model (a relatively simple idealised model of certain financial markets) are relatively easy to calculate — a desirable property of financial models — and are very useful for derivatives traders, especially those who seek to hedge their portfolios from adverse changes in market conditions. For this reason ...
While moneyness is a function of both spot and strike, usually one of these is fixed, and the other varies. Given a specific option, the strike is fixed, and different spots yield the moneyness of that option at different market prices; this is useful in option pricing and understanding the Black–Scholes formula.