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81
Randomization and the American Put
 The Review of Financial Studies
, 1998
"... Conference. In particular, I am grateful to an unknown RFS referee, Kerry Back, Michael Brennan, Darrell Du e, ..."
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Cited by 105 (1 self)
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Conference. In particular, I am grateful to an unknown RFS referee, Kerry Back, Michael Brennan, Darrell Du e,
Pricing and Hedging American Options: A Recursive Integration Method
 Review of Financial Studies
, 1996
"... In this paper, we present a new method for pricing and hedging American options along with an efficient implementation procedure. The proposed method is efficient and accurate in computing both option values and various option hedge parameters. We demonstrate the computational accuracy and efficienc ..."
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Cited by 57 (3 self)
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In this paper, we present a new method for pricing and hedging American options along with an efficient implementation procedure. The proposed method is efficient and accurate in computing both option values and various option hedge parameters. We demonstrate the computational accuracy and efficiency of this numerical procedure in relation to other competing approaches. We also suggest how the method can be applied to the case of any American option for which a closedform solution exists for the corresponding European option. A variety of financial products such as fixedincome derivatives, mortgagebacked securities and corporate securities have earlyexercise or Americanstyle features that significantly influence their valuation and hedging. Considerable interest exists, therefore, in both academic and practitioner circles, in methods of valuation and hedging Americanstyle options that are conceptually sound, as well as efficient in their implementation. It has been recognized early in the ...
Continuoustime methods in finance: A review and an assessment
 Journal of Finance
, 2000
"... I survey and assess the development of continuoustime methods in finance during the last 30 years. The subperiod 1969 to 1980 saw a dizzying pace of development with seminal ideas in derivatives securities pricing, term structure theory, asset pricing, and optimal consumption and portfolio choices. ..."
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Cited by 52 (0 self)
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I survey and assess the development of continuoustime methods in finance during the last 30 years. The subperiod 1969 to 1980 saw a dizzying pace of development with seminal ideas in derivatives securities pricing, term structure theory, asset pricing, and optimal consumption and portfolio choices. During the period 1981 to 1999 the theory has been extended and modified to better explain empirical regularities in various subfields of finance. This latter subperiod has seen significant progress in econometric theory, computational and estimation methods to test and implement continuoustime models. Capital market frictions and bargaining issues are being increasingly incorporated in continuoustime theory. THE ROOTS OF MODERN CONTINUOUSTIME METHODS in finance can be traced back to the seminal contributions of Merton ~1969, 1971, 1973b! in the late 1960s and early 1970s. Merton ~1969! pioneered the use of continuoustime modeling in financial economics by formulating the intertemporal consumption and portfolio choice problem of an investor in a stochastic dynamic programming setting.
Pricing American Options Under Variance
 Gamma, J. Comp. Finance
"... We derive a form of the partial integrodifferential equation (PIDE) for pricing American options under variance gamma (VG) process. We then develop a numerical algorithm to solve for values of American options under variance gamma model. In this study, we compare the exercise boundary and early exe ..."
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Cited by 47 (2 self)
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We derive a form of the partial integrodifferential equation (PIDE) for pricing American options under variance gamma (VG) process. We then develop a numerical algorithm to solve for values of American options under variance gamma model. In this study, we compare the exercise boundary and early exercise premia between geometric VG law and geometric Brownian motion (GBM). We find that GBM premia are understated and hence we conclude that further work is necessary in developing fast efficient algorithms for solving PIDE’s with a view to calibrating stochastic processes to a surface of American option prices. 1
On the American option problem
 Math. Finance
, 2005
"... We show how the changeofvariable formula with local time on curves derived recently in [17] can be used to prove that the optimal stopping boundary for the American put option can be characterized as the unique solution of a nonlinear integral equation arising from the early exercise premium repre ..."
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Cited by 39 (9 self)
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We show how the changeofvariable formula with local time on curves derived recently in [17] can be used to prove that the optimal stopping boundary for the American put option can be characterized as the unique solution of a nonlinear integral equation arising from the early exercise premium representation. This settles the question raised in [15] (dating back to [13]). 1.
