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 risk-neutral information process. Several efficient numerical techniques exist for pricing American securities depending on one or few (up to 3) risk sources. They are either lattice-based techniques or finite difference approximations of the Black-Scholes diffusion equation. However, these methods cannot be used for high-dimensional problems, since their memory requirement is exponential in the

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We consider the problem of pricing path-dependent contingent claims. Classically, this problem can be cast into the Black-Scholes valuation framework through inclusion of the path-dependent variables into the state space. This leads to solving a degenerate advection-diffusion Partial Differential Equation (PDE). Standard Finite Difference (FD) methods are known to be inadequate for solving such degenerate PDE. Hence, path-dependent European claims are typically priced through Monte-Carlo simulation. To date, there is no numerical method for pricing path-dependent American claims. We first establish necessary and sufficient conditions amenable to a Lie algebraic characterization, under which degenerate diffusions can be reduced to lower-dimensional non-degenerate diffusions on a sub-manifold 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...

. 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 Ho--Lee-- 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 ...

: 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 asset-liability and risk-capital 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 (...

Given standard diffusion-based option pricing assumptions and a set of traded European option quotes and their pay-offs 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 time-dependent volatility function. Our problem differs from the standard direct problem in whichvolatilities and maturitypay-offs 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 difference-based 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 time-dependent volatility function. Specifically, we build on the foundation of re...

It is common for firms to issue or purchase options on the firm’s own stock. Examples include convertible bonds, warrants, call options as employee compensation, or the sale of put options as part of share repurchase programs. This paper shows that option positions with implicit borrowing—such as put sales and call purchases—are tax-disadvantaged relative to the equivalent synthetic option with explicit borrowing. Conversely, option positions with implicit lending—such as compensation calls—are tax-advantaged. We also show that firms are better off from a tax perspective issuing bifurcated convertible bonds— bonds plus warrants—rather than an otherwise equivalent standard convertible. The put option sales which have been popular with some firms are like issuing debt with non-deductible interest and thus have a tax cost. For example, we estimate that in 1999 the tax cost to Microsoft of written puts was about $80m per year.

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 mean-reverting. 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 continuous-time SV models are provided. Classical arbitrage-free option pricing is performed on the tree, and provides answers that are close to market prices of options on the SP500 or on blue-chip stocks. We compare our results to non-random 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, Monte-Carlo method, options market, option pricing, particle method, random tree, stochastic filtering, stochastic volatility. 1

This paper studies a general equilibrium model of an economy with production under uncertainty in which firms’ capital (ownership) structures creates a moral hazard problem for their managers. The concept of an equilibrium with rational, competitive price perceptions (RCPP) is introduced, in which investors correctly anticipate the optimal effort of entrepreneurs by observing their financial decisions, and entrepreneurs are aware that investors use their financial decisions as signals. The competitive element in the equilibrium valuation of firms comes from the fact that entrepreneurs cannot affect the market price of risks. It is shown that under appropriate spanning assumptions an RCPP is constrained Pareto optimal. Furthermore, if sufficiently many options are traded, then full optimality can be obtained despite the moral hazard problem: options serve both to increase the span of the market and to provide incentives for entrepreneurs.

INTRODUCTION European barrier options are path-dependent options in which the existence of the European options depends on whether the underlying asset price has touched a barrier level during the option life. They have emerged as significant products for hedging and investment in foreign exchange, equity, and commodity markets since the late 1980s, largely in the overthe -counter (OTC) markets. The only barrier option that is traded on the options exchanges is the capped index spread on the S&P 100 and S&P 500. The cap is automatically exercised with a fixed profit when the underlying price closes beyond the barrier level. Thus the option is canceled with a fixed return at the barrier level. Its pricing and hedging are discussed by Chance (1994). An example of a barrier option is an up-and-out put. An investor could buy an up-and-out U.S. dollar put (Japanese yen call) instead of an ordinary U.S. dollar put to hedge the value of U.S. dollar versus Japanese yen. The put optio