Functional vs Banach space stochastic calculus & strong-viscosity solutions to semilinear parabolic path-dependent PDEs.
|
|
- Junior Holt
- 5 years ago
- Views:
Transcription
1 Functional vs Banach space stochastic calculus & strong-viscosity solutions to semilinear parabolic path-dependent PDEs Andrea Cosso LPMA, Université Paris Diderot joint work with Francesco Russo ENSTA, ParisTech 7th General AMaMeF and Swissquote Conference September 8th, 2015
2 Outline 1 Functional vs Banach space stochastic calculus Functional Itô calculus via regularization Comparing the two approaches 2 Path-dependent PDE Path-dependent SDE Strict solutions 3 Strong-viscosity solutions Towards a weaker notion of solution Definition, existence and uniqueness
3 Outline 1 Functional vs Banach space stochastic calculus Functional Itô calculus via regularization Comparing the two approaches 2 Path-dependent PDE Path-dependent SDE Strict solutions 3 Strong-viscosity solutions Towards a weaker notion of solution Definition, existence and uniqueness
4 Functional Itô & Banach space stochastic calculus Functional Itô calculus is an extension of classical Itô calculus designed ad hoc for functionals F (t, X +t, X t ) depending on time t, past and present values of the process X. B. Dupire (2009) Functional Itô calculus. Portfolio Research Paper, Bloomberg. R. Cont & D.-A. Fournié (2013) Functional Itô calculus and stochastic integral representation of martingales. Annals of Probability, 41 (1), Banach space stochastic calculus also gives an expansion of F (t, X +t, X t ), but considering the path X +t = (X s+t ) s [ T,0] as an element of the Banach space B = C([ T, 0]) (B can be a generic separable Banach space). C. Di Girolami & F. Russo (2010) Infinite dimensional stochastic calculus via regularization and applications. Ph.D. Thesis, preprint inria C. Di Girolami & F. Russo (2014) Generalized covariation for Banach space valued processes, Itô formula and applications. Osaka J. Math., 51 (3),
5 First step: functional Itô calculus via regularization Using regularization techniques, instead of discretization techniques of Föllmer type, is not the only issue. We also investigate other possible improvements of functional Itô calculus: To define functional derivatives, we do not need to extend a functional from C([ T, 0]) to D([ T, 0]), but to a space C ([ T, 0]) which gets stuck as much as possible to the natural space C([ T, 0]). Time and path plays two distinct roles in our setting. = we define the horizontal derivative independently of the time derivative.
6 The space C ([ T, 0]): motivation In the classical literature on path-dependent SDEs, it is usual to consider the state space L 2 ([ T, 0]) R( past present Here we take where C b ([ T, 0[) R( past present C b ([ T, 0[) = { f : [ T, 0[ R: f is bounded and continuous}.
7 Definition Definition C ([ T, 0]) : set of bounded functions η : [ T, 0] R continuous on [ T, 0[, equipped with an inductive topology which induces the following convergence η n n in C ([ T,0]) η if: (i) η n C. (ii) sup t K η n (t) η(t) 0, compact set K [ T, 0[. (iii) η n (0) η(0).
8 Remarks C([ T, 0]) is dense in C ([ T, 0]), when endowed with the topology of C ([ T, 0]). Examples of continuous functionals: (a) U(η) = g(η(t 1 ),..., η(t n )), with T t 1 < < t n 0 and g : R n R continuous. (b) U(η) = [ T,0] ϕ(t)d η(t), with ϕ: [ T, 0] R a càdlàg bounded variation function. The functional U(η) = sup t [ T,0] η(t) is not continuous.
9 Functional derivatives Definition Let u: C ([ T, 0]) R and η C ([ T, 0]). (i) Horizontal derivative at η: D H u(η) u(η( )1 [ T,0[ + η(0)1 {0} ) u(η( ε)1 [ T,0[ + η(0)1 {0} ) := lim. ε 0 + ε (ii) First-order vertical derivative at η: D V u(η) := a ũ(η [ T,0[, η(0)). (iii) Second-order vertical derivative at η: D V V u(η) := 2 aaũ(η [ T,0[, η(0)). ũ(γ, a) := u(γ1 [ T,0[ + a1 {0} ), (γ, a) C b ([ T, 0[) R.
10 The space C 1,2 (([0, T ] past) present) Definition U : [0, T ] C([ T, 0]) R is in C 1,2 (([0, T ] past) present)) if: U admits a (necessarily unique) continuous extension u: [0, T ] C ([ T, 0]) R. t u, D H u, D V u, D V V u exist and are continuous. Then we define on [0, T ] C([ T, 0]): D H U := D H u, D V U := D V u, D V V U := D V V u.
11 Functional Itô s formula Theorem Let U be in C 1,2 (([0, T ] past) present) and X = (X t ) t [0,T ] be a real continuous finite quadratic variation process. U(t, X t ) = U(0, X 0 ) + + t 0 t 0 ( t U(s, X s ) + D H U(s, X s ) ) ds D V U(s, X s )d X s t 0 D V V U(s, X s )d[x] s for all 0 t T, where X = (X t ) t denotes the window process associated with X, defined by X t := {X t+s, s [ T, 0]}.
12 Comparing the two approaches Identification of the functional derivatives Our aim is to prove formulae which allow to express functional derivatives in terms of differential operators arising in the Banach space stochastic calculus. Notation: we denote by D U the Fréchet derivative of U, which can be written as D U(η)ϕ = ϕ(x)d dx U(η) = ϕ(x) ( Ddx U(η)+Dδ 0 U(η)δ 0 (dx) ) [ T,0] [ T,0] for some uniquely determined finite signed Borel measure D dx U(η) on [ T, 0]. Vertical derivative Horizontal derivative D V U(η) = D δ 0 U(η) D H U(η) =?
