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Optimal stopping problems can be found in areas of statistics, economics, and mathematical finance (related to the pricing of American options). A key example of an optimal stopping problem is the secretary problem.
Poisson process, an example of a jump process; Continuous-time Markov chain (CTMC), an example of a jump process and a generalization of the Poisson process; Counting process, an example of a jump process and a generalization of the Poisson process in a different direction than that of CTMCs; Interacting particle system, an example of a jump ...
A European call valued using the Black–Scholes pricing equation for varying asset price and time-to-expiry . In this particular example, the strike price is set to 1. The Black–Scholes formula calculates the price of European put and call options. This price is consistent with the Black–Scholes equation.
The call owner can exercise the option, putting up cash to buy the stock at the strike price. Or the owner can simply sell the option at its fair market value to another buyer before it expires.
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 Black–Scholes PDE. Once in this form, a ...
The Poisson random measure with intensity measure is a family of random variables {} defined on some probability space (,,) such that i) ∀ A ∈ A , N A {\displaystyle \forall A\in {\mathcal {A}},\quad N_{A}} is a Poisson random variable with rate μ ( A ) {\displaystyle \mu (A)} .
The solution of the PDE gives the value of the option at any earlier time, [{,}]. To solve the PDE we recognize that it is a Cauchy–Euler equation which can be transformed into a diffusion equation by introducing the change-of-variable transformation
In probability theory and statistics, the Poisson distribution (/ ˈ p w ɑː s ɒ n /; French pronunciation:) is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant mean rate and independently of the time since the last event. [1]