# Thesis Topics in Mathematical Finance

## Interest Rate Models with Stochastic Volatility

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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## Static Hedging

"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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## American Options

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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## Economics/Efficiency of Betting Markets

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