Lower and upper bounds of martingale measure densities in continuous time markets
|
|
- Hilda Dawson
- 5 years ago
- Views:
Transcription
1 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 presentation based on a joint work with Inga B. Eide
2 Outlines 1. Market modeling: EMM and no-arbitrage pricing principle 2. Framework: Claims and price operators 3. No-arbitrage pricing and representation theorems 4. EMM and extension theorems for operators 5. A version of the fundamental theorem of asset pricing References
3 1. Market modeling Market modeling is based on a probability space (Ω, F, P) identifying the possible future scenarios. The probability measure P is derived from DATA and/or EXPERTS BELIEFS of the possible scenarios and the possible dynamics of the random phenomenon. On the other hand the modeling of asset pricing is connected with the idealization of a FAIR MARKET. This is based on the principle of no-arbitrage and its relation with a risk neutral probability measure P 0, under which all discounted prices are (local) martingales with respect to the evolution of the market events (for this reason P 0 is also called martingale measure). The probability space (Ω, F, P 0 ) provides an efficient mathematical framework.
4 Fundamental theorem of asset pricing Naturally, we would like the models of mathematical finance to be both consistent with data analysis and to be mathematically feasible. This topic was largely investigated for quite a long time yielding the various versions of the fundamental theorem of asset pricing. The basic statement is: For a given market model on (Ω, F, P) and the flow of market events F = {F t F, t [0, T ]} (T > 0), satisfying some assumptions, there exists a martingale measure P 0 such that P 0 P, i.e. P 0 (A) = 0 P(A) = 0, A F T. Cf. e.g. Delbaen, Harrison, Kreps, Pliska, Schachermayer.
5 No-arbitrage pricing principle Mathematically the existence of an equivalent martingale measure (EMM) implies the absence of arbitrage opportunities, this embodying the economical fact that in a fair market there should be no possibility of earning riskless profit. In fact the principle of no-arbitrage provides the basic pricing rule in mathematical finance: For any claim X, achievable at time t and purchased at time s, its fair price x st (X ) is given by x st (X ) = E 0 [ Rs R t X F s ]. Here R t, t [0, T ], represents some riskless investment always achievable and always available on the market (the numéraire).
6 The martingale measure P 0 used in the no-arbitrage evaluation is not necessarily unique: the equivalent martingale measure is unique if and only if the market is complete, i.e. if all claims are attainable in the market. In an incomplete market, if the claim X is attainable, the no-arbitrage evaluation of the price is independent of the choice of the martingale measure applied, thus the price is unique. But, if X is not-attainable, then the no-arbitrage principle does not give a unique price, but a whole range of prices that are equally valid from the no-arbitrage point of view. Many authors have been engaged in the study of how to select a martingale measure to be used. The approaches have been different.
7 Selection of one measure that either is in some sense optimal or whose use is justified by specific arguments in incomplete markets. Without aim or possibility to be complete we mention the minimal martingale measure and variance-optimal martingale measure which are both in some sense minimizing the distance to the physical measure (ref. e.g. Schweizer 2001). The Esscher measure is motivated by utility arguments to justify its use and it is also proved that it is also structure-preserving when applied to Lévy driven models (ref. e.g. Delbaen et al. 1989, Gerber et al. 1996). Instead of searching for the unique optimal equivalent martingale measure, one can try to characterize probability measures that are in some sense reasonable. This is the case of restrictions of the set of EMM in such a way that not only arbitrage opportunities are ruled out, but also deals that are too good as in the case of bounds on the Sharpe ratio (the ratio of the risk premium to the volatility). Ref. e.g. Cochrane et al. (2000), Björk et al. (2006), Staum (2006).
