H. Geman and M. Yor :
“Quelques relations entre processus de Bessel, options asiatiques et fonctions confluentes hypergéométriques ”
[Some relations between Bessel processes, Asian options and confluent hypergeometric functions ],
C. R. Acad. Sci., Paris, Sér. I
314 : 6
(1992 ),
pp. 471–474 .
Translated into English in Exponential functionals of Brownian motion and related processes (2001)
MR
1154389 Zbl
0759.60084 article
People
BibTeX
@article {key1154389m,
AUTHOR = {Geman, H\'elyette and Yor, Marc},
TITLE = {Quelques relations entre processus de
{B}essel, options asiatiques et fonctions
confluentes hyperg\'eom\'etriques [Some
relations between {B}essel processes,
{A}sian options and confluent hypergeometric
functions]},
JOURNAL = {C. R. Acad. Sci., Paris, S\'er. I},
FJOURNAL = {Comptes Rendus de l'Acad\'emie des Sciences.
S\'erie I. Math\'ematique},
VOLUME = {314},
NUMBER = {6},
YEAR = {1992},
PAGES = {471--474},
NOTE = {Translated into English in \textit{Exponential
functionals of Brownian motion and related
processes} (2001). MR:1154389. Zbl:0759.60084.},
ISSN = {0764-4442},
}
H. Geman and M. Yor :
“Bessel processes, Asian options, and perpetuities Math. Finance
3 : 4
(October 1993 ),
pp. 349–375 .
Zbl
0884.90029 article
Abstract
People
BibTeX
Using Bessel processes, one can solve several open problems involving the integral of an exponential of Brownian motion. This point will be illustrated with three examples. The first one is a formula for the Laplace transform of an Asian option which is “out of the money.” The second example concerns volatility misspecification in portfolio insurance strategies, when the stochastic volatility is represented by the Hull and White model. The third one is the valuation of perpetuities or annuities under stochastic interest rates within the Cox–Ingersoll–Ross framework. Moreover, without using time changes or Bessel processes, but only simple probabilistic methods, we obtain further results about Asian options: the computation of the moments of all orders of an arithmetic average of geometric Brownian motion; the property that, in contrast with most of what has been written so far, the Asian option may be more expensive than the standard option (e.g., options on currencies or oil spreads); and a simple, closed-form expression of the Asian option price when the option is “in the money,” thereby illuminating the impact on the Asian option price of the revealed underlying asset price as time goes by. This formula has an interesting resemblance with the Black–Scholes formula, even though the comparison cannot be carried too far.
@article {key0884.90029z,
AUTHOR = {Geman, H\'elyette and Yor, Marc},
TITLE = {Bessel processes, {A}sian options, and
perpetuities},
JOURNAL = {Math. Finance},
FJOURNAL = {Mathematical Finance},
VOLUME = {3},
NUMBER = {4},
MONTH = {October},
YEAR = {1993},
PAGES = {349--375},
DOI = {10.1111/j.1467-9965.1993.tb00092.x},
NOTE = {Zbl:0884.90029.},
ISSN = {0960-1627},
}
H. Geman and M. Yor :
“Pricing and hedging double-barrier options: A probabilistic approach Math. Finance
6 : 4
(October 1996 ),
pp. 365–378 .
Zbl
0915.90016 article
Abstract
People
BibTeX
Barrier options have become increasingly popular over the last few years. Less expensive than standard options, they may provide the appropriate hedge in a number of risk management strategies. In the case of a single-barrier option, the valuation problem is not very difficult (see [Merton 1973] and [Goldman et al. 1979]). the situation where the option gets knocked out when the underlying instrument hits either of two well-defined boundaries is less straightforward. Kunitomo and Ikeda [1992] provide a pricing formula expressed as the sum of an infinite series whose convergence is studied through numerical procedures and suggested to be rapid. We follow a methodology which proved quite successful in the case of Asian options (see Geman and Yor [1992, 1993]) and which has its roots in some fundamental properties of Brownian motion. This methodology permits the derivation of a simple expression of the Laplace transform of the double-barrir price with respect to its maturity date. the inversion of the Laplace transform using techniques developed by Geman and Eydeland [1995], is then fairly easy to perform.