Martingale approach to pricing perpetual American options on two stocks
 Mathematical Finance
, 1996
"... ABSTRACT The method of Esscher transforms is a tool for valuing options on a stock, if the logarithm of the stock price is governed by a stochastic process with stationary and independent increments. The price of a derivative security is calculated as the expectation, with respect to the riskneutr ..."
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Cited by 35 (3 self)
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ABSTRACT The method of Esscher transforms is a tool for valuing options on a stock, if the logarithm of the stock price is governed by a stochastic process with stationary and independent increments. The price of a derivative security is calculated as the expectation, with respect to the riskneutral Esscher measure, of the discounted payoffs. Applying the optional sampling theorem we derive a simple, yet general formula for the price of a perpetual American put option on a stock whose downward movements are skipfree. Similarly, we obtain a formula for the price of a perpetual American call option on a stock whose upward movements are skipfree. Under the classical assumption, that the stock price is a geometric Brownian motion, the general perpetual American contingent claim is analysed, and formulas for the perpetual downandout call option and Russian option are obtained. The martingale approach avoids the use of differential equations and provides additional insight. We also explain the relationship between Samuelson's high contact condition and the first order condition for optimality.
A mathematical analysis for the optimal exercise boundary of American put options, preprint February 8
"... Abstract. We study a free boundary problem arising from American put options. In particular we prove existence and uniqueness for this problem and we derive, and prove rigorously, high order asymptotic expansions for the early exercise boundary near expiry. We provide four approximations for the bou ..."
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Cited by 34 (2 self)
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Abstract. We study a free boundary problem arising from American put options. In particular we prove existence and uniqueness for this problem and we derive, and prove rigorously, high order asymptotic expansions for the early exercise boundary near expiry. We provide four approximations for the boundary: one is explicit and is valid near expiry (weeks); two others are implicit involving inverse functions and are accurate for longer time to expiry (months); the fourth is an ODE initial value problem which is very accurate for all times to expiry, is extremely stable, and hence can be solve instantaneously on any computer. We further provide an ode iterative scheme which can reach its numerical fixed point in five iterations for all time to expiry. We also provide a large time (equivalent to regular expiration times but large interest rate and/or volatility) behavior of the exercise boundary. To demonstrate the accuracy of our approximations, we present the results of a numerical simulation. 1.
An Iteration Procedure for Solving Integral Equations Related to Optimal Stopping Problems
, 2006
"... ..."
The trap of complacency in predicting the maximum
 Ann. Probab
, 2007
"... Given a standard Brownian motion B µ = (B µ t)0≤t≤T with drift µ ∈ R and letting S µ t = max0≤s≤t B µ s for 0 ≤ t ≤ T, we consider the optimal prediction problem: V = inf E(B ..."
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Cited by 28 (16 self)
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Given a standard Brownian motion B µ = (B µ t)0≤t≤T with drift µ ∈ R and letting S µ t = max0≤s≤t B µ s for 0 ≤ t ≤ T, we consider the optimal prediction problem: V = inf E(B
Optimal Exercise of Executive Stock Options and Implications for Firm Cost
, 2010
"... This paper conducts a comprehensive study of the optimal exercise policy for an executive stock option and its implications for option cost, average life, and alternative valuation concepts. The paper is the first to provide analytical results for an executive with general concave utility. Wealthier ..."
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Cited by 23 (3 self)
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This paper conducts a comprehensive study of the optimal exercise policy for an executive stock option and its implications for option cost, average life, and alternative valuation concepts. The paper is the first to provide analytical results for an executive with general concave utility. Wealthier or less riskaverse executives exercise later and create greater option cost. However, option cost can decline with volatility. We show when there exists a single exercise boundary, yet demonstrate the possibility of a split continuation region. We also show that, for CRRA utility, the option cost does not converge to the BlackScholes value as the correlation between the stock and the market portfolio converges to one. We compare our model’s option cost with the modified BlackScholes approximation typically used in practice, and show that the