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},
}
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},
}
D. B. Madan and M. Yor :
“Making Markov martingales meet marginals: With explicit constructions Bernoulli
8 : 4
(2002 ),
pp. 509–536 .
MR
1914701 Zbl
1009.60037 article
Abstract
People
BibTeX
We present three generic constructions of martingales that all have the Markov property with known and prespecified marginal densities. These constructions are further investigated for the special case when the prespecified marginals satisfy the scaling property and hence the only datum needed for the construction is the density at unit time. Interesting relations with stochastic orders are presented, along with numerous examples of the resulting martingales.
@article {key1914701m,
AUTHOR = {Madan, Dilip B. and Yor, Marc},
TITLE = {Making {M}arkov martingales meet marginals:
{W}ith explicit constructions},
JOURNAL = {Bernoulli},
FJOURNAL = {Bernoulli. Official Journal of the Bernoulli
Society for Mathematical Statistics
and Probability},
VOLUME = {8},
NUMBER = {4},
YEAR = {2002},
PAGES = {509--536},
URL = {https://projecteuclid.org/euclid.bj/1078681382},
NOTE = {MR:1914701. Zbl:1009.60037.},
ISSN = {1350-7265},
}
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},
}
D. Madan and M. Yor :
“On the Itô–Tanaka formula for strictly local martingales: Does it need a correction term? Obwerwolfach Rep.
1 : 2
(2004 ),
pp. 1365–1366 .
Report no. 26/2004. Brief summary of lecture given by Yor at the mini-workshop “Local time-space calculus with applications ,” Oberwolfach, Germany, 16–22 May 2004.
article
People
BibTeX
@article {key68281724,
AUTHOR = {Madan, D. and Yor, M.},
TITLE = {On the {I}t\^o--{T}anaka formula for
strictly local martingales: {D}oes it
need a correction term?},
JOURNAL = {Obwerwolfach Rep.},
FJOURNAL = {Obwerwolfach Reports},
VOLUME = {1},
NUMBER = {2},
YEAR = {2004},
PAGES = {1365--1366},
URL = {http://www.riss.kr/link?id=O42412553},
NOTE = {Report no. 26/2004. Brief summary of
lecture given by Yor at the mini-workshop
``Local time-space calculus with applications
'', Oberwolfach, Germany, 16--22 May
2004.},
ISSN = {1660-8933},
}
D. Madan and M. Yor :
Mimicking the one-dimensional marginals of the Heston model ,
January 2004 .
unpublished
People
BibTeX
@unpublished {key84158599,
AUTHOR = {Madan, D. and Yor, M.},
TITLE = {Mimicking the one-dimensional marginals
of the {H}eston model},
MONTH = {January},
YEAR = {2004},
}
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},
}
D. Madan and M. Yor :
CGMY and Meixner subordinators are absolutely continuous with respect to one sided stable subordinators .
Preprint ,
January 2006 .
ArXiv
math/0601173 techreport
Abstract
People
BibTeX
@techreport {keymath/0601173a,
AUTHOR = {Madan, Dilip and Yor, Marc},
TITLE = {C{GMY} and {M}eixner subordinators are
absolutely continuous with respect to
one sided stable subordinators},
TYPE = {preprint},
MONTH = {January},
YEAR = {2006},
NOTE = {ArXiv:math/0601173.},
}
D. B. Madan and M. Yor :
“Itô’s integrated formula for strict local martingales 157–170
in
In memoriam Paul-André Meyer: Séminaire de probabilités, XXXIX
[In memoriam Paul-André Meyer: Thirty-ninth probability seminar ].
Edited by M. Émery and M. Yor .
Lecture Notes in Mathematics .
Springer (Berlin ),
2006 .
MR
2276895 Zbl
1133.60025 incollection
Abstract
People
BibTeX
For \( F:\mathbb{R}\to\mathbb{R} \) a \( C^2 \) function, and more generally a difference of convex functions, \( (S_t \) , \( t\geq 0) \) a continuous strict local martingale taking values in \( \mathbb{R}_+ \) , we investigate under which condition the stochastic integral appearing in Itô’s formula applied to \( F(S_t) \) , \( t\geq 0 \) , is a true martingale and, if not, how it may be corrected to become one.