@article {key0915.90016z,
AUTHOR = {Geman, H\'elyette and Yor, Marc},
TITLE = {Pricing and hedging double-barrier options:
{A} probabilistic approach},
JOURNAL = {Math. Finance},
FJOURNAL = {Mathematical Finance},
VOLUME = {6},
NUMBER = {4},
MONTH = {October},
YEAR = {1996},
PAGES = {365--378},
DOI = {10.1111/j.1467-9965.1996.tb00122.x},
NOTE = {Zbl:0915.90016.},
ISSN = {0960-1627},
}
M. Yor, M. Chesney, H. Geman, and M. Jeanblanc-Picqué :
“Some combinations of Asian, Parisian and barrier options ,”
pp. 61–87
in
Mathematics of derivative securities
(Cambridge, UK, January–June 1995 ).
Edited by M. Dempster and S. R. Pleska .
Publications of the Newton Institute 15 .
Cambridge University Press ,
1997 .
MR
1491368 Zbl
0911.90036 incollection
People
BibTeX
@incollection {key1491368m,
AUTHOR = {Yor, M. and Chesney, M. and Geman, H.
and Jeanblanc-Picqu\'e, M.},
TITLE = {Some combinations of {A}sian, {P}arisian
and barrier options},
BOOKTITLE = {Mathematics of derivative securities},
EDITOR = {Dempster, M.A.H. and Pleska, Stanley
R.},
SERIES = {Publications of the Newton Institute},
NUMBER = {15},
PUBLISHER = {Cambridge University Press},
YEAR = {1997},
PAGES = {61--87},
NOTE = {(Cambridge, UK, January--June 1995).
MR:1491368. Zbl:0911.90036.},
ISSN = {1366-2651},
ISBN = {9780521584241},
}
H. Geman and M. Yor :
“Stochastic time changes in catastrophe option pricing Insurance Math. Econom.
21 : 3
(December 1997 ),
pp. 185–193 .
MR
1614517 Zbl
0894.90046 article
Abstract
People
BibTeX
Catastrophe insurance derivatives (Futures and options) were introduced in December 1992 by the Chicago Board of Trade in order to offer insurers new ways of hedging their underwriting risk. Only CAT options and combinations of options such as call spreads are traded today, and the ISO index has been replaced by the PCS index. Otherwise, the economic goal of these instruments continues to be for insurers an alternative to reinsurance and for portfolio managers a new class of assets to invest in.
The pricing methodology of these derivatives relies on some crucial elements:
the choice of the stochastic modelling of the aggregate reported claim index dynamics (since the terminal value of this index defines the pay-off of the CAT options);
the decision of a financial versus actuarial approach to the valuation;
the number of sources of randomness in the model and the determination of a “martingale measure” for insurance and reinsurance instruments.
We represent in this paper the dynamics of the aggregate claim index by the sum of a geometric Brownian motion which accounts for the randomness in the reporting of the claims and a Poisson process which accounts for the occurrence of catastrophes (only catastrophic claims are incorporated in the index). Geman [1994] and Cummins and Geman [1995] took this modelling for the instantaneous claim process. Our choice here is closer to the classical actuarial representation while preserving the quasi-completeness of insurance derivative markets obtained by applying the Delbaen and Haezendonck [1989] methodology to the class of layers of reinsurance replicating the call spreads. Moreover, we obtain semi-analytical solutions for the CAT options and call spreads by extending to the jump-diffusion case the method of the Laplace transform and stochastic time changes introduced in Geman and Yor [1993, 1996] in order to price financial path-dependent options through the properties of excursion theory.
@article {key1614517m,
AUTHOR = {Geman, Helyette and Yor, Marc},
TITLE = {Stochastic time changes in catastrophe
option pricing},
JOURNAL = {Insurance Math. Econom.},
FJOURNAL = {Insurance: Mathematics \& Economics},
VOLUME = {21},
NUMBER = {3},
MONTH = {December},
YEAR = {1997},
PAGES = {185--193},
DOI = {10.1016/S0167-6687(97)00017-6},
NOTE = {MR:1614517. Zbl:0894.90046.},
ISSN = {0167-6687},
}
H. Geman and M. Yor :
“Some relations between Bessel processes, Asian options and confluent hypergeometric functions 49–54
in
M. Yor :
Exponential functionals of Brownian motion and related processes .
