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Abstract:
The various macro econometrics model for inflation are helpless when it comes to the pricing of inflation derivatives. The only article targeting inflation option pricing, the Jarrow Yildirim model, relies on non observable data. This makes the estimation of the model parameters a non trivial problem. In addition, their framework do not examine any relationship between the most liquid inflation derivatives instruments: the year to year and zero coupon swap. To fill this gap, we see how to derive a model on inflation, based on traded and liquid market instrument. Applying the same strategy as the one for a market model on interest rates, we derive no-arbitrage relationship between zero coupon and year to year swaps. We explain how to compute the convexity adjustment and what relationship the volatility surface should satisfy. Within this framework, it becomes much easier to estimate model parameters and to price inflation derivatives in a consistent way.

Inflation index, forward, zero-coupon, year-on-year, volatility cube, convexity adjustment

Abstract:
Despite the recent growth of inflation linked derivatives market, the publicly available literature is very small. The various macro econometrics models are helpless when it comes to pricing inflation derivatives. The only freely accessible model, the Jarrow and Yildirim [4], relies on non observable data such as real yields. This makes this model hard to calibrate. In addition, it does not provide simple connection between liquid instruments like year on year, zero coupon swap and the modeling of the corresponding CPI correlation. To fill this gap, we adapt a market model to inflation. This can be seen as a simple translation of the Libor market model to inflation. We see how volatilities of year on year, zero coupon swap and the integrated CPI correlation are related. Hence, out of the three, only two are independent and these two provide the latter. We derive an upper and lower bound for the non independent parameter leading to coherence tests. We conclude by convexity correction formula for year on year rates, emphasizing the impact of correlation between interest and inflation rates.

Inflation derivatives, year on year, zero coupon swap, CPI, convexity correction

Abstract:
With the growing competition on the inflation derivatives market and the resulting tightening of trading margins, it has become crucial to include seasonality in inflation models. In this paper, after reviewing how to estimate seasonality component on CPI data, we examine its impact on the pricing of various inflation linked derivatives.

Inflation, seasonality, decomposition scheme, trend

Abstract:
Current Monte Carlo pricing engines may face computational challenge for the Greeks, because of not only their time consumption but also their poor convergence when using a finite difference estimate with a brute force perturbation. The same story may apply to conditional expectation. In this short paper, following Fournie et al. (1999), we explain how to tackle this issue using Malliavin calculus to smooth the payoff to estimate. We discuss the relationship with the likelihood ratio method of Broadie and Glasserman (1996). We show on numerical results the efficiency of this method and discuss when it is appropriate or not to use it. We see how to apply this method to the Heston model.

Monte-Carlo, Greeks, Conditional expectation, Malliavin Calculus, Likelihood Ratio, Homogeneity, Heston

Abstract:
This paper underlines the strong seasonality effect of the inflation market and shows how to include it in the pricing of inflation-linked products.

inflation derivatives, seasonality

Abstract:
This paper explains how to calculate convexity adjustment for interest rates derivatives when assuming a deterministic time dependent volatility, using martingale theory. The motivation of this paper lies in two directions. First, we set up a proper no-arbitrage framework illustrated by a relationship between yield rate drift and bond price. Second, making approximation, we come to a closed formula with specification of the error term. Earlier works (Brotherton et al. (1993) and Hull (1997)) assumed constant volatility and could not specify the approximation error. As an application, we examine the convexity bias between CMS and forward swap rates

Abstract:
This paper presents an approximated formula of the convexity adjustment of Constant Maturity Swap rates, using Wiener Chaos expansion, for multi-factor lognormal zero coupon models.

We derive closed formulae for CMS bond and swap and apply results to various well-known one-factor models (Ho and Lee (1986), Amin and Jarrow (1992), Hull and White (1990), Mercurio and Moraleda (1996)). Quasi Monte Carlo simulations confirm the efficiency of the approximation. Its precision relies on the importance of second and higher order terms.

Abstract:
It is well known that the cost of a call and put option is equal to its intrinsic value plus the cost of a stop loss strategy. This stop loss strategy can be re-expressed in terms of the local time. It provides easily closed forms solution for model like Black Scholes [8] or [3]. This paper examines the theory of local time for stochastic volatility models and in particular the SABR model [5]. It gives an approximated formula for the local time in SABR and shows that this model can be valued using a Black Scholes formula but where all the terms are complex number. This formula turns out to be more robust for low and high strikes. This solves in particular the problem of valuing the whole smile in SABR as required in the replication method for CMS and the copula integration for CMS spread options.

local time, stochastic volatility models, SABR, Black Scholes

Abstract:
In this paper, we assume that log returns can be modelled by a Levy process. We give explicit formulae for option prices by means of the Fourier transform. We explain how to infer the characteristics of the Levy process from option prices.
This enables us to generate an implicit volatility surface implied by market data. This model is of particular interest since it extends the seminal Black Scholes [1973] model consistently with volatility smile.