13 Identification of D H U: definition of χ 0 χ 0 subspace of M([ T, 0] 2 ): µ M([ T, 0] 2 ) belongs to χ 0 if µ(dx, dy) = g 1 (x, y)dxdy + λ 1 δ 0 (dx) δ 0 (dy) + g 2 (x)dx λ 2 δ 0 (dy) + λ 3 δ 0 (dx) g 3 (y)dy + g 4 (x)δ y (dx) dy, with g 1 L 2 ([ T, 0] 2 ), g 2, g 3 L 2 ([ T, 0]), g 4 L ([ T, 0]), λ 1, λ 2, λ 3 R.
14 Identification of D H U Theorem Let η C([ T, 0]) be such that the quadratic variation [η] on [ T, 0] exists. Let U : C([ T, 0]) R be C 2 -Fréchet such that: (i) D 2 U : C([ T, 0]) χ 0. (ii) Dx 2,Diag U(η) ( g 4 ) has a [η]-zero set of discontinuity (e.g., if it is countable). (iii) There exist continuous extensions of U and D 2 dx dy U u: C ([ T, 0]) R, D 2 dx dy u: C ([ T, 0]) χ 0. (iv) The horizontal derivative D H U(η) exists at η. Then D H U(η) = [ T,0] D dx U(η)d+ η(x) 1 2 [ T,0] Dx 2,Diag U(η)d[η](x)
15 Outline 1 Functional vs Banach space stochastic calculus Functional Itô calculus via regularization Comparing the two approaches 2 Path-dependent PDE Path-dependent SDE Strict solutions 3 Strong-viscosity solutions Towards a weaker notion of solution Definition, existence and uniqueness
16 Semilinear parabolic path-dependent PDE Consider the semilinear parabolic path-dependent PDE on [0, T ] C([ T, 0]): t U + D H U + b(t, η)d V U σ(t, η)2 D V V U + F (t, η, U, σ(t, η)d V U) = 0, U(T, η) = H(η). Standing Assumption (A). b, σ, F, H are Borel measurable functions satisfying, for some positive constants C and m, b(t, η) b(t, η ) + σ(t, η) σ(t, η ) C η η, F (t, η, y, z) F (t, η, y, z ) C ( y y + z z ), b(t, 0) + σ(t, 0) C, F (t, η, 0, 0) + H(η) C ( 1 + η m), for all t [0, T ], η, η C([ T, 0]), y, y, z, z R.
17 Path-dependent SDE For every (t, η) [0, T ] C([ T, 0]), consider the path-dependent SDE: { dx s = b(s, X s )dt + σ(s, X s )dw s, s [t, T ], X s = η(s t), s [ T + t, t]. W is a real Brownian motion on (Ω, F, P). F is the completion of the natural filtration generated by W. Proposition (t, η) [0, T ] C([ T, 0]),! (up to indistinguishability) F-adapted continuous process X t,η = (Xs t,η ) s [ T +t,t ] solution to the path-dependent SDE. Moreover, for any p 1 there exists a positive constant C p such that [ E sup s [ T +t,t ] X t,η s p] ( C p 1 + η p ).
18 Strict solutions Definition A map U in C 1,2 (([0, T [ past) present) and C([0, T ] C([ T, 0])), satisfying the path-dependent PDE, is called a strict solution. Notation S 2 (t, T ), t T, the set of real càdlàg adapted processes Y = (Y s ) t s T such that [ Y 2 := E sup Y S 2 (t,t ) s 2] <. t s T H 2 (t, T ), t T, the set of real predictable processes Z = (Z s ) t s T such that Z 2 H 2 (t,t ) := E [ T t ] Z s 2 ds <.
19 Strict solutions: Feynman-Kac formula & uniqueness Theorem Let U : [0, T ] C([ T, 0]) R be a strict solution to the path-dependent PDE, satisfying the polynomial growth condition U(t, η) C ( 1 + η m ), (t, η) [0, T ] C([ T, 0]), for some positive constant m. Then, we have where (Ys t,η, Z t,η U(t, η) = Y t,η t, (t, η) [0, T ] C([ T, 0]), s ) s = (U(s, X t,η s ), σ(s, X t,η s )D V U(s, X t,η s )1 [t,t [ (s)) s with (Y t,η, Z t,η ) S 2 (t, T ) H 2 (t, T ), is the solution to the Backward Stochastic Differential Equation (BSDE) T T Ys t,η = H(X t,η T ) + F (r, X t,η r, Yr t,η, Zr t,η )dr Zr t,η dw r. s s
20 Strict solutions: existence (I) Theorem Suppose that b, σ, F, H are cylindrical and smooth, i.e. ( b(t, η) = b ϕ 1(x + t)d η(x),..., [ t,0] ( σ(t, η) = σ ϕ 1(x + t)d η(x),..., [ t,0] ( F (t, η, y, z) = F t, ϕ 1(x + t)d η(x),..., where [ t,0] [ t,0] [ t,0] ( H(η) = H ϕ 1(x + T )d η(x),..., [ T,0] [ t,0] ) ϕ N (x + t)d η(x) [ T,0] ) ϕ N (x + t)d η(x) ) ϕ N (x + t)d η(x), y, z ) ϕ N (x + T )d η(x) (i) b, σ, F, H are continuous and satisfy Assumption (A) with x R N in place of η. (ii) b and σ are of class C 3 with partial derivatives from order 1 up to order 3 bounded.