@incollection {key2276895m,
AUTHOR = {Madan, Dilip B. and Yor, Marc},
TITLE = {It\^o's integrated formula for strict
local martingales},
BOOKTITLE = {In memoriam {P}aul-{A}ndr\'e {M}eyer:
{S}\'eminaire de probabilit\'es, {XXXIX}
[In memoriam {P}aul-{A}ndr\'e {M}eyer:
{T}hirty-ninth probability seminar]},
EDITOR = {\'Emery, M. and Yor, M.},
SERIES = {Lecture Notes in Mathematics},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {2006},
PAGES = {157--170},
DOI = {10.1007/978-3-540-35513-7_13},
NOTE = {MR:2276895. Zbl:1133.60025.},
ISSN = {1874},
ISBN = {9783540309949},
}
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},
}
M. Yor :
“Some remarkable properties of gamma processes 37–47
in
Advances in mathematical finance
(College Park, MD, 29 September–1 October 1 2006 ).
Edited by M. C. Fu, R. A. Jarrow, J.-Y. Yen, and R. J. Elliott .
Applied Numerical Harmonic Analysis .
Birkhäuser (Boston, MA ),
2007 .
Paper presented at the mathematical finance conference in honor of the 60th birthday of Dilip B. Madan.
MR
2359361 Zbl
1156.60030 incollection
Abstract
People
BibTeX
A number of remarkable properties of gamma processes are gathered in this paper, including realisation of their bridges, absolute continuity relationships, realisation of a gamma process as an inverse local time, and the effect of a gamma process as a time change. Some of them are put in perspective with their Brownian counterparts.
@incollection {key2359361m,
AUTHOR = {Yor, Marc},
TITLE = {Some remarkable properties of gamma
processes},
BOOKTITLE = {Advances in mathematical finance},
EDITOR = {Fu, Michael C. and Jarrow, Robert A.
and Yen, Ju-Yi and Elliott, Robert J.},
SERIES = {Applied Numerical Harmonic Analysis},
PUBLISHER = {Birkh\"auser},
ADDRESS = {Boston, MA},
YEAR = {2007},
PAGES = {37--47},
DOI = {10.1007/978-0-8176-4545-8_3},
NOTE = {(College Park, MD, 29 September--1 October
1 2006). Paper presented at the mathematical
finance conference in honor of the 60th
birthday of Dilip B. Madan. MR:2359361.
Zbl:1156.60030.},
ISSN = {2296-5009},
ISBN = {9780817645441},
}
M. Yor :
“A note about Selberg’s integrals in relation with the beta-gamma algebra 49–58
in
Advances in mathematical finance
(College Park, MD, 29 September–1 October 1 2006 ).
Edited by M. C. Fu, R. A. Jarrow, J.-Y. Yen, and R. J. Elliott .
Applied Numerical Harmonic Analysis .
Birkhäuser (Boston, MA ),
2007 .
Paper presented at the mathematical finance conference in honor of the 60th birthday of Dilip B. Madan.
MR
2359362 Zbl
1160.33002 incollection
Abstract
People
BibTeX
To prove their formulae for the moments of the characteristic polynomial of the generic matrix of \( U(N) \) , Keating and Snaith [2000] (see also Keating [2004]) use Selberg’s integrals as a ‘black box’. In this note, we point out some identities in law which are equivalent to the expressions of Selberg’s integrals and which involve beta, gamma, and normal variables. However, this is a mere probabilistic translation of Selberg’s results, and does not provide an independent proof of them. An outcome of some of these translations is that certain logarithms of (Vandermonde) random discriminants are self-decomposable, which hinges on the self-decomposability of the logarithms of the beta \( (a,b) \) (\( 2a + b \geq 1 \) ) and gamma (\( a \gt 0 \) ) variables. Such selfdecomposability properties have been of interest in some joint papers with D. Madan.
@incollection {key2359362m,
AUTHOR = {Yor, Marc},
TITLE = {A note about {S}elberg's integrals in
relation with the beta-gamma algebra},
BOOKTITLE = {Advances in mathematical finance},
EDITOR = {Fu, Michael C. and Jarrow, Robert A.
and Yen, Ju-Yi and Elliott, Robert J.},
SERIES = {Applied Numerical Harmonic Analysis},
PUBLISHER = {Birkh\"auser},
ADDRESS = {Boston, MA},
YEAR = {2007},
PAGES = {49--58},
DOI = {10.1007/978-0-8176-4545-8_4},
NOTE = {(College Park, MD, 29 September--1 October
1 2006). Paper presented at the mathematical
finance conference in honor of the 60th
birthday of Dilip B. Madan. MR:2359362.
