American option pricing under GARCH by a Markov chain approximation

Duan, Jin‐Chuan; Simonato, Jean‐Guy

doi:10.1016/s0165-1889(00)00003-8

Cited by 116 publications

(85 citation statements)

References 22 publications

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“…(3)- (5) in solving for the probabilities. Increasing n makes inequality (11) harder to satisfy for those states with a large h 2 t , whose existence has been confirmed earlier for GARCH.…”

Section: Inequalities (6) Implysupporting

confidence: 55%

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On accurate and provably efficient GARCH option pricing algorithms

Lyuu

2005

Quantitative Finance

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The trinomial-tree GARCH option pricing algorithms of Ritchken and Trevor (1999) and Cakici and Topyan (2000) are claimed to be efficient and accurate. However, this thesis proves that both algorithms generate trees that explode exponentially when the number of partitions per day, n, exceeds a typically small number determined by the model parameters. Worse, when explosion happens, the tree cannot grow beyond a certain maturity, making it useless for pricing derivatives with a longer maturity. Meanwhile, a small n has accuracy problems and does not prevent explosion. This thesis then presents a trinomial-tree GARCH option pricing algorithm that solves the above problems.The algorithm provably does not have the short-maturity problem. Furthermore, the tree size is guaranteed to be quadratic if n is less than a threshold easily determined by the model parameters. This result for the first time puts a tree-based GARCH option pricing algorithm in the same complexity class as binomial and trinomial trees under the Black-Scholes model. Extensive numerical evaluation is conducted to confirm the analytical results and the accuracy of the algorithm.

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“…(3)- (5) in solving for the probabilities. Increasing n makes inequality (11) harder to satisfy for those states with a large h 2 t , whose existence has been confirmed earlier for GARCH.…”

Section: Inequalities (6) Implysupporting

confidence: 55%

“…Other GARCH option pricing techniques include the Markov chain approximation of Duan and Simonato (2001), the Edgeworth tree approximation of Duan et al (2002), and analytical approximations as in Heston and Nandi (2000).…”

Section: Introductionmentioning

confidence: 99%

On accurate and provably efficient GARCH option pricing algorithms

Lyuu

2005

Quantitative Finance

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“…Though these works assume that the log return process log(S t /S t−1 ) follows some type of discrete-time Markov or semi-Markov process, the tree models that are proposed differ from the MT model in one important regard: starting from any vertex of the tree, the magnitudes of the up and down jumps are al-ways the same. The same is true in models where a Markov chain is used to approximate the true underlying process-see [5], for instance. In the MT model, if we start from a vertex such that the jump leading to that vertex was an upward jump, then we have different up/down magnitudes as compared with a vertex such that the jump leading to that vertex was a downward jump.…”

Section: Past Workmentioning

confidence: 90%

Markov Tree Options Pricing

Bhat

Kumar²

2010

Proceedings of the 2009 SIAM Conference on “Mathematics for Industry”

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This paper questions one of the fundamental assumptions made in options pricing: that the daily returns of a stock are independent and identically distributed (IID). We apply an estimation procedure to years of daily return data for all stocks in the French CAC-40 index. We find six stocks whose log returns are best modeled by a first-order Markov chain, not an IID sequence. We further propose the Markov tree (MT) model, a modification of the standard binomial options pricing model, that takes into account this first-order Markov behavior. Empirical tests reveal that, for the six stocks found earlier, the MT model's option prices agree very closely with market prices. IntroductionIn the Black-Scholes model for the price of a European option, one of the main assumptions is that the price of the underlying asset follows a geometric Brownian motion [8]. If S t is the underlying asset price at time t, one assumes dS t = μS t dt + σS t dW t , where μ and σ are constants and W t is a Brownian motion. For fixed t > 0, define X n = log(S (n+1)t /S nt ). Then

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“…Before option valuation in the context of the two aforementioned Lévy processes, we will propose how to construct a homogeneous Markov chain in discrete time. Duan and Simonato (2001) has already argued that this chain can converge to the true density of a stochastic process as the number of states increases over the discrete time points. In order to make a Markov chain under the Lévy processes of our interest, let us at first define a column vector whose elements are n logarithmic stock prices possible in the future:…”

Section: Construction Of a Markovmentioning

confidence: 97%

“…, a scaling factor, is an increasing function of n, but should satisfy some mild partition conditions that p(n) < n for all positive integer n (Duan and Simonato, 2001). The transition probability matrix (M) describes the transition probabilities from one state (an underlying asset price) to the other over a discrete unit time.…”

Section: Construction Of a Markovmentioning

confidence: 99%

Valuation of European and American Option Prices Under the Levy Processes with a Markov Chain Approximation

Han

2013

Management Science and Financial Engineering

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This paper suggests a numerical method for valuation of European and American options under the two Lévy Processes, Normal Inverse Gaussian Model and the Variance Gamma model. The method is based on approximation of underlying asset price using a finite-state, time-homogeneous Markov chain. We examine the effectiveness of the proposed method with simulation results, which are compared with those from the existing numerical method, the latticebased method.

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American option pricing under GARCH by a Markov chain approximation

Cited by 116 publications

References 22 publications

On accurate and provably efficient GARCH option pricing algorithms

On accurate and provably efficient GARCH option pricing algorithms

Markov Tree Options Pricing

Valuation of European and American Option Prices Under the Levy Processes with a Markov Chain Approximation

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