Springer Finance .
Springer (Berlin ),
2001 .
English translation of an article published in C. R. Acad. Sci., Paris, Sér. I 314 :6 (1992)
incollection
Abstract
People
BibTeX
A closed formula is obtained for the Laplace transform of moments of certain exponential functionals of Brownian motion with drift, which give the price of some financial options, so-called Asian options. A second equivalent formula is presented, which is the translation, in this context, of some intertwining properties of Bessel processes or confluent hypergeometric functions.
@incollection {key70981829,
AUTHOR = {Geman, H\'elyette and Yor, Marc},
TITLE = {Some relations between {B}essel processes,
{A}sian options and confluent hypergeometric
functions},
BOOKTITLE = {Exponential functionals of {B}rownian
motion and related processes},
SERIES = {Springer Finance},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {2001},
PAGES = {49--54},
DOI = {10.1007/978-3-642-56634-9_4},
NOTE = {English translation of an article published
in \textit{C. R. Acad. Sci., Paris,
S\'er. I} \textbf{314}:6 (1992).},
ISSN = {1616-0533},
ISBN = {9783540659433},
}
H. Geman, D. B. Madan, and M. Yor :
“Time changes for Lévy processes Math. Finance
11 : 1
(2001 ),
pp. 79–96 .
MR
1807849 Zbl
0983.60082 article
Abstract
People
BibTeX
The goal of this paper is to consider pure jump Lévy processes of finite variation with an infinite arrival rate of jumps as models for the logarithm of asset prices. These processes may be written as time-changed Brownian motion. We exhibit the explicit time change for each of a wide class of Lévy processes and show that the time change is a weighted price move measure of time. Additionally, we present a number of Lévy processes that are analytically tractable, in their characteristic functions and Lévy densities, and hence are relevant for option pricing.
@article {key1807849m,
AUTHOR = {Geman, H\'elyette and Madan, Dilip B.
and Yor, Marc},
TITLE = {Time changes for {L}\'evy processes},
JOURNAL = {Math. Finance},
FJOURNAL = {Mathematical Finance. An International
Journal of Mathematics, Statistics and
Financial Economics},
VOLUME = {11},
NUMBER = {1},
YEAR = {2001},
PAGES = {79--96},
DOI = {10.1111/1467-9965.00108},
NOTE = {MR:1807849. Zbl:0983.60082.},
ISSN = {0960-1627},
}
M. Yor :
Exponential functionals of Brownian motion and related processes Springer Finance .
Springer (Berlin ),
2001 .
With an introductory chapter by Hélyette Geman.
MR
1854494 Zbl
0999.60004 book
People
BibTeX
@book {key1854494m,
AUTHOR = {Yor, Marc},
TITLE = {Exponential functionals of {B}rownian
motion and related processes},
SERIES = {Springer Finance},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {2001},
PAGES = {x+205},
DOI = {10.1007/978-3-642-56634-9},
URL = {http://www.springer.com/978-3-540-65943-3},
NOTE = {With an introductory chapter by H\'elyette
Geman. MR:1854494. Zbl:0999.60004.},
ISSN = {1616-0533},
ISBN = {9783540659433},
}
H. Geman, D. B. Madan, and M. Yor :
“Asset prices are Brownian motion: Only in business time 103–146
in
Quantitative analysis in financial markets: Collected papers of the New York University mathematical finance seminar
(New York, 1995–1998 ),
vol. 2 .
Edited by M. Avellaneda .
World Scientific (River Edge, NJ ),
2001 .
MR
1886692 Zbl
1134.91019 incollection
Abstract
People
BibTeX
This paper argues that asset price processes arising from market clearing conditions should be modeled as pure jump processes, with no continuous martingale component. However, we show that continuity and normality can always be obtained after a time change. We study various examples of time changes and show that in all cases they are related to measures of economic activity. For the most general class of processes, the time change is a size-weighted sum of order arrivals. The paper provides a number of new processes for modeling prices. Characteristic functions for these processes are also given in closed form.