Zbl:1160.33002.},
ISSN = {2296-5009},
ISBN = {9780817645441},
}
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},
}
D. Madan, B. Roynette, and M. Yor :
“Option prices as probabilities Financ. Res. Lett.
5 : 2
(June 2008 ),
pp. 79–87 .
article
Abstract
People
BibTeX
Four distribution functions are associated with call and put prices seen as functions of their strike and maturity. The random variables associated with these distributions are identified when the process for moneyness defined as the stock price relative to the forward price is a positive local martingale with no positive jumps that tends to zero at infinity. Results on calls require moneyness to be a continuous martingale as well. It is shown that for puts the distributions in the strike are those for the remaining supremum while for calls, they relate to the remaining infimum. In maturity we see the distribution functions for the last passage times of moneyness to strike.
@article {key57450396,
AUTHOR = {Madan, D. and Roynette, B. and Yor,
M.},
TITLE = {Option prices as probabilities},
JOURNAL = {Financ. Res. Lett.},
FJOURNAL = {Finance Research Letters},
VOLUME = {5},
NUMBER = {2},
MONTH = {June},
YEAR = {2008},
PAGES = {79--87},
DOI = {10.1016/j.frl.2008.02.002},
ISSN = {1544-6123},
}
D. Madan, B. Roynette, and M. Yor :
“Unifying Black–Scholes type formulae which involve Brownian last passage times up to a finite horizon Asia-Pac. Financ. Mark.
15 : 2
(June 2008 ),
pp. 97–115 .
Zbl
1163.91414 article
Abstract
People
BibTeX
The authors recently discovered some interesting relations between the Black–Scholes formula and last passage times of the Brownian exponential martingales, which invites one to seek analogous results for last passage times up to a finite horizon. This is achieved in the present paper, where Yuri’s formula, as originally presented in Akahori et al. (On the pricing of options written on the last exit time, 2008), is also derived.
@article {key1163.91414z,
AUTHOR = {Madan, D. and Roynette, B. and Yor,
M.},
TITLE = {Unifying {B}lack--{S}choles type formulae
which involve {B}rownian last passage
times up to a finite horizon},
JOURNAL = {Asia-Pac. Financ. Mark.},
FJOURNAL = {Asia-Pacific Financial Markets},
VOLUME = {15},
NUMBER = {2},
MONTH = {June},
YEAR = {2008},
PAGES = {97--115},
DOI = {10.1007/s10690-008-9068-y},
NOTE = {Zbl:1163.91414.},
ISSN = {1387-2834},
}
D. Madan, B. Roynette, and M. Yor :
“Put option prices as joint distribution functions in strike and maturity: The Black–Scholes case Int. J. Theor. Appl. Finance
12 : 8
(2009 ),
pp. 1075–1090 .
MR
2598932 Zbl
1183.91179 article
Abstract
People
BibTeX
For a large class of \( \mathbb{R}_+ \) valued, continuous local martingales \( (M_t \) , \( t\geq 0) \) , with \( M_0 = 1 \) and \( M_{\infty} = 0 \) , the put quantity:
\[ \Pi_M(K,t) = E((K-M_t)^+) \]
turns out to be the distribution function in both variables \( K \) and \( t \) , for \( K\leq 1 \) and \( t\geq 0 \) , of a probability \( \gamma_M \) on \( [0,1]{\times} [0, \infty) \) . In this paper, the first in a series of three, we discuss in detail the case where
\[ M_t = \mathcal{E}_t := \exp\bigl(B_t - \tfrac{t}{2}\bigr) ,\]
for \( (B_t \) , \( t\geq 0) \) a standard Brownian motion.
