Thesis Topics in Mathematical Finance
Sometimes interest rates fluctuate a lot, sometimes not so much.
Volatility is stochastic.
A standard reference for stochastic volatility interest rates models
is
Longstaff, Francis and Eduardo Schwartz (1992)
Interest Rate Volatility and the Term
Structure: A Two-Factor General Equilibrium Model, Journal
of Finance, Vol. 47(4), pp. 1259-1282.
Empirical evidence is presented in
Andersen, Torben G. and Jesper Lund (1997),
Estimating continuous-time stochastic volatility models of the
short-term interest rate, Journal of Econometrics, Vol
77(2), pp. 343-377.
The Longstaff-Schwartz model is a special case of the general
multi-dimensional affine factor models. We know the one-dimensional
case from Bjørk. "Models are affine if and only if drift and
sigma^2 are affine. Bond prices can be found by solving Ricatti
equations." The same is just about true in higher dimensions
(although it may not be eminently clear how "higher dimensions" can pop up),
but things get a lot more complicated. Such models are treated in
Duffie, Darrell and Rui Kan (1996), "A Yield Factor Model of Interest
Rates", Mathematical Finance, Vol. 6, pp. 379-406,
and more recently in
Dai, Qiang and Kenneth Singleton (2000),
Specification Analysis of Affine Term Structure Models,
Journal of Finance, Vol. 55(5), pp. 1043-1978.
A particularly interesting analysis (theoretically and empirically)
is given in
Collin-Dufresne, Pierre and Robert Goldstein (2002),
Do Bonds Span the Fixed Income Markets? Theory and Evidence for
Unspanned Stochastic Volatility, Journal of Finance, Vol. 57(4).
Thesis-wise this C-D&G article is something you can "build
around". I'd suggest reading it
& then backtracking into the earlier literature on a "need-to-know" basis.
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"One man's derivative is another man's underlying."
The traditional approach to pricing and hedging options has
been to construct a dynamically adjusted portfolio involving
the underlying (stock) and some largely risk-free asset.
Nowadays very liquid markets exist for "plain vanilla"
options (put- and call-options) with a variety of characteristics
(expirations and strike-prices). In many cases such options
are much more natural hedging instruments for "exotic"
options.
An easy introduction to static hedging is
Derman, Emmanuel, Deniz Ergener and Iraj Kani (1995)
Static Options Replication,
Journal of Derivatives , Vol. 2(Summer).
In this paper static hedges for barrier options are contructed
in Black-Scholes or binomial models through calendar-spreads.
The master of static hedging is Peter Carr.
In
Carr, Peter and Andrew Chou (1997), Breaking Barriers , Risk Magazine.
it is shown to statically hedge basic barrier options by trading
options with different strikes. The trick is to adjust the
terminal payoff function of the (knock-in) barrier security in
such a way that the simple contingent claim with this payoff
function has the same price as the barrier security.
The same authors also have a
paper on complex barrier options and lookbacks.
Carr & Picron (1998),
Static Hedging of Timing Risk, Journal of Derivatives,
gives results for options with rebates (and non-zero interest rates and dividends), and
Carr, Peter, Katrina Ellis, and Vishal Gupta (1998),
"Static Hedging of Exotic Options", Journal of Finance, 53(3),
1165-91
shows the static hedging articles also appear
in top journals.
The Carr-articles work with the Black-Scholes model, and end up
with quite explicit formulas for "which & how many puts and calls
to buy".
The article
Andersen, Leif, Jesper Andreasen, and David Eliezer, (2002),
"Static Replication of Barrier Options:
Some General Results , Journal of Computational Finance, 5,
1-25,
is "the full Monty" within a PDE-framework: Rebates, interest
rates and dividends, general knock-out regions, state and time
dependent volatility.
Light numerics will get you a long way with this topic (at least if
you stay away from that last article).
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An evergreen topic with a huge literature.
Below I mention but a few articles. If I do not give full reference it means
that you can find it in Chapter 9 of
Musiela and Rutkowski's
Martingale Methods in Financial Modelling. That's a nice survey,
so you should read that anyway.
