Research Projects List

Monis Quantitative Analysis Team
(February 16, 1999)

Here is the resume of different possible research directions identified by the team:
 


Candidates are warmly encouraged to develop, in the limits of the constraints imposed by the internship, a critical approach of problems. Moreover in an research activity it is important to enhance the synergy by discussions with colleagues. For that purpose we have set up a world-wide discussion group hosted by www.mailbase.ac.uk. Any interested individual can subscribe at the following address:

http://www.mailbase.ac.uk/lists/finance-and-physics/

Some useful information on numerical simulations can be obtained from the lectures notes of Prof. M. Caffarel by clicking here (only in French).
Prototyping of models can be performed on the available platforms of Symbolic and Algebraic Computation (Maple V, Matlab). Or by implementing algorithms in C/C++.


1) Malliavin Calculus for Monte Carlo methods.

Introduction

Malliavin calculus has been applied to financial problems at different levels [1,2]. >From a mathematical perspective, the standard problem of portfolio analysis solved by Black-Scholes is well defined in terms of Backward Stochastic Differential Equations (BSDE). While the general theory of BSDE gives that this standard problem has a unique solution, it says little about how to find this solution explicitly. Malliavin calculus allows the rigorous minded mathematician to obtain an answer to standard questions such as: What is the option price and the optimal strategy to use. Thus in this first approach, Malliavin calculus appears as the natural calculus to use for the study of financial problems.
A direct implication of this statement is that Malliavin calculus provides a hedging formula more fundamental than other ones. Thus it can be used to find fundamental methods of pricing and hedging, as described for example in [3].
An other use of Malliavin calculus, which is the one of interest in this project, is to use it to devise efficient Monte Carlo methods for calculating the price and sensitivities of financial products. These are computed respectively as expected values of functional of brownian motions and their differentials.

The Malliavin calculus defines the derivative of functions on Wiener space and can be seen as a theory of integration by part in this space. Thanks to Malliavin calculus, we can show that the hedge factor, or differentials of the price, can be computed as the expectation, under a risk neutral probability measure, of the option payoff multiplied by a weight. A good description of this method is given in [4,5].

Project

The aim of the present project is to apply Malliavin calculus to compute the differentials of a wide set of options. This requires at first an extension of the work of  Fournie et al. to overcome difficulties associated with non differentiable payoffs. Followed by an implementation of the method (which will be used uniquely to prove the good behaviour of the proposed method). Among the most determinant characteristics we will focus on we have accuracy and speed.

In order to precise our goal, we will expand on the previous paragraph. At Monis we have an application called Generalized Monte Carlo (GMC) which perform, as its name suggest, a Monte Carlo simulation to compute the price of an option as well as its differentials, commonly called greeks. In order to compute the greeks, GMC use a finite difference approximation (e.g (F(x+d)-F(x-d))/2d where F(x) is the option price (i.e computed as a mean average over the set of sample paths). Thus greeks calculations require roughly twice the amount of time to calculate than the price itself. Using Malliavin calculus we expect to be able to reduce this calculation time as well improve the stability of the greeks.

A particularly interesting feature of GMC is that the option payoff is specified by the user using a "high level programming language", thus we are looking to an implementation of Malliavin calculus which is, to a great extend, option independent. A difficulty to overcome, in order to achieve this goal, is to find an appropriate solution to the treatment of singularity in the options payoff. A good exempla is presented in Fournie and al., which show how the discontinuity of a call payoff can influence the computation.A prototype program should be implemented and used as a test for the various methods found to treat singularities as well as to estimate the accuracy and speed of convergence.

Finally, a report (leading to a further publication) should be provided, with a clear description of the algorithms. This report should be clear enough to allow for a direct implementation of the methods in the existing Monis software.

References

[1] D. Nualart, Malliavin calculus and related topics, Probability and its applications, Springer Verlag 1995. Back to Text.
[2] B. Oskendal, An introduction to Malliavin Calculus with application to economics, Lecture Notes, 1997. Back to Text.
[3] E. Barucci and  M.E. Mancino, Wiener chaos and Hermite polynomial expansion for pricing and hedging contingent claims, Preprint. Back to Text.
[4] E. Fournie, J.M. Lasry, J. Lebuchoux, P.L. Lions and N. Touzi, An application of Malliavin calculus to Monte Carlo methods in finance, Ceremade 9726, 1997. Back to Text.
[5] E. Fournie, J.M Lasry, J. Lebuchoux and P.L. Lions, Applications of Malliavin Calculus  to Monte Carlo Methods in Finance II, Ceremade 9901, 1999. Back to text.


2) Study and application of Extreme Value Theory to risk estimation and pricing of financial instruments: CAT-Bonds/Insurance/Weather Derivatives.

