Past Nomura Seminar

Given a deterministically time-changed Brownian motion Z starting from 1, whose time-change V (t) satisfies V (t) > t for all t > 0, we perform an explicit construction of a process X which is Brownian motion in its own filtration and that hits zero for the first time at V (S), where S := inf {t > 0 : Z_t = 0}. We also provide the semimartingale decomposition of X under the filtration jointly generated by X and Z. Our construction relies on a combination of enlargement of filtration and filtering techniques. The resulting process X may be viewed as the analogue of a 3-dimensional Bessel bridge starting from 1 at time 0 and ending at 0 at the random time V (S). We call this a dynamic Bessel bridge since S is not known at time 0 but is slowly revealed in time by observing Z. Our study is motivated by insider trading models with default risk. (this is a joint work with Luciano Campi and Albina Danilova)

Leveraged ETFs are funds that target a multiple of the daily return of a reference asset; eg UYG (Proshares) targets twice the daily return of XLF (Financial SPDR) and SKF targets minus twice the daily return of XLF.

 It is well known that these leveraged funds have exposure to realized volatility. In particular, the relation between the leveraged and the unleveraged funds over a given time-horizon (larger than 1 day) is uncertain and will depend on the realized volatility. This talk examines this phenomenon theoretically and empirically first, and then uses this to price options on leveraged ETFs in terms of the prices of options on the underlying ETF. The resulting model allows to model the volatility skews of the leveraged and unleveraged funds in relation to each other and therefore suggest an arbitrage relation that could prove useful for traders and risk-managers.

We argue that a natural extension of the well known structural credit risk framework of Black and Cox is to model both the firm's assets and liabilities as correlated geometric Brownian motions. This financially reasonable assumption leads to a unification of equity derivatives (written on the stock price), and credit securities like bonds and credit default swaps (CDS), nesting the Black-Cox credit model with a particular stochastic volatility model for the stock. As we will see, it yields reasonable pricing performance with acceptable computational efficiency. However, it has been well understood how to extend a credit framework like this quite dramatically by the trick of time- changing the Brownian motions. We will find that the resulting two factor time-changed Brownian motion framework can encompass well known equity models such as the variance gamma model, and at the same time reproduce the stylized facts about default stemming from structural models of credit. We will end with some encouraging calibration results for a dataset of equity and credit derivative prices written on Ford Motor Company.

We discuss a backward stochastic differential equation, (BSDE), approach to a risk-based, optimal investment problem of an insurer. A simplified continuous-time economy with two investment vehicles, namely, a fixed interest security and a share, is considered.

The insurer's risk process is modeled by a diffusion approximation to a compound Poisson risk process. The goal of the insurer is to select an optimal portfolio so as to minimize the risk described by a convex risk measure of his/her terminal wealth. The optimal investment problem is then formulated as a zero-sum stochastic differential game between the insurer and the market. The BSDE approach is used to solve the game problem. This leads to a simple and natural approach for the existence and uniqueness of an optimal strategy of the game problem without Markov assumptions. Closed-form solutions to the optimal strategies of the insurer and the market are obtained in some particular cases.

We derive a general methodology to approximate the law of the average of the marginal of diffusion processes. The average is computed w.r.t. a general parameter that is involved in the diffusion dynamics. Our approach is suitable to compute expectations of functions of arithmetic or geometric means. In the context of small SDE coefficients, we establish an expansion, which terms are explicit and easy to compute. We also provide non asymptotic error bounds. Applications to the pricing of basket options, Asian options or commodities options are then presented. This talk is based on a joint work with M. Miri.

We study exponential Levy models with change-point which is a random variable, independent from initial Levy processes. On canonical space with initially enlarged filtration we describe all equivalent martingale measures for change-

point model and we give the conditions for the existence of f-minimal equivalent martingale measure. Using the connection between utility maximisation and f-divergence minimisation, we obtain a general formula for optimal strategy in change-point case for initially enlarged filtration and also for progressively enlarged filtration when the utility is exponential. We illustrate our results considering the Black-Scholes model with change-point.

