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- Chebyshev Interpolation for Parametric Option Pricing | Applied Financial Mathematics

Pricing 28 May 2015 Kategorie Research Seminars TU Berlin Room MA 041 Straße des 17 Juni 136 10623 Berlin 5 p m Kathrin Glau Technische Universität München Function approximation with Chebyshev polynomials is a well established and thoroughly investigated method within the field of numerical analysis The method enjoys attractive convergence properties and its implementation is straightforward We propose to apply tensorized Chebyshev interpolation to computing Parametric Option Prices POP This allows us to exploit the recurrent nature of the pricing problem in an efficient reliable and general way For a large variety of option types and affine asset models we prove that the convergence rate of the method is exponential if there is a single varying parameter and of any arbitrary polynomial order in the multivariate case Numerical experiments confirm these findings and show that the method achieves a significant gain in efficiency Our numerical examples include European and path dependent basket options in Levy models and the Heston model News Computer klüger als der Mensch Das vergessene Werkzeug der Ökonomie Research Projects funded under HU s strategic partnership programs with Princeton University and NUS Young researcher worksho p d fine job opportunities d fine continuously offers job opportunities

Original URL path: http://horst.qfl-berlin.de/chebyshev-interpolation-parametric-option-pricing (2016-04-24)

Open archived version from archive - Central Limit Theorem for additive functionals of some Markov processes: anomalous results | Applied Financial Mathematics

mathbb R with unique invariant probability mu and some additive functional S t sum k 1 t f X k or S t int 0 t f X s ds for some mu centered f If f in mathbb L 2 mu the expected appropriate normalization is sqrt Var S t expected to be of order sqrt t and the expected limit is then a standard gaussian If f in mathbb L p mu 1 p 2 one expects in some cases some stable limit after appropriate normalization It turns out that the mixing rate of the process equivalently the rate of convergence to equilibrium is of particular importance for these results to hold true We shall recall some of the main recent and less recent results in this direction and explain how the mixing rate enters into the game We shall also discuss a particular class of examples for which depending on whether the convergence to equilibrium is quick enough or not anomalous limit with some variance breaking or anomalous normalization appear At the level of the invariance principle instead of the simple CLT theorem the expected limiting process becomes a fractional Brownian motion instead of the usual one

Original URL path: http://horst.qfl-berlin.de/central-limit-theorem-additive-functionals-some-markov-processes-anomalous-results (2016-04-24)

Open archived version from archive - Moral Hazard in Dynamic Risk Management | Applied Financial Mathematics

17 Juni 136 10623 Berlin 4 p m Dylan Possamai Université Paris Dauphine CEREMADE We consider a contracting problem in which a principal hires an agent to manage a risky project When the agent chooses volatility components of the output process and the principal observes the output continuously the principal can compute the quadratic variation of the output but not the individual components This leads to moral hazard with respect to the risk choices of the agent Using a very recent theory of singular changes of measures for Ito processes we formulate the principal agent problem in this context and solve it in the case of CARA preferences In that case the optimal contract is linear in these factors the contractible sources of risk including the output the quadratic variation of the output and the cross variations between the output and the contractible risk sources Thus path dependent contracts naturally arise when there is moral hazard with respect to risk management This is a joint work with Nizar Touzi CMAP Ecole Polytechnique and Jaksa Cvitanic Caltech News Computer klüger als der Mensch Das vergessene Werkzeug der Ökonomie Research Projects funded under HU s strategic partnership programs with Princeton University and

Original URL path: http://horst.qfl-berlin.de/moral-hazard-dynamic-risk-management (2016-04-24)

Open archived version from archive - The KPZ (Kardar-Parisi-Zhang) equation and its universality class | Applied Financial Mathematics

m Herbert Spohn TU München The one dimensional KPZ equation is a stochastic PDE which describes the dynamics of surface growth It is one representative of a much larger universality class I will discuss a few models in this class and explain how they are connected They all share the common feature to be stochastic integrable News Computer klüger als der Mensch Das vergessene Werkzeug der Ökonomie Research Projects funded

Original URL path: http://horst.qfl-berlin.de/kpz-kardar-parisi-zhang-equation-and-its-universality-class (2016-04-24)