Levy process, Fourier and Laplace transform, Smile

Abstract:
Current Monte Carlo pricing engines may face computational challenge for the Greeks, because of not only their time consumption but also their poor convergence when using a finite difference estimate with a brute force perturbation. The same story may apply to conditional expectation. In this short paper, following Fournie et al. (1999), we explain how to tackle this issue using Malliavin calculus to smooth the payoff to estimate. We discuss the relationship with the likelihood ratio method of Broadie and Glasserman (1996). We show on numerical results the efficiency of this method and discuss when it is appropriate or not to use it. We see how to apply this method to the Heston model.

Monte-Carlo, Greeks, Conditional expectation, Malliavin Calculus, Likehood Ratio, Homogeneity, Heston, Stochastic volatility, Calibration

Abstract:
This paper presents an efficient methodology for the discrete Asian options consistent with different types of underlying densities, especially non-normal returns as suggested by the empirical literature (Mandelbrot (1963) and Fama (1965)). Based on Fast Fourier Transform, the method is an enhanced version of the algorithm of Carverhil and Clewlow (1992). The contribution of this paper is to improve their algorithm and to adapt it to non-lognormal densities. This enables us to examine the impact of fat-tailed distribution on price as well as on delta. We find evidence that fat tails lead to wider jumps in the delta.

Fast Fourier Transform, Asian options, Convolution, Fat-Tails.

Abstract:
Traditional methods for the computation of the Greeks with Monte Carlo simulations converge very slowly for strongly discontinuous payoff options. As a solution, Fournie et al. (1999) and Benhamou (2000) suggested the use of Malliavin weighted scheme especially for options depending on a finite set of dates. This paper extends their works to continuous time Asian options. We illustrate results for the case of the Black diffusion.

Abstract:
This paper studies the impact of stochastic interest rates for local volatility hybrids. Our research shows that it is possible to explicitly determine the bias between the local volatility of a model with stochastic interest rates and the local volatility of the same model, but with deterministic interest rates as a function between the correlation of the stochastic interest rates and the digital at the local strike. The paper will show that this bias can be expressed in a simpler form under the assumption of a diffusion of the stochastic interest rates, enabling us to compute a fast calibration for a hybrid model with stochastic interest rates. This bias leads to a decrease in the value of the local volatility as a result of the induced volatility caused by the stochastic drift. Numerical results illustrate the importance of the bias and confirm that some stochastic noise arises from the stochastic drift.

local volatility, stochastic interest rates, hybrid, Malliavin calculus

Abstract:
Favored by the Security Exchange Commission, Electronics Communication Networks (ECNs) have grown as alternative trading systems that enable to bypass the markets makers on the stock markets and allow investors to directly compensate and execute their orders with more discretion and at a lower cost. In this paper we underline the fragile character of the current ECNs and question their competitive advantages through empirical evidences. We find a rationale for market makers and ECNs' excessive spreads and overreactions. The use of network theory highlights notions of critical mass, open interface and alliances. Moreover, since competition between market makers and ECNs is based on volume, the emergence of ECNs has been mainly possible because of the growth of the American stock market. Furthermore, strategies of new ECNs are built on anticipated future growth. Should the market shrink, ECNs would rapidly be forced to merge and most of them would disappear.

Abstract:
This paper presents an efficient method for pricing discrete Asian options. Its contribution to the existing literature consists in targeting at smile and non proportional dividend effects. Using an homogeneity property, we show how to reduce an n +1 dimensional problem to a 2 or 3 dimensional one. We derive a PDE for the Asian option and solve it with the standard Crank Nicholson method. The dimension reduction imposes us to interpolate and extrapolate our conditional price at each fixing date. Within a deterministic volatility structure consistent with the smile, the homogeneity property is roughly conserved, thanks to a vega correction term. This allows us to stay in a two dimensional framework as in the Black Scholes case. We examine different numerical specifications of our finite difference (interpolation method, grid boundaries, time and space steps) as well as the extension to the case of non proportional discrete dividends, using a jump condition. We benchmark our results with Quasi Monte-Carlo simulation and a multi-dimensional PDE.