8 In various stochastic phenomena the impact of some events is crucial both for its own being and for the market effects triggered. As example, think of the devastating effects of natural catastrophies (e.g. earthquakes, hurricanes, etc.) and epidemies (e.g. SARS, bird flu, etc.). These events, though devastating, occur with small, but still positive probabilities. Having this in mind, the choice of a reasonable EMM should take in to account a proper evaluation of small probabilities, i.e. P(A) small P 0 (A) small. Note in fact that the assessment under P of the risk of these events incurring can be seriously misjugded under a P 0 only equivalent to P. This is particularly relevant for the evaluation of insurance linked securities.
9 Goal We study the existence of EMM P 0 with densities dp0 dp pre-considered lower and upper bounds: 0 < m dp0 dp M < P-a.s. We have to stress that these bounds are random variables: m = m(ω), ω Ω, M = M(ω), ω Ω. lying within
10 A bit on related literature: In Roklin and Schachermayer (2006), we follow the study on the existence of a martingale measure with lower bounded density. In Kabanov and Stricker (2001) (see also Ràzonyi (2002)) densities are bounded from above. Here the study shows that the set of equivalent σ-martingale measures with density in L (F) is dense (in total variation) in the set of equivalent σ-martingale measures. Here, we consider lower and upper bounds for martingale measure densities simultaneously.
11 2. Framework We consider a continuous time market model without friction for the time interval [0, T ] (T > 0) on the complete probability space (Ω, F, P). The flow of information is described by the filtration F := {F t F, t [0, T ]} augmented of the P-zero events, right-continuous and such that F = F T. Claims. For any fixed t, the achievable claims in the market payable at t constitue the convex sub-cone of the cone L + t L + p (F t ) L + p (F t ) := {X L p (Ω, F t, P) : X 0}, p [1, ).
12 N.B. The case L + t = L + p (F t ) for all t [0, T ] corresponds to a complete market. Otherwise the market is incomplete, i.e. L + t L + p (F t ) for at least a t [0, T ].
13 N.B. The case L + t = L + p (F t ) for all t [0, T ] corresponds to a complete market. Otherwise the market is incomplete, i.e. L + t L + p (F t ) for at least a t [0, T ]. N.B. If borrowing and short-selling is admitted, then the variety of claims is a linear sub-space L t L p (F t ) : L t := L + t L + t, i.e. X L t can be represented as X = X X with X, X L + t.
14 Price operators If X L + t is available at time s : s t, then its price ˆx st (X ) is an F s -measurable random variable such that ˆx st (X ) < P a.s.
15 Price operators If X L + t is available at time s : s t, then its price ˆx st (X ) is an F s -measurable random variable such that ˆx st (X ) < P a.s. N.B. If short-selling is admitted, the price operators are defined on the sub-space L t L p (F t ) according to ˆx st (X ) := ˆx st (X ) ˆx st (X ) for any X L t : X = X X with X, X L + t.
16 Any s, t : s t, the price operator ˆx st (X ), X L + t, satisfies: It is strictly monotone, i.e. for any X, X L + t ˆx st (X ) ˆx st (X ), X X, ˆx st (X ) > ˆx st (X ), X > X available at s The notation represents the standard point-wise P-a.s., while > means that, in addition to P-a.s., the point-wise relation > holds on an event of positive P-measure. It is additive, i.e. for any X, X L + t available at s ˆx st (X + X ) = ˆx st (X ) + ˆx st (X ). It is F s -homogeneous, i.e. for any X L + t F s -multiplier λ such that λx L + t, then available at s and any ˆx st (λx ) = λˆx st (X ). We set ˆx tt (X ) = X, X t +.
17 Definition. The price operator ˆx st (X ), X L + t, is tame if ˆx st (X ) L p (F s ), X L + t, i.e. ˆx st (X ) p := ( E(ˆx st (X ) ) p ) 1/p <. N.B. This definition is motivated by forthcoming arguments on time-consistency of the price operators. In view of the forthcoming no-arbitrage arguments we consider only tame price operators.
