Optimal investment and contingent claim valuation in illiquid markets
|
|
- Godwin Scott Bond
- 6 years ago
- Views:
Transcription
1 and contingent claim valuation in illiquid markets Teemu Pennanen King s College London Ari-Pekka Perkkiö Technische Universität Berlin 1 / 35
2 In most models of mathematical finance, there is at least one perfectly liquid asset that can be bought and sold in unlimited amounts at a fixed unit price. transaction costs (if any) are proportional to traded quantities. In practice, however, much of trading consists of exchanging sequences of cash-flows (coupon-paying bonds, dividends, swaps,...) unit prices depend nonlinearly on traded amounts. We use elementary convex analysis to extend certain fundamental theorems on optimal investment and contingent claim valuation to illiquid markets and general swap contracts. 2 / 35
3 Illiquidity Hodges, Neuberger, Optimal replication of contingent claims under transaction costs, Rev. Fut. Markets, Dalang, Morton, Willinger, Equivalent martingale measures and no-arbitrage in stochastic securities market models, Stoch. and Stoch. Rep., Artzner, Delbaen and Koch-Medona, Risk measures and efficient use of capital, Astin Bulletin, Hilli, Koivu, Pennanen, Cash-flow based valuation of pension liabilities, European Actuarial Journal, Pennanen, Superhedging in illiquid markets, Math. Finance, 2011 Pennanen, and contingent claim valuation in illiquid markets, Finance and Stochastics, to appear. Pennanen, Perkkiö, Stochastic programs without duality gaps, Mathematical Programming, Pennanen, Perkkiö, Convex duality in optimal investment and contingent claim valuation in illiquid markets, manuscript. 3 / 35
4 Limit order book of TDC A/S on 12 January 2005 at 13:58:19.43 in Copenhagen Stock Exchange Bid Ask Price Quantity Price Quantity / 35
5 The corresponding marginal price curve. Negative quantity corresponds to a sale PRICE QUANTITY 5 / 35
6 Consider a financial market where a finite set J of assets can be traded at t = 0,...,T. Let (Ω,F,(F t ) T,P) be a filtered probability space. The cost (in cash) of buying a portfolio x R J at time t in state ω will be denoted by S t (x,ω). We will assume that S t (,ω) is convex with S t (0,ω) = 0, S t (x, ) is F t -measurable. Such a sequence (S t ) will be called a convex cost process. 6 / 35
7 Example 1 (Liquid markets) If s = (s t ) T is an (F t ) T -adapted R J -valued price process, then the functions S t (x,ω) = s t (ω) x define a convex cost process. Example 2 (Jouini and Kallal, 1995) If (s a t) T and (s b t) T are (F t ) T -adapted with s b s a, then the functions { s a S t (x,ω) = t(ω)x if x 0, s b t(ω)x if x 0 define a convex cost process. 7 / 35
8 Example 3 (Çetin and Rogers, 2007) If s = (s t ) T is an (F t ) T -adapted process and ψ is a lower semicontinuous convex function on R with ψ(0) = 0, then the functions S t (x,ω) = x 0 +s t (ω)ψ(x 1 ) define a convex cost process. Example 4 (Dolinsky and Soner, 2013) If s = (s t ) T is (F t ) T -adapted and G t (x, ) are F t -measurable functions such that G t (,ω) are finite and convex, then the functions S t (x,ω) = x 0 +s t (ω) x 1 +G t (x 1,ω) define a convex cost process. 8 / 35
9 We allow for portfolio constraints requiring that the portfolio held over (t,t+1] in state ω has to belong to a set D t (ω) R J. We assume that D t (ω) are closed and convex with 0 D t (ω). {ω Ω D t (ω) U } F t for every open U R J. 9 / 35
10 Models where D t (ω) is independent of (t,ω) have been studied e.g. in [Cvitanić and Karatzas, 1992] and [Jouini and Kallal, 1995]. In [Napp, 2003], D t (ω) = {x R d M t (ω)x K}, where K R L is a closed convex cone and M t is an F t -measurable matrix. General constraints have been studied in [Evstigneev, Schürger and Taksar, 2004], [Rokhlin, 2005] and [Czichowsky and Schweizer, 2012]. 10 / 35
11 Let c M := {(c t ) T c t L 0 (Ω,F t,p)} and consider the problem T minimize V t (S t ( x t )+c t ) over x N D N D = {(x t ) T x t L 0 (Ω,F t,p;r J ), x t D t, x T = 0}, V t : L 0 R are convex, nondecreasing and V t (0) = 0. Example 5 If V t = δ L 0 for t < T, the problem can be written minimize V T (S T ( x T )+c T ) over x N D subject to S t ( x t )+c t 0, t = 0,...,T / 35
12 Example 6 (Markets with a numeraire) When S t (x,ω) = x 0 + S t ( x,ω) and D t (ω) = R D t (ω), the problem can be written as ( T minimize V T S t ( x t )+ T c t ) over x N D. When S t ( x,ω) = s t (ω) x, T S t ( x t ) = T s t x t = T 1 x t s t / 35