- Well-definedness of the problem: Bensoussan (1984),
Karatzas (1989), Myneni (1992). Even in complete(ly) discrete
models (as for instance I&F-teori) American securities usually
aren't rigorously defined & analyzed (there's no need
to because "it's obvious").
- PDE/free boundary-formulation & solution: McKean (1965), van
Moerbeke (1976),
Brennan &Schwartz (1979),
Cox, Ross & Rubinstein (1979) (or any text-book for American option pricing in
binomial models). Note that this is earlier than
the rigorous formulation.
- Analytical representation/approximation:
Barone-Adesi & Whaley (1987), Kim (1990).
- Recent contributions/approaches:
- Randomized Models. These are advocated by for instance
Peter Carr and
Dietmar Leisen .
- Pricing American Options by Monte Carlo Simulation.
(Despite what one might think, there no connection (or is there?) to
'randomized models'.) Conventional wisdom was that
American options couldn't be priced by simulation. (Take a
minute to see why that's a natural conclusion.) That's chaged.
There is a lot a papers by Paul Glasserman,
Brodie & Glasserman, for instance.
These can be a bit hard to grasp.
No such problems with the ingenious least-squares
idea of
Longstaff & Schwartz . Their approach has created a number of "follow-up" papers, see for instance
idea of
this one and
CAF working papers 90 and 112-3 by Lars Stentoft.
The strategy is clear: Read a number of articles describing some
techniques; start with the most recent ones &
"back-track". Describe. Implement. Compare. It is possible to
keep the thesis completely theoretical (i.e. no computers & numbers),
but it is more natural to involve some numerical work. In fact,
depending on how deep you go into the PDE-approach, you can get
"as heavy numerics as you like".
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If you take look, you'll see that even top-class journals (J. of
Political Economy, J. of Financial Economics, Journal of Finance,
...) quite frequently publish articles on betting markets.
The "official" reason is that
betting markets are quite close to the idealized
markets with uncertainty we usually analyze (eg. there is a clear
terminal at which the outcome is decided), so we may get valuable
information about decisions under uncertainty. The real reason is
of course that betting markets are fun. (If you don't think so,
then this is not the topic for you).
A survey can be found in:
Hausch, Donald B. and William T. Ziemba
"Efficiency of Sports and Lottery Betting Markets", Chapter 18 in
Jarrow et al. (eds.)
Handbooks in OR & MS Vol. 9.
One very common feature (try a Google-search)
in the analyses of betting markets is the
so-called "favorite/longshot-bias". High odds are too low & vice
versa. Or in other words: You get a higher rate of return from
betting on favorites than on "longshots". This is the exact
opposite of how things usually work in financial markets, where low
risk means low expected return , while very risky (in some sense)
investments must have a high expected rate of return.
There is an obvious explanation: Agents in betting markets are
risk-lovers, not risk-averse -- why else would they enter the
market at all? This means that our usual inequalities are
reversed. This explanation goes back at least to
Weitzman. With many risk-loving bettors and bookbalancing
bookmakers you can make your own nice "equilibrium odds curves".
But there are other explanations. A paper by
Hurley and McDonough shows transactions costs & short-selling
contraints may do the trick (in parimutuel betting at least).
In an interesting series of papers (with a microeconomic approach)
Hyun Song Shin shows how the bias may arise because some bettors are
better informed than others ("insider trading") -- and bookmakers
know that. See Shin 1991 (the basic idea)
Shin 1992 (an extension to more than 2
possible outcomes & competing bookmakers; a lot messier but same
qualitative result) and Shin 1993
(empirical evidence).
A "fundamental analysis" approach to finding abnormal returns ("good
odds/bets") is given by Dixon and
Coles. They "simply" make a statistical model to
fit/predict outcomes of football matches & compares to (mis)quoted
odds.
What about data? Well, you can probably find a lot on the web
(try some Google-searches),
but I might be able to get my hands on "some of the really good stuff"
through my connections with Betbrain.
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Rolf Poulsen
Last modified: Thu Sep 19 13:22:07 MET DST 2002