18 Discounting To be able to compare prices over time, we consider a numéraire which represents the chosen unit of measurement for money. This is an asset always available with payoff R t > 0 P-a.s. for every t and R s = ˆx st (R t ), s t, R s R s p 0, s s. We set R 0 = 1. Hence it is natural to assume that, for any X L + t X R s L + t, s t and we also assume that for every s t there exist c st, C st > 0 constants such that c st R t R s C st.
19 N.B. We have that 1 = ˆx st(r t ) R s = ˆx st ( Rt R s ) L + s. Thus 1 L + t for all t [0, T ]. Definition. The discounted price operator is defined as: Naturally, x st (R t ) = 1, s t. x st (X ) := ˆx st(x ) R s = ˆx st ( X R s ), s t. N.B. The discounted price operator x st inherits the properties of strict monotonicity, additivity and F s -homogeneity from ˆx st. Thus it is itself a price operator. Moreover, the discounted price operator is tame if and only if the price operator is tame.
20 Here we consider the family of price operators of X L + t, t T, ˆx st (X ), 0 s t, x st (X ), 0 s t. Definition. The family of the prices above is right-continuous at s if X is available for some interval of time [s, s + δ] (δ > 0) and ˆx s t(x ) ˆx st (X ) p 0, s s. N.B. The family of discounted prices is right-continuous if and only if the family of the original prices is. Definition. Let T [0, T ]. The family x st, s, t T : s t, of tame discounted price operators x st (X ), X L + t, is time-consistent (in T ) if for all s, u, t T : s u t x st (X ) = x su ( xut (X ) ), for all X L + t such that x ut (X ) L + u.
21 Comment. This axiomatic approach to price processes is inspired by risk measure theory. The requirements of monotonicity, additivity, and homogenuity are related to the concept of coherent risk measures. The additional assumption of strict monotonicity, is related to the concept of a relevant risk measure.
22 3. No-arbitrage pricing and representation theorems We can reformulate the basic statement of the fundamental theorem saying that: The absence of arbitrage is ensured by the existence of an EMM P 0, such that the discounted prices x st (X ), X L + t, admit the representation x st (X ) = E 0 [X F s ], X L + t. Moreover, for any t [0, T ] and X L + t the price process x st (X ), 0 s t is a martingale with respect to the measure P 0 and the filtration F. Definition. A probability measure Q P is tame if, for every t [0, T ], we have that E Q [X F t ] L p (F t ), X L p (F T ).
23 Facts. Let us consider P 0 P and tame probability measure. The study of the operator conditional expectation : E 0 [ F s ] : L p (F t ) L p (F s ) shows that it is: tame, strictly monotone, linear, and F s -homogeneous. Hence it has all the properties of a tame price operator on the whole L + p (F s ). Moreover, the family of conditional expectations is time-consistent: E 0 [X F s ] = E 0 [E 0 [X F u ] F s ], X L p (F t ), 0 s u t, and also right-continuous. Quite remarkably, it turns out that the converse is also true: any tame price operator x st (X ), X L p (F t ), admits representation as conditional expectation with respect to an equivalent martingale measure.
24 Lemma. For s, t [0, T ] : s t fixed, the operator x st (X ), X L p (F t ), is monotone linear F s -homogeneous if and only if it admits representation (1) x st (X ) = E 0 st[x F s ], X L p (F t ), with respect to a probability measure Pst(A) 0 = f st (ω)p(dω), A F t, A where f st L + 1 q (F t ), q + 1 p = 1. Moreover the operator is strictly monotone if and only if f st > 0 P-a.s. In addition, the operator (1) is bounded (continuous) if and only if { essupe[fst q F s ] <, p (1, ) essupf st <, p = 1 and is tame if and only if P 0 st is tame.
25 The results above are restricted to the two fixed time points s t. Now we keep s fixed and we compare the representations for different time points u t. Theorem. Let s, t [0, T ]: s t. Assume that the operators x su (X ), X L p (F u ), s u t, are tame price operators constituting a time-consistent family. Then, for all u [s, t], the representation (2) x su (X ) = E 0 st[x F s ], X L p (F u ), holds in terms of the measure Pst 0 defined on (Ω, F t ). Moreover Pst 0 Fu = Psu, 0 for all u [s, t].
