Applied mathematics-mathematical finance-option pricing

In this paper, we study a multi-period portfolio selection model in which a generic class of probability distributions is assumed for the returns of the risky asset. An investor with a power utility function rebalances a portfolio comprising a risk-free and risky asset at the beginning of each time period in order to maximize expected utility of terminal wealth. Trading the risky asset incurs a cost that is proportional to the value of the transaction. At each time period, the optimal investment strategy involves buying or selling the risky asset to reach the boundaries of a certain no-transaction region. In the limit of small transaction costs, dynamic programming and perturbation analysis are applied to obtain explicit approximations to the optimal boundaries and optimal value function of the portfolio at each stage of a multi-period investment process of any length.

The main objective of the work described is to find a pricing model for weather derivatives with payouts depending on temperature. Historical data are used to suggest a stochastic process that describes the evolution of the temperature. Since temperature is a non-tradable quantity, unique prices of contracts in an incomplete market are obtained using the market price of risk. Numerical examples of prices of some contracts are presented, using an approximation formula as well as Monte Carlo simulations.

This paper determines first‐passage time distributions with a twofold emphasis on the dynamics of the state variables and interest rate uncertainty. Underlyings follow two‐dimensional geometric Brownian motions, Ornstein–Uhlenbeck processes or Poisson jump‐diffusion processes, and boundaries are either fixed or indexed on risk‐free bonds. Forward‐neutral changes of numeraire enable one to derive generic valuation expressions, while changing time allows one to determine closed‐form solutions for geometric Brownian motions and moving barriers. In turn, the latter formulas are used to reduce the variance of Monte Carlo simulations in the case of jump‐diffusion processes, by means of the control variate method.

The adoption of copula functions is suggested in order to price bivariate contingent claims. Copulas enable the marginal distributions extracted from vertical spreads in the options markets to be imbedded in a multivariate pricing kernel. It is proved that such a kernel is a copula function, and that its super-replication strategy is represented by the Frechet bounds. Applications provided include

In this paper models with transaction costs for pricing of European options using mixed fractional Brownian motion (fbm) and Partial differential equation (PDE) model are considered. Investigation on price sensitivity to volatility and formulation of asymptotic strategy for replicating self financing assets are also made. Simulation experiments for the models are run to obtain the call and put prices using fbm with Hurst parameter 1 3 H  and the Crank-Nicolson method for the numerical solution to the PDE model. It is found that stock prices increase with time to maturity dates, whereas, the call prices decrease steadily as time approaches maturity dates in conformity with theta hedging strategy. In the recent times, development of financial instruments for pricing options are on increasing and the use of mixed fractional Brownian motion (fbm) for simulation of derivatives are not too common. In this paper, simulation experiments are designed and implemented using codes in MATLAB for the models considered. We obtained the call and put prices using fbm with Hurst parameter 1 3 H  and the Crank-Nicolson method for the numerical solution to the PDE model using fbm realization. Investigation on price sensitivity to volatility and formulation of asymptotic strategy for replicating self financing assets are also made.

We consider in this article the arbitrage free pricing of double knock-out barrier options with payoffs that are arbitrary functions of the underlying asset, where we allow exponentially time-varying barrier levels in an otherwise standard Black-Scholes model. Our approach, reminiscent of the method of images of electromagnetics, considerably simplifies the derivation of analytical formulae for this class of exotics by reducing the pricing of any double-barrier problem to that of pricing a related European option. We illustrate the method by reproducing the well-known formulae of for the standard knock-out double-barrier call and put options. We give an explanation for the rapid rate of convergence of the doubly infinite sums for affine payoffs in the stock price, as encountered in the pricing of double-barrier call and put options first observed by .

In this paper we present a mean-reverting jump diffusion model for the electricity spot price and derive the corresponding forward in closed-form. Based on historical spot data and forward data from England and Wales we calibrate the model and present months, quarters, and seasons-ahead forward surfaces.