@article {key2598932m,
AUTHOR = {Madan, D. and Roynette, B. and Yor,
M.},
TITLE = {Put option prices as joint distribution
functions in strike and maturity: {T}he
{B}lack--{S}choles case},
JOURNAL = {Int. J. Theor. Appl. Finance},
FJOURNAL = {International Journal of Theoretical
and Applied Finance},
VOLUME = {12},
NUMBER = {8},
YEAR = {2009},
PAGES = {1075--1090},
DOI = {10.1142/S0219024909005580},
NOTE = {MR:2598932. Zbl:1183.91179.},
ISSN = {0219-0249},
}
D. B. Madan and M. Yor :
“The S&P 500 index as a Sato process travelling at the speed of the VIX Appl. Math. Finance
18 : 3–4
(2011 ),
pp. 227–244 .
MR
2818165 Zbl
1239.91186 article
Abstract
People
BibTeX
The logarithm of SPX is modeled as a Sato process running at a speed proportional to the current level of the VIX. When the logarithm of the VIX is an exponential compound Poisson process with drift one may obtain exact expressions for the prices of equity options taken at an independent exponential maturity. The parameters for the Lévy process are calibrated from VIX options while the parameters for the Sato process driving the stock may be calibrated from market option prices taken at an independent exponential maturity. Results confirm that both the option surface and the VIX time changed Sato process volatilities, skews and term volatility spreads are responsive to the VIX level and the VIX option surface.
@article {key2818165m,
AUTHOR = {Madan, Dilip B. and Yor, Marc},
TITLE = {The {S}\&{P} 500 index as a {S}ato process
travelling at the speed of the {VIX}},
JOURNAL = {Appl. Math. Finance},
FJOURNAL = {Applied Mathematical Finance},
VOLUME = {18},
NUMBER = {3--4},
YEAR = {2011},
PAGES = {227--244},
DOI = {10.1080/1350486X.2010.486558},
NOTE = {MR:2818165. Zbl:1239.91186.},
ISSN = {1350-486X},
}
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},
}
D. B. Madan and M. Yor :
“Moments of Wiener integrals for subordinators Electron. Commun. Probab.
17
(2012 ).
Article no. 55, 8 pp.
MR
2999983 Zbl
1329.60165 article
Abstract
People
BibTeX
@article {key2999983m,
AUTHOR = {Madan, Dilip B. and Yor, Marc},
TITLE = {Moments of {W}iener integrals for subordinators},
JOURNAL = {Electron. Commun. Probab.},
FJOURNAL = {Electronic Communications in Probability},
VOLUME = {17},
YEAR = {2012},
DOI = {10.1214/ECP.v17-2206},
NOTE = {Article no. 55, 8 pp. MR:2999983. Zbl:1329.60165.},
ISSN = {1083-589X},
}
E. Eberlein, D. Madan, M. Pistorius, and M. Yor :
“A simple stochastic rate model for rate equity hybrid products Appl. Math. Finance
20 : 5–6
(2013 ),
pp. 461–488 .
MR
3169863 Zbl
1396.91780 article
Abstract
People
BibTeX
A positive spot rate model driven by a gamma process and correlated with equity is introduced and calibrated via closed forms for the joint characteristic function for the rate \( r \) , its integral \( y \) and the logarithm of the stock price \( s \) under the \( T \) -forward measure. The law of the triple \( (r,y,s) \) is expressed as a nonlinear transform of three independent processes, a gamma process, a variance gamma process and a Wiener integral with respect to the Dirichlet process. The generalized Stieltjes transform of the Wiener integral with respect to the Dirichlet process is derived in closed form. Inversion of this transform using Schwarz (2005, The generalized Stieltjes transform and its inverse, Journal of Mathematical Physics , 46(1), doi: 10.1063/1.1825077) makes large step simulations possible. Valuing functions are built and hedged using quantization and high dimensional interpolation methods. The hedging objective is taken to be capital minimization as described by Carr, Madan and Vicente Alvarez (2011, Markets, profits, capital, leverage and returns, Journal of Risk , 14(1), pp. 95–122).
@article {key3169863m,
AUTHOR = {Eberlein, Ernst and Madan, Dilip and
Pistorius, Martijn and Yor, Marc},
TITLE = {A simple stochastic rate model for rate
equity hybrid products},
JOURNAL = {Appl. Math. Finance},
FJOURNAL = {Applied Mathematical Finance},
VOLUME = {20},
NUMBER = {5--6},
YEAR = {2013},
PAGES = {461--488},
DOI = {10.1080/1350486X.2013.770240},
NOTE = {MR:3169863. Zbl:1396.91780.},
ISSN = {1350-486X},
}
E. Eberlein, D. Madan, M. Pistorius, W. Schoutens, and M. Yor :
“Two price economies in continuous time Ann. Finance
10 : 1
(February 2014 ),
pp. 71–100 .
