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  2. Black–Scholes equation - Wikipedia

    en.wikipedia.org/wiki/BlackScholes_equation

    Black and Scholes' insight was that the portfolio represented by the right-hand side is riskless: thus the equation says that the riskless return over any infinitesimal time interval can be expressed as the sum of theta and a term incorporating gamma.

  3. Black–Scholes model - Wikipedia

    en.wikipedia.org/wiki/BlackScholes_model

    Further, the BlackScholes equation, a partial differential equation that governs the price of the option, enables pricing using numerical methods when an explicit formula is not possible. The BlackScholes formula has only one parameter that cannot be directly observed in the market: the average future volatility of the underlying asset ...

  4. Finite difference methods for option pricing - Wikipedia

    en.wikipedia.org/wiki/Finite_difference_methods...

    The discrete difference equations may then be solved iteratively to calculate a price for the option. [4] The approach arises since the evolution of the option value can be modelled via a partial differential equation (PDE), as a function of (at least) time and price of underlying; see for example the BlackScholes PDE. Once in this form, a ...

  5. Vanna–Volga pricing - Wikipedia

    en.wikipedia.org/wiki/Vanna–Volga_pricing

    It consists of adjusting the BlackScholes theoretical value (BSTV) by the cost of a portfolio which hedges three main risks associated to the volatility of the option: the Vega, the Vanna and the Volga. The Vanna is the sensitivity of the Vega with respect to a change in the spot FX rate:

  6. Monte Carlo methods for option pricing - Wikipedia

    en.wikipedia.org/wiki/Monte_Carlo_methods_for...

    For example, for bond options [3] the underlying is a bond, but the source of uncertainty is the annualized interest rate (i.e. the short rate). Here, for each randomly generated yield curve we observe a different resultant bond price on the option's exercise date; this bond price is then the input for the determination of the option's payoff.

  7. Short-rate model - Wikipedia

    en.wikipedia.org/wiki/Short-rate_model

    Tree returning the OAS (black vs red): the short rate is the top value; the development of the bond value shows pull-to-par clearly . A short-rate model, in the context of interest rate derivatives, is a mathematical model that describes the future evolution of interest rates by describing the future evolution of the short rate, usually written .

  8. Black's approximation - Wikipedia

    en.wikipedia.org/wiki/Black's_approximation

    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 BlackScholes formula (hereinafter, "BS Formula") provides an explicit equation for the value of a call option on a non-dividend paying stock. In ...

  9. Itô's lemma - Wikipedia

    en.wikipedia.org/wiki/Itô's_lemma

    Itô's lemma can be used to derive the BlackScholes 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