In the setting of the Black-Scholes option pricing market model, the seller of a European option must trade continuously in time. This is, of course, unrealistic from the practical viewpoint. He must then follow a discrete trading strategy. However, it does not seem natural to hedge at deterministic times regardless of moves of the spot price. In this paper, it is supposed that the hedger trades at a fixed number N of rebalancing (stopping) times. The problem (P N ) of selecting the optimal hedging times and ratios which allow to minimize the variance of replication error is considered. For given N rebalancing, we identify the discrete optimal hedging strategy for this criterion. The problem (P N ) is then transformed into a multidimensional optimal stopping problem with boundary constraint. The restrictive problem (P N BS ) of selecting the optimal rebalancing for the same criterion is also considered when the ratios are given by Black-Scholes. Using the vector-valued optimal stopping theory, the existence is shown of an optimal sequence of rebalancing for each one of the problems (P N ) and (P N BS ). It is also shown that they are asymptotically equivalents when the number of rebalances becomes large and an optimality criterion is stated for the problem (P N ). The same study is made when we impose more realistic restrictions on the hedging times. In the special case of two rebalances, the problem (P 2 BS ) is solved and the problems (P 2 BS ) and (P 2 ) are transformed into two optimal stopping problems. This transformation is useful for numerical purposes.

The recent liberalization of the electricity and gas markets has resulted in the growth of energy exchanges and modelling problems. In this paper, we modelize jointly gas and electricity spot prices using a mean-reverting model which fits the correlations structures for the two commodities. The dynamics are based on Ornstein processes with parameterized diffusion coefficients. Moreover, using the empirical distributions of the spot prices, we derive a class of such parameterized diffusions which captures the most salient statistical properties: stationarity, spikes and heavy-tailed distributions.

We consider the pricing of a range of volatility derivatives, including volatility and variance swaps. Under risk-neutral valuation we provide closed form formulae for volatility-average and variance swaps for a variety of diffusion and jump-diffusion models for volatility. We describe a general partial differential equation framework for derivatives that have an extra dependence on an average of the volatility. We

The Beal Conjecture, A x + B y = C z , is analyzed as of a proof based on selfsame multiples through addition and the presentation of counterexamples. The Beal Conjecture requests the presentation of counterexamples based upon selfsame multiplication, when in fact such counterexamples do not exist. Counterexamples do exist through selfsame addition A x + B y = z(C), where it is possible to present relations of equivalency that have terms in positive integers with no common prime factor. The analysis in this essay presents an explanation of the equation based upon its stipulated algebraic notation.

Barrick Gold Corporation is the largest gold mining company in the world, with its headquarters in Toronto, Ontario, Canada. It has mines spread across the world like in Argentina, Australia, Canada, Chile, USA, Saudi Arabia, etc.
In this analysis I collected the closing price of ABX from 2016-01-05, to 2017-01-05, in order to evaluate the price of two options, expiration day fixed at 20th of January 2017, one call and one put, both with strike price fixed at 17.00$.
In order to do this, we’ll proceed collecting the actual market price of our options, and then we’ll compare the results given by the Black & Scholes formula. After listing the problems related with the B&S, we’ll try some statistical methods which allows for jumps like the Levy process.
Finally we’ll see a different way of modelling our prices. We’ll compare the results obtained with the Monte Carlo simulation, with that offered by the Fast Fourier Transform.

Weather influences our daily lives and choices and has an enormous impact on corporate revenues and earnings. Weather derivatives differ from most derivatives in that the underlying weather cannot be traded and their market is relatively illiquid. The weather derivative market is therefore incomplete. This paper implements a pricing methodology for weather derivatives that can increase the precision of measuring weather risk. We have applied continous autoregressive models (CAR) with seasonal variation to model the temperature in Berlin and with that to get the explicite nature of non-arbitrage prices for temperature derivatives. We infer the implied market price from Berlin cumulative monthly temperature futures that are traded at the Chicago Mercantile Exchange (CME), which is an important parameter of the associated equivalent martingale measures used to price and hedge weather future/options in the market. We propose to study the market price of risk, not only as a piecewise constant linear function, but also as a time dependent object. In all of the previous cases, we found that the market price of weather risk is different from zero and shows a seasonal structure. With the extract information we price other exotic options, such as cooling/heating degree day temperatures and non-standard maturity contracts.

The paper considers the pricing of a range of volatility derivatives, including volatility and variance swaps and swaptions. Under risk-neutral valuation closed-form formulae for volatility-average and variance swaps for a variety of diffusion and jump-diffusion models for volatility are ...

The “plain vanilla” swap is an agreement to exchange interest rate payments on nominally identical principal.  In the plain vanilla swap a floating interest rate is swapped for a fixed rate.  These swaps are made between corporations with differing interest rate risk over the period of time under consideration. Swaps can be used to transform a liability into a fix rate loan or visa-versa. Financial intermediaries are involved to make the exchange more efficient, but bank fees involved.