26 Summary and the following steps. Whenever we have a time-consistent family of tame price operators x st (X ), 0 s t T, defined on the whole cone X L + p (F t ), we have an EMM. This is always true in markets that are complete. However, in general, operators are defined on the sub-cones L + t L + p (F t ). Then the existence of an EMM is linked to the admissibility of an extension of the price operator from the sub-cones to the corresponding cones. Need to give conditions (necessary and sufficient) for the extension of operators.
27 4. EMM and extension theorems for operators Let m st, M st L + q (F t) ( 1 p + 1 q = 1) such that 0 < m st M st, P-a.s. and { ess sup E[Mst F q s ] <, 1 < p <, ess sup M st <, p = 1. Theorem. For s, t [0, T ]: s t, fixed. The price operator x st (X ), X L + t, lying in the sandwich E[Xm st F s ] x st (X ) E[XM st F s ], X L + t, admits a tame strictly monotone linear F s -homogeneous extension x st (X ), X L + p (F t ), defined on the whole cone L + p (F t ) if and only if the sandwich condition E[Y m st F s ] + x st (X ) x st (X ) + E[Y M st F s ] holds for all X, X L + t X + Y X + Y. and Y, Y L + p (F t ) such that
28 The sandwich extension theorem is an extension theorem for operators lying in a given sandwich. The theorem fits in the Banach lattice framework generalizing the König extension theorem for functionals. N.B. The extension x st (X ), X L + p (F t ), if existing, is sandwich preserving, i.e. E[Xm st F s ] x st (X ) E[XM st F s ], X L + p (F t ). N.B. The extension, if existing, admits representation x st (X ) = E 0 st[x F s ] = E[Xf st F s ] N.B. The density f st lies in the sandwich ( f st := dp0 st dp ). 0 < m st f st M st P a.s.
29 5. A version of the fundamental theorem of asset pricing Let m, M L q (F T ) ( 1 p + 1 q = 1) such that 0 < m M, P-a.s. and { ess sup E[M q F s ] <, 1 < p <, ess sup M <, p = 1. For any s t define m st := (E[m F 0 ]) t s T E[m F t ] E[m F s ], M st := (E[M F 0 ]) t s T E[M F t ] E[M F s ]. In particular we have that m = m 0T, M = M 0T. N.B. For any s t, we have m 0T = m 0s m st m tt. Analogously for M.
30 Theorem. A tame martingale measure P 0 on (Ω, F T ) equivalent to P such that the density f := dp0 dp lies in the sandwich 0 < m f M P a.s. exists if and only if for all s, t [0, T ]: s t, the price operators x st (X ), X L + t, satisfy the sandwich condition E[Y m st F s ] + x st (X ) x st (X ) + E[Y M st F s ] for all X, X L + t and Y, Y L + p (F t ) such that X + Y X + Y. N.B. In this case, we have also 0 < m st E[f F t] E[f F s ] M st.
31 Sketch of proof. Necessary condition. Consider the set of EMM: P := {P 0 dp0 dp = f, m f M; s, t [0, T ], s t E[f F 0 ] and the approximating set: P (T ) := x st (X ) = E 0 [X F s ], X L + t, {P 0 dp0 dp = f, m f M; s T, t [s, T ] E[f F 0 ] where T is some partition of [0, T ] of the form x st (X ) = E 0 [X F s ], X L + t, T = {s 0, s 1,..., s K }, with 0 = s 0 < s 1 < < s K = T. }, },
32 Further, we consider a sequence {T n } n=1 of increasingly refined partitions, such that T n T n+1 and mesh(t n ) 0 as n. Clearly P (Tn+1) P (Tn). Then the proof proceeds with the following steps: A. P (T ) is non-empty for any finite partition T, B. the infinite intersection n=1 P(Tn) is non-empty, and C. any P 0 n=1 P(Tn) is also in P.