21 Strict solutions: existence (II) Theorem (cont d) (iii) For all t [0, T ], F (t,,, ) C 3 (R N ) and moreover we assume the validity of the properties below. (a) F (t,, 0, 0) belongs to C 3 and its third order partial derivatives satisfy a polynomial growth condition uniformly in t. (b) D y F, Dz F are bounded on [0, T ] R N R R, as well as their derivatives of order one and second with respect to x 1,..., x N, y, z. (iv) H C 3 (R N ) and its third order partial derivatives satisfy a polynomial growth condition. (v) ϕ 1,..., ϕ N C 2 ([0, T ]). Then, the map U given by U(t, η) = Y t,η t, (t, η) [0, T ] C([ T, 0]), is the unique strict solution to the path-dependent PDE.
22 Outline 1 Functional vs Banach space stochastic calculus Functional Itô calculus via regularization Comparing the two approaches 2 Path-dependent PDE Path-dependent SDE Strict solutions 3 Strong-viscosity solutions Towards a weaker notion of solution Definition, existence and uniqueness
23 Towards a weaker notion of solution Consider the lookback-type payoff: H(η) = sup η(x), η C([ T, 0]). x [ T,0] In this case, we expect that the map U(t, η) = E [ H(W t,η T )], (t, η) [0, T ] C([ T, 0]) is virtually a solution to the path-dependent PDE: t U + D H U DV V U = 0, U(T, η) = H(η). Unfortunately, U is not continuous with respect to the topology of C ([ T, 0]), therefore it can not be a strict solution. U is a strong-viscosity solution.
24 Strong-viscosity solutions: introduction Various definitions of viscosity-type solutions for path-dependent PDEs have been given. We recall in particular: I. Ekren, C. Keller, N. Touzi, and J. Zhang (2014) On viscosity solutions of path dependent PDEs. Annals of Probability, 42 (1), We propose a notion of viscosity-type solution with the following peculiarities: it is a purely analytic object; it can be easily adapted to more general equations than classical partial differential equations. We call it strong-viscosity solution to distinguish it from the classical notion of viscosity solution and from the definition introduced by Ekren, Keller, Touzi, Zhang.
25 Strong-viscosity solutions: path-dependent case Path-dependent PDE on [0, T ] C([ T, 0]): t U + D H U + b(t, η)d V U σ(t, η)2 D V V U + F (t, η, U, σ(t, η)d V U) = 0, U(T, η) = H(η). Standing Assumption (A). b, σ, F, H are Borel measurable functions satisfying, for some positive constants C and m, b(t, η) b(t, η ) + σ(t, η) σ(t, η ) C η η, F (t, η, y, z) F (t, η, y, z ) C ( y y + z z ), b(t, 0) + σ(t, 0) C, F (t, η, 0, 0) + H(η) C ( 1 + η m), for all t [0, T ], η, η C([ T, 0]), y, y, z, z R.
26 Definition (I) Definition A function U : [0, T ] C([ T, 0]) R is called a strong-viscosity solution to the path-dependent PDE if there exists a sequence (U n, H n, F n, b n, σ n ) n of Borel measurable functions satisfying: (i) For some positive constants C and m, b n (t, η) b n (t, η ) + σ n (t, η) σ n (t, η ) C η η F n (t, η, y, z) F n (t, η, y, z ) C( y y + z z ) b n (t, 0) + σ n (t, 0) C U n (t, η) + H n (η) + F n (t, η, 0, 0) C ( 1 + η m ) for all t [0, T ], η, η C([ T, 0]), y, y R, z, z R. Moreover, the functions U n (t, ), H n ( ), F n (t,,, ), n N, are equicontinuous on compact sets, uniformly with respect to t [0, T ].
27 Definition (II) Definition (cont d) (ii) U n is a strict solution to t U n + D H U n + b n (t, η)d V U n σ n(t, η) 2 D V V U n + F n (t, η, U n, σ n (t, η)d V U n ) = 0, (t, η) [0, T [ C([ T, 0]), U n (T, η) = H n (η), η C([ T, 0]). (iii) (U n, H n, F n, b n, σ n ) n converges pointwise to (U, H, F, b, σ) as n.
28 Feynman-Kac formula & uniqueness Theorem Let U : [0, T ] C([ T, 0]) R be a strong-viscosity solution to the path-dependent PDE. Then, we have where (Y t,η s Y t,η s U(t, η) = Y t,η t, (t, η) [0, T ] C([ T, 0]), = U(s, X t,η s the BSDE Y t,η s, Zs t,η ) s [t,t ] S 2 (t, T ) H 2 (t, T ), with ), is the unique solution in S 2 (t, T ) H 2 (t, T ) to = H(X t,η T ) + T s F (r, X t,η r, Y t,η r T, Zr t,η )dr Zr t,η dw r, s for all t s T. In particular, there exists at most one strong-viscosity solution to the path-dependent PDE.
29 Existence Consider the path-dependent heat equation { t U + D H U DV V U = 0, (t, η) [0, T [ C([ T, 0]), U(T, η) = H(η), η C([ T, 0]). Theorem Suppose that H is continuous. Then, the map U(t, η) = E [ H(W t,η T )], for all (t, η) [0, T ] C([ T, 0]), is the unique strong-viscosity solution to the path-dependent heat equation. H can be in particular the lookback-type payoff H(η) = sup η(x), η C([ T, 0]). x [ T,0]
30 Thank you!
Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations.
Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 1/81 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type
More informationEquivalence between Semimartingales and Itô Processes
International Journal of Mathematical Analysis Vol. 9, 215, no. 16, 787-791 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ijma.215.411358 Equivalence between Semimartingales and Itô Processes
More informationPATH-PEPENDENT PARABOLIC PDES AND PATH-DEPENDENT FEYNMAN-KAC FORMULA
PATH-PEPENDENT PARABOLIC PDES AND PATH-DEPENDENT FEYNMAN-KAC FORMULA CNRS, CMAP Ecole Polytechnique Bachelier Paris, january 8 2016 Dynamic Risk Measures and Path-Dependent second order PDEs, SEFE, Fred
More informationNon-semimartingales in finance
Non-semimartingales in finance Pricing and Hedging Options with Quadratic Variation Tommi Sottinen University of Vaasa 1st Northern Triangular Seminar 9-11 March 2009, Helsinki University of Technology
More informationHedging under Arbitrage
Hedging under Arbitrage Johannes Ruf Columbia University, Department of Statistics Modeling and Managing Financial Risks January 12, 2011 Motivation Given: a frictionless market of stocks with continuous
More informationConstructing Markov models for barrier options
Constructing Markov models for barrier options Gerard Brunick joint work with Steven Shreve Department of Mathematics University of Texas at Austin Nov. 14 th, 2009 3 rd Western Conference on Mathematical
More informationSTOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL
STOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL YOUNGGEUN YOO Abstract. Ito s lemma is often used in Ito calculus to find the differentials of a stochastic process that depends on time. This paper will introduce
More informationLecture 4. Finite difference and finite element methods
Finite difference and finite element methods Lecture 4 Outline Black-Scholes equation From expectation to PDE Goal: compute the value of European option with payoff g which is the conditional expectation
More informationABOUT THE PRICING EQUATION IN FINANCE
ABOUT THE PRICING EQUATION IN FINANCE Stéphane CRÉPEY University of Evry, France stephane.crepey@univ-evry.fr AMAMEF at Vienna University of Technology 17 22 September 2007 1 We derive the pricing equation
More informationAMH4 - ADVANCED OPTION PRICING. Contents
AMH4 - ADVANCED OPTION PRICING ANDREW TULLOCH Contents 1. Theory of Option Pricing 2 2. Black-Scholes PDE Method 4 3. Martingale method 4 4. Monte Carlo methods 5 4.1. Method of antithetic variances 5
More informationFunctional Ito calculus. hedging of path-dependent options
and hedging of path-dependent options Laboratoire de Probabilités et Modèles Aléatoires CNRS - Université de Paris VI-VII and Columbia University, New York Background Hans Föllmer (1979) Calcul d Itô sans
More informationEconomathematics. Problem Sheet 1. Zbigniew Palmowski. Ws 2 dw s = 1 t
Economathematics Problem Sheet 1 Zbigniew Palmowski 1. Calculate Ee X where X is a gaussian random variable with mean µ and volatility σ >.. Verify that where W is a Wiener process. Ws dw s = 1 3 W t 3
More informationMartingale Transport, Skorokhod Embedding and Peacocks
Martingale Transport, Skorokhod Embedding and CEREMADE, Université Paris Dauphine Collaboration with Pierre Henry-Labordère, Nizar Touzi 08 July, 2014 Second young researchers meeting on BSDEs, Numerics
More informationHedging under arbitrage
Hedging under arbitrage Johannes Ruf Columbia University, Department of Statistics AnStAp10 August 12, 2010 Motivation Usually, there are several trading strategies at one s disposal to obtain a given
More informationM5MF6. Advanced Methods in Derivatives Pricing
Course: Setter: M5MF6 Dr Antoine Jacquier MSc EXAMINATIONS IN MATHEMATICS AND FINANCE DEPARTMENT OF MATHEMATICS April 2016 M5MF6 Advanced Methods in Derivatives Pricing Setter s signature...........................................
More informationSPDE and portfolio choice (joint work with M. Musiela) Princeton University. Thaleia Zariphopoulou The University of Texas at Austin
SPDE and portfolio choice (joint work with M. Musiela) Princeton University November 2007 Thaleia Zariphopoulou The University of Texas at Austin 1 Performance measurement of investment strategies 2 Market
More informationContinuous-time Stochastic Control and Optimization with Financial Applications
Huyen Pham Continuous-time Stochastic Control and Optimization with Financial Applications 4y Springer Some elements of stochastic analysis 1 1.1 Stochastic processes 1 1.1.1 Filtration and processes 1
More informationStochastic Calculus, Application of Real Analysis in Finance
, Application of Real Analysis in Finance Workshop for Young Mathematicians in Korea Seungkyu Lee Pohang University of Science and Technology August 4th, 2010 Contents 1 BINOMIAL ASSET PRICING MODEL Contents
More informationReplication and Absence of Arbitrage in Non-Semimartingale Models
Replication and Absence of Arbitrage in Non-Semimartingale Models Matematiikan päivät, Tampere, 4-5. January 2006 Tommi Sottinen University of Helsinki 4.1.2006 Outline 1. The classical pricing model:
More informationAre stylized facts irrelevant in option-pricing?