This paper considers the pricing of contingent claims using an approach developed and used in insurance pricing. The approach is of interest and signi…cance because of the increased integration of insurance and …nancial markets and also because insurance related risks are trading in …nancial markets as a result of securitisation and new contracts on futures exchanges. This approach uses probability distortion functions as the dual of the utility functions used in …nancial theory. The pricing formula is the same as the Black-Scholes formula for contingent claims when the underlying asset price is log-normal. The paper compares the probability distortion function approach with that based on …nancial theory. The theory underlying the approaches is set out and limitations on the use of the insurance based approach are illustrated. We extend the probability distortion approach to the pricing of contingent claims for more general assumptions than those used for Black-Scholes option pricing.

We formulate and analyze an inverse problem using derivatives prices to obtain an implied filtering density on volatility's hidden state. Stochastic volatility is the unobserved state in a hidden Markov model (HMM) and can be tracked using Bayesian filtering. However, derivative data can be considered as conditional expectations that are already observed in the market, and which can be used as input to an inverse problem whose solution is an implied conditional density on volatility. Our analysis relies on a specification of the martingale change of measure, which we refer to as separability. This specification has a multiplicative component that behaves like a risk premium on volatility uncertainty in the market. When applied to SPX options data, the estimated model and implied densities produce variance-swap rates that are consistent with the VIX volatility index. The implied densities are relatively stable over time and pick up some of the monthly effects that occur due to the options' expiration, indicating that the volatility-uncertainty premium could experience cyclic effects due to the maturity date of the options.

The aim of this article is to provide a systematic analysis of the conditions such that Fourier transform valuation formulas are valid in a general framework; i.e. when the option has an arbitrary payoff function and depends on the path of the asset price process. An interplay between the conditions on the payoff function and the process arises naturally. We also extend these results to the multi-dimensional case, and discuss the calculation of Greeks by Fourier transform methods. As an application, we price options on the minimum of two assets in Lévy and stochastic volatility models. 2000 Mathematics Subject Classification. 91B28; 42B10; 60G48. Key words and phrases. option valuation; Fourier transform; semimartingales; Lévy processes; stochastic volatility models; options on several assets. We would like to thank Friedrich Hubalek and Gabriel Maresch for valuable discussions. K. G. would like to thank the DFG for financial support through project EB66/11-1, and the Austrian Science Fund (FWF) for an invitation under grant P18022. A. P. gratefully acknowledges the financial support from the Austrian Science Fund (FWF grant Y328, START Prize). We would like to thank the two anonymous referees for their careful reading of the manuscript and their valuable suggestions that have improved the paper.

In this paper we will consider a setting where a large number of agents are trading commodity bundles. Assuming that agents of the same type have a certain utility attached to each transaction, we construct a statistical equilibrium which in turn implies prices on the different commodities. Our basic question is then the following: Assume that some commodities come out with prices that are socially unacceptable. Is it possible to change these prices systematically if a new type of agents is paid to enter the market? In the paper we will consider explicit examples where this can be done.

Wireless sensor networks (WSNs) have attracted a wide range of disciplines where close interactions with the physical world are essential. The distributed sensing capabilities and the ease of deployment provided by a wireless communication paradigm make WSNs an important component of our daily lives. By providing distributed, real-time information from the physical world, WSNs extend the reach of current cyber infrastructures to the physical world. In this paper, we present a detailed explanation of wireless sensor networks architecture and protocol. Our aim is to provide a contemporary look at the current state of the art in WSNs and discuss the still-open research issues in this field.

The main objective of this work is to find a pricing model for weather derivatives with payouts depending on temperature. We use historical data to first suggest a stochastic process that describes the evolution of the temperature. Since temperature is a non-tradable quantity, we obtain unique prices of contracts in an incomplete market, using the market price of risk. Numerical examples of prices of some contracts are presented, using an approximation formula as well as Monte Carlo simulations. * Kungsgatan 37 2tr, SE-111 56 Stockholm.

This paper unifies the classical theory of stochastic dominance and investor preferences with the recent literature on risk measures applied to the choice problem faced by investors. First we summarize the main stochastic dominance rules used in the finance literature. Then we discuss the connection with the theory of integral stochastic orders and we introduce orderings consistent with investors' preferences. Thus, we classify them, distinguishing several categories of orderings associated with different classes of investors. Finally we show how we can use risk measures and orderings consistent with some preferences to determine the investors' optimal choices.