13 We denote the optimal value function by T ϕ(c) = inf V t (S t ( x t )+c t ). x N D Note that ϕ(c) = inf d C V(c d), where V(c) := T V t(c t ) and C := {c M x N D : S t ( x t )+c t 0 t} is the set of claims that can be superhedged without cost. 13 / 35
14 The recession cone C = {c M c+αc C c C, α > 0} of C consists of claims that can be superhedged without cost in unlimited amounts. If C is a cone, then C = C. Lemma 7 The function ϕ : M R is convex and ϕ( c+c) ϕ( c) c M, c C. In particular, ϕ is constant on the linear space C ( C ). 14 / 35
15 Example 8 (The classical model) Consider the classical perfectly liquid market model where C = {c M x N : T c t T 1 x t s t+1 } and C = C. We have c C ( C ) if there is an x N such that T c t = T 1 x t s t+1. The converse holds under the no-arbitrage condition. 15 / 35
16 of contingent claims In incomplete markets, the hedging argument for valuation of contingent claims has two natural generalizations: reservation value: How much capital do we need to cover our liabilities at an acceptable level of risk? indifference price: What is the least price we can sell a financial product for without increasing our risk? The former is important in accounting, financial reporting and supervision (and in the Black Scholes Merton model). The latter is more relevant in trading. In complete markets, reservation values and indifference prices coincide. 16 / 35
17 Reservation value We define the reservation value for a liability c M by π 0 (c) = inf{α R ϕ(c αp 0 ) 0} where p 0 = (1,0,...,0). π 0 can be interpreted much like a risk measure in [Artzner, Delbaen, Eber and Heath, 1999]. However, we have not assumed the existence of a cash-account so π 0 is defined on sequences of cash-flows. If V = δ M, we have ϕ = δ C and π 0 (c) = π 0 sup(c) := inf{α R c αp 0 C}. Let π 0 inf (c) = π0 sup( c). 17 / 35
18 Reservation value Theorem 9 The reservation value π 0 is convex and nondecreasing with respect to C. We have π 0 π 0 sup and if π 0 (0) 0, then π 0 inf(c) π 0 (c) π 0 sup(c) with equalities throughout if c αp 0 C ( C) for α R. π 0 is translation invariant : if c M is replicable with initial capital α: c αp 0 C ( C ), then π 0 (c+c ) = π 0 (c)+α. In complete markets, c αp 0 C ( C) for some α R so π 0 (c) is independent of preferences and views. 18 / 35
19 Swap contracts In a swap contract, an agent receives a sequence p M of premiums and delivers a sequence c M of claims. Examples: Swaps with a fixed leg : p = (1,...,1), random c. In credit derivatives (CDS, CDO,...) and other insurance contracts, both p and c are random. Traditionally in mathematical finance, p = (1,0,...,0) and c = (0,...,0,c T ). Claims and premiums live in the same space M = {(c t ) T c t L 0 (Ω,F t,p;r)}. 19 / 35
20 Swap contracts If we already have liabilities c M, then π( c,p;c) := inf{α R ϕ( c+c αp) ϕ( c)} gives the least swap rate that would allow us to enter a swap contract without worsening our financial position. Similarly, π b ( c,p;c) := sup{α R ϕ( c c+αp) ϕ( c)} = π( c,p; c) gives the greatest swap rate we would need on the opposite side of the trade. When p = (1,0,...,0) and c = (0,...,0,c T ), we get a nonlinear version of the indifference price of [Hodges and Neuberger, 1989]. 20 / 35
21 Swap contracts Define the super- and subhedging swap rates, π sup (p;c) = inf{α c αp C }, π inf (p;c) = sup{α αp c C }. If C is a cone and p = (1,0,...,0), we recover the super- and subhedging costs π 0 sup and π 0 inf. Theorem 10 If π( c,p;0) 0, then π inf (p;c) π b ( c,p;c) π( c,p;c) π sup (p;c) with equalities if c αp C ( C ) for some α R. Agents with identical views, preferences and financial position have no reason to trade with each other. Prices are independent of such subjective factors when c αp C ( C ) for some α R. If in addition, p = p 0, then swap rates coincide with reservation values. 21 / 35
22 Swap contracts Example 11 (The classical model) Consider the classical perfectly liquid market model where C = {c M x N : T c t T 1 x t s t+1 } and C = C. The condition c αp C ( C ) holds if there exist α R and x N such that T c t = α T p t + T 1 x t s t+1. The converse holds under the no-arbitrage condition. 22 / 35
23 Given a market model (S,D), let S t (x,ω) = sup α>0 S t (αx,ω) α and D t (ω) = α>0αd t (ω). If S is sublinear and D is conical, then S = S and D = D Theorem 12 Assume that V(c) = E T V t(c t ), where V t are bounded from below. If the cone L := {x N D S t ( x t ) 0} is a linear space, then C is closed and ϕ is lower semicontinuous in L 0. The lower bound can be replaced by RAE; [Perkkiö, 2014]. 23 / 35
