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66
Numerical Valuation of High Dimensional Multivariate American Securities
, 1994
"... We consider the problem of pricing an American contingent claim whose payoff depends on several sources of uncertainty. Using classical assumptions from the Arbitrage Pricing Theory, the theoretical price can be computed as the maximum over all possible early exercise strategies of the discounted ..."
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Cited by 95 (0 self)
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We consider the problem of pricing an American contingent claim whose payoff depends on several sources of uncertainty. Using classical assumptions from the Arbitrage Pricing Theory, the theoretical price can be computed as the maximum over all possible early exercise strategies of the discounted expected cash flows under the modified riskneutral information process. Several efficient numerical techniques exist for pricing American securities depending on one or few (up to 3) risk sources. They are either latticebased techniques or finite difference approximations of the BlackScholes diffusion equation. However, these methods cannot be used for highdimensional problems, since their memory requirement is exponential in the
From the bird’s eye to the microscope: A survey of new stylized facts of the intradaily foreign exchange markets
, 1997
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On the pricing of contingent claims under constraints
 Annals of Applied Probability
, 1996
"... We discuss the problem of pricing contingent claims, such as European calloptions, based on the fundamental principle of “absence of arbitrage ” and in the presence of constraints on portfolio choice, e.g. incomplete markets and markets with shortselling constraints. Under such constraints, we sho ..."
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Cited by 40 (2 self)
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We discuss the problem of pricing contingent claims, such as European calloptions, based on the fundamental principle of “absence of arbitrage ” and in the presence of constraints on portfolio choice, e.g. incomplete markets and markets with shortselling constraints. Under such constraints, we show that there exists an arbitragefree interval which contains the celebrated BlackScholes price (corresponding to the unconstrained case); no price in the interior of this interval permits arbitrage, but every price outside the interval does. In the case of convex constraints, the endpoints of this interval are characterized in terms of auxiliary stochastic control problems, in the manner of Cvitanić & Karatzas (1993). These characterizations lead to explicit computations, or bounds, in several interesting cases. Furthermore, a unique fair price ˆp is selected inside this interval, based on utility maximization and “marginal rate of substitution ” principles; again, characterizations are provided for ˆp, and these lead to very explicit computations. All these results are also extended to treat the problem of pricing contingent claims in the presence of a higher interest rate for borrowing. In the special case of a European calloption in a market with constant coefficients, the endpoints of the arbitragefree interval are the BlackScholes prices corresponding to the two different interest rates; and the fair price coincides with that of
Pricing of American PathDependent Contingent Claims
, 1994
"... We consider the problem of pricing pathdependent contingent claims. Classically, this problem can be cast into the BlackScholes valuation framework through inclusion of the pathdependent variables into the state space. This leads to solving a degenerate advectiondiffusion Partial Differential Eq ..."
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Cited by 39 (1 self)
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We consider the problem of pricing pathdependent contingent claims. Classically, this problem can be cast into the BlackScholes valuation framework through inclusion of the pathdependent variables into the state space. This leads to solving a degenerate advectiondiffusion Partial Differential Equation (PDE). Standard Finite Difference (FD) methods are known to be inadequate for solving such degenerate PDE. Hence, pathdependent European claims are typically priced through MonteCarlo simulation. To date, there is no numerical method for pricing pathdependent American claims. We first establish necessary and sufficient conditions amenable to a Lie algebraic characterization, under which degenerate diffusions can be reduced to lowerdimensional nondegenerate diffusions on a submanifold of the underlying asset space. We apply these results to pathdependent options. Then, we describe a new numerical technique, called Forward Shooting Grid (FSG) method, that efficiently copes with de...
Competitive solutions for online financial problems
 ACM Comput. Surv
, 1998
"... This article surveys results concerning online algorithms for solving problems related to the management of money and other assets. In particular, the survey focuses on search, replacement, and portfolio selection problems. ..."
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Cited by 14 (0 self)
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This article surveys results concerning online algorithms for solving problems related to the management of money and other assets. In particular, the survey focuses on search, replacement, and portfolio selection problems.
Actuarial versus Financial Pricing of Insurance
 Risk Finance
, 1996
"... : 1 Introduction This paper grew out of various recent discussions with academics and practitioners around the theme of the interplay between insurance and finance. Some issues were:  The increasing collaboration between insurance companies and banks.  The emergence of finance related insuran ..."