Hedging interest rate exposures using interest rate futures contracts requires some knowledge of the volatility function of the interest rates. Use of historical data as well as interest rate options like caps and swaptions to estimate this volatility function have been proposed in the literature. In this paper the interest rate futures price is modelled within an arbitrage-free framework for

This paper concerns the pricing of American options with stochastic stopping time constraints expressed in terms of the states of a Markov process. Following the ideas of Menaldi et al., we transform the constrained into an unconstrained optimal stopping problem. The transformation replaces the original payoff by the value of a generalized barrier option. We also provide a Monte Carlo method to numerically calculate the option value for multidimensional Markov processes. We adapt the Longstaff-Schwartz algorithm to solve the stochastic Cauchy-Dirichlet problem related to the valuation problem of the barrier option along a set of simulated trajectories of the underlying Markov process. * We thank Walter Farkas for helpful comments and suggestions. Markus Leippold acknowledges the financial support of the Swiss National Science Foundation (NCCR FINRISK).

Use is made of the duality property of random walks to develop a numerical method for the valuation of discrete-time lookback options. This method leads to a recursive numerical integration procedure which is fast, accurate and easy to implement.

A stock loan is a contract whereby a stockholder uses shares as collateral to borrow money from a bank or financial institution. In , this contract is modeled as a perpetual American option with a time varying strike and analyzed in detail within a risk-neutral framework. In this paper, we extend the valuation of such loans to an incomplete market setting, which takes into account the natural trading restrictions faced by the client. When the maturity of the loan is infinite, we use a time-homogeneous utility maximization problem to obtain an exact formula for the value of the loan fee to be charged by the bank. For loans of finite maturity, we characterize the fee using variational inequality techniques. In both cases we show analytically how the fee varies with the model parameters and illustrate the results numerically.

This article discusses the properties of the minimal entropy martingale measure in relation to the problem of real options valuations in multinomial lattices. The methods are used to determine the postponement option embedded in developable land and the value of an expansion option. The methods presented provide an easy procedure that can easily be used to valuate real options in higher dimensional multinomial lattices. This helps us to go beyond the commonly used binomial models in the valuation process.

Index funds that track a benchmark, such as the market cap-weighted S&P 500 index, tend to have portfolio holdings biased towards slower-growth large-cap equities that result in the fund’s under-performance, especially in economic downturns. We develop a rigorous quantitative framework that allows dynamic-rebalancing of the allocations such that portfolio exposure in a market segment can change periodically based on economic activity, measured via a set of macro-economic and financial indicators. The method incorporates potential shifts in the economic state, and the likelihood thereof, to determine the fund’s risk orientation optimally in tracking or not tracking the benchmark index. That is, the greater the likelihood of a stronger economic state, the higher the degree of tracking the market index; however, a lack of confidence in the economic state results in a more index-neutral portfolio composition. The proposed smart indexing optimal strategy generates superior risk-adjusted ...

This paper considers the pricing of contingent claims using an approach developed and used in insurance pricing. The approach is of interest and signi…cance because of the increased integration of insurance and …nancial markets and also because insurance related risks are trading in …nancial markets as a result of securitisation and new contracts on futures exchanges. This approach uses probability distortion functions as the dual of the utility functions used in …nancial theory. The pricing formula is the same as the Black-Scholes formula for contingent claims when the underlying asset price is log-normal. The paper compares the probability distortion function approach with that based on …nancial theory. The theory underlying the approaches is set out and limitations on the use of the insurance based approach are illustrated. We extend the probability distortion approach to the pricing of contingent claims for more general assumptions than those used for Black-Scholes option pricin...

Market by order (MBO) data-a detailed feed of individual trade instructions for a given stock on an exchange-is arguably one of the most granular sources of microstructure information. While limit order books (LOBs) are implicitly derived from it, MBO data is largely neglected by current academic literature which focuses primarily on LOB modelling. In this paper, we demonstrate the utility of MBO data for forecasting high-frequency price movements, providing an orthogonal source of information to LOB snapshots and expanding the universe of alpha discovery. We provide the first predictive analysis on MBO data by carefully introducing the data structure and presenting a specific normalisation scheme to consider level information in order books and to allow model training with multiple instruments. Through forecasting experiments using deep neural networks, we show that while MBO-driven and LOB-driven models individually provide similar performance, ensembles of the two can lead to improvements in forecasting accuracy-indicating that MBO data is additive to LOB-based features.