Open archived version from archive - Pricing under Rough Volatility | Applied Financial Mathematics

MA 041 Straße des 17 Juni 136 10623 Berlin 5 p m Christian Bayer WIAS Berlin From an analysis of the time series of volatility using recent high frequency data Gatheral Jaisson and Rosenbaum SSRN 2509457 2014 showed that log volatility behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0 1 at any reasonable time scale The resulting Rough Fractional Stochastic Volatility RFSV model is remarkably consistent with financial time series data We now show how the RFSV model can be used to price claims on both the underlying and integrated volatility We analyze in detail a simple case of this model the rBergomi model In particular we find that the rBergomi model fits the SPX volatility markedly better than conventional Markovian stochastic volatility models and with fewer parameters Finally we show that actual SPX variance swap curves seem to be consistent with model forecasts with particular dramatic examples from the weekend of the collapse of Lehman Brothers and the Flash Crash News Computer klüger als der Mensch Das vergessene Werkzeug der Ökonomie Research Projects funded under HU s strategic partnership programs with Princeton University and NUS Young researcher worksho p d fine job opportunities

Original URL path: http://horst.qfl-berlin.de/pricing-under-rough-volatility (2016-04-24)

Open archived version from archive - Sensitivity of Optimal Comsumption Streams | Applied Financial Mathematics

m Martin Herdegen ETH Zürich We study the sensitivity of optimal consumption streams with respect to perturbations of the random endowment We show that to the leading order any consumption correction for the perturbed endowment is still optimal as long as the budget constraint is binding More importantly we also establish the optimal correction at the next to leading order This can be computed in two steps First one has to find the optimal correction for a deterministic perturbation This only involves the risk tolerance process of the unperturbed problem and yields a risk tolerance martingale If the risk tolerance process is deterministic e g in the case of a deterministic unperturbed endowment the latter is constant In a second step one can then calculate the optimal correction for any random perturbation This is given by an explicit formula containing only the conditional expectation of the terminal cumulative perturbation under an equivalent measure induced by the risk tolerance martingale the risk tolerance martingale and the risk tolerance process itself Talk based on a joint work with Johannes Muhle Karbe ETH Zürich News Computer klüger als der Mensch Das vergessene Werkzeug der Ökonomie Research Projects funded under HU s strategic partnership

Original URL path: http://horst.qfl-berlin.de/sensitivity-optimal-comsumption-streams (2016-04-24)

Open archived version from archive - Markovian stochastic control with f-expectation | Applied Financial Mathematics

This requires some special techniques of stochastic analysis and backward stochastic differential equations to handle the difficulties arising from the nonlinearity of the expectation Using this result and properties of reflected backward stochastic differential equations we prove that the value function of our mixed control problem is a viscosity solution of a nonlinear Hamilton Jacobi Bellman variational inequality Uniqueness of the viscosity solution is obtained under additional assumptions Illustrating examples

Original URL path: http://horst.qfl-berlin.de/tba-10 (2016-04-24)

Open archived version from archive - Asymptotic indifference pricing in Lévy models | Applied Financial Mathematics

in practice due to the high computational cost of solving the non linear partial integro differential equation associated to the indifference price In this work we develop closed form approximations to exponential utility indifference prices in exponential Lévy models To this end we first establish a new non asymptotic approximation of the indifference price which extends earlier results on small risk aversion asymptotics of this quantity Next we use this formula to derive a closed form approximation of the indifference price by treating the Lévy model as a perturbation of the Black Scholes model This extends the methodology introduced in a recent paper for smooth linear functionals of Lévy processes Ales C erny Stephan Denkl and Jan Kallsen Hedging in Le vy models and the time step equivalent of jumps ArXiv September 2013 to nonlinear and non smooth functionals Our closed formula represents the indifference price as a linear combination of the Black Scholes price and correction terms which depend on tractable characteristics of the underlying Lévy process such as skewness and kurtosis and the derivatives of the Black Scholes price As a by product we obtain a simple explicit formula for the spread between the buyer s and the

Original URL path: http://horst.qfl-berlin.de/tba-13 (2016-04-24)

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