Minicourses
Damiano Brigo: Nonlinear valuation under credit gap risk, collateral initial and
variation margins, funding costs and multiple curves
The market for financial products and derivatives
reached an outstanding notional size of 708 USD Trillions in 2011,
amounting to ten times the planet gross domestic product. Even
discounting double counting, derivatives appear to be an important
part of the world economy and have played a key role in the onset of
the financial crisis in 2007.
After briefly reviewing the
Nobel-awarded option pricing paradigm by Black Scholes and Merton,
hinting at precursors such as Bachelier and DeFinetti, we explain how
the self-financing condition and Ito's formula lead to the Black
Scholes Partial Differential Equation (PDE) for basic option payoffs.
We hint at the Feynman Kac theorem and explain how the
risk neutral measure of no arbitrage theory may be related to this.
Following this quick introduction, we describe the changes triggered
by post 2007 events. We re-discuss the valuation theory assumptions
and introduce valuation under counterparty credit risk, collateral
posting, initial and variation margins, and funding costs. We explain
model dependence induced by credit effects, hybrid features,
contagion, payout uncertainty, and nonlinear effects due to
replacement closeout at default and possibly asymmetric borrowing and
lending rates in the margin interest and in the funding strategy for
the hedge of the relevant portfolio. Nonlinearity manifests itself in
the valuation equations taking the form of semi-linear PDEs or
Backward SDEs. We discuss existence and uniqueness of solutions for
these equations.
We present an invariance theorem showing that the
final valuation equations do not depend on unobservable risk free
rates, that become purely instrumental variables. Valuation is thus
based only on real market rates and processes. We also present a high
level analysis of the consequences of nonlinearities, both from the
point of view of methodology and from an operational angle, including
deal/entity/aggregation dependent valuation probability measures and
the role of banks treasuries.
Finally, we hint at how one may connect
these developments to interest rate theory under multiple discount
curves, thus building a consistent valuation framework encompassing
most post-2007 effects. (slides of the course)
Extra
slides of Fields institute lectures
Video
Fields institute
seminar video
Papers
[1] Damiano Brigo, Counterparty Risk FAQ: Credit VaR, PFE, CVA, DVA, Closeout, Netting, Collateral, Re-Hypothecation, WWR, Basel, Funding, CCDS and Margin Lending
[2] Andrea Pallavicini, Daniele Perini, Damiano Brigo, Funding, Collateral and Hedging: Uncovering the Mechanics and the Subtleties of Funding Valuation Adjustments
[3] Andrea Pallavicini, Damiano Brigo, Interest-Rate Modelling in Collateralized Markets: Multiple Curves, Credit-Liquidity Effects, CCPs
[4] Damiano Brigo, Andrea Pallavicini, CCP Cleared or Bilateral CSA Trades with Initial/Variation Margins Under Credit, Funding and Wrong-Way Risks: A Unified Valuation Approach
[5] Damiano Brigo, Qing Daphne Liu, Andrea Pallavicini, David Sloth, Nonlinear Valuation Under Collateral, Credit Risk and Funding Costs: A Numerical Case Study Extending Black-Scholes
Ludger Rüschendorf: Dependence, risk bounds, optimal allocations and portfolios
The main focus in this course is on the description of the influence of dependence in multivariate stochastic models for risk vectors. In particular we are interested in the description of the impact of dependence on the formulation of risk bounds, on the range of portfolio risk measures on problems of optimal risk allocation (diversification), and the construction of optimal portfolios.
In more detail:
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We will point out general methodological tools for dependence modeling and analysis. In particular we discuss Hoeffding-Fréchet bounds and mass transportation and their impact on the representation of risk measures for portfolio vectors, on the characterization of worst case dependence structures, and, in particular, on possible generalizations of comonotonic dependence structures.
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We introduce to recent developments on obtaining (sharp) risk bounds for the Value at Risk and other risk functional of joint portfolios. A field of active recent development is the inclusion of partial dependence information to obtain improved risk bounds. In particular we also consider the question of model risk as, for example, apparent in several of the popular and much used credit risk models.