Cornerstone asset pricing models, such as capital asset pricing model (CAPM) and arbitrage pricing theory (APT), yield theoretical predictions about the relationship between expected returns and exposure to systematic risk, as measured by beta(s). Numerous studies have investigated the empirical validity of these models. We show that even if no relationship holds between true expected returns and betas in the population, the existence of low-probability extreme outcomes induces a spurious correlation between the sample means and the sample betas. Moreover, the magnitude of this purely spurious correlation is similar to the empirically documented correlation, and the regression slopes and intercepts are very similar as well. This result does not necessarily constitute evidence against the theoretical asset pricing models, but it does shed new light on previous empirical results, and it points to an issue that should be carefully considered in the empirical testing of these models. The analysis points to the dangers of relying on simple least squares regression for drawing conclusions about the validity of equilibrium pricing models.

Accordind to the article we are going to demonstrate the appearance of explosions in three quantities in interest rate models with log-normally distributed rates in discrete time. (1) The expectation of the money market account in the Black, Derman, Toy model, (2) the prices of Eurodollar futures contracts in a model with log-normally distributed rates in the terminal measure and (3) the prices of Eurodollar futures contracts in the one-factor log-normal Libor market model (LMM). We derive exact upper and lower bounds on the prices and on the standard deviation of the Monte Carlo pricing of Eurodollar futures in the one factor log-normal Libor market model. These bounds explode at a non-zero value of volatility, and thus imply a limitation on the applicability of the LMM and on its Monte Carlo simulation to sufficiently low volatilities.

The European sovereign debt crisis, started in the second half of 2011, has posed the problem for asset managers, trades and risk managers to assess sovereign default risk. In the reduced form framework, it is necessary to understand the interrelationship between creditworthiness of a sovereign, its intensity to default and the correlation with the exchange rate between the bond's currency and the currency in which the CDS spread are quoted. To do this, we propose a hybrid sovereign risk model in which the intensity of default is based on the jump to default extended CEV model. We analyze the differences between the default intensity under the domestic and foreign measure and we compute the default-survival probabilities in the bond's currency measure. We also give an approximation formula to CDS spread obtained by perturbation theory and provide an efficient method to calibrate the model to CDS spread quoted by the market. Finally, we test the model on real market data by several calibration experiments to confirm the robustness of our method.

Margrabe provides a pricing formula for an exchange option where the distributions of both stock prices are log-normal with correlated Wiener components. Merton has provided a formula for the price of a European call option on a single stock where the stock price process contains a continuous Poisson jump component, in addition to a continuous log-normally distributed component. We use Merton's analysis to extend Margrabe's results to the case of exchange options where both stock price processes also contain compound Poisson jump components. A Radon-Nikodým derivative process that induces the change of measure from the market measure to an equivalent martingale measure is introduced. The choice of parameters in the Radon-Nikodým derivative allows us to price the option under different financial-economic scenarios. We also consider American style exchange options and provide a probabilistic intepretation of the early exercise premium.

In a discrete setting, we develop a model for pricing a contingent claim in incomplete markets. Since hedging opportunities influence the price of a contingent claim, we first introduce the optimal hedging strategy assuming that a contingent claim has been issued: a strategy implemented by investing initial wealth plus the selling price is optimal if it maximizes the expected utility of the agent's net payoff, which is the difference between the outcome of the hedging portfolio and the payoff of the claim. Next, we introduce the 'reservation price' as a subjective valuation of a contingent claim. This is defined as the minimum price that makes the issue of the claim preferable to stay put given that, once the claim has been written, the writer hedges it according to the expected utility criterion. We define the reservation price both for a short position (reservation selling price) and for a long position (reservation buying price) in the claim. When the contingent claim is redundant, both the selling and the buying price collapse in the usual Arrow-Debreu (or Black-Scholes) price. If the claim is non-redundant, then the reservation prices are interior points of the bid-ask interval. We provide also two numerical examples with different utility functions and contingent claims. Some qualitative properties of the reservation price are shown. tis for helpful discussions and comments and the participants to seminars in Purdue University, Seminari di Giardino Giusti -Università di Verona, Dipartimento di Matematica per le Decisioni -Università di Firenze. We are indebted to Fulvio Ortu for the notion of reservation price. All remaining errors are ours.

