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- Finite difference methods for option pricing - Wikipedia
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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 derivativessuch as Asian options.
In other cases, the source of uncertainty may be at a remove. For example, for bond options  the underlying is a bondbut the source of uncertainty is the annualized interest rate i. 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.
Method[ edit ] As above, the PDE is expressed in a discretized form, using finite differencesand the evolution in the option price is then modelled using a lattice with corresponding dimensions : time runs from 0 to maturity; and price runs from 0 to a "high" value, such that the option is deeply in or out of the money. The option is then valued as follows:  Maturity values are simply the difference between the exercise price of the option and the value of the underlying at each point. Values at the boundaries - i. Here, using a technique such as Crank—Nicolson or the explicit method : the PDE is discretized per the technique chosen, such that the value at each lattice point is specified as a function of the value at later and adjacent points; see Stencil numerical analysis ; the value at each point is then found using the technique in question.
The same approach is used in valuing swaptions where the value of the underlying swap is also a function of the evolving interest rate. Whereas these options are more commonly valued using lattice based modelsas above, for path dependent interest rate derivatives — such as CMOs — simulation is the primary technique employed.
Monte Carlo Methods allow for a compounding in the uncertainty.
In all such models, correlation between the underlying sources of risk is also incorporated; see Cholesky decomposition Monte Carlo simulation. Further complications, such as the impact of commodity prices or inflation on the underlying, can also be introduced.
Since simulation can accommodate complex problems of this sort, it is often used in analysing real options  where management's decision at any point is a function of multiple underlying variables. Simulation can similarly be used to value options where the payoff depends on the value of multiple underlying assets  such as a Basket option or Rainbow option.
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Here, correlation between asset returns is likewise metode de stabilire a prețurilor opțiunilor. Additionally, the stochastic process of the underlying s may be specified so as to exhibit jumps or mean reversion or both; this feature makes simulation the primary valuation method applicable to energy derivatives.
For example, in models incorporating stochastic volatilitythe volatility of the underlying changes with time; see Heston model. Bermudan or American options.
Monte Carlo methods for option pricing
It was first introduced by Jacques Carriere in The value is defined as the least squares regression against market price of the option value at that state and time -step. Option value for this regression is defined as the value of exercise possibilities dependent on market price plus the value of the timestep value which that exercise would result in defined in the previous step of the process.
This second step can be done with multiple price paths to add a stochastic effect to the procedure. The technique is thus widely used in valuing path dependent structures like lookback- and Asian options  and in real options analysis.
They are, in a sense, a method of last resort;  see further under Monte Carlo methods in finance.
De-a lungul istoriei, prețurile au fost stabilite de către cumpărători și vânzători în urmă negocierilor care au avut loc între ei.
With faster computing capability this computational constraint is less of a concern.