Are stylized facts irrelevant in option-pricing? Kyiv, June 19-23, 2006 Tommi Sottinen, University of Helsinki Based on a joint work No-arbitrage pricing beyond semimartingales with C. Bender, Weierstrass
More informationRisk Neutral Measures
CHPTER 4 Risk Neutral Measures Our aim in this section is to show how risk neutral measures can be used to price derivative securities. The key advantage is that under a risk neutral measure the discounted
More informationModel-independent bounds for Asian options
Model-independent bounds for Asian options A dynamic programming approach Alexander M. G. Cox 1 Sigrid Källblad 2 1 University of Bath 2 CMAP, École Polytechnique 7th General AMaMeF and Swissquote Conference
More informationLocal vs Non-local Forward Equations for Option Pricing
Local vs Non-local Forward Equations for Option Pricing Rama Cont Yu Gu Abstract When the underlying asset is a continuous martingale, call option prices solve the Dupire equation, a forward parabolic
More informationAn overview of some financial models using BSDE with enlarged filtrations
An overview of some financial models using BSDE with enlarged filtrations Anne EYRAUD-LOISEL Workshop : Enlargement of Filtrations and Applications to Finance and Insurance May 31st - June 4th, 2010, Jena
More informationUtility indifference valuation for non-smooth payoffs on a market with some non tradable assets
Utility indifference valuation for non-smooth payoffs on a market with some non tradable assets - Joint work with G. Benedetti (Paris-Dauphine, CREST) - Luciano Campi Université Paris 13, FiME and CREST
More informationValuation of derivative assets Lecture 6
Valuation of derivative assets Lecture 6 Magnus Wiktorsson September 14, 2017 Magnus Wiktorsson L6 September 14, 2017 1 / 13 Feynman-Kac representation This is the link between a class of Partial Differential
More information1.1 Basic Financial Derivatives: Forward Contracts and Options
Chapter 1 Preliminaries 1.1 Basic Financial Derivatives: Forward Contracts and Options A derivative is a financial instrument whose value depends on the values of other, more basic underlying variables
More informationMartingales & Strict Local Martingales PDE & Probability Methods INRIA, Sophia-Antipolis
Martingales & Strict Local Martingales PDE & Probability Methods INRIA, Sophia-Antipolis Philip Protter, Columbia University Based on work with Aditi Dandapani, 2016 Columbia PhD, now at ETH, Zurich March
More informationContinuous Time Finance. Tomas Björk
Continuous Time Finance Tomas Björk 1 II Stochastic Calculus Tomas Björk 2 Typical Setup Take as given the market price process, S(t), of some underlying asset. S(t) = price, at t, per unit of underlying
More informationForward Monte-Carlo Scheme for PDEs: Multi-Type Marked Branching Diffusions
Forward Monte-Carlo Scheme for PDEs: Multi-Type Marked Branching Diffusions Pierre Henry-Labordère 1 1 Global markets Quantitative Research, SOCIÉTÉ GÉNÉRALE Outline 1 Introduction 2 Semi-linear PDEs 3
More informationLimit Theorems for the Empirical Distribution Function of Scaled Increments of Itô Semimartingales at high frequencies
Limit Theorems for the Empirical Distribution Function of Scaled Increments of Itô Semimartingales at high frequencies George Tauchen Duke University Viktor Todorov Northwestern University 2013 Motivation
More informationMinimal Variance Hedging in Large Financial Markets: random fields approach
Minimal Variance Hedging in Large Financial Markets: random fields approach Giulia Di Nunno Third AMaMeF Conference: Advances in Mathematical Finance Pitesti, May 5-1 28 based on a work in progress with
More informationRohini Kumar. Statistics and Applied Probability, UCSB (Joint work with J. Feng and J.-P. Fouque)
Small time asymptotics for fast mean-reverting stochastic volatility models Statistics and Applied Probability, UCSB (Joint work with J. Feng and J.-P. Fouque) March 11, 2011 Frontier Probability Days,
More informationModel-independent bounds for Asian options
Model-independent bounds for Asian options A dynamic programming approach Alexander M. G. Cox 1 Sigrid Källblad 2 1 University of Bath 2 CMAP, École Polytechnique University of Michigan, 2nd December,
More informationHow to hedge Asian options in fractional Black-Scholes model
How to hedge Asian options in fractional Black-Scholes model Heikki ikanmäki Jena, March 29, 211 Fractional Lévy processes 1/36 Outline of the talk 1. Introduction 2. Main results 3. Methodology 4. Conclusions
More informationLogarithmic derivatives of densities for jump processes
Logarithmic derivatives of densities for jump processes Atsushi AKEUCHI Osaka City University (JAPAN) June 3, 29 City University of Hong Kong Workshop on Stochastic Analysis and Finance (June 29 - July
More informationLocal Volatility Dynamic Models
René Carmona Bendheim Center for Finance Department of Operations Research & Financial Engineering Princeton University Columbia November 9, 27 Contents Joint work with Sergey Nadtochyi Motivation 1 Understanding
More informationVolatility Smiles and Yield Frowns
Volatility Smiles and Yield Frowns Peter Carr NYU CBOE Conference on Derivatives and Volatility, Chicago, Nov. 10, 2017 Peter Carr (NYU) Volatility Smiles and Yield Frowns 11/10/2017 1 / 33 Interest Rates
More informationBROWNIAN MOTION Antonella Basso, Martina Nardon