24 Example 13 In the classical perfectly liquid market model L = {x N s t x t 0, x T = 0}, so the linearity condition becomes the no-arbitrage condition and we recover the key lemma from [Schachermayer, 1992]. Example 14 When D R J, the linearity condition becomes the robust no-arbitrage condition: there exists a positively homogeneous arbitrage-free cost process S with S t (x,ω) S t (x,ω) x R J, S t (x,ω) < S t (x,ω) x / lins t (,ω); see [Schachermayer, 2004]. 24 / 35
25 The linearity condition can hold even under arbitrage. Example 15 If S t (x,ω) > 0 for x / R J, then L = {0}. Example 16 In [Çetin and Rogers, 2007], S t (x,ω) = x 0 +s t (ω)ψ(x 1 ) and S t (x,ω) = x 0 +s t (ω)ψ (x 1 ). When infψ = 0 and supψ = we have ψ = δ R, so the condition in Example 15 holds. Example 17 If S t (,ω) = s t (ω) x for a componentwise strictly positive price process s and D t (ω) R J + (infinite short selling is prohibited), then L = {0}. 25 / 35
26 Proposition 18 Assume that ϕ is proper and lower semicontinuous. The conditions ϕ (p 0 ) > 0, π 0 (0) >, π 0 (c) > for all c M, are equivalent and imply that π 0 is proper and lower semicontinuous on M and that the infimum π 0 (c) = inf{α ϕ(c αp 0 ) 0} is attained for every c M. 26 / 35
27 Proposition 19 Assume that ϕ is proper and lower semicontinuous. Then, for every c domϕ and p M, the conditions ϕ (p) > 0, π( c,p;0) >, π( c,p;c) > for all c M, are equivalent and imply that π( c,p; ) is proper and lower semicontinuous on M and that the infimum π( c,p;c) = inf{α ϕ( c+c αp) ϕ( c)} is attained for every c M. 27 / 35
28 Let M p = {c M c t L p (Ω,F t,p;r)}. The bilinear form T c,y := E c t y t puts M 1 and M in separating duality. The conjugate of a function f on M 1 is defined by f (y) = sup c M 1 { c,y f(c)}. If f is proper, convex and lower semicontinuous, then f(y) = sup y M { c,y f (y)}. 28 / 35
29 We assume from now on that T V(c) = E V t (c t ) for convex random functions V t : R Ω R with V t (0) = 0. Theorem 20 If S t (x, ) L 1 for all x R J, then ϕ (y) = V (y)+σ C (y) where V (y) = E T V t (y t ) and σ C (y) = sup c C c,y. Moreover, T σ C (y) = inf v N 1E [(y t S t ) (v t )+σ Dt (E[ v t+1 F t ])] where the infimum is attained for all y M. 29 / 35
30 Example 21 If S t (ω,x) = s t (ω) x and D t (ω) is a cone, C = {y M E[ (y t+1 s t+1 ) F t ] D t}. Example 22 If S t (ω,x) = sup{s x s [s b t(ω),s a t(ω)]} and D t (ω) = R J, then C = {y M ys is a martingale for some s [s b,s a ]}. Example 23 In the classical model, C consists of positive multiples of martingale densities. 30 / 35
31 Theorem 24 Assume the linearity condition, the Inada condition V t = δ R and that p 0 / C and infϕ < 0. Then π 0 (c) = sup y M { c,y σ C (y) σ B (y) y 0 = 1}, where B = {c M 1 V(c) 0}. In particular, when C is conical and V is positively homogeneous, π 0 (c) = sup y M { c,y y C B, y 0 = 1}. Extends good deal bounds to sequences of cash-flows. 31 / 35
32 Theorem 25 Assume the linearity condition, the Inada condition and that p / C and infϕ < ϕ( c). Then π( c,p;c) = sup y M { c,y σc (y) σ B( c) (y) p,y = 1 }, where B( c) = {c M 1 V( c+c) ϕ( c)}. In particular, if C is conical, π( c,p;c) = sup y M { c,y σb( c) (y) u C, p,y = 1 }. 32 / 35
33 Example 26 In the classical model, with p = (1,0,...,0) and V t = δ R for t < T, we get { π( c,p;c) = sup c,y σa( c) (y) p,y = 1 } y M { ) } = sup Q Q T ( E Q dq ( c t +c t ) σ B( c) E t dp T = sup sup Q Q α>0e Q{ ( c t +c t ) α [ V T( dq dp /α) ϕ( c) ] } where Q is the set of absolutely continuous martingale measures; see [Biagini, Frittelli, Grasselli, 2011] for a continuous-time version. 33 / 35
34 Theorem 27 (FTAP) Assume that S is finite-valued and that D R J. Then the following are equivalent 1. S satisfies the robust no-arbitrage condition. 2. There is a strictly consistent price system: adapted processes y and s such that y > 0, s t ridoms t and ys is a martingale. In the classical linear market model, ridoms t = {1, s t } so we recover the Dalang Morton Willinger theorem. The robust no-arbitrage condition means that there exists a sublinear arbitrage-free cost process S with dom S t ridoms t. 34 / 35