33 Example Let L + T := {αx + β : α, β 0} be the set of claims, i.e. α represents the fraction of the claim X = (z z 0 )N(T, dz), N(T, dz) Poi(T ν(dz)), z>z 0 and β is the amount in a money market account with zero-interest. Note that X can be interpreted as an insurance policy covering all losses exceeding the deductible z 0 in the time span [0, T ]. Let the price at time 0 of X be given by the expected value principle, i.e. x 0T (X ) = (1 + δ)ex = (1 + δ)t (z z 0 )ν(dz). z>z 0 Let the bounds for the possible density be given by: δt ν(u) m = e M = (1 + δ) N(T,U) e δt ν(u), U := (z 0, )
34 Then both and P 0 1 {N(T, dz) = n} = (1 + δ) n e δt ν(dz) P{N(T, dz) = n} P 0 2 {N(T, dz) = n} = { 1 P{N(T, dz) = n}, z z, (1 + γ) n e γt ν(dz) P{N(T, dz) = n}, z > z, for z > z 0 and γ := δex T R z (z z 0)ν(dz), are EMM. However, the sandwich condition shows that while for P 0 1 we have m dp0 1 dp M P a.s., for P2 0 the relation is not true.
35 References This presentation was based on: G. Di Nunno, Inga B. Eide. Events of small but positive probability and a fundamental theorem of asset pricing. E-print, University of Oslo Relevant related references: S. Albeverio, G. Di Nunno, and Y. A. Rozanov. Price operators analysis in L p-spaces. Acta Applicandae Mathematicae, 89:85 108, T. Björk and I. Slinko. Towards a general theory of good deal bounds. Review of Finance, 10: , J. H. Cochrane and J. Saá-Requejo. Beyond arbitrage: good-deal asset price bounds in incomplete markets. Journal of Political Economy, 101:79 119, F. Delbaen and J. Haezendonck. A martingale approach to premium calculation principles in an arbitrage free market. Insurance: Mathematics and Economics, 8: , F. Delbaen and W. Schachermayer. A general version of the fundamental theorem of asset pricing. Math. Ann., 300: , F. Delbaen and W. Schachermayer. The Mathematics of Arbitrage. Springer, G. Di Nunno. Some versions of the fundamental teorem of asset pricing. Preprint 13/02, available at G. Di Nunno. Hölder equality for conditional expectations with applications to linear monotone operators. Theor. Probability Appl., 48: , B. Fuchssteiner and W. Lusky. Convex Cones. North-Holland, H. U. Gerber and E. S. W. Shiu. Actuarial bridges to dynamic hedging and option pricing. Insurance: Mathematics and Economics, 18: , J.M. Harrison and S. Pliska. A stochastic calculus model of continuous trading: Complete markets. Stochastic processes and their applications, 15: , 1983.
36 Y. Kabanov and C. Stricker. On equivalent martingale measures with bounded densities, pages Lecture Notes in Math Springer, O. Kreps. Arbitrage and equilibrium in economics with infinitely many commodities. Journal of Math. Econom., 8:15 35, M. Rásonyi. A note on martingale measures with bounded densities. Tr. Mat. Inst. Steklova, 237: , D. Rokhlin and W. Schachermayer. A note on lower bounds of martingale measure densities. Illinois Journal of Mathematics, 50: , M. Schweizer. A guided tour through quadratic hedging approaches. In E. Jouini, J. Cvitanic, and M. Musiela, editors, Option Pricing, Interest Rates and Risk Management, pages Cambridge University Press, J. Staum. Fundamental theorems of asset pricing for good-deal bounds. Mathematical Finance, 50: , V. R. Young. Premium principles. In Encyclopedia of Actuarial Science. J. Wiley & Sons, Ltd, 2004.