BROWNIAN MOTION Antonella Basso, Martina Nardon basso@unive.it, mnardon@unive.it Department of Applied Mathematics University Ca Foscari Venice Brownian motion p. 1 Brownian motion Brownian motion plays
More informationHedging of Contingent Claims under Incomplete Information
Projektbereich B Discussion Paper No. B 166 Hedging of Contingent Claims under Incomplete Information by Hans Föllmer ) Martin Schweizer ) October 199 ) Financial support by Deutsche Forschungsgemeinschaft,
More informationDrunken Birds, Brownian Motion, and Other Random Fun
Drunken Birds, Brownian Motion, and Other Random Fun Michael Perlmutter Department of Mathematics Purdue University 1 M. Perlmutter(Purdue) Brownian Motion and Martingales Outline Review of Basic Probability
More informationStochastic Dynamical Systems and SDE s. An Informal Introduction
Stochastic Dynamical Systems and SDE s An Informal Introduction Olav Kallenberg Graduate Student Seminar, April 18, 2012 1 / 33 2 / 33 Simple recursion: Deterministic system, discrete time x n+1 = f (x
More informationKing s College London
King s College London University Of London This paper is part of an examination of the College counting towards the award of a degree. Examinations are governed by the College Regulations under the authority
More informationExponential utility maximization under partial information
Exponential utility maximization under partial information Marina Santacroce Politecnico di Torino Joint work with M. Mania AMaMeF 5-1 May, 28 Pitesti, May 1th, 28 Outline Expected utility maximization
More informationNumerical schemes for SDEs
Lecture 5 Numerical schemes for SDEs Lecture Notes by Jan Palczewski Computational Finance p. 1 A Stochastic Differential Equation (SDE) is an object of the following type dx t = a(t,x t )dt + b(t,x t
More informationCONVERGENCE OF OPTION REWARDS FOR MARKOV TYPE PRICE PROCESSES MODULATED BY STOCHASTIC INDICES
CONVERGENCE OF OPTION REWARDS FOR MARKOV TYPE PRICE PROCESSES MODULATED BY STOCHASTIC INDICES D. S. SILVESTROV, H. JÖNSSON, AND F. STENBERG Abstract. A general price process represented by a two-component
More informationIntroduction to Probability Theory and Stochastic Processes for Finance Lecture Notes
Introduction to Probability Theory and Stochastic Processes for Finance Lecture Notes Fabio Trojani Department of Economics, University of St. Gallen, Switzerland Correspondence address: Fabio Trojani,
More informationOptimal asset allocation under forward performance criteria Oberwolfach, February 2007
Optimal asset allocation under forward performance criteria Oberwolfach, February 2007 Thaleia Zariphopoulou The University of Texas at Austin 1 References Indifference valuation in binomial models (with
More informationValuation of derivative assets Lecture 8
Valuation of derivative assets Lecture 8 Magnus Wiktorsson September 27, 2018 Magnus Wiktorsson L8 September 27, 2018 1 / 14 The risk neutral valuation formula Let X be contingent claim with maturity T.
More informationPath Dependent British Options
Path Dependent British Options Kristoffer J Glover (Joint work with G. Peskir and F. Samee) School of Finance and Economics University of Technology, Sydney 18th August 2009 (PDE & Mathematical Finance
More informationLecture 3: Review of mathematical finance and derivative pricing models
Lecture 3: Review of mathematical finance and derivative pricing models Xiaoguang Wang STAT 598W January 21th, 2014 (STAT 598W) Lecture 3 1 / 51 Outline 1 Some model independent definitions and principals
More informationArbitrage of the first kind and filtration enlargements in semimartingale financial models. Beatrice Acciaio
Arbitrage of the first kind and filtration enlargements in semimartingale financial models Beatrice Acciaio the London School of Economics and Political Science (based on a joint work with C. Fontana and
More informationReplication under Price Impact and Martingale Representation Property
Replication under Price Impact and Martingale Representation Property Dmitry Kramkov joint work with Sergio Pulido (Évry, Paris) Carnegie Mellon University Workshop on Equilibrium Theory, Carnegie Mellon,
More informationLast Time. Martingale inequalities Martingale convergence theorem Uniformly integrable martingales. Today s lecture: Sections 4.4.1, 5.
MATH136/STAT219 Lecture 21, November 12, 2008 p. 1/11 Last Time Martingale inequalities Martingale convergence theorem Uniformly integrable martingales Today s lecture: Sections 4.4.1, 5.3 MATH136/STAT219
More informationRMSC 4005 Stochastic Calculus for Finance and Risk. 1 Exercises. (c) Let X = {X n } n=0 be a {F n }-supermartingale. Show that.
1. EXERCISES RMSC 45 Stochastic Calculus for Finance and Risk Exercises 1 Exercises 1. (a) Let X = {X n } n= be a {F n }-martingale. Show that E(X n ) = E(X ) n N (b) Let X = {X n } n= be a {F n }-submartingale.
More informationVolatility Smiles and Yield Frowns
Volatility Smiles and Yield Frowns Peter Carr NYU IFS, Chengdu, China, July 30, 2018 Peter Carr (NYU) Volatility Smiles and Yield Frowns 7/30/2018 1 / 35 Interest Rates and Volatility Practitioners and
More informationInterest rate models in continuous time
slides for the course Interest rate theory, University of Ljubljana, 2012-13/I, part IV József Gáll University of Debrecen Nov. 2012 Jan. 2013, Ljubljana Continuous time markets General assumptions, notations
More informationFinance II. May 27, F (t, x)+αx f t x σ2 x 2 2 F F (T,x) = ln(x).