Abstract:
This paper presented a new technique for the simulation of the Greeks (i.e. price sensitivities to parameters), efficient for strongly discontinuous payoff options. The use of Malliavin calculus, by means of an integration by parts, enables to shift the differentiation operator from the payoff function to the diffusion kernel, introducing a weighting function.(Fournie et al. (1999)). Expressing the weighting function as a Skorohod integral, we show how to characterize the integrand with necessary and sufficient conditions, giving a complete description of weighting function solutions. Interestingly, for adapted process, the Skorohod integral turns to be the classical Ito integral.

Abstract:
Using Malliavin calculus techniques, we derive an analytical formula for the price of European options, for any model including local volatility and Poisson jump process. We show that the accuracy of the formula depends on the smoothness of the payoff function. Our approach relies on an asymptotic expansion related to small diffusion and small jump frequency/size. Our formula has excellent accuracy (the error on implied Black-Scholes volatilities for call option is smaller than 2 bp for various strikes and maturities). Additionally, model calibration becomes very rapid.

asymptotic expansion, Malliavin calculus, volatility skew and smile, small diffusion process, small jump frequency/size

Abstract:
Because of its very general formulation, the local volatility model does not have an analytical solution for European options. In this article, we present a new methodology to derive closed form solutions for the price of any European options. The formula results from an asymptotic expansion, terms of which are Black-Scholes price and related Greeks. The accuracy of the formula depends on the payoff smoothness and it converges with very few terms.

local volatility model, European options, asymptotic expansion, Malliavin calculus, small diffusion process, CEV model

Abstract:
The use of the Heston model is still challenging because it has a closed formula only when the parameters are constant [Hes93] or piecewise constant [MN03]. Hence, using a small volatility of volatility expansion and Malliavin calculus techniques, we derive an accurate analytical formula for the price of vanilla options for any time dependent Heston model (the accuracy is less than a few bps for various strikes and maturities). In addition, we establish tight error estimates. The advantage of this approach over Fourier based methods is its rapidity (gain by a factor 100 or more), while maintaining a competitive accuracy. From the approximative formula, we also derive some corollaries related first to equivalent Heston models (extending some work of Piterbarg on stochastic volatility models [Pit05]) and second, to the calibration procedure in terms of ill-posed problems.

asymptotic expansion, Malliavin calculus, small volatility of volatility, time dependent Heston model

Abstract:
Using the Stein numerical method, introduced by El Karoui and Jiao {ElKJ} and El Karoui, Jiao and Kurtz {ElKJK}, we compare, in terms of accuracy and efficiency, the pricing of the basket default swaps (NTDs and CDO Tranches). In the Factor Copula Model framework, we compare the following copula functions: 1 factor and 3 factors Gaussian copula, Clayton copula, Marshall-Olkin copula, Double-t copula and Student copula. Stein numerical method is also compared with the Recursive method of Hull and White, with the Probability Generating Function method (an exact Fourier transform like method) and with the Monte Carlo method.

Stein method, copula, CDO, credit derivatives

Abstract:
With the success of variable annuities, insurance companies are piling up large risks in terms of both equity and fixed income assets. These risks should be properly modeled as the resulting dynamic hedging strategy is very sensitive to the modeling assumptions. The current literature has been largely focusing on simple variations around Black-Scholes model with basic interest rates term structure models. However, in a more realistic world, one should account for both Stochastic Volatility and Stochastic Interest rates. In this paper, we examine the combine effect of a Heston-type model for the underlying asset with a HJM affine stochastic interest rates model and apply it to the pricing of GMxB (GMIB, GMDB, GMAB and GMWB). We see that stochastic volatility and stochastic interest rates have an impact on the resulting fair value of the contract and the resulting fair fee as well as mainly on the vega hedge. Interestingly, using a stochastic volatility model leads to scenarios with high level of volatility for long maturities resulting in a higher contract value and a resulting fair fee. We also see that the impact of stochastic interet rates and volatility is more pronounced on the vega hedge than on the delta hedge.

variable annuity, stochastic volatility, stochastic interest rates

Abstract:
With the success of variable annuities, insurance companies are piling up large risks in terms of both equity and fixed income assets. These risks should be properly modeled as the resulting dynamic hedging strategy is very sensitive to the modeling assumptions. The current literature has been largely focusing on simple variations around Black-Scholes model with basic interest rates term structure models. However, in a more realistic world, one should account for both Stochastic Volatility and Stochastic Interest rates. In this paper, we examine the combine effect of a Heston-type model for the underlying asset with a HJM affine stochastic interest rates model and apply it to the pricing of GMxB (GMIB, GMDB, GMAB and GMWB). We see that stochastic volatility and stochastic interest rates have an impact on the resulting fair value of the contract and the resulting fair fee as well as mainly on the vega hedge. Interestingly, using a stochastic volatility model leads to scenarios with high level of volatility for long maturities resulting in a higher contract value and a resulting fair fee. We also see that the impact of stochastic interet rates and volatility is more pronounced on the vega hedge than on the delta hedge.