Lower 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 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 informationLecture 8: Introduction to asset pricing
THE UNIVERSITY OF SOUTHAMPTON Paul Klein Office: Murray Building, 3005 Email: p.klein@soton.ac.uk URL: http://paulklein.se Economics 3010 Topics in Macroeconomics 3 Autumn 2010 Lecture 8: Introduction
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 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 informationFundamental Theorems of Asset Pricing. 3.1 Arbitrage and risk neutral probability measures
Lecture 3 Fundamental Theorems of Asset Pricing 3.1 Arbitrage and risk neutral probability measures Several important concepts were illustrated in the example in Lecture 2: arbitrage; risk neutral probability
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 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 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 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 informationLecture 8: Asset pricing
BURNABY SIMON FRASER UNIVERSITY BRITISH COLUMBIA Paul Klein Office: WMC 3635 Phone: (778) 782-9391 Email: paul klein 2@sfu.ca URL: http://paulklein.ca/newsite/teaching/483.php Economics 483 Advanced Topics
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 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 informationPricing Exotic Options Under a Higher-order Hidden Markov Model
Pricing Exotic Options Under a Higher-order Hidden Markov Model Wai-Ki Ching Tak-Kuen Siu Li-min Li 26 Jan. 2007 Abstract In this paper, we consider the pricing of exotic options when the price dynamic
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 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 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 informationArbitrage and Asset Pricing
Section A Arbitrage and Asset Pricing 4 Section A. Arbitrage and Asset Pricing The theme of this handbook is financial decision making. The decisions are the amount of investment capital to allocate to
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 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 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 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 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 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 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 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 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 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 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 informationA Note on the No Arbitrage Condition for International Financial Markets
A Note on the No Arbitrage Condition for International Financial Markets FREDDY DELBAEN 1 Department of Mathematics Vrije Universiteit Brussel and HIROSHI SHIRAKAWA 2 Department of Industrial and Systems
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 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 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 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 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 informationA class of coherent risk measures based on one-sided moments
A class of coherent risk measures based on one-sided moments T. Fischer Darmstadt University of Technology November 11, 2003 Abstract This brief paper explains how to obtain upper boundaries of shortfall
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 informationOptimal investment and contingent claim valuation in illiquid markets
and contingent claim valuation in illiquid markets Teemu Pennanen King s College London Ari-Pekka Perkkiö Technische Universität Berlin 1 / 35 In most models of mathematical finance, there is at least
More informationPricing Dynamic Solvency Insurance and Investment Fund Protection
Pricing Dynamic Solvency Insurance and Investment Fund Protection Hans U. Gerber and Gérard Pafumi Switzerland Abstract In the first part of the paper the surplus of a company is modelled by a Wiener process.
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 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 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 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 informationON THE FUNDAMENTAL THEOREM OF ASSET PRICING. Dedicated to the memory of G. Kallianpur
Communications on Stochastic Analysis Vol. 9, No. 2 (2015) 251-265 Serials Publications www.serialspublications.com ON THE FUNDAMENTAL THEOREM OF ASSET PRICING ABHAY G. BHATT AND RAJEEVA L. KARANDIKAR
More informationPerformance Measurement with Nonnormal. the Generalized Sharpe Ratio and Other "Good-Deal" Measures