Finance II May 27, 25 1.-15. All notation should be clearly defined. Arguments should be complete and careful. 1. (a) Solve the boundary value problem F (t, x)+αx f t x + 1 2 σ2 x 2 2 F (t, x) x2 =, F
More informationConstructive martingale representation using Functional Itô Calculus: a local martingale extension
Mathematical Statistics Stockholm University Constructive martingale representation using Functional Itô Calculus: a local martingale extension Kristoffer Lindensjö Research Report 216:21 ISSN 165-377
More informationBACHELIER FINANCE SOCIETY. 4 th World Congress Tokyo, Investments and forward utilities. Thaleia Zariphopoulou The University of Texas at Austin
BACHELIER FINANCE SOCIETY 4 th World Congress Tokyo, 26 Investments and forward utilities Thaleia Zariphopoulou The University of Texas at Austin 1 Topics Utility-based measurement of performance Utilities
More informationThe stochastic calculus
Gdansk A schedule of the lecture Stochastic differential equations Ito calculus, Ito process Ornstein - Uhlenbeck (OU) process Heston model Stopping time for OU process Stochastic differential equations
More informationOptimal robust bounds for variance options and asymptotically extreme models
Optimal robust bounds for variance options and asymptotically extreme models Alexander Cox 1 Jiajie Wang 2 1 University of Bath 2 Università di Roma La Sapienza Advances in Financial Mathematics, 9th January,
More informationStochastic calculus Introduction I. Stochastic Finance. C. Azizieh VUB 1/91. C. Azizieh VUB Stochastic Finance
Stochastic Finance C. Azizieh VUB C. Azizieh VUB Stochastic Finance 1/91 Agenda of the course Stochastic calculus : introduction Black-Scholes model Interest rates models C. Azizieh VUB Stochastic Finance
More informationStochastic Partial Differential Equations and Portfolio Choice. Crete, May Thaleia Zariphopoulou
Stochastic Partial Differential Equations and Portfolio Choice Crete, May 2011 Thaleia Zariphopoulou Oxford-Man Institute and Mathematical Institute University of Oxford and Mathematics and IROM, The University
More informationMAS452/MAS6052. MAS452/MAS Turn Over SCHOOL OF MATHEMATICS AND STATISTICS. Stochastic Processes and Financial Mathematics
t r t r2 r t SCHOOL OF MATHEMATICS AND STATISTICS Stochastic Processes and Financial Mathematics Spring Semester 2017 2018 3 hours t s s tt t q st s 1 r s r t r s rts t q st s r t r r t Please leave this
More informationKing s College London
King s College London University Of London This paper is part of an examination of the College counting towards the award of a degree. Examinations are governed by the College Regulations under the authority
More informationDoubly reflected BSDEs with jumps and generalized Dynkin games
Doubly reflected BSDEs with jumps and generalized Dynkin games Roxana DUMITRESCU (University Paris Dauphine, Crest and INRIA) Joint works with M.C. Quenez (Univ. Paris Diderot) and Agnès Sulem (INRIA Paris-Rocquecourt)
More informationThe British Russian Option
The British Russian Option Kristoffer J Glover (Joint work with G. Peskir and F. Samee) School of Finance and Economics University of Technology, Sydney 25th June 2010 (6th World Congress of the BFS, Toronto)
More informationExam Quantitative Finance (35V5A1)
Exam Quantitative Finance (35V5A1) Part I: Discrete-time finance Exercise 1 (20 points) a. Provide the definition of the pricing kernel k q. Relate this pricing kernel to the set of discount factors D
More informationPAPER 27 STOCHASTIC CALCULUS AND APPLICATIONS
MATHEMATICAL TRIPOS Part III Thursday, 5 June, 214 1:3 pm to 4:3 pm PAPER 27 STOCHASTIC CALCULUS AND APPLICATIONS Attempt no more than FOUR questions. There are SIX questions in total. The questions carry
More information1 Implied Volatility from Local Volatility
Abstract We try to understand the Berestycki, Busca, and Florent () (BBF) result in the context of the work presented in Lectures and. Implied Volatility from Local Volatility. Current Plan as of March
More information25857 Interest Rate Modelling
25857 UTS Business School University of Technology Sydney Chapter 20. Change of Numeraire May 15, 2014 1/36 Chapter 20. Change of Numeraire 1 The Radon-Nikodym Derivative 2 Option Pricing under Stochastic
More informationBasic Concepts and Examples in Finance
Basic Concepts and Examples in Finance Bernardo D Auria email: bernardo.dauria@uc3m.es web: www.est.uc3m.es/bdauria July 5, 2017 ICMAT / UC3M The Financial Market The Financial Market We assume there are
More informationMESURES DE RISQUE DYNAMIQUES DYNAMIC RISK MEASURES
from BMO martingales MESURES DE RISQUE DYNAMIQUES DYNAMIC RISK MEASURES CNRS - CMAP Ecole Polytechnique March 1, 2007 1/ 45 OUTLINE from BMO martingales 1 INTRODUCTION 2 DYNAMIC RISK MEASURES Time Consistency
More informationAn Introduction to Stochastic Calculus
An Introduction to Stochastic Calculus Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 2-3 Haijun Li An Introduction to Stochastic Calculus Week 2-3 1 / 24 Outline
More informationStochastic Differential equations as applied to pricing of options
Stochastic Differential equations as applied to pricing of options By Yasin LUT Supevisor:Prof. Tuomo Kauranne December 2010 Introduction Pricing an European call option Conclusion INTRODUCTION A stochastic
More informationVariance Reduction for Monte Carlo Simulation in a Stochastic Volatility Environment
Variance Reduction for Monte Carlo Simulation in a Stochastic Volatility Environment Jean-Pierre Fouque Tracey Andrew Tullie December 11, 21 Abstract We propose a variance reduction method for Monte Carlo
More informationThe Use of Importance Sampling to Speed Up Stochastic Volatility Simulations
The Use of Importance Sampling to Speed Up Stochastic Volatility Simulations Stan Stilger June 6, 1 Fouque and Tullie use importance sampling for variance reduction in stochastic volatility simulations.
More informationApplication of Stochastic Calculus to Price a Quanto Spread
Application of Stochastic Calculus to Price a Quanto Spread Christopher Ting http://www.mysmu.edu/faculty/christophert/ Algorithmic Quantitative Finance July 15, 2017 Christopher Ting July 15, 2017 1/33
More informationThere are no predictable jumps in arbitrage-free markets
There are no predictable jumps in arbitrage-free markets Markus Pelger October 21, 2016 Abstract We model asset prices in the most general sensible form as special semimartingales. This approach allows
More informationConvergence of Discretized Stochastic (Interest Rate) Processes with Stochastic Drift Term.