Variable annuity, Stochastic Volatility, Stochastic Interest rates

Abstract:
The main result of this article is the presentation of the Distribution Match Method. This method applies to a general multi-factor pricing model under assumption of normal law drift. The idea is to find an equivalent one-factor model for European options. The equivalent model admits a weak solution, which has the same one-dimensional marginal probability distribution. Moreover, the one-dimensional distribution can be explicitly calculated under certain condition. This result can consequently induct closed formula for the future price and European option price. We apply these results to two well known commodity models, the Gabillon and the Gibson Schwartz model, to provide the price for the future price and a closed formula for the European options.

Gabillon model, Gibson Schwartz model, commodity, one factor weak solution, Distribution Match Method

Abstract:
This report presents different hybrid models and their application to the pricing of exotic products. A Market Model combining a risk-free term structure and a defaultable one for one underlying is first developped, with the application to the pricing of some exotic products, especially designed for the hedging needs of pension funds (1). Recalling the basis of the underlying HJM model (2) will then give us the possibility to extend some results to the modelling of the credit migration process (3), and to a multi-name framework (4). Along the lines, some links with Equity diffusions are covered.

Exotics, Hybrids, HJM, BGM, Markov Chains, transition matrix, Longstaff Schwarz, probability measure, forward neutral, survival neutral

Abstract:
This paper presents new approximation formulae of European options in a local volatility model with stochastic interest rates. This is a companion paper to our work on perturbation methods for local volatility models http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1275872 for the case of stochastic interest rates. The originality of this approach is to model the local volatility of the discounted spot and to obtain accurate approximations with tight estimates of the error terms. This approach can also be used in the case of stochastic dividends or stochastic convenience yields. We finally provide numerical results to illustrate the accuracy with real market data.

asymptotic expansion, local volatility model, HJM framework, Hull and White model, Malliavin calculus, small diffusion process, CEV model

Abstract:
This paper reexamines the Malliavin weighting functions introduced by Fournie et al. (1999) as a new method for efficient and fast computations of the Greeks. Reexpressing the weighting function generator in terms of its Skorohod integrand, we show that these weighting functions have to satisfy necessary and sufficient conditions expressed as conditional expectations. We then derive the weighting function with the smallest total variance. This is of particular interest as it bridges the method of Malliavin weights and the one of likelihood ratio, as introduced by Broadie and Glasserman (1996). The likelihood ratio is precisely the weighting function with the smallest total variance. We finally examine when to use the Malliavin method and when to prefer finite difference.

Monte Carlo, Quasi-Monte Carlo, Greeks, Malliavin Calculus, Likehood Ratio

Abstract:
This book provides a broad description of the financial derivatives business from a practitioner's point of view, with a particular emphasis on fixed income derivatives, a specific development on fixed income derivatives and a practical approach to the field. With particular emphasis on the concrete usage of mathematical models, numerical methods and the pricing methodology, this book is an essential reading for anyone considering a career in derivatives either as a trader, a quant or a structurer.

Derivatives, Fixed Income, Mathematical Modeling, Finance Structuring

Abstract:
Using the Stein numerical method, introduced by El Karoui and Jiao and El Karoui et al., the paper compares, in terms of accuracy and efficiency, the pricing of the basket default swaps (NTDs and CDO tranches). In the factor copula model framework, the paper compares the following copula functions: one-factor and three-factors Gaussian copula, Clayton copula, Marshall-Olkin copula, Double-t copula and Student copula. Stein numerical method is also compared with the Recursive method of Hull and White, probability generating function method (an exact Fourier transform like method) and the Monte Carlo method.

Abstract:
The inflation market, despite its exploding growth remains not very liquid except for vanilla derivatives. And unlike interest rates, it does not yet offer liquid and reliable swaption prices. Key objective of this paper is to compute inflation swaption volatility from liquid year on year volatility. Following a similar method as for the derivation of swaption volatility from interest cap in a market model, we derive a closed form solution for the inflation swaption volatility. We examine various more simple methods to compare our relative value computation. Using this estimate of the inflation volatility, we compute not only inflation swaption but also options on real yield and inflation spreads.

Inflation swaption, year-on-year volatility, implied volatility, correlation, real yield option