MR
3159206 Zbl
1298.91086 article
Abstract
People
BibTeX
Static and discrete time pricing operators for two price economies are reviewed and then generalized to the continuous time setting of an underlying Hunt process. The continuous time operators define nonlinear partial integro-differential equations that are solved numerically for the three valuations of bid, ask and expectation. The operators employ concave distortions by inducing a probability into the infinitesimal generator of a Hunt process. This probability is then distorted. Two nonlinear operators based on different approaches to truncating small jumps are developed and termed \( QV \) for quadratic variation and \( NL \) for normalized Lévy. Examples illustrate the resulting valuations. A sample book of derivatives on a single underlier is employed to display the gap between the bid and ask values for the book and the sum of comparable values for the components of the book.
@article {key3159206m,
AUTHOR = {Eberlein, Ernst and Madan, Dilip and
Pistorius, Martijn and Schoutens, Wim
and Yor, Marc},
TITLE = {Two price economies in continuous time},
JOURNAL = {Ann. Finance},
FJOURNAL = {Annals of Finance},
VOLUME = {10},
NUMBER = {1},
MONTH = {February},
YEAR = {2014},
PAGES = {71--100},
DOI = {10.1007/s10436-013-0228-3},
NOTE = {MR:3159206. Zbl:1298.91086.},
ISSN = {1614-2446},
}
E. Eberlein, D. B. Madan, M. Pistorius, and M. Yor :
“Bid and ask prices as non-linear continuous time G-expectations based on distortions Math. Financ. Econ.
8 : 3
(June 2014 ),
pp. 265–289 .
MR
3212643 Zbl
1307.91086 article
Abstract
People
BibTeX
@article {key3212643m,
AUTHOR = {Eberlein, Ernst and Madan, Dilip B.
and Pistorius, Martijn and Yor, Marc},
TITLE = {Bid and ask prices as non-linear continuous
time {G}-expectations based on distortions},
JOURNAL = {Math. Financ. Econ.},
FJOURNAL = {Mathematics and Financial Economics},
VOLUME = {8},
NUMBER = {3},
MONTH = {June},
YEAR = {2014},
PAGES = {265--289},
DOI = {10.1007/s11579-014-0117-1},
NOTE = {MR:3212643. Zbl:1307.91086.},
ISSN = {1862-9679},
}
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},
}
D. B. Madan and M. Yor :
“On valuing stochastic perpetuities using new long horizon stock price models distinguishing booms, busts, and balanced markets Math. Finance
26 : 2
(2016 ),
pp. 296–328 .
MR
3481306 Zbl
1348.91272 article
Abstract
People
BibTeX
For longer horizons, assuming no dividend distributions, models for discounted stock prices in balanced markets are formulated as conditional expectations of nontrivial terminal random variables defined at infinity. Observing that extant models fail to have this property, new models are proposed. The new concept of a balanced market proposed here permits a distinction between such markets and unduly optimistic or pessimistic ones. A tractable example is developed and termed the discounted variance gamma model. Calibrations to market data provide empirical support. Additionally, procedures are presented for the valuation of path dependent stochastic perpetuities. Evidence is provided for long dated equity linked claims paying coupon for time spent by equity above a lower barrier, being underpriced by extant models relative to the new discounted ones. Given the popularity of such claims, the resulting mispricing could possibly take some corrections. Furthermore for these new discounted models, implied volatility curves do not flatten out at the larger maturities.
@article {key3481306m,
AUTHOR = {Madan, Dilip B. and Yor, Marc},
TITLE = {On valuing stochastic perpetuities using
new long horizon stock price models
distinguishing booms, busts, and balanced
markets},
JOURNAL = {Math. Finance},
FJOURNAL = {Mathematical Finance. An International
Journal of Mathematics, Statistics and
Financial Economics},
VOLUME = {26},
NUMBER = {2},
YEAR = {2016},
PAGES = {296--328},
DOI = {10.1111/mafi.12056},
NOTE = {MR:3481306. Zbl:1348.91272.},
ISSN = {0960-1627},
}