35 Summary Financial contracts often involve sequences of cash-flows. Reservation values and indifference swap rates/prices can (and should) be derived from hedging arguments. In practice (incomplete markets), valuations are subjective: they depend on views, risk preferences, trading expertise and the current financial position of an agent. Much of classical asset pricing theory can be extended to convex models of illiquid markets. The mathematics and computational techniques for hedging and pricing in illiquid markets combine techniques from stochastics and convex analysis. 35 / 35
Asset valuation and optimal investment
Asset valuation and optimal investment Teemu Pennanen Department of Mathematics King s College London 1 / 57 Optimal investment and asset pricing are often treated as separate problems (Markovitz vs. Black
More informationOptimal investment and contingent claim valuation in illiquid markets
Optimal investment and contingent claim valuation in illiquid markets Teemu Pennanen May 18, 2014 Abstract This paper extends basic results on arbitrage bounds and attainable claims to illiquid markets
More informationConvex duality in optimal investment under illiquidity
Convex duality in optimal investment under illiquidity Teemu Pennanen August 16, 2013 Abstract We study the problem of optimal investment by embedding it in the general conjugate duality framework of convex
More informationMarkets with convex transaction costs
1 Markets with convex transaction costs Irina Penner Humboldt University of Berlin Email: penner@math.hu-berlin.de Joint work with Teemu Pennanen Helsinki University of Technology Special Semester on Stochastics
More informationSuperhedging in illiquid markets
Superhedging in illiquid markets to appear in Mathematical Finance Teemu Pennanen Abstract We study superhedging of securities that give random payments possibly at multiple dates. Such securities are
More informationArbitrage and deflators in illiquid markets
Finance and Stochastics manuscript No. (will be inserted by the editor) Arbitrage and deflators in illiquid markets Teemu Pennanen Received: date / Accepted: date Abstract This paper presents a stochastic
More informationLECTURE 4: BID AND ASK HEDGING
LECTURE 4: BID AND ASK HEDGING 1. Introduction One of the consequences of incompleteness is that the price of derivatives is no longer unique. Various strategies for dealing with this exist, but a useful
More informationOn the Lower Arbitrage Bound of American Contingent Claims
On the Lower Arbitrage Bound of American Contingent Claims Beatrice Acciaio Gregor Svindland December 2011 Abstract We prove that in a discrete-time market model the lower arbitrage bound of an American
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 informationarxiv: v1 [q-fin.pr] 11 Oct 2008
arxiv:0810.2016v1 [q-fin.pr] 11 Oct 2008 Hedging of claims with physical delivery under convex transaction costs Teemu Pennanen February 12, 2018 Abstract Irina Penner We study superhedging of contingent
More informationbased on two joint papers with Sara Biagini Scuola Normale Superiore di Pisa, Università degli Studi di Perugia
Marco Frittelli Università degli Studi di Firenze Winter School on Mathematical Finance January 24, 2005 Lunteren. On Utility Maximization in Incomplete Markets. based on two joint papers with Sara Biagini
More informationA model for a large investor trading at market indifference prices
A model for a large investor trading at market indifference prices Dmitry Kramkov (joint work with Peter Bank) Carnegie Mellon University and University of Oxford 5th Oxford-Princeton Workshop on Financial
More informationGuarantee valuation in Notional Defined Contribution pension systems
Guarantee valuation in Notional Defined Contribution pension systems Jennifer Alonso García (joint work with Pierre Devolder) Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA) Université
More information6: MULTI-PERIOD MARKET MODELS
6: MULTI-PERIOD MARKET MODELS Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2016 M. Rutkowski (USydney) 6: Multi-Period Market Models 1 / 55 Outline We will examine
More informationOn Utility Based Pricing of Contingent Claims in Incomplete Markets
On Utility Based Pricing of Contingent Claims in Incomplete Markets J. Hugonnier 1 D. Kramkov 2 W. Schachermayer 3 March 5, 2004 1 HEC Montréal and CIRANO, 3000 Chemin de la Côte S te Catherine, Montréal,
More informationLower and upper bounds of martingale measure densities in continuous time markets
Lower and upper bounds of martingale measure densities in continuous time markets Giulia Di Nunno CMA, Univ. of Oslo Workshop on Stochastic Analysis and Finance Hong Kong, June 29 th - July 3 rd 2009.
