Stochastic models in finance
- Authors: Mazengera, Hassan
- Date: 2017
- Subjects: Finance -- Mathematical models , C++ (Computer program language) , GARCH model , Lebesgue-Radon-Nikodym theorems , Radon measures , Stochastic models , Stochastic processes , Stochastic processes -- Computer programs , Martingales (Mathematics) , Pricing -- Mathematical models
- Language: English
- Type: text , Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/162724 , vital:40976
- Description: Stochastic models for pricing financial securities are developed. First, we consider the Black Scholes model, which is a classic example of a complete market model and finally focus on Lévy driven models. Jumps may render the market incomplete and are induced in a model by inclusion of a Poisson process. Lévy driven models are more realistic in modelling of asset price dynamics than the Black Scholes model. Martingales are central in pricing, especially of derivatives and we give them the desired attention in the context of pricing. There are an increasing number of important pricing models where analytical solutions are not available hence computational methods come in handy, see Broadie and Glasserman (1997). It is also important to note that computational methods are also applicable to models with analytical solutions. We computationally value selected stochastic financial models using C++. Computational methods are also used to value or price complex financial instruments such as path dependent derivatives. This pricing procedure is applied in the computational valuation of a stochastic (revenue based) loan contract. Derivatives with simple pay of functions and models with analytical solutions are considered for illustrative purposes. The Black-Scholes P.D.E is complex to solve analytically and finite difference methods are widely used. Explicit finite difference scheme is considered in this thesis for computational valuation of derivatives that are modelled by the Black-Scholes P.D.E. Stochastic modelling of asset prices is important for the valuation of derivatives: Gaussian, exponential and gamma variates are simulated for the valuation purposes.
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- Date Issued: 2017
Analytic pricing of American put options
- Authors: Glover, Elistan Nicholas
- Date: 2009
- Subjects: Options (Finance) -- Prices -- Mathematical models , Derivative securities -- Prices -- Mathematical models , Finance -- Mathematical models , Martingales (Mathematics)
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5566 , http://hdl.handle.net/10962/d1002804 , Options (Finance) -- Prices -- Mathematical models , Derivative securities -- Prices -- Mathematical models , Finance -- Mathematical models , Martingales (Mathematics)
- Description: American options are the most commonly traded financial derivatives in the market. Pricing these options fairly, so as to avoid arbitrage, is of paramount importance. Closed form solutions for American put options cannot be utilised in practice and so numerical techniques are employed. This thesis looks at the work done by other researchers to find an analytic solution to the American put option pricing problem and suggests a practical method, that uses Monte Carlo simulation, to approximate the American put option price. The theory behind option pricing is first discussed using a discrete model. Once the concepts of arbitrage-free pricing and hedging have been dealt with, this model is extended to a continuous-time setting. Martingale theory is introduced to put the option pricing theory in a more formal framework. The construction of a hedging portfolio is discussed in detail and it is shown how financial derivatives are priced according to a unique riskneutral probability measure. Black-Scholes model is discussed and utilised to find closed form solutions to European style options. American options are discussed in detail and it is shown that under certain conditions, American style options can be solved according to closed form solutions. Various numerical techniques are presented to approximate the true American put option price. Chief among these methods is the Richardson extrapolation on a sequence of Bermudan options method that was developed by Geske and Johnson. This model is extended to a Repeated-Richardson extrapolation technique. Finally, a Monte Carlo simulation is used to approximate Bermudan put options. These values are then extrapolated to approximate the price of an American put option. The use of extrapolation techniques was hampered by the presence of non-uniform convergence of the Bermudan put option sequence. When convergence was uniform, the approximations were accurate up to a few cents difference.
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- Date Issued: 2009