Introduction

Due to the increasing complexity of financial instruments, even more complicated tools to manage the risk have to be put into place. The securitisation of risk and alternative risk transfer ask for a coherent and realistic valuation of extreme fluctuations. The conventional approach in parametric risk estimation in finance is to use a "normal" distribution, this choice makes the mathematics more easy but has the problem to badly describe tail characteristics of  real distributions. Indeed, extreme fluctuations are characterized by their intrinsic scarcity. Therefore a probabilistic approach from a frequentistic point of view is sorted out, and the conventional "Gaussian" approach often led to dangerous unrealistic risk estimates.  Nevertheless results from EVT, like e.g. the Fisher-Tippet theorem, can be used to overcome this difficulty. Indeed, under very general assumptions, probability distributions of extreme events (which from a frequentistic approach can be estimated only with an infinite number of observations) can be described by a generalised Pareto distribution. And for a large class of underlying distributions it is possible to determine the Hill estimator, which is needed to fix the real excess distribution. The interesting point is that the Hill estimator can be estimated by the market dynamics with a limited number of observations. It is therefore possible, given a time horizon and an upper bound for the jump, to provide the probability of having a jump bigger than a stated bound.
 
 
Rate  Hill estimator 99.6% Confidence Level 99.92% Confidence Level
USD-DEM 4.19 1.7% 2.5%
USD-JPY 4.40 1.7% 2.4%
USD-CHF 4.13 1.8% 2.7%
USD-ITL 3.57 1.9% 3.4%

 

In simple terms, using real time data with a sparseness compatible with our forecast horizon, we could be able to asess a realistic probability of exceeding a fixed boundary level. This value clearly depends on the confidence level, i.e. on the quantile we need as shown in the previous table. Extreme value theory offers an important set of techniques and estimators for quantifying the boundaries between different gain/loss classes.

Furthermore the Hill estimator could be an indirect indicator of the market liquidity. By analysing high frequency data, under time aggregation hypothesis  it will be possible to detect trends in the liquidity evolution (i.e. Bid/Offer spread).
 

Project

The candidate will be asked to familiarise himself with the basics of Extreme Value Theory  (e.g. P. Embrecht et al., WEB site in finance at ETHZ, Richard Davis & Thomas Mikosch), and to apply it  to the analysis of different time series (e.g. Market Indexes, CAC 40, Futures on CAC 40, or if possible  meteorological data).  An area of application and interest is the valuation of properly securitised products in the realm of catastrophe insurance like: CAT futures and CBOT where securitisation is achieved through the construction of derivatives written on a newly constructed industry wide loss-ratio index. Good overviews stressing the financial engineering of these products are Doherty (1997), Tilley (1997), Schmock (1997), and Embrechts, Samorodnitsky and Resnick (1998).

Bibliography


[1] P.Embrecht, C. Kluppelberg, T. Mikosch, " Modelling Extremal Events", in Applications of Mathematics, Ed. Springer.  Back to Text.
[2] N.A. Doherty in Financial innovation in the management catastrophe risk. Joint Day Proceedings, XXVIIIth International ASTIN Colloquium and 7th International AIFR Colloquium, Cairns (Australia), 1-26. (1997).  Back to Text.
[3] J.A. Tilley in The securitisation of catastrophic property risks. Joint Day Proceedings, XXVIIIth International Colloquium and 7th International AIFR Colloquium, Cairns (Australia), 27-53. (1997). Back to Text.
[4] U. Schmock Estimating the value of the Wincat coupons of the Winterthur Insurance Convertible Bond. Joint Day Proceedings, XXVIIIth International ASTIN Colloquium and 7th International AIFR Colloquium, Cairns (Australia), 231-259, (1997). Back to Text.
[5] Embrecht et al.,  "Living on the edge", RISK 11, 96-100, (1998).Back to Text.


4) PDE solutions, comparison with existing trees.

Introduction

This project is concerned with the pricing of Convertible Bonds using Partial Differential Equations methods. In the following, the model is presented  and a possible PDE method is suggested to use for its evaluation. In the method choice, a particular emphasis should be given to the calculation of derivatives as well to the treatment of discontinuity.

Model

A bond is a contract, paid up-front, that yields a known amount on a known date in the future, the maturity date T. The bond may also pay a known cash dividend, the coupon, at fixed times during the life of the contract. Bonds may be issued both by governments or companies. The main purpose of a bond issue is the raising of capital.