Performance Measurement with Nonnormal Distributions: the Generalized Sharpe Ratio and Other "Good-Deal" Measures Stewart D Hodges forcsh@wbs.warwick.uk.ac University of Warwick ISMA Centre Research Seminar
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 informationSensitivity of American Option Prices with Different Strikes, Maturities and Volatilities
Applied Mathematical Sciences, Vol. 6, 2012, no. 112, 5597-5602 Sensitivity of American Option Prices with Different Strikes, Maturities and Volatilities Nasir Rehman Department of Mathematics and Statistics
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 informationDiscrete time interest rate models
slides for the course Interest rate theory, University of Ljubljana, 2012-13/I, part II József Gáll University of Debrecen, Faculty of Economics Nov. 2012 Jan. 2013, Ljubljana Introduction to discrete
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 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 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 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 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 informationA No-Arbitrage Theorem for Uncertain Stock Model
Fuzzy Optim Decis Making manuscript No (will be inserted by the editor) A No-Arbitrage Theorem for Uncertain Stock Model Kai Yao Received: date / Accepted: date Abstract Stock model is used to describe
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 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 informationA note on sufficient conditions for no arbitrage
Finance Research Letters 2 (2005) 125 130 www.elsevier.com/locate/frl A note on sufficient conditions for no arbitrage Peter Carr a, Dilip B. Madan b, a Bloomberg LP/Courant Institute, New York University,
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 informationOptimal Option Pricing via Esscher Transforms with the Meixner Process
Communications in Mathematical Finance, vol. 2, no. 2, 2013, 1-21 ISSN: 2241-1968 (print), 2241 195X (online) Scienpress Ltd, 2013 Optimal Option Pricing via Esscher Transforms with the Meixner Process
More informationRisk Minimization Control for Beating the Market Strategies
Risk Minimization Control for Beating the Market Strategies Jan Večeř, Columbia University, Department of Statistics, Mingxin Xu, Carnegie Mellon University, Department of Mathematical Sciences, Olympia
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 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 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 informationOPTION PRICE WHEN THE STOCK IS A SEMIMARTINGALE
DOI: 1.1214/ECP.v7-149 Elect. Comm. in Probab. 7 (22) 79 83 ELECTRONIC COMMUNICATIONS in PROBABILITY OPTION PRICE WHEN THE STOCK IS A SEMIMARTINGALE FIMA KLEBANER Department of Mathematics & Statistics,
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 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 informationOptimal Order Placement
Optimal Order Placement Peter Bank joint work with Antje Fruth OMI Colloquium Oxford-Man-Institute, October 16, 2012 Optimal order execution Broker is asked to do a transaction of a significant fraction
More informationNon replication of options
Non replication of options Christos Kountzakis, Ioannis A Polyrakis and Foivos Xanthos June 30, 2008 Abstract In this paper we study the scarcity of replication of options in the two period model of financial
More informationThe Value of Information in Central-Place Foraging. Research Report
The Value of Information in Central-Place Foraging. Research Report E. J. Collins A. I. Houston J. M. McNamara 22 February 2006 Abstract We consider a central place forager with two qualitatively different
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 information4 Martingales in Discrete-Time
4 Martingales in Discrete-Time Suppose that (Ω, F, P is a probability space. Definition 4.1. A sequence F = {F n, n = 0, 1,...} is called a filtration if each F n is a sub-σ-algebra of F, and F n F n+1
More informationFunctional vs Banach space stochastic calculus & strong-viscosity solutions to semilinear parabolic path-dependent PDEs.
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,
More informationForecast Horizons for Production Planning with Stochastic Demand
Forecast Horizons for Production Planning with Stochastic Demand Alfredo Garcia and Robert L. Smith Department of Industrial and Operations Engineering Universityof Michigan, Ann Arbor MI 48109 December
More informationAuction Prices and Asset Allocations of the Electronic Security Trading System Xetra
Auction Prices and Asset Allocations of the Electronic Security Trading System Xetra Li Xihao Bielefeld Graduate School of Economics and Management Jan Wenzelburger Department of Economics University of
More informationExponential utility maximization under partial information and sufficiency of information