Key words and phrases: f-divergence, exponential Levy models, change-point, optimal portfolio

MSC 2010 subject classifications: 60G46, 60G48, 60G51, 91B70

We show how to solve optimization problems in the presence of proportional transaction costs by determining a shadow price, which is a solution to the dual problem. Put differently, this is a fictitious frictionless market evolving within the bid-ask spread, that leads to the same optimization problem as in the original market with transaction costs. In addition, we also discuss how to obtain asymptotic expansions of arbitrary order for small transaction costs. This is joint work with Stefan Gerhold, Paolo Guasoni, and Walter Schachermayer.

We present three examples of credit products whose valuation poses challenging modeling problems related to armageddon scenarios and extreme losses, analyzing their behaviour pre- and in-crisis.

The products are Credit Index Options (CIOs), Collateralized Debt Obligations (CDOs), and Credit Valuation Adjustment (CVA) related products. We show that poor mathematical treatment of possibly vanishing numeraires in CIOs and lack of modes in the tail of the loss distribution in CDOs may lead to inaccurate valuation, both pre- and especially in crisis. We also consider the limits of copula models in trying to represent systemic risk in credit intensity models. We finally enlarge the picture and comment on a number of common biases in the public perception of modeling in relationship with the crisis.

Non-linear backward stochastic differential equations (BSDEs in

short) were firstly introduced by Pardoux and Peng (\cite{PP1990},

1990), who proved the existence and uniqueness of the adapted solution, under smooth square integrability assumptions on the coefficient and the terminal condition, and when the coefficient $g(t,\omega ,y,z)$ is Lipschitz in $(y,z)$ uniformly in $(t,\omega

)$. From then on, the theory of backward stochastic differential equations (BSDE) has been widely and rapidly developed. And many problems in mathematical finance can be treated as BSDEs. The natural connection between BSDE and partial differential equations (PDE) of parabolic and elliptic types is also important applications. In this talk, we study a new developement of BSDE, 

BSDE with contraint and reflecting barrier.

The existence and uniqueness results are presented and we will give some application of this kind of BSDE at last.

In the Heston stochastic volatility model, the variance process is given by a mean-reverting square-root process. It is known that its transition probability density can be represented by a non-central chi-square density. There are fundamental differences in the behaviour of the variance process depending on the number of degrees of freedom: if the number of degrees of freedom is larger or equal to 2, the zero boundary is unattainable; if it is smaller than 2, the zero boundary is attracting and attainable.

We focus on the attainable zero boundary case and in particular the case when the number of degrees of freedom is smaller than 1, typical in foreign exchange markets. We prove a new representation for the density based on powers of generalized Gaussian random variables. Further we prove that Marsaglia's polar method extends to the generalized Gaussian distribution, providing an exact and efficient method for generalized Gaussian sampling. Thus, we establish a new exact and efficient method for simulating the Cox-Ingersoll-Ross process for an attracting and attainable zero boundary, and thus establish a new simple method for simulating the Heston model.

We demonstrate our method in the computation of option prices for parameter cases that are described in the literature as challenging and practically relevant.

The static two price economy of conic finance is first employed to

define capital, profit, and subsequently return and leverage. Examples

illustrate how profits are negative on claims taking exposure to loss

and positive on claims taking gain exposure. It is argued that though

markets do not have preferences or objectives of their own, competitive

pressures lead markets to become capital minimizers or leverage

maximizers. Yet within a static context one observes that hedging

strategies must then depart from delta hedging and incorporate gamma

adjustments. Finally these ideas are generalized to a dynamic context

where for dynamic conic finance, the bid and ask price sequences are

seen as nonlinear expectation operators associated with the solution of

particular backward stochastic difference equations (BSDE) solved in

discrete time at particular tenors leading to tenor specific or

equivalently liquidity contingent pricing. The drivers of the associated

BSDEs are exhibited in complete detail.

We present theory and numerics of affine processes and several of their applications in finance. The theory is appealing due to methods from probability theory, analysis and geometry. Applications are diverse since affine processes combine analytical tractability with a high flexibility to model stylized facts like heavy tails or stochastic volatility.

In this presentation we discuss the Heston model with stochastic interest rates driven by Hull-White or Cox-Ingersoll-Ross processes.