Convertible bonds are bonds involving a dual option. On one hand the holder has the option to exchange the bonds for the company's stock at certain time in the future. The amount of stock obtained in exchange for one bond is called the exchange ratio n(t). On the other hand, the issuer has the right to buy back the bonds. The price at which the bonds can be bought back is the call price CP(t). The holder has the right to convert the bonds once they have been called, the call feature is therefore often a way of forcing conversion at a time earlier than the holder would otherwise choose. Furthermore, the convertible bond is sometimes puttable, i.e. the holder has the right to sell it back to the company at a known put price PP(t).

We consider a two factor model to compute the price of the convertible bond. The two factors are the company stock price S(t) and the interest rate r(t). Furthermore a third stochastic component is introduced in the conversion ratio alpha(t) by equity resets. As a consequence the bond price is a function V(S,r,alpha,t) of the three stochastic components.

The company's stock price is assumed to follows a geometric Brownian motion and the interest rate r(t) is described by the extended Vasicek model (the two process are correlated).
 

Project

The starting point of this project is the PDE which is to be solved with particular attention to the method for handling the boundary conditions. For instance, the boundaries related to the equity reset feature reduces the PDE to a first order hyperbolic equation which much be discretised carefully to avoid spurious oscillations.

We will discretise equation using a Galerkin finite element method for the diffusion terms (all the second order derivatives).The convection terms (involving first order derivatives in S, alpha and r will be discretised using a finite volume approach. One popular method, already applied in finance, is the flux limiter of van Leer. A van Leer limiter is known to produce an oscillation free solutionAn automatic time step selection method will be used between sampling dates (resets, dividends and coupons).An unstructured grid of triangular elements will be used. The possibility of inserting new nodes at arbitrary locations in the computational domain is an important feature particularly useful for financial applications. Two methods for handling the problem of the early exercise feature associated with American options will be considered: (1) One method is to view the problem as a linear complementary problem and then use a projected SOR technique for solving the discrete algebraic equations. (2) An alternative method is to view the problem as a nonlinear algebraic system, where the nonlinear constraint can be imposed using a penalty method. The resulting system of nonlinear algebraic equations is then solved using Newton iteration.

This project is mainly a numerical analysis study of PDE applied to finance. Its main interest stands on the finding of a good and efficient PDE method to deal with the financial constraints. From a technical point of view, the PDE is linear thus all the difficulty is in the treatment of boundary conditions, which can be discontinous. An operational constraint is the request of a solution in times not exceeding few minutes. The PDE method is competing with lattice based method as well as with Monte Carlo methods, thus an advance implementation making use of adaptive techniques is imperative for the success of this project.

References

[1] P. Wilmott, S. Howison and J. Dewynne,  The mathematics of financial derivatives,  Cambridge university Press, 1997.

[2] R.J. LeVeque, Numerical Methods for Conservation Laws,  Birkauser, 1990.

[3] R. Zvan, P.A. Forsyth and K.R. Vetzal, A finite element approach to the pricing of discrete lookbacks with stochastic volatility, preprint.
 


5) Interacting agents models and definition of crash precursors:

Introduction

In our research planning we are interested to investigate interacting agent models. The idea is to apply  techniques of statistical physics to construct a model of an evolving stock market driven by  individuality (noise traders) and general expectation (fundamentalist). Indeed, the frequency of large variations in stock prices raise doubts on existing models, which all fail in accounting for non-Gaussian statistics. From a less quantitative aspect, but with a possible valuable impact from a forecasting point of view, it could be interesting to understand market dynamics and to identify crash precursors from a fundamental structure of the market evolution. In a general view financial markets seem to exhibits a scale invariant behaviour [2]. If for a physicist, scale invariance means the absence of characteristic scale, from a business  point of view, it means the existence of catastrophic risk which can bankrupt a company. It is rather tempting to identify the mechanism responsible for the scale invariance with what physicists call Self-Organised Criticality (SOC) [3] i.e. the expression of an underlying unstable dynamical critical point. General examples of  SOC systems are: sand-piles, earthquakes, forest fires, etc. E.g. in the sand-pile case the pile grows until the structure become unstable and an avalanche is initiated. In this way, the pile reaches a stationary critical state, characterised by a critical slope, in which additional grains of sand will fall off the pile via avalanches distributed in lifetime and size according to a power law.