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Cited by 14 (1 self)
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: 1 Introduction This paper grew out of various recent discussions with academics and practitioners around the theme of the interplay between insurance and finance. Some issues were:  The increasing collaboration between insurance companies and banks.  The emergence of finance related insurance products, as there are catastrophe futures and options, PCS options, index linked policies, . . .  The deregulation of various (national) insurance markets.  The discussions around risk management methodology for financial institu tions (think of the various Basle Committee Reports).  The evolution from a more liability modelling oriented industry (insurance) to a more global financial industry involving assetliability and riskcapital based modelling.  The emergence of financial engineering as a new profession, its interplay with actuarial training and research. Besides these more general issues, specific questions were recently discussed in papers like Gerber and Shiu (...
NonParametric Estimation of an Implied Volatility Surface
, 1998
"... Given standard diffusionbased option pricing assumptions and a set of traded European option quotes and their payoffs at maturity,we identify a unique and stable set of diffusion coefficients or volatilities. Effectively, we invert a set of option prices into a state and timedependent volatility ..."
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Cited by 14 (0 self)
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Given standard diffusionbased option pricing assumptions and a set of traded European option quotes and their payoffs at maturity,we identify a unique and stable set of diffusion coefficients or volatilities. Effectively, we invert a set of option prices into a state and timedependent volatility function. Our problem differs from the standard direct problem in whichvolatilities and maturitypayoffs are known and the associated option values are calculated. Specifically, our approach, which is based on a small parameter expansion of the option value function, is a finite differencebased procedure. This approach builds on previous work which has followed Tikhonov's treatmentofintegral equations of the Fredholm or convolution type. An implementation of our approach with CBOE S&P 500 option data is also discussed. In this paper, we address the general problem of inverting option prices into a stateand timedependent volatility function. Specifically, we build on the foundation of re...
A Term Structure Model And The Pricing Of Interest Rate Derivative
, 1993
"... . The paper developes a general arbitrage free model for the term structure of interest rates. The principal model is formulated in a discrete time structure. It differs substantially from the HoLee Model (1986) and does not generate negative spot and forward rates. The results for the continuou ..."
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Cited by 11 (1 self)
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. The paper developes a general arbitrage free model for the term structure of interest rates. The principal model is formulated in a discrete time structure. It differs substantially from the HoLee Model (1986) and does not generate negative spot and forward rates. The results for the continuous time limit support this. The probability distribution with finite support is derived for the spot rate return. The model permits the arbitrage free valuation of bond options and interest rate options and produces dynamic portfolio strategies to duplicate these contracts. Introduction The uncertainty of future interest rate movements is a serious aspect to financial decision making. Investment decisions are often very sensitive to changes of the term structure. Therefore the management of interest rate uncertainty is an important subject and it is necessary to analyse financial innovation which are designed to deal with the interest rate risk. Examples of such instruments are put and call ...
Stochastic volatility: option pricing using a multinomial recombining tree
, 2006
"... We treat the problem of option pricing under the Stochastic Volatility (SV) model: the volatility of the underlying asset is a function of an exogenous stochastic process, typically assumed to be meanreverting. Assuming that only discrete past stock information is available, we adapt an interacting ..."
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Cited by 11 (5 self)
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We treat the problem of option pricing under the Stochastic Volatility (SV) model: the volatility of the underlying asset is a function of an exogenous stochastic process, typically assumed to be meanreverting. Assuming that only discrete past stock information is available, we adapt an interacting particle stochastic filtering algorithm due to Del Moral, Jacod and Protter (Del Moral et al., 2001) to estimate the SV, and construct a quadrinomial tree which samples volatilities from the SV filter’s empirical measure approximation at time 0. Proofs of convergence of the tree to continuoustime SV models are provided. Classical arbitragefree option pricing is performed on the tree, and provides answers that are close to market prices of options on the SP500 or on bluechip stocks. We compare our results to nonrandom volatility models, and to models which continue to estimate volatility after time 0. We show precisely how to calibrate our incomplete market, choosing a specific martingale measure, by using a benchmark option. Key words and phrases: incomplete markets, MonteCarlo method, options market, option pricing, particle method, random tree, stochastic filtering, stochastic volatility. 1