More informationArbitrage Theory without a Reference Probability: challenges of the model independent approach
Arbitrage Theory without a Reference Probability: challenges of the model independent approach Matteo Burzoni Marco Frittelli Marco Maggis June 30, 2015 Abstract In a model independent discrete time financial
More informationViability, Arbitrage and Preferences
Viability, Arbitrage and Preferences H. Mete Soner ETH Zürich and Swiss Finance Institute Joint with Matteo Burzoni, ETH Zürich Frank Riedel, University of Bielefeld Thera Stochastics in Honor of Ioannis
More informationConditional Certainty Equivalent
Conditional Certainty Equivalent Marco Frittelli and Marco Maggis University of Milan Bachelier Finance Society World Congress, Hilton Hotel, Toronto, June 25, 2010 Marco Maggis (University of Milan) CCE
More informationPathwise Finance: Arbitrage and Pricing-Hedging Duality
Pathwise Finance: Arbitrage and Pricing-Hedging Duality Marco Frittelli Milano University Based on joint works with Matteo Burzoni, Z. Hou, Marco Maggis and J. Obloj CFMAR 10th Anniversary Conference,
More informationCONSISTENCY AMONG TRADING DESKS
CONSISTENCY AMONG TRADING DESKS David Heath 1 and Hyejin Ku 2 1 Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, USA, email:heath@andrew.cmu.edu 2 Department of Mathematics
More informationSpot and forward dynamic utilities. and their associated pricing systems. Thaleia Zariphopoulou. UT, Austin
Spot and forward dynamic utilities and their associated pricing systems Thaleia Zariphopoulou UT, Austin 1 Joint work with Marek Musiela (BNP Paribas, London) References A valuation algorithm for indifference
More informationEuropean Contingent Claims
European Contingent Claims Seminar: Financial Modelling in Life Insurance organized by Dr. Nikolic and Dr. Meyhöfer Zhiwen Ning 13.05.2016 Zhiwen Ning European Contingent Claims 13.05.2016 1 / 23 outline
More informationIndices of Acceptability as Performance Measures. Dilip B. Madan Robert H. Smith School of Business
Indices of Acceptability as Performance Measures Dilip B. Madan Robert H. Smith School of Business An Introduction to Conic Finance A Mini Course at Eurandom January 13 2011 Outline Operationally defining
More informationMATH3075/3975 FINANCIAL MATHEMATICS TUTORIAL PROBLEMS
MATH307/37 FINANCIAL MATHEMATICS TUTORIAL PROBLEMS School of Mathematics and Statistics Semester, 04 Tutorial problems should be used to test your mathematical skills and understanding of the lecture material.
More informationLower and upper bounds of martingale measure densities in continuous time markets
Lower and upper bounds of martingale measure densities in continuous time markets Giulia Di Nunno Workshop: Finance and Insurance Jena, March 16 th 20 th 2009. presentation based on a joint work with Inga
More informationMathematical Finance in discrete time
Lecture Notes for Mathematical Finance in discrete time University of Vienna, Faculty of Mathematics, Fall 2015/16 Christa Cuchiero University of Vienna christa.cuchiero@univie.ac.at Draft Version June
More informationRobust hedging with tradable options under price impact
- Robust hedging with tradable options under price impact Arash Fahim, Florida State University joint work with Y-J Huang, DCU, Dublin March 2016, ECFM, WPI practice is not robust - Pricing under a selected
More information4: SINGLE-PERIOD MARKET MODELS
4: SINGLE-PERIOD MARKET MODELS Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2016 M. Rutkowski (USydney) Slides 4: Single-Period Market Models 1 / 87 General Single-Period
More informationMartingale Pricing Theory in Discrete-Time and Discrete-Space Models
IEOR E4707: Foundations of Financial Engineering c 206 by Martin Haugh Martingale Pricing Theory in Discrete-Time and Discrete-Space Models These notes develop the theory of martingale pricing in a discrete-time,
More informationOptimizing S-shaped utility and risk management: ineffectiveness of VaR and ES constraints
Optimizing S-shaped utility and risk management: ineffectiveness of VaR and ES constraints John Armstrong Dept. of Mathematics King s College London Joint work with Damiano Brigo Dept. of Mathematics,
More informationCLAIM HEDGING IN AN INCOMPLETE MARKET
Vol 18 No 2 Journal of Systems Science and Complexity Apr 2005 CLAIM HEDGING IN AN INCOMPLETE MARKET SUN Wangui (School of Economics & Management Northwest University Xi an 710069 China Email: wans6312@pubxaonlinecom)
More informationHow do Variance Swaps Shape the Smile?
How do Variance Swaps Shape the Smile? A Summary of Arbitrage Restrictions and Smile Asymptotics Vimal Raval Imperial College London & UBS Investment Bank www2.imperial.ac.uk/ vr402 Joint Work with Mark
More informationUmut Çetin and L. C. G. Rogers Modelling liquidity effects in discrete time
Umut Çetin and L. C. G. Rogers Modelling liquidity effects in discrete time Article (Accepted version) (Refereed) Original citation: Cetin, Umut and Rogers, L.C.G. (2007) Modelling liquidity effects in
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 informationMean-Variance Hedging under Additional Market Information
Mean-Variance Hedging under Additional Market Information Frank hierbach Department of Statistics University of Bonn Adenauerallee 24 42 53113 Bonn, Germany email: thierbach@finasto.uni-bonn.de Abstract
More informationMartingale invariance and utility maximization