Albeit the general conditions under which a physical system exhibits SOC are largely unknown, some facts have however been established:

The second point is one of the most neglected in mathematical modelling in finance since market and market players are considered as totally independent and  not influencing each other. However, recent results in that direction are reported in [4].
It is rather tempting to apply a picture of a dynamical critical point, similar to what has been found recently for earthquakes,  to predicts financial crashes. Dynamical critical points exhibits a characteristic log-periodic signatures. Evidence of which has been found by Sornette et al. [5] in the analysis of the two major crashes of this century: October 1927 (Dow Jones) and October 1987  (S&P500). They found for these two cases, using historical data before the crashes, a critical time (i.e. the market crash) in very good agreement with the real timing of the events. They suggest that this agreement reflects the fundamental co-operative nature of the behaviour of stock markets.
In general co-operative behaviours of complex systems cannot be reduced to a decomposition of elementary causes. A crash emerges naturally as an intrinsic signature of the functioning of the market. Indeed there is a need to insert in market models the effect of a positive feed-back interactions in which traders exchange information according to a hierarchical structure [5].  As a general comment, it is interesting to note the existing similarities between log-periodic structure observed in the market [5] and Elliot waves [6], a technique which is strongly rooted in the financial analysisí folklore.
Crash statistics are very poor and the natural excitation, that any kind of evidence of a crash precursor can produce, should be always moderated by critical views.  Albeit it is tempting to see financial crashes as a critical phenomena described by statistical mechanics where, in a particular situation, all the subparts of the system react co-operatively, there are only few experimental situations where these scenarios can be tested. In that sense, any empirical finding, is not statistically significant and could be even more dangerous than the crash itself if its conclusions are used to asess a strategy (L.Laloux et al.  [7]) based on an ex ante prediction. To use such empirical results, the lack of statistically relevant set of experimental configurations must be compensated by a a priori knowledge of the market conditions. Any conclusion on a market crash prediction, based on empirical results, must be handled as a Bayesian inference on magnitude and time-scale on the next catastrophic event given the knowledge of the past.

Project

In this framework is rather tempting to apply a picture of a dynamical critical point where the system is spontaneously driven towards a critical dynamical state. This approach has already been analysed by several authors, from P. Back et al. to Lux and Marchesi  (Nature 397, 498 - 500 (1999)), see also Johansen et al., Vandewalle et al. [1].  Indeed, evidence of scaling properties in financial prices, similar to those characterising systems of a large number of interacting particles, drive the idea that main observed features in financial data can be explained by a multi-agent model of financial market.

The candidate will be asked to familiarise himself with the existing literature on the subject, and to test the theory on real data. It means that he will be asked to create a working prototype of the model, and its implementation to test it against real market data. It is clear that the previous literature constitutes a starting point and any original contribution will be appreciated on its own right.

Bibliography


[1] Eur. Phys. J. B4, 139-141 (1998). Back to text to text

[2] R. Cont, Proceedings of the CNRS Workshop on scale Invariance, Les Houches, 1997 Back to text

[3] D. Sornette in "Physics of Complexity" Editions Frontieres. Back to text to text

[4] J. Cvitanic, in "Mathematics in Derivative Securities" Cambridge University Press and references therein. Back to text to text

[6] A. Prost and R. Prechter, "Elliot Waves Principle", New Classic Library, 1985 Back to text to text.


6) MC generator for interest rates

Introduction

In view of developing a new Monte Carlo generator for interest rates financial products, it will be interesting to analyse the relation between different models and real data. The primary objective, will be to clarify the roles played by various features of models and their parameters in the pricing of bonds and related assets (see e.g. D. Backus), analysing a database of  monthly spot rates (continuously-compounded zero-coupon yields) and forward rates from the McCulloch-Kwon dataset (binary "zipped" file, 487k) - high density data are also available. The universe of bond pricing is populated by a variety of models used by academics and practitioners alike.  Since the theory of bond prices is essentially the choice of a price kernel, i.e. the stochastic process governing prices of state-contingent claims, different models differ by the description of the underlying process. It follows that assumptions used to develop coherent pricing criteria for interest rate dependent securities may differ from the statistical description of the movements of real interest rates (see R. Cont).  Indeed, as pointed out by Boucheaud et al., and R. Cont, some empirical observations on the deformation of the term structure cannot be explained by classical arbitrage-free models.

Project

Starting from a statistical description of the real data, it will be interesting to perform a comparative study of the most fashionable models used by practitioners, e.g. Black, Derman and Toy; Black-Karasinski; Heat, Jarrow and  Morton,  and identify possible incoherence between theories and observations. As a comparative test, a multi-affine analysis [1] can be performed on the available data to investigate the existence of scaling properties of the available data. The obtained result can then be compared with the analysis of synthetic interest rates time series obtained via Monte Carlo simulation assuming a given choice of a price kernel. Finally a phenomenological approach via a principal component analysis could by applied following the method of  Boucheaud et al and the infinite dimensional approach of R. Cont.

Bibliography


[1] N. Vandewalle and M. Ausloos, Eur. Phys. J. B 4, 257-261 (1998). Back to Text.



London February 16th, 1999.

Dr Gabriele Susinno
Dr Marco Rigo
Quantitative Analysts at Monis Software Ltd.