We present a model for pricing and hedging derivative securities and option portfolios in an environment where the volatility is not known precisely, but is assumed instead to lie between two extreme values min and max . These bounds could be inferred from extreme values of the implied volatilities of liquid options, or from high-low peaks in historical stock-or option-implied volatilities. They can be viewed as de ning a con dence interval for future volatility values. We show that the extremal non-arbitrageable prices for the derivative asset which arise as the volatility paths vary in such a band can be described by a non-linear PDE, which we call the Black-Scholes-Barenblatt equation. In this equation, the \pricing" volatility is selected dynamically from the two extreme values min , max , according to the convexity of the valuefunction. A simple algorithm for solving the equation by nite-di erencing or a trinomial tree is presented. We show that this model captures the importance of diversi cation in managing derivatives positions. It can be used systematically to construct e cient hedges using other derivatives in conjunction with the underlying asset. y

The passport option is a call option on the balance of a trading account. The option holder retains the gain from trading, while the writer is liable for the loss. We establish pricing equations for various passport options including the multi-asset passport and those with discrete trading constraints. The results are typically known as the Hamilton-Jacobi-Bellman equations and multiple layers of free boundary partial di erential equations for a sequence of optimal stopping times. Also we examine the gain by selling passport options to utility maximising investors and to investors who guess the market from imperfect information.

Weather influences our daily lives and choices and has an enormous impact on corporate revenues and earnings. Weather derivatives differ from most derivatives in that the underlying weather cannot be traded and their market is relatively illiquid. The weather derivative market is therefore incomplete. This paper implements a pricing methodology for weather derivatives that can increase the precision of measuring weather risk. We have applied continous autoregressive models (CAR) with seasonal variation to model the temperature in Berlin and with that to get the explicite nature of non-arbitrage prices for temperature derivatives. We infer the implied market price from Berlin cumulative monthly temperature futures that are traded at the Chicago Mercantile Exchange (CME), which is an important parameter of the associated equivalent martingale measures used to price and hedge weather future/options in the market. We propose to study the market price of risk, not only as a piecewise constant linear function, but also as a time dependent object. In all of the previous cases, we found that the market price of weather risk is different from zero and shows a seasonal structure. With the extract information we price other exotic options, such as cooling/heating degree day temperatures and non-standard maturity contracts.

Weather influences our daily lives and choices and has an enormous impact on corporate revenues and earnings. Weather derivatives differ from most derivatives in that the underlying weather cannot be traded and their market is relatively illiquid. The weather derivative market is therefore incomplete. This paper implements a pricing methodology for weather derivatives that can increase the precision of measuring weather risk. We have applied continous autoregressive models (CAR) with seasonal variation to model the temperature in Berlin and with that to get the explicite nature of non-arbitrage prices for temperature derivatives. We infer the implied market price from Berlin cumulative monthly temperature futures that are traded at the Chicago Mercantile Exchange (CME), which is an important parameter of the associated equivalent martingale measures used to price and hedge weather future/options in the market. We propose to study the market price of risk, not only as a piecewise constant linear function, but also as a time dependent object. In all of the previous cases, we found that the market price of weather risk is different from zero and shows a seasonal structure. With the extract information we price other exotic options, such as cooling/heating degree day temperatures and non-standard maturity contracts.

Financial time series exhibit two different type of non linear correlations: (i) volatility autocorrelations that have a very long range memory, on the order of years, and (ii) asymmetric return-volatility (or 'leverage') correlations that are much shorter ranged. Different stochastic volatility models have been proposed in the past to account for both these correlations. However, in these models, the decay of the correlations is exponential, with a single time scale for both the volatility and the leverage correlations, at variance with observations. We extend the linear Ornstein-Uhlenbeck stochastic volatility model by assuming that the mean reverting level is itself random. We find that the resulting three-dimensional diffusion process can account for different correlation time scales. We show that the results are in good agreement with a century of the Dow Jones index daily returns , with the exception of crash days.

We propose a numerical method to price corporate bonds based on the model of default risk developed by Madan and Unal in [23]. In particular using a perturbation approach we derive two semi-explicit formulae that allows to approximate the survival probability of the firm issuing the bond very efficiently. More precisely we consider both the first-order and the secondorder power series expansions of the survival probability in powers of the model parameter c. The zero-order coefficient of the series is evaluated using an exact analytical formula. The first-order and the second-order coefficients of the series are computed using an approximation algorithm based on the Laplace transform.

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 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.