Martingale invariance and utility maximization Thorsten Rheinlander Jena, June 21 Thorsten Rheinlander () Martingale invariance Jena, June 21 1 / 27 Martingale invariance property Consider two ltrations
More informationComputing Bounds on Risk-Neutral Measures from the Observed Prices of Call Options
Computing Bounds on Risk-Neutral Measures from the Observed Prices of Call Options Michi NISHIHARA, Mutsunori YAGIURA, Toshihide IBARAKI Abstract This paper derives, in closed forms, upper and lower bounds
More informationOn Using Shadow Prices in Portfolio optimization with Transaction Costs
On Using Shadow Prices in Portfolio optimization with Transaction Costs Johannes Muhle-Karbe Universität Wien Joint work with Jan Kallsen Universidad de Murcia 12.03.2010 Outline The Merton problem The
More informationPricing and hedging in incomplete markets
Pricing and hedging in incomplete markets Chapter 10 From Chapter 9: Pricing Rules: Market complete+nonarbitrage= Asset prices The idea is based on perfect hedge: H = V 0 + T 0 φ t ds t + T 0 φ 0 t ds
More informationMathematics in Finance
Mathematics in Finance Steven E. Shreve Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213 USA shreve@andrew.cmu.edu A Talk in the Series Probability in Science and Industry
More informationINTRODUCTION TO ARBITRAGE PRICING OF FINANCIAL DERIVATIVES
INTRODUCTION TO ARBITRAGE PRICING OF FINANCIAL DERIVATIVES Marek Rutkowski Faculty of Mathematics and Information Science Warsaw University of Technology 00-661 Warszawa, Poland 1 Call and Put Spot Options
More informationECOLE POLYTECHNIQUE CENTRE DE MATHÉMATIQUES APPLIQUÉES UMR CNRS PALAISEAU CEDEX (FRANCE). Tél: Fax:
ECOLE POLYTECHNIQUE CENTRE DE MATHÉMATIQUES APPLIQUÉES UMR CNRS 7641 91128 PALAISEAU CEDEX (FRANCE). Tél: 01 69 33 41 50. Fax: 01 69 33 30 11 http://www.cmap.polytechnique.fr/ Bid-Ask Dynamic Pricing in
More informationLaw of the Minimal Price
Law of the Minimal Price Eckhard Platen School of Finance and Economics and Department of Mathematical Sciences University of Technology, Sydney Lit: Platen, E. & Heath, D.: A Benchmark Approach to Quantitative
More informationPricing theory of financial derivatives
Pricing theory of financial derivatives One-period securities model S denotes the price process {S(t) : t = 0, 1}, where S(t) = (S 1 (t) S 2 (t) S M (t)). Here, M is the number of securities. At t = 1,
More information3 Arbitrage pricing theory in discrete time.
3 Arbitrage pricing theory in discrete time. Orientation. In the examples studied in Chapter 1, we worked with a single period model and Gaussian returns; in this Chapter, we shall drop these assumptions
More informationMATH 5510 Mathematical Models of Financial Derivatives. Topic 1 Risk neutral pricing principles under single-period securities models
MATH 5510 Mathematical Models of Financial Derivatives Topic 1 Risk neutral pricing principles under single-period securities models 1.1 Law of one price and Arrow securities 1.2 No-arbitrage theory and
More informationPRICING CONTINGENT CLAIMS: A COMPUTATIONAL COMPATIBLE APPROACH
PRICING CONTINGENT CLAIMS: A COMPUTATIONAL COMPATIBLE APPROACH Shaowu Tian Department of Mathematics University of California, Davis stian@ucdavis.edu Roger J-B Wets Department of Mathematics University
More information- Introduction to Mathematical Finance -
- Introduction to Mathematical Finance - Lecture Notes by Ulrich Horst The objective of this course is to give an introduction to the probabilistic techniques required to understand the most widely used
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 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 informationA generalized coherent risk measure: The firm s perspective
Finance Research Letters 2 (2005) 23 29 www.elsevier.com/locate/frl A generalized coherent risk measure: The firm s perspective Robert A. Jarrow a,b,, Amiyatosh K. Purnanandam c a Johnson Graduate School
More informationRisk Measures and Optimal Risk Transfers
Risk Measures and Optimal Risk Transfers Université de Lyon 1, ISFA April 23 2014 Tlemcen - CIMPA Research School Motivations Study of optimal risk transfer structures, Natural question in Reinsurance.
More informationOptimizing S-shaped utility and risk management
Optimizing S-shaped utility and risk management Ineffectiveness of VaR and ES constraints John Armstrong (KCL), Damiano Brigo (Imperial) Quant Summit March 2018 Are ES constraints effective against rogue
More informationPrice functionals with bid ask spreads: an axiomatic approach
Journal of Mathematical Economics 34 (2000) 547 558 Price functionals with bid ask spreads: an axiomatic approach Elyès Jouini,1 CEREMADE, Université Paris IX Dauphine, Place De Lattre-de-Tossigny, 75775
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 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 informationOn Asymptotic Power Utility-Based Pricing and Hedging
On Asymptotic Power Utility-Based Pricing and Hedging Johannes Muhle-Karbe ETH Zürich Joint work with Jan Kallsen and Richard Vierthauer LUH Kolloquium, 21.11.2013, Hannover Outline Introduction Asymptotic
More informationOptimization Models in Financial Mathematics
Optimization Models in Financial Mathematics John R. Birge Northwestern University www.iems.northwestern.edu/~jrbirge Illinois Section MAA, April 3, 2004 1 Introduction Trends in financial mathematics
