Pricing american options under discrete and continuous time setting

Download
index.pdf
2013
Kozpınar, Sinem
Show full item record
260
views
162
downloads
In this thesis, pricing of American options are analyzed in discrete and continuous time markets. We first discuss the discrete-time valuation of American options assuming that the underlying asset pays no dividend during the life of the option. In this setting, we uniquely price American options by introducing the Snell envelope and optimal stopping time problem. We prove the main results studied in Lamberton and Lapeyre (1996) in details. In addition, we show that the price of an American call option with no dividend is equal to the price of its European counterpart. Then, we extend this results to the continuous-time for both dividend and no dividend case. Following Black-Scholes model, we present two different techniques to price American options: martingale pricing technique and variational inequalities. Under martingale pricing approach, we take the expectation of discounted payoff process and determine the stopping time that maximizes this expected value. Then, we derive a pricing formula for both dividend and no dividend case. We also show that an early exercise is not optimal for American call options without dividend. We observed that this rule is not valid for American call options on a dividend paying underlying asset. Then, we introduce the variational inequalities that an American option satisfies and investigate the regular solutions of this inequalities. However, these approaches generally do not admit a closed-form solution for the price of American options. Therefore, we give a brief introduction to the finite difference and PSOR methods and adapt these methods for American options. Finally, a numerical application is done by comparing the e fficiencies of these methods. Moreover, the impact of Black-Scholes parameters strike price, volatility and dividend yield on the price of American options are also investigated. The thesis ends with a conclusion and an outlook to future studies.
Stochastic analysis., Stochastic differential equations., Time-domain analysis., Malliavin calculus.
http://etd.lib.metu.edu.tr/upload/12616424/index.pdf
https://hdl.handle.net/11511/23095
Graduate School of Applied Mathematics, Thesis

Suggestions

Pricing European and American options under Heston model using discontinuous Galerkin finite elements
Kozpınar, Sinem; Karasözen, Bülent (2020-11-01)
This paper deals with pricing of European and American options, when the underlying asset price follows Heston model, via the interior penalty discontinuous Galerkin finite element method (dGFEM). The advantages of dGFEM space discretization with Rannacher smoothing as time integrator with nonsmooth initial and boundary conditions are illustrated for European vanilla options, digital call and American put options. The convection dominated Heston model for vanishing volatility is efficiently solved utilizing...
Computation of the greeks in black-scholes-merton and stochastic volatility models using malliavin calculus
Yılmaz, Bilgi; Yolcu Okur, Yeliz; Department of Financial Mathematics (2014)
The computation of the Greeks of options is an essential aspect of financial mathematics. The investors use the information gained from this aspect for hedging purposes or to decide whether to invest in an option or not. However, computation of the Greeks is not straightforward in some cases due to technical difficulties. For instance, the value function of some options are complicated or moreover in some cases they might not have a closed form solution which makes the computation of their Greeks cumbersome...
On the methods of pricing American options: case study
AYDOGAN, Burcu; Aksoy, Umit; Uğur, Ömür (2018-01-01)
In this study, a comparative analysis of numerical and approximation methods for pricing American options is performed. Binomial and finite difference approximations are discussed; furthermore, Roll-Geske-Whaley, Barone-Adesi and Whaley and Bjerksund-Stensland analytical approximations as well as the least-squares Monte Carlo method of Longstaff and Schwartz are presented. Applicability and efficiency in almost all circumstances, numerical solutions of the corresponding free boundary problem is emphasized. ...
Application of stochastic volatility models with jumps to BIST options
Rahiminejat, Monireh; Sezer, Ali Devin; Department of Financial Mathematics (2017)
This thesis gives a derivation of call and put option pricing formulas under stochastic volatility models with jumps; the precise model is a combination of Merton and Heston models. The derivation is based on the computation of the characteristic function of the underlying process. We use the derived formulas to fit the model to options written on two stocks in the BIST30 index covering the first two months of 2017. The fit is done by minimizing a weighted $L_2$ distance between the observed prices and the ...
Computation of the Delta of European options under stochastic volatility models
Yolcu Okur, Yeliz; Sayer, Tilman; Yılmaz, Bilgi; Inkaya, B. Alper (2018-06-01)
We employ Malliavin calculus techniques to compute the Delta of European type options in the presence of stochastic volatility. We obtain a general formula for the Malliavin weight and apply the derived formula to the well known models of Stein-Stein and Heston in order to show the numerical accuracy and efficiency of our approach.
Citation Formats
S. Kozpınar, “Pricing american options under discrete and continuous time setting,” M.S. - Master of Science, Middle East Technical University, 2013.
METU IIS - Integrated Information Service open@metu.edu.tr