@incollection {key1886692m,
AUTHOR = {Geman, Helyette and Madan, Dilip B.
and Yor, Marc},
TITLE = {Asset prices are {B}rownian motion:
{O}nly in business time},
BOOKTITLE = {Quantitative analysis in financial markets:
{C}ollected papers of the {N}ew {Y}ork
{U}niversity mathematical finance seminar},
EDITOR = {Avellaneda, M.},
VOLUME = {2},
PUBLISHER = {World Scientific},
ADDRESS = {River Edge, NJ},
YEAR = {2001},
PAGES = {103--146},
DOI = {10.1142/9789812810663_0004},
NOTE = {(New York, 1995--1998). MR:1886692.
Zbl:1134.91019.},
ISBN = {9810242263},
}
P. Carr, H. Geman, D. B. Madan, and M. Yor :
“The fine structure of asset returns: An empirical investigation J. Business
75 : 2
(April 2002 ),
pp. 305–332 .
article
Abstract
People
BibTeX
We investigate the importance of diffusion and jumps in a new model for asset returns. In contrast to standard models, we allow for jump components displaying finite or infinite activity and variation. Empirical investigations of time series indicate that index dynamics are devoid of a diffusion component, which may be present in the dynamics of individual stocks. This leads to the conjecture, confirmed on options data, that the risk-neutral process should be free of a diffusion component. We conclude that the statistical and risk-neutral processes for equity prices are pure jump processes of infinite activity and finite variation.
@article {key51213502,
AUTHOR = {Carr, P. and Geman, H. and Madan, D.
B. and Yor, M.},
TITLE = {The fine structure of asset returns:
{A}n empirical investigation},
JOURNAL = {J. Business},
FJOURNAL = {The Journal of Business},
VOLUME = {75},
NUMBER = {2},
MONTH = {April},
YEAR = {2002},
PAGES = {305--332},
URL = {http://www.jstor.org/stable/10.1086/338705},
ISSN = {0021-9398},
}
H. Geman, D. B. Madan, and M. Yor :
“Stochastic volatility, jumps and hidden time changes Finance Stoch.
6 : 1
(January 2002 ),
pp. 63–90 .
MR
1885584 Zbl
1006.60026 article
Abstract
People
BibTeX
Stochastic volatility and jumps are viewed as arising from Brownian subordination given here by an independent purely discontinuous process and we inquire into the relation between the realized variance or quadratic variation of the process and the time change. The class of models considered encompasses a wide range of models employed in practical financial modeling. It is shown that in general the time change cannot be recovered from the composite process and we obtain its conditional distribution in a variety of cases. The implications of our results for working with stochastic volatility models in general is also described. We solve the recovery problem, i.e., the identification the conditional law for a variety of cases, the simplest solution being for the gamma time change when this conditional law is that of the first hitting time process of Brownian motion with drift attaining the level of the variation of the time changed process. We also introduce and solve in certain cases the problem of stochastic scaling. A stochastic scalar is a subordinator that recovers the law of a given subordinator when evaluated at an independent and time scaled copy of the given subordinator. These results are of importance in comparing price quality delivered by alternate exchanges.
@article {key1885584m,
AUTHOR = {Geman, H\'elyette and Madan, Dilip B.
and Yor, Marc},
TITLE = {Stochastic volatility, jumps and hidden
time changes},
JOURNAL = {Finance Stoch.},
FJOURNAL = {Finance and Stochastics},
VOLUME = {6},
NUMBER = {1},
MONTH = {January},
YEAR = {2002},
PAGES = {63--90},
DOI = {10.1007/s780-002-8401-3},
NOTE = {MR:1885584. Zbl:1006.60026.},
ISSN = {0949-2984},
}
C. Donati-Martin, H. Matsumoto, and M. Yor :
“The law of geometric Brownian motion and its integral, revisited: Application to conditional moments 221–243
in
Mathematical finance — Bachelier Congress, 2000
(Paris, 29 June–1 July 2000 ).
Edited by H. Geman, D. Madan, S. R. Pliska, and T. Vorst .
Springer Finance .
Springer (Berlin ),
2002 .
MR
1960566 Zbl
1030.91029 incollection
People
BibTeX
@incollection {key1960566m,
AUTHOR = {Donati-Martin, Catherine and Matsumoto,
Hiroyuki and Yor, Marc},
TITLE = {The law of geometric {B}rownian motion
and its integral, revisited: {A}pplication
to conditional moments},
BOOKTITLE = {Mathematical finance---{B}achelier {C}ongress,
2000},
EDITOR = {Geman, Helyette and Madan, Dilip and
Pliska, Stanley R. and Vorst, Ton},
SERIES = {Springer Finance},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {2002},
PAGES = {221--243},
DOI = {10.1007/978-3-662-12429-1_11},
NOTE = {(Paris, 29 June--1 July 2000). MR:1960566.