More informationarxiv: v3 [q-fin.pr] 22 Dec 2013
arxiv:1107.5720v3 [q-fin.pr] 22 Dec 2013 An algorithm for calculating the set of superhedging portfolios in markets with transaction costs Andreas Löhne, Birgit Rudloff July 2, 2018 Abstract We study the
More informationModel Free Hedging. David Hobson. Bachelier World Congress Brussels, June University of Warwick
Model Free Hedging David Hobson University of Warwick www.warwick.ac.uk/go/dhobson Bachelier World Congress Brussels, June 2014 Overview The classical model-based approach Robust or model-independent pricing
More informationLECTURE 2: MULTIPERIOD MODELS AND TREES
LECTURE 2: MULTIPERIOD MODELS AND TREES 1. Introduction One-period models, which were the subject of Lecture 1, are of limited usefulness in the pricing and hedging of derivative securities. In real-world
More informationPortfolio Optimisation under Transaction Costs
Portfolio Optimisation under Transaction Costs W. Schachermayer University of Vienna Faculty of Mathematics joint work with Ch. Czichowsky (Univ. Vienna), J. Muhle-Karbe (ETH Zürich) June 2012 We fix a
More informationFollow links for Class Use and other Permissions. For more information send to:
COPYRIGHT NOTICE: Costis Skiadas: Asset Pricing Theory is published by Princeton University Press and copyrighted, 2009, by Princeton University Press. All rights reserved. No part of this book may be
More informationMartingale Optimal Transport and Robust Finance
Martingale Optimal Transport and Robust Finance Marcel Nutz Columbia University (with Mathias Beiglböck and Nizar Touzi) April 2015 Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance
More informationAmerican Foreign Exchange Options and some Continuity Estimates of the Optimal Exercise Boundary with respect to Volatility
American Foreign Exchange Options and some Continuity Estimates of the Optimal Exercise Boundary with respect to Volatility Nasir Rehman Allam Iqbal Open University Islamabad, Pakistan. Outline Mathematical
More informationLarge Deviations and Stochastic Volatility with Jumps: Asymptotic Implied Volatility for Affine Models
Large Deviations and Stochastic Volatility with Jumps: TU Berlin with A. Jaquier and A. Mijatović (Imperial College London) SIAM conference on Financial Mathematics, Minneapolis, MN July 10, 2012 Implied
More informationBasic Arbitrage Theory KTH Tomas Björk
Basic Arbitrage Theory KTH 2010 Tomas Björk Tomas Björk, 2010 Contents 1. Mathematics recap. (Ch 10-12) 2. Recap of the martingale approach. (Ch 10-12) 3. Change of numeraire. (Ch 26) Björk,T. Arbitrage
More informationChanges of the filtration and the default event risk premium
Changes of the filtration and the default event risk premium Department of Banking and Finance University of Zurich April 22 2013 Math Finance Colloquium USC Change of the probability measure Change of
More informationRisk Neutral Pricing. to government bonds (provided that the government is reliable).
Risk Neutral Pricing 1 Introduction and History A classical problem, coming up frequently in practical business, is the valuation of future cash flows which are somewhat risky. By the term risky we mean
More informationOptimal Portfolio Liquidation with Dynamic Coherent Risk
Optimal Portfolio Liquidation with Dynamic Coherent Risk Andrey Selivanov 1 Mikhail Urusov 2 1 Moscow State University and Gazprom Export 2 Ulm University Analysis, Stochastics, and Applications. A Conference
More informationRobust Portfolio Choice and Indifference Valuation
and Indifference Valuation Mitja Stadje Dep. of Econometrics & Operations Research Tilburg University joint work with Roger Laeven July, 2012 http://alexandria.tue.nl/repository/books/733411.pdf Setting
More informationON UTILITY-BASED PRICING OF CONTINGENT CLAIMS IN INCOMPLETE MARKETS
Mathematical Finance, Vol. 15, No. 2 (April 2005), 203 212 ON UTILITY-BASED PRICING OF CONTINGENT CLAIMS IN INCOMPLETE MARKETS JULIEN HUGONNIER Institute of Banking and Finance, HEC Université delausanne
More informationThe Birth of Financial Bubbles
The Birth of Financial Bubbles Philip Protter, Cornell University Finance and Related Mathematical Statistics Issues Kyoto Based on work with R. Jarrow and K. Shimbo September 3-6, 2008 Famous bubbles
More informationResearch Article Optimal Hedging and Pricing of Equity-LinkedLife Insurance Contracts in a Discrete-Time Incomplete Market
Journal of Probability and Statistics Volume 2011, Article ID 850727, 23 pages doi:10.1155/2011/850727 Research Article Optimal Hedging and Pricing of Equity-LinkedLife Insurance Contracts in a Discrete-Time
More informationHans-Fredo List Swiss Reinsurance Company Mythenquai 50/60, CH-8022 Zurich Telephone: Facsimile:
Risk/Arbitrage Strategies: A New Concept for Asset/Liability Management, Optimal Fund Design and Optimal Portfolio Selection in a Dynamic, Continuous-Time Framework Part III: A Risk/Arbitrage Pricing Theory
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 informationOption Pricing with Delayed Information
Option Pricing with Delayed Information Mostafa Mousavi University of California Santa Barbara Joint work with: Tomoyuki Ichiba CFMAR 10th Anniversary Conference May 19, 2017 Mostafa Mousavi (UCSB) Option
More informationUNIFORM BOUNDS FOR BLACK SCHOLES IMPLIED VOLATILITY