Exponential utility maximization under partial information and sufficiency of information Marina Santacroce Politecnico di Torino Joint work with M. Mania WORKSHOP FINANCE and INSURANCE March 16-2, Jena
More informationThe ruin probabilities of a multidimensional perturbed risk model
MATHEMATICAL COMMUNICATIONS 231 Math. Commun. 18(2013, 231 239 The ruin probabilities of a multidimensional perturbed risk model Tatjana Slijepčević-Manger 1, 1 Faculty of Civil Engineering, University
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 informationNo-arbitrage Pricing Approach and Fundamental Theorem of Asset Pricing
No-arbitrage Pricing Approach and Fundamental Theorem of Asset Pricing presented by Yue Kuen KWOK Department of Mathematics Hong Kong University of Science and Technology 1 Parable of the bookmaker Taking
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 informationThe Notion of Arbitrage and Free Lunch in Mathematical Finance
The Notion of Arbitrage and Free Lunch in Mathematical Finance Walter Schachermayer Vienna University of Technology and Université Paris Dauphine Abstract We shall explain the concepts alluded to in the
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 informationLecture 7: Bayesian approach to MAB - Gittins index
Advanced Topics in Machine Learning and Algorithmic Game Theory Lecture 7: Bayesian approach to MAB - Gittins index Lecturer: Yishay Mansour Scribe: Mariano Schain 7.1 Introduction In the Bayesian approach
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 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 informationMartingales. by D. Cox December 2, 2009
Martingales by D. Cox December 2, 2009 1 Stochastic Processes. Definition 1.1 Let T be an arbitrary index set. A stochastic process indexed by T is a family of random variables (X t : t T) defined on a
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 informationLIST OF PUBLICATIONS
LIST OF PUBLICATIONS Miklós Rásonyi PhD thesis [R0] M. Rásonyi: On certain problems of arbitrage theory in discrete-time financial market models. PhD thesis, Université de Franche-Comté, Besançon, 2002.
More informationProblem Set 3. Thomas Philippon. April 19, Human Wealth, Financial Wealth and Consumption
Problem Set 3 Thomas Philippon April 19, 2002 1 Human Wealth, Financial Wealth and Consumption The goal of the question is to derive the formulas on p13 of Topic 2. This is a partial equilibrium analysis
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 information( 0) ,...,S N ,S 2 ( 0)... S N S 2. N and a portfolio is created that way, the value of the portfolio at time 0 is: (0) N S N ( 1, ) +...
No-Arbitrage Pricing Theory Single-Period odel There are N securities denoted ( S,S,...,S N ), they can be stocks, bonds, or any securities, we assume they are all traded, and have prices available. Ω
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 informationOption Pricing under Delay Geometric Brownian Motion with Regime Switching
Science Journal of Applied Mathematics and Statistics 2016; 4(6): 263-268 http://www.sciencepublishinggroup.com/j/sjams doi: 10.11648/j.sjams.20160406.13 ISSN: 2376-9491 (Print); ISSN: 2376-9513 (Online)
More informationOptimally Thresholded Realized Power Variations for Lévy Jump Diffusion Models
Optimally Thresholded Realized Power Variations for Lévy Jump Diffusion Models José E. Figueroa-López 1 1 Department of Statistics Purdue University University of Missouri-Kansas City Department of Mathematics
More informationCharacterization of the Optimum
ECO 317 Economics of Uncertainty Fall Term 2009 Notes for lectures 5. Portfolio Allocation with One Riskless, One Risky Asset Characterization of the Optimum Consider a risk-averse, expected-utility-maximizing
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 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 informationFin 501: Asset Pricing Fin 501:
Lecture 3: One-period Model Pricing Prof. Markus K. Brunnermeier Slide 03-1 Overview: Pricing i 1. LOOP, No arbitrage 2. Forwards 3. Options: Parity relationship 4. No arbitrage and existence of state
More informationOptimal Stopping Rules of Discrete-Time Callable Financial Commodities with Two Stopping Boundaries
The Ninth International Symposium on Operations Research Its Applications (ISORA 10) Chengdu-Jiuzhaigou, China, August 19 23, 2010 Copyright 2010 ORSC & APORC, pp. 215 224 Optimal Stopping Rules of Discrete-Time
More informationLecture 17. The model is parametrized by the time period, δt, and three fixed constant parameters, v, σ and the riskless rate r.
Lecture 7 Overture to continuous models Before rigorously deriving the acclaimed Black-Scholes pricing formula for the value of a European option, we developed a substantial body of material, in continuous
More information