Zbl:1030.91029.},
ISSN = {1616-0533},
ISBN = {9783540677819},
}
P. Carr, H. Geman, D. B. Madan, and M. Yor :
“Stochastic volatility for Lévy processes Math. Finance
13 : 3
(2003 ),
pp. 345–382 .
EFA 2002 Berlin meetings, presented paper.
MR
1995283 Zbl
1092.91022 article
Abstract
People
BibTeX
Three processes reflecting persistence of volatility are initially formulated by evaluating three Lévy processes at a time change given by the integral of a mean-reverting square root process. The model for the mean-reverting time change is then generalized to include non-Gaussian models that are solutions to Ornstein–Uhlenbeck equations driven by one-sided discontinuous Lévy processes permitting correlation with the stock. Positive stock price processes are obtained by exponentiating and mean correcting these processes, or alternatively by stochastically exponentiating these processes. The characteristic functions for the log price can be used to yield option prices via the fast Fourier transform. In general mean-corrected exponentiation performs better than employing the stochastic exponential. It is observed that the mean-corrected exponential model is not a martingale in the filtration in which it is originally defined. This leads us to formulate and investigate the important property of martingale marginals where we seek martingales in altered filtrations consistent with the one-dimensional marginal distributions of the level of the process at each future date.
@article {key1995283m,
AUTHOR = {Carr, Peter and Geman, H\'elyette and
Madan, Dilip B. and Yor, Marc},
TITLE = {Stochastic volatility for {L}\'evy processes},
JOURNAL = {Math. Finance},
FJOURNAL = {Mathematical Finance},
VOLUME = {13},
NUMBER = {3},
YEAR = {2003},
PAGES = {345--382},
DOI = {10.1111/1467-9965.00020},
NOTE = {EFA 2002 Berlin meetings, presented
paper. MR:1995283. Zbl:1092.91022.},
ISSN = {0960-1627},
}
P. Carr, H. Geman, D. B. Madan, and M. Yor :
“From local volatility to local Lévy models Quant. Finance
4 : 5
(October 2004 ),
pp. 581–588 .
MR
2241297 article
Abstract
People
BibTeX
@article {key2241297m,
AUTHOR = {Carr, Peter and Geman, H\'elyette and
Madan, Dilip B. and Yor, Marc},
TITLE = {From local volatility to local {L}\'evy
models},
JOURNAL = {Quant. Finance},
FJOURNAL = {Quantitative Finance},
VOLUME = {4},
NUMBER = {5},
MONTH = {October},
YEAR = {2004},
PAGES = {581--588},
DOI = {10.1080/14697680400024921},
URL = {https://ssrn.com/abstract=957170},
NOTE = {MR:2241297.},
ISSN = {1469-7688},
}
M. Atlan, H. Geman, and M. Yor :
Options on hedge funds under the high water mark rule .
Preprint ,
October 2005 .
ArXiv
math/0510497 techreport
Abstract
People
BibTeX
The rapidly growing hedge fund industry has provided individual and institutional investors with new investment vehicles and styles of management. It has also brought forward a new form of performance contract: hedge fund managers receive incentive fees which are typically a fraction of the fund net asset value (NAV) above its starting level–a rule known as high water mark. Options on hedge funds are becoming increasingly popular, in particular because they allow investors with limited capital to get exposure to this new asset class. The goal of the paper is to propose a valuation of plain-vanilla options on hedge funds which accounts for the high water market rule. Mathematically, this valuation leads to an interesting use of local times of Brownian motion. Option prices are numerically computed by inversion of their Laplace transforms.
@techreport {keymath/0510497a,
AUTHOR = {Atlan, Marc and Geman, H\'elyette and
Yor, Marc},
TITLE = {Options on hedge funds under the high
water mark rule},
TYPE = {preprint},
MONTH = {October},
YEAR = {2005},
NOTE = {ArXiv:math/0510497.},
}
P. Carr, H. Geman, D. B. Madan, and M. Yor :
“Pricing options on realized variance Finance Stoch.