UNIFORM BOUNDS FOR BLACK SCHOLES IMPLIED VOLATILITY MICHAEL R. TEHRANCHI UNIVERSITY OF CAMBRIDGE Abstract. The Black Scholes implied total variance function is defined by V BS (k, c) = v Φ ( k/ v + v/2
More informationOption Pricing and Hedging with Small Transaction Costs
Option Pricing and Hedging with Small Transaction Costs Jan Kallsen Johannes Muhle-Karbe Abstract An investor with constant absolute risk aversion trades a risky asset with general Itôdynamics, in the
More information3.2 No-arbitrage theory and risk neutral probability measure
Mathematical Models in Economics and Finance Topic 3 Fundamental theorem of asset pricing 3.1 Law of one price and Arrow securities 3.2 No-arbitrage theory and risk neutral probability measure 3.3 Valuation
More informationStochastic Processes and Stochastic Calculus - 9 Complete and Incomplete Market Models
Stochastic Processes and Stochastic Calculus - 9 Complete and Incomplete Market Models Eni Musta Università degli studi di Pisa San Miniato - 16 September 2016 Overview 1 Self-financing portfolio 2 Complete
More informationOn robust pricing and hedging and the resulting notions of weak arbitrage
On robust pricing and hedging and the resulting notions of weak arbitrage Jan Ob lój University of Oxford obloj@maths.ox.ac.uk based on joint works with Alexander Cox (University of Bath) 5 th Oxford Princeton
More informationOn the law of one price
Noname manuscript No. (will be inserted by the editor) On the law of one price Jean-Michel Courtault 1, Freddy Delbaen 2, Yuri Kabanov 3, Christophe Stricker 4 1 L.I.B.R.E., Université defranche-comté,
More informationOptimal construction of a fund of funds
Optimal construction of a fund of funds Petri Hilli, Matti Koivu and Teemu Pennanen January 28, 29 Introduction We study the problem of diversifying a given initial capital over a finite number of investment
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 informationThe super-replication theorem under proportional transaction costs revisited
he super-replication theorem under proportional transaction costs revisited Walter Schachermayer dedicated to Ivar Ekeland on the occasion of his seventieth birthday June 4, 2014 Abstract We consider a
More informationStochastic Dynamics of Financial Markets
Stochastic Dynamics of Financial Markets A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Humanities 2013 Mikhail Valentinovich Zhitlukhin School
More informationOrder book resilience, price manipulations, and the positive portfolio problem
Order book resilience, price manipulations, and the positive portfolio problem Alexander Schied Mannheim University PRisMa Workshop Vienna, September 28, 2009 Joint work with Aurélien Alfonsi and Alla
More informationsample-bookchapter 2015/7/7 9:44 page 1 #1 THE BINOMIAL MODEL
sample-bookchapter 2015/7/7 9:44 page 1 #1 1 THE BINOMIAL MODEL In this chapter we will study, in some detail, the simplest possible nontrivial model of a financial market the binomial model. This is a
More informationBlack-Scholes and Game Theory. Tushar Vaidya ESD
Black-Scholes and Game Theory Tushar Vaidya ESD Sequential game Two players: Nature and Investor Nature acts as an adversary, reveals state of the world S t Investor acts by action a t Investor incurs
More informationCHAPTER 12. Hedging. hedging strategy = replicating strategy. Question : How to find a hedging strategy? In other words, for an attainable contingent
CHAPTER 12 Hedging hedging dddddddddddddd ddd hedging strategy = replicating strategy hedgingdd) ddd Question : How to find a hedging strategy? In other words, for an attainable contingent claim, find
More informationAn Explicit Example of a Shadow Price Process with Stochastic Investment Opportunity Set
An Explicit Example of a Shadow Price Process with Stochastic Investment Opportunity Set Christoph Czichowsky Faculty of Mathematics University of Vienna SIAM FM 12 New Developments in Optimal Portfolio
More informationArbitrage Theory. The research of this paper was partially supported by the NATO Grant CRG
Arbitrage Theory Kabanov Yu. M. Laboratoire de Mathématiques, Université de Franche-Comté 16 Route de Gray, F-25030 Besançon Cedex, FRANCE and Central Economics and Mathematics Institute of the Russian
More informationOptimal Risk Transfer
Optimal Risk Transfer Pauline Barrieu and Nicole El Karoui February, 1st 2004 (Preliminary version) Abstract We develop a methodology to optimally design a financial issue to hedge non-tradable risk on
More informationPricing and hedging in the presence of extraneous risks
Stochastic Processes and their Applications 117 (2007) 742 765 www.elsevier.com/locate/spa Pricing and hedging in the presence of extraneous risks Pierre Collin Dufresne a, Julien Hugonnier b, a Haas School
More informationVALUATION OF FLEXIBLE INSURANCE CONTRACTS
Teor Imov r.tamatem.statist. Theor. Probability and Math. Statist. Vip. 73, 005 No. 73, 006, Pages 109 115 S 0094-90000700685-0 Article electronically published on January 17, 007 UDC 519.1 VALUATION OF
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 informationMartingale Approach to Pricing and Hedging
Introduction and echniques Lecture 9 in Financial Mathematics UiO-SK451 Autumn 15 eacher:s. Ortiz-Latorre Martingale Approach to Pricing and Hedging 1 Risk Neutral Pricing Assume that we are in the basic
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 information