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In certain cases, it is possible to solve for an exact formula, such as in the case of a European call, which was done by Black and Scholes. The solution is conceptually simple. Since in the Black–Scholes model, the underlying stock price follows a geometric Brownian motion, the distribution of , conditional on its price at time , is a log ...
Black–Scholes cannot be applied directly to bond securities because of pull-to-par. As the bond reaches its maturity date, all of the prices involved with the bond become known, thereby decreasing its volatility, and the simple Black–Scholes model does not reflect this process.
This section outlines moneyness measures from simple but less useful to more complex but more useful. [6] Simpler measures of moneyness can be computed immediately from observable market data without any theoretical assumptions, while more complex measures use the implied volatility, and thus the Black–Scholes model.
Since the underlying random process is the same, for enough price paths, the value of a european option here should be the same as under Black–Scholes. More generally though, simulation is employed for path dependent exotic derivatives, such as Asian options. In other cases, the source of uncertainty may be at a remove.
In finance, Black's approximation is an approximate method for computing the value of an American call option on a stock paying a single dividend. It was described by Fischer Black in 1975. [1] 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 ...
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.
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In the Black–Scholes model, the theoretical value of a vanilla option is a monotonic increasing function of the volatility of the underlying asset. This means it is usually possible to compute a unique implied volatility from a given market price for an option. This implied volatility is best regarded as a rescaling of option prices which ...