Geometric Brownian motion is the most widely used stochastic process in financial modeling. Its properties are buildingblocks for modelling a wide variety of exotic options and other derivative contract.
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| The Binomial tree is a discretized description of geometric Brownian motion which is often used to describe asset behavior. The structure is a recombining tree where the asset S can move either up or down. full text » |
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| This is the Jarrow and Rudd version of the Binomial tree. The Binomial tree is a discretized description of geometric Brownian motion which is often used to describe asset behavior. The structure is a recombining tree where the asset S can move either up or down. full text » |
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| This is the Tian version of the Binomial tree. The Binomial tree is a discretized description of geometric Brownian motion which is often used to describe asset behavior. The structure is a recombining tree where the asset S can move either up or down. full text » |
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| This is the Trigeorgis version of the Binomial tree. The Binomial tree is a discretized description of geometric Brownian motion which is often used to describe asset behavior. The structure is a recombining tree where the asset S can move either up or down. full text » |
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| These equations describe the correlation in time of geometric Brownian motion.
When looking at a two moments in time (''t'' and ''T'') of a specific price-path we can observe two basic facts when comparing the paths between now and ''t'' and ''T'' respectively.
* the part up to ''t'' of the two paths are identical
* the part after ''t'' is uncorrelated with the part up to ''t''
Using these two fact, the results are easy to verify.
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| This equation calculates the expected value of a geometric Brownian motion raised to the power of p of the part of the geometric Brownian motion that is below K at time T. This equation is used in "Pricing and Hedging Power Options" by Ronald C. Heynen and Harry M. Kat. full text » |
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| This equation calculates the expected value of a geometric Brownian motion raised to the power of p of the part of the geometric Brownian motion that is above K at time T. This equation is used in "Pricing and Hedging Power Options" by Ronald C. Heynen and Harry M. Kat. full text » |
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| The geometric average in time of geometric Brownian motion is lognormal distributed.
These equations express the two parameters of the lognormal distribution as a function of the continuosly sampled geometric average. Averaging starts at ''t'' and stops at ''T''.
An application is the Vorst model for Asian options. In this model the arithmetic average of the Asian options is approximated with a geometric average.
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| The geometric average in time of geometric Brownian motion is lognormal distributed.
These equations express the two parameters of the lognormal distribution as a function of the discretely sampled geometric average. The first averaging point is ''t1'', the last ''tn'', and the total number of averaging points is ''n''.
An application is the Vorst model for Asian options. In this model the arithmetic average of the Asian options is approximated with a geometric average.
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| The Geometric Brownian describes the most widely used model in finance. It is used to simulate the stochastic behaviour of stocks, currencies, futures.
The value of this process is strick positive, St cannot get below zero. full text » |
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| The probability that the high and the low of an underlying are within the range [L,H]. The underlying behavior is geometric Brownian motion with a yield (drift) \mu, volatility \sigma, and has an initial value of S_0 full text » |
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| Probability density of geometric Brownian motion hitting a barrier for the first time at T. full text » |
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| This equation gives the probability density function of an underlying S at some future time t. The underlying behavior is geometric Brownian motion, has a present value S0, a yield (drift) of \mu, and volatility \sigma.
The probability density function is has a lognormal distribution with mean \mu. full text » |
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| Cumulative probability function of the high of an underlying between now and some future time T. The underlying behavior is geometric Brownian motion. full text » |
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| Cumulative probability function of the high of an underlying between now and some future time t. The underlying behavior is geometric Brownian motion. full text » |
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| Simulating geometric Brownian motion. This equation is the exact solution of the geometrix brownian motion SDE. full text » |
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| Simulating geometric Brownian motion with a cash dividend before T. full text » |
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| Simulating geometric Brownian motion with a stock dividend. full text » |
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| Simulating geometric Brownian motion with multiple cash dividend in the simulation step to T. full text » |
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| The Trinomial tree is a discretized description of geometric Brownian motion which is often used to describe asset behavior. The structure is a recombining tree where the asset S can move up, mid or down. full text » |
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