We present approximations in the Heston-Hull-White hybrid model, so that a characteristic function can be derived and derivative pricing can be efficiently done using the Fourier Cosine expansion technique.

This pricing method, called the COS method, is explained in some detail.

We furthermore discuss the effect of the approximations in the hybrid model on the instantaneous correlations, and check the influence of the correlation between stock and interest rate on the implied volatilities.

Abstract: describe several results on the convergence of approximation schemes for possibly degenerate, linear or nonlinear parabolic equations which apply in particular to equations arising in option pricing or portfolio management. We address both the questions of the convergence and the rate of convergence.

The talk will be devoted to criteria of absence of arbitrage opportunities under small transaction costs for a family of multi-asset models of financial market.

In this talk a number of identities will be discussed that relate to the event of level crossing of certain types of stochastic processes. Some of these identities are surprisingly simple and have interpretations in surplus modelling of insurance portfolios, the design of taxation schemes, optimal dividend strategies and the pricing of barrier options.

A stock loan is a contract between two parties: the lender, usually a bank or other financial institution providing a loan, and the borrower, represented by a client who owns one share of a stock used as collateral for the loan. Several reasons might motivate the client to get into such a deal. For example he might not want to sell his stock or even face selling restrictions, while at the same time being in need of available funds to attend to another financial operation. In Xia and Zhou (2007), a stock loan is modeled as a perpetual American option with a time varying strike and analyzed in detail within the Black-Scholes 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 obtain an exact formula for the value of the loan fee to be charged by the bank based on a result in Henderson (2007). For loans of finite maturity, we characterize its value using a variational inequality first presented in Oberman and Zariphopoulou (2003). In both cases we show analytically how the fee varies with the model parameters and illustrate the results numerically. This is joint work with Cesar G. Velez (Universidad Nacional de Colombia).

The theory and computation of convex measures of financial risk has been a very active area of Financial Mathematics, with a rich history in a short number of years. The axioms specify sensible properties that measures of risk should possess (and which the industry's favourite, value-at-risk, does not). The most common example is related to the expectation of an exponential utility function.

A basic application is hedging, that is taking off-setting positions, to optimally reduce the risk measure of a portfolio. In standard continuous-time models with dynamic hedging, this leads to nonlinear PDE problems of HJB type. We discuss so-called static-dynamic hedging of exotic options under convex risk measures, and specifically the existence and uniqueness of an optimal position. We illustrate the computational challenge when we move away from the risk measure associated with exponential utility.

Joint work with Aytac Ilhan (Goldman Sachs) and Mattias Jonsson (University of Michigan).

We introduce and study ergodic multidimensional diffusion processes interacting through their ranks; these interactions lead to invariant measures which are in broad agreement with stability properties of large equity markets over long time-periods.

The models we develop assign growth rates and variances that depend on both the name (identity) and the rank (according to capitalization) of each individual asset.

Such models are able realistically to capture critical features of the observed stability of capital distribution over the past century, all the while being simple enough to allow for rather detailed analytical study.

The methodologies used in this study touch upon the question of triple points for systems of interacting diffusions; in particular, some choices of parameters may permit triple (or higher-order) collisions to occur. We show, however, that such multiple collisions have no effect on any of the stability properties of the resulting system. This is accomplished through a detailed analysis of intersection local times.

The theory we develop has connections with the analysis of Queueing Networks in heavy traffic, as well as with models of competing particle systems in Statistical Mechanics, such as the Sherrington-Kirkpatrick model for spin-glasses.

"We investigate a construction of a Skorokhod embedding due to Root (1969), which has been the subject of recent interest for applications in Mathematical Finance (Dupire, Carr & Lee), where the construction has applications for model-free pricing and hedging of variance derivatives. In this context, there are two related questions: firstly of the construction of the stopping time, which is related to a free boundary problem, and in this direction, we expand on work of Dupire and Carr & Lee; secondly of the optimality of the construction, which is originally due to Rost (1976). In the financial context, optimality is connected to the construction of hedging strategies, and by giving a novel proof of the optimality of the Root construction, we are able to identify model-free hedging strategies for variance derivatives. Finally, we will present some evidence on the numerical performance of such hedges. (Joint work with Jiajie Wang)"