9 : 4
(October 2005 ),
pp. 453–475 .
MR
2213777 Zbl
1096.91022 article
Abstract
People
BibTeX
Models which hypothesize that returns are pure jump processes with independent increments have been shown to be capable of capturing the observed variation of market prices of vanilla stock options across strike and maturity. In this paper, these models are employed to derive in closed form the prices of derivatives written on future realized quadratic variation. Alternative work on pricing derivatives on quadratic variation has alternatively assumed that the underlying returns process is continuous over time. We compare the model values of derivatives on quadratic variation for the two types of models and find substantial differences.
@article {key2213777m,
AUTHOR = {Carr, Peter and Geman, H\'elyette and
Madan, Dilip B. and Yor, Marc},
TITLE = {Pricing options on realized variance},
JOURNAL = {Finance Stoch.},
FJOURNAL = {Finance and Stochastics},
VOLUME = {9},
NUMBER = {4},
MONTH = {October},
YEAR = {2005},
PAGES = {453--475},
DOI = {10.1007/s00780-005-0155-x},
NOTE = {MR:2213777. Zbl:1096.91022.},
ISSN = {0949-2984},
}
P. Carr, H. Geman, D. B. Madan, and M. Yor :
“Self-decomposability and option pricing Math. Finance
17 : 1
(2007 ),
pp. 31–57 .
MR
2281791 Zbl
1278.91157 article
Abstract
People
BibTeX
The risk-neutral process is modeled by a four parameter self-similar process of independent increments with a self-decomposable law for its unit time distribution. Six different processes in this general class are theoretically formulated and empirically investigated. We show that all six models are capable of adequately synthesizing European option prices across the spectrum of strikes and maturities at a point of time. Considerations of parameter stability over time suggest a preference for two of these models. Currently, there are several option pricing models with 6–10 free parameters that deliver a comparable level of performance in synthesizing option prices. The dimension reduction attained here should prove useful in studying the variation over time of option prices.
@article {key2281791m,
AUTHOR = {Carr, Peter and Geman, H\'elyette and
Madan, Dilip B. and Yor, Marc},
TITLE = {Self-decomposability and option pricing},
JOURNAL = {Math. Finance},
FJOURNAL = {Mathematical Finance. An International
Journal of Mathematics, Statistics and
Financial Economics},
VOLUME = {17},
NUMBER = {1},
YEAR = {2007},
PAGES = {31--57},
DOI = {10.1111/j.1467-9965.2007.00293.x},
NOTE = {MR:2281791. Zbl:1278.91157.},
ISSN = {0960-1627},
}
H. Geman, D. B. Madan, and M. Yor :
“Probing option prices for information Methodol. Comput. Appl. Probab.
9 : 1
(March 2007 ),
pp. 115–131 .
MR
2364984 Zbl
1157.60067 article
Abstract
People
BibTeX
We present a methodology for extracting information from option prices when the market is viewed as knowledgeable. By expanding the information filtration judiciously and determining conditional characteristic functions for the log of the stock price, we obtain option pricing formulae which when fit to market data may reveal this information. In particular, we consider probing option prices for knowledge of the future stock price, instantaneous volatility, and the asymptotic dividend stream. Additionally the bridge laws developed are also useful for simulation based on stratified sampling that conditions on the terminal values of paths.
@article {key2364984m,
AUTHOR = {Geman, H\'elyette and Madan, Dilip B.
and Yor, Marc},
TITLE = {Probing option prices for information},
JOURNAL = {Methodol. Comput. Appl. Probab.},
FJOURNAL = {Methodology and Computing in Applied
Probability},
VOLUME = {9},
NUMBER = {1},
MONTH = {March},
YEAR = {2007},
PAGES = {115--131},
DOI = {10.1007/s11009-006-9005-3},
NOTE = {MR:2364984. Zbl:1157.60067.},
ISSN = {1387-5841},
}
M. Atlan, H. Geman, D. B. Madan, and M. Yor :
“Correlation and the pricing of risks Ann. Finance
3 : 4
(October 2007 ),
pp. 411–453 .