Convergence of Discretized Stochastic (Interest Rate) Processes with Stochastic Drift Term. G. Deelstra F. Delbaen Free University of Brussels, Department of Mathematics, Pleinlaan 2, B-15 Brussels, Belgium
More informationParameter sensitivity of CIR process
Parameter sensitivity of CIR process Sidi Mohamed Ould Aly To cite this version: Sidi Mohamed Ould Aly. Parameter sensitivity of CIR process. Electronic Communications in Probability, Institute of Mathematical
More informationMSc Financial Engineering CHRISTMAS ASSIGNMENT: MERTON S JUMP-DIFFUSION MODEL. To be handed in by monday January 28, 2013
MSc Financial Engineering 2012-13 CHRISTMAS ASSIGNMENT: MERTON S JUMP-DIFFUSION MODEL To be handed in by monday January 28, 2013 Department EMS, Birkbeck Introduction The assignment consists of Reading
More informationSTOCHASTIC INTEGRALS
Stat 391/FinMath 346 Lecture 8 STOCHASTIC INTEGRALS X t = CONTINUOUS PROCESS θ t = PORTFOLIO: #X t HELD AT t { St : STOCK PRICE M t : MG W t : BROWNIAN MOTION DISCRETE TIME: = t < t 1
More informationComputational Finance
Path Dependent Options Computational Finance School of Mathematics 2018 The Random Walk One of the main assumption of the Black-Scholes framework is that the underlying stock price follows a random walk
More informationOn the pricing equations in local / stochastic volatility models
On the pricing equations in local / stochastic volatility models Hao Xing Fields Institute/Boston University joint work with Erhan Bayraktar, University of Michigan Kostas Kardaras, Boston University Probability
More informationOption pricing in the stochastic volatility model of Barndorff-Nielsen and Shephard
Option pricing in the stochastic volatility model of Barndorff-Nielsen and Shephard Indifference pricing and the minimal entropy martingale measure Fred Espen Benth Centre of Mathematics for Applications
More informationInsider information and arbitrage profits via enlargements of filtrations
Insider information and arbitrage profits via enlargements of filtrations Claudio Fontana Laboratoire de Probabilités et Modèles Aléatoires Université Paris Diderot XVI Workshop on Quantitative Finance
More informationChapter 3: Black-Scholes Equation and Its Numerical Evaluation
Chapter 3: Black-Scholes Equation and Its Numerical Evaluation 3.1 Itô Integral 3.1.1 Convergence in the Mean and Stieltjes Integral Definition 3.1 (Convergence in the Mean) A sequence {X n } n ln of random
More informationKØBENHAVNS UNIVERSITET (Blok 2, 2011/2012) Naturvidenskabelig kandidateksamen Continuous time finance (FinKont) TIME ALLOWED : 3 hours
This question paper consists of 3 printed pages FinKont KØBENHAVNS UNIVERSITET (Blok 2, 211/212) Naturvidenskabelig kandidateksamen Continuous time finance (FinKont) TIME ALLOWED : 3 hours This exam paper
More informationHedging Credit Derivatives in Intensity Based Models
Hedging Credit Derivatives in Intensity Based Models PETER CARR Head of Quantitative Financial Research, Bloomberg LP, New York Director of the Masters Program in Math Finance, Courant Institute, NYU Stanford
More informationS t d with probability (1 p), where
Stochastic Calculus Week 3 Topics: Towards Black-Scholes Stochastic Processes Brownian Motion Conditional Expectations Continuous-time Martingales Towards Black Scholes Suppose again that S t+δt equals
More informationL 2 -theoretical study of the relation between the LIBOR market model and the HJM model Takashi Yasuoka
Journal of Math-for-Industry, Vol. 5 (213A-2), pp. 11 16 L 2 -theoretical study of the relation between the LIBOR market model and the HJM model Takashi Yasuoka Received on November 2, 212 / Revised on
More informationParameters Estimation in Stochastic Process Model
Parameters Estimation in Stochastic Process Model A Quasi-Likelihood Approach Ziliang Li University of Maryland, College Park GEE RIT, Spring 28 Outline 1 Model Review The Big Model in Mind: Signal + Noise
More informationPortfolio optimization problem with default risk
Portfolio optimization problem with default risk M.Mazidi, A. Delavarkhalafi, A.Mokhtari mazidi.3635@gmail.com delavarkh@yazduni.ac.ir ahmokhtari20@gmail.com Faculty of Mathematics, Yazd University, P.O.
More informationThe Black-Scholes Equation using Heat Equation
The Black-Scholes Equation using Heat Equation Peter Cassar May 0, 05 Assumptions of the Black-Scholes Model We have a risk free asset given by the price process, dbt = rbt The asset price follows a geometric
More informationM.I.T Fall Practice Problems
M.I.T. 15.450-Fall 2010 Sloan School of Management Professor Leonid Kogan Practice Problems 1. Consider a 3-period model with t = 0, 1, 2, 3. There are a stock and a risk-free asset. The initial stock
More informationIn Discrete Time a Local Martingale is a Martingale under an Equivalent Probability Measure
In Discrete Time a Local Martingale is a Martingale under an Equivalent Probability Measure Yuri Kabanov 1,2 1 Laboratoire de Mathématiques, Université de Franche-Comté, 16 Route de Gray, 253 Besançon,
More informationStochastic Calculus - An Introduction
Stochastic Calculus - An Introduction M. Kazim Khan Kent State University. UET, Taxila August 15-16, 17 Outline 1 From R.W. to B.M. B.M. 3 Stochastic Integration 4 Ito s Formula 5 Recap Random Walk Consider
More informationPAPER 211 ADVANCED FINANCIAL MODELS
MATHEMATICAL TRIPOS Part III Friday, 27 May, 2016 1:30 pm to 4:30 pm PAPER 211 ADVANCED FINANCIAL MODELS Attempt no more than FOUR questions. There are SIX questions in total. The questions carry equal
More information