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We give an introduction to the use of positive resp. negative dependence in order to construct improved (optimal) risk allocations and optimal portfolios. Applications to real data show the considerable potential of these relatively recent construction methods.
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Based on extreme value theory the notion of extreme risk index (ERI) is introduced and based on it a new method of portfolio optimization is introduced taking into account the inherent dependence between the data. This ERI based approach is compared with the Markowitz approach in an empirical study of S&P500 data.
References
[1]
Rüschendorf, L.: Mathematical Risk Analysis, Springer (2013).
Other papers on Financial mathematics and risk measures.
Special invited lectures
Christian Bender:
Primal-dual Monte-Carlo methods for nonlinear pricing problems
Nonlinear option pricing problems arise e.g. in the presence of early exercise features, model risk, or credit/funding value adjustments. Many of these problem lead (possibly after a time discretization is performed) to a backward dynamic program. In this lecture we first review the primal-dual method of Andersen and Broadie for the construction of confidence intervals on the price of a Bermudan option by Monte Carlo simulation. We then explain how this approach can be extended to a wide class of convex dynamic programs.The results are illustrated by various numerical examples including European option pricing under uncertain volatility and Bermudan option pricing under funding risk. (slides)
References
[1] Andersen and Broadie (2004), Primal-Dual Simulation Algorithm for Pricing
Multidimensional American Options. Management Sci. 50, 1222-1234.
[2] Bender, Schweizer, and Zhou (2014), A Primal Dual Algorithm for BSDEs. preprint.
[3] Rogers (2002), Monte Carlo Valuation of American Options. Math. Finance 12, 271-286.
Freddy Delbaen:
Monetary utility functions with the CxLS (convex level sets) property
Monetary utility functions are -- except for the expected value -- not of von Neumann-Morgenstern type. In case the utility function has convex level sets in the set of probability measures on the real line, we can give some characterisation that comes close to the vN-M form. For coherent utility functions this was solved by Ziegel. The general concave case under the extra assumptions of weak compactness, was solved by Stephan Weber. In the general case the utility functions are only semi continuous. Using the fact that law determined utility functions are monotone with respect to convex ordering, we can overcome most of the technical problems. The characterisation is similar to Weber's theorem except that we need vN-M utility functions that take the value $-\infty$. Having convex level sets can be seen as a weakened form of the independence axiom in the vN-M theorem.
Matheus Grasselli:
A stock-flow consistent macroeconomic model for asset price bubbles
In this talk I first describe a stock-flow consistent model for an economy with households, firms, and
banks in the form of a three-dimensional dynamical system for wages, employment, and firm debt. This is
then extended by a fourth variable representing the flow of borrowing that is used purely for speculation on
an existing financial asset, rather than productive capital investment. Finally, the system is augmented by introducing
a price dynamics for the financial asset in the form of a standard geometric Brownian motion plus a downward
jump modelled as a non-homogenous Poisson process whose intensity is an increasing function of the
speculative ratio. The compensator for this downward jump then leads to the super-exponential growth characteristic
of asset price bubbles. Moreover, when the bubble bursts the cost of borrowing in the real economy increases, leading
to a feedback mechanism from the asset price dynamics to the original system. This is joint work with Bernardo Costa Lima
and Adrien Nguyen Huu. (slides)
Short lectures
Kees de Graaf:
Efficient computation of CVA sensitivities in the finite difference Monte-Carlo method for portfolios of FX-options
In this talk, we discuss the recently developed efficient method to measure the Expected Exposure (EE) over the lifetime of options. This is done by combining the scenario generation of the Monte Carlo (MC) method with the option valuation based on solving a Partial Differential Equation (PDE) on a grid. The results are accurate and computationally efficient. After introducing the method we will show how to extend this method by applying it on multiple options and show how to compute first (delta) and second order (gamma) sensitivities. Our results show that the method is accurate and computationally more efficient than a traditional bump and revalue routine. As a first application we asses the impact on CVA, delta and gamma of adding a barrier option to a portfolio of a call and a put option.
Reference
Cornelis S.L. De Graaf, Drona Kandhai, Peter M.A. Sloot,
Efficient Estimation of Sensitivities for Counterparty Credit Risk with the Finite Difference Monte-Carlo Method.