Zbl
1233.91320 article
Abstract
People
BibTeX
Given a pricing kernel we investigate the class of risks that are not priced by this kernel. Risks are random payoffs written on underlying uncertainties that may themselves either be random variables, processes, events or information filtrations. A risk is said to be not priced by a kernel if all derivatives on this risk always earn a zero excess return, or equivalently the derivatives may be priced without a change of measure. We say that such risks are not kernel priced. It is shown that reliance on direct correlation between the risk and the pricing kernel as an indicator for the kernel pricing of a risk can be misleading. Examples are given of risks that are uncorrelated with the pricing kernel but are kernel priced. These examples lead to new definitions for risks that are not kernel priced in correlation terms. Additionally we show that the pricing kernel itself viewed as a random variable is strongly negatively kernel priced implying in particular that all monotone increasing functions of the kernel receive a negative risk premium. Moreover the equivalence class of the kernel under increasing monotone transformations is unique in possessing this property.
@article {key1233.91320z,
AUTHOR = {Atlan, Marc and Geman, H\'elyette and
Madan, Dilip B. and Yor, Marc},
TITLE = {Correlation and the pricing of risks},
JOURNAL = {Ann. Finance},
FJOURNAL = {Annals of Finance},
VOLUME = {3},
NUMBER = {4},
MONTH = {October},
YEAR = {2007},
PAGES = {411--453},
DOI = {10.1007/s10436-006-0063-x},
NOTE = {Zbl:1233.91320.},
ISSN = {1614-2446},
}
P. Carr, H. Geman, D. B. Madan, and M. Yor :
“Options on realized variance and convex orders Quant. Finance
11 : 11
(2011 ),
pp. 1685–1694 .
MR
2850996 Zbl
1277.91164 article
Abstract
People
BibTeX
@article {key2850996m,
AUTHOR = {Carr, Peter and Geman, Helyette and
Madan, Dilip B. and Yor, Marc},
TITLE = {Options on realized variance and convex
orders},
JOURNAL = {Quant. Finance},
FJOURNAL = {Quantitative Finance},
VOLUME = {11},
NUMBER = {11},
YEAR = {2011},
PAGES = {1685--1694},
DOI = {10.1080/14697680903397675},
NOTE = {MR:2850996. Zbl:1277.91164.},
ISSN = {1469-7688},
}
H. Geman and M. Jeanblanc :
“Marc Yor: A beautiful mind has disappeared Stochastic Process. Appl.
124 : 6
(June 2014 ),
pp. v–vii .
MR
3188345 Zbl
1294.01043 article
People
BibTeX
@article {key3188345m,
AUTHOR = {Geman, Helyette and Jeanblanc, Monique},
TITLE = {Marc {Y}or: {A} beautiful mind has disappeared},
JOURNAL = {Stochastic Process. Appl.},
FJOURNAL = {Stochastic Processes and their Applications},
VOLUME = {124},
NUMBER = {6},
MONTH = {June},
YEAR = {2014},
PAGES = {v--vii},
DOI = {10.1016/j.spa.2014.02.006},
NOTE = {MR:3188345. Zbl:1294.01043.},
ISSN = {0304-4149},
}
M. Atlan, D. Madan, and H. Geman :
“Marc Yor and mathematical finance ,”
pp. 79–90
in
Marc Yor: La passion du mouvement brownien
[Marc Yor: The passion of Brownian motion ].
Edited by J. Bertoin, M. Jeanblanc, J.-F. Le Gall, and Z. Shi .
Société Mathématique de France (Paris ),
2015 .
Gazette des Mathématiciens and Matapli special issue.
incollection
People
BibTeX
@incollection {key10247119,
AUTHOR = {Atlan, Marc and Madan, Dilip and Geman,
H\'elyette},
TITLE = {Marc {Y}or and mathematical finance},
BOOKTITLE = {Marc {Y}or: {L}a passion du mouvement
brownien [Marc {Y}or: {T}he passion
of {B}rownian motion]},
EDITOR = {Bertoin, Jean and Jeanblanc, Monique
and Le Gall, Jean-Fran\c{c}ois and Shi,
Zhan},
PUBLISHER = {Soci\'et\'e Math\'ematique de France},
ADDRESS = {Paris},
YEAR = {2015},
PAGES = {79--90},
NOTE = {Gazette des Math\'ematiciens and Matapli
special issue.},
ISBN = {9782856298015},
}
