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Information and Option Pricings
 Journal of Quantitative Finance
, 2001
"... How can one relate stock fluctuations and informationbased human activities ? We present a model of an incomplete market by adjoining the BlackScholes exponential Brownian motion model for stock fluctuations with a hidden Markov process, which represents the state of information in the investors' ..."
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Cited by 15 (3 self)
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How can one relate stock fluctuations and informationbased human activities ? We present a model of an incomplete market by adjoining the BlackScholes exponential Brownian motion model for stock fluctuations with a hidden Markov process, which represents the state of information in the investors' community. The drift and volatility parameters take different values depending on the state of this hidden Markov process. Standard option pricing procedure under this model becomes problematic. Yet, with an additional economic assumption, we provide an explicit closedform formula for the arbitragefree price of the European call option. Our model can be discretized via a Skorohod imbedding technique. We conclude with an example of a simulation of IBM stock, which shows that, not surprisingly, information does affect the market. AMS classification: 60J65; 60J10; Tel: (914) 945 2348; Fax: (914) 945 3434; Email: xinguo@us.ibm.com. 1 Keywords: BlackScholes; Hidden Markov processes; Inside information; Arbitrage; Equivalent martingale measure 1 Motivation We begin our discussion with the classic case of BreX, a Canadian gold mining company. The mineral company stumbled on what looked like a huge gold cache in Indonesia. Consequently, the stock price skyrocketed for a while. Then, all of a sudden, the volatility increased by orders of magnitude due to heavy inside tradings. The reason turned out to be that a privileged few were aware of the fraudulent gold assays performed by the company. The honeymoon was over and the stocks crashed when this news became public (cf. Figure 1, New York Times, May 5, 1997). BreX perished. Figure 1: The rise and fall of BreX. (Source: New York Times, May 5, 1997.) One of the morals of the story is: volatility increased when there was in...
Closedform solutions for perpetual American Put options with regime switching
 SIAM Journal on Applied Mathematics
"... Abstract. This paper studies an optimal stopping time problem for pricing perpetual American put options in a regime switching model. An explicit optimal stopping rule and the corresponding value function in a closed form are obtained using the “modified smooth fit ” technique. The solution is then ..."
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Cited by 11 (0 self)
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Abstract. This paper studies an optimal stopping time problem for pricing perpetual American put options in a regime switching model. An explicit optimal stopping rule and the corresponding value function in a closed form are obtained using the “modified smooth fit ” technique. The solution is then compared with the numerical results obtained via a dynamic programming approach and also with a twopoint boundaryvalue differential equation (TPBVDE) method. Key words. Markov chain, optimal stopping time, American options, regime switching, modified smooth fit principle AMS subject classifications. 90A09, 60J27 DOI. 10.1137/S0036139903426083 1. Introduction. Given a probability space (Ω, F,P), consider a process X(t) which satisfies (in a strong sense) a stochastic differential equation of the following form: (1) dX(t)=X(t)µ ɛ(t)dt + X(t)σ ɛ(t)dW(t), X(0) = x,
Some Optimal Stopping Problems With NonTrivial Boundaries for Pricing Exotic Options
 J. Appl. Probab
, 2001
"... We solve the following three optimal stopping problems for dierent kinds of options, based on the BlackScholes model of stock uctuations: (i) The perpetual lookback American option for the running maximum of the stock price during the life of the option. This problem is more dicult than the closely ..."
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Cited by 7 (1 self)
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We solve the following three optimal stopping problems for dierent kinds of options, based on the BlackScholes model of stock uctuations: (i) The perpetual lookback American option for the running maximum of the stock price during the life of the option. This problem is more dicult than the closely related one for the Russian option and we show that for a class of utility functions the free boundary is governed by a nonlinear ordinary dierential equation. (ii) A new type of stock option for a company, where the company provides a guaranteed minimum as an added incentive in case the market appreciation of the stock is low, thereby making the option more attractive to the employee. We show that the value of this option is given by solving a nonalgebraic equation. (iii) A new call option for the option buyer who is riskaverse and gets to choose, a priori, a xed constant l as a \hedge" on a possible downturn of the stock price, where the buyer gets the maximum of l and the price at ...
A RegimeSwitching Model for European Option Pricing
 STOCHASTIC PROCESSES, OPTIMIZATION, AND CONTROL THEORY APPLICATIONS IN FINANCIAL ENGINEEARING, QUEUEING NETWORKS, AND MANUFACTURING
, 2006
"... We study the pricing of Europeanstyle options, with the rate of return and the volatility of the underlying asset depending on the market mode or regime that switches among a finite number of states. This regimeswitching model is formulated as a geometric Brownian motion modulated by a finites ..."
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Cited by 7 (1 self)
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We study the pricing of Europeanstyle options, with the rate of return and the volatility of the underlying asset depending on the market mode or regime that switches among a finite number of states. This regimeswitching model is formulated as a geometric Brownian motion modulated by a finitestate Markov chain. With a Girsanovlike change of measure, we derive the option price using riskneutral valuation, along with a system of partial differential equations that govern the option price, with smoothed boundary conditions. We also develop a numerical approach to compute the pricing formula, using a successive approximation scheme with a geometric rate of convergence. Numerical examples demonstrate the presence of the volatility smile and volatility term structure with a simple, twostate regimeswitching model.
On perpetual American put valuation and firstpassage in a regimeswitching model with jumps,” Finance and Stochastics
, 2008
"... In this paper we consider the problem of pricing a perpetual American put option in an exponential regimeswitching Lévy model. For the case of the (dense) class of phasetype jumps and finitely many regimes we derive an explicit expression for the value function. The solution of the corresponding f ..."
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Cited by 7 (3 self)
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In this paper we consider the problem of pricing a perpetual American put option in an exponential regimeswitching Lévy model. For the case of the (dense) class of phasetype jumps and finitely many regimes we derive an explicit expression for the value function. The solution of the corresponding first passage problem under a statedependent level rests on a path transformation and a new matrix WienerHopf factorization result for this class of processes.
Applied Stochastic Processes, 2003, 554, Rm 552, Mon 6:209PM
"... d the book and do the problems you will have mastered all that you will need to get a good grade. An important class of stochastic processes, or "random functions", X(t), satisfy E[X(t + s)jX(t)] = X(t) (1) which is called the martingale hypothesis; we say that the stochastic process is a martinga ..."
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d the book and do the problems you will have mastered all that you will need to get a good grade. An important class of stochastic processes, or "random functions", X(t), satisfy E[X(t + s)jX(t)] = X(t) (1) which is called the martingale hypothesis; we say that the stochastic process is a martingale. The reason people believe that stock prices follow the martingale law (1) is rather deep: people believe that all knowledge of the future has already been incorporated into the price of the stock so that the price is a "martingale" and no trends remain. That is, the expected value of the price at any time in the future is simply the present value. They believe that arbitrageurs have already correctly found the right value and they have removed all profits; this assumes the stock market is allknowing and wise and there is no way to obtain any additional information. To make the notion of martingale more precise we need to know some more probability theory. A random variable is a funct
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"... A BlackScholes market is considered in which the underlying economy, as modelled by the parameters and volatility of the processes, switches between a finite number of states. The switching is modelled by a hidden Markov chain. European options are priced and a BlackScholes equation obtained. The ..."
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A BlackScholes market is considered in which the underlying economy, as modelled by the parameters and volatility of the processes, switches between a finite number of states. The switching is modelled by a hidden Markov chain. European options are priced and a BlackScholes equation obtained. The approximate valuation of American options due to BaroneAdesi and Whaley is extended to this setting.