Shashi Jain:
The Stochastic Grid Bundling Method: Application to exposure calculations
Since the credit crisis of 2008, managing counterparty credit risk has become an integrated part of derivative trading desks' day-to-day activities. Banks use Monte Carlo methods to simulate the future values of the portfolio of derivatives with a counterparty. For derivatives that are priced using Monte Carlo methods might require a nested simulation in order to compute their values for all the future scenarios.
In this talk we present the Stochastic Grid Bundling Method (SGBM) - as an alternative to the traditional Monte Carlo based option pricing methods - to efficiently simulate the future values of such derivatives, while avoiding expensive nested simulations. We analyze results computed using SGBM for Bermudan and European options (such as swaptions, and basket of equity options).
Reference
Shashi Jain, Cornelis W. Oosterlee, The Stochastic Grid Bundling Method: Efficient Pricing of Bermudan Options and their Greeks.
Andrei Lalu:
Asset returns with self-exciting jumps: option pricing and time-varying jump risk premia
We develop a semi-closed-form option pricing approach in the context of a parametric model for asset returns with clustered jumps. The stochastic jump intensity in the model self-excites as a result of jumps occurring, so the model can accommodate jump clustering, a phenomenon which is empirically relevant when markets are in turmoil. Assuming an arbitrage free pricing kernel, we develop a procedure to filter out the latent state variables and estimate model parameters via the generalized method of moments. The moment conditions are based on the model's conditional characteristic function. Using a long time-series of S&P 500 options prices we estimate model parameters and conduct inference on the variance- and jump- risk premiums specified in the pricing kernel. We find strong evidence in favor of self-excitation in the jump intensity process for the S&P 500 index. Lastly, we find that the in-sample and out-of-sample option pricing performance of our model exceeds that of alternative models with time-varying jump intensity specifications.
Daniël Linders:
Basket option pricing and implied correlation in a Lévy copula model
We introduce the Lévy copula model and consider the problem of finding accurate approximations for the price of a basket option. The basket is a weighted sum of dependent stock prices and its distribution function is unknown or too complex to work with. Therefore, we replace the random variable describing the basket price at maturity by a random variable with a more simple structure. Moreover, the Carr-Madan formula can be used to determine approximate basket option prices.
In a second part of the paper we show how implied volatility and implied correlation can be defined in our Lévy copula model. In our model, each stock price is described by a volatility parameter and the marginal parameters can be calibrated separately from the correlation parameters using single name option prices. However, the available market prices for basket options together with our newly designed basket option pricing formula enables us to determine implied Lévy correlation estimates. We observe that implied correlation depends on the strike and the so-called implied Lévy correlation smile is flatter than its Gaussian counterpart. The standard technique to price non-traded basket options (or other multi-asset derivatives), is by interpolating on the implied correlation curve. In the Gaussian copula model, this can sometimes lead to non-meaningful correlation values. We show that the Lévy version of the implied correlation solves (at least to some extent) this problem. Joint work with Wim Schoutens. (slides)
Reference
Daniël Linders, Wim Schoutens (2015),
Basket option pricing and implied correlation in a
Lévy copula model.
Poster presentation
Matteo Burzoni (Università degli studi di Milano): On Robustness of Arbitrage under Model Uncertainty
In a model independent discrete time financial market (i.e. with no reference probability measures a priori fixed),
we discuss the notion of Arbitrage and we investigate the richness of the family of probability measures for which
the price process is a martingale. We show how different notions of Arbitrage can be studied under the same
general framework, in particular, by specifying a class of non-negligible sets S. Properties of elements of the class
S reflect into intrinsic properties of the class of polar sets with respect to the set of martingale measures.
In particular for S being the open sets we show that the absence of arbitrage opportunities, with respect to an opportune
filtration enlargement, guarantees the existence of full support martingale measures. We also provide a dual representation
in terms of weakly open sets of probability measures, which highlights the robust nature of our approach.
Based on a joint work with M. Frittelli, M. Maggis.
Reference
Matteo Burzoni, Marco Frittelli, Marco Maggis (2014), Robust Arbitrage under Uncertainty in Discrete Time.
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