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

- Malliavin Calculus for Monte Carlo generators.
- Study and application of Extreme Value Theory (EVT) to risk estimation and pricing of financial instruments.
- Time Series related projects: volatility estimators via wavelet decomposition (Haar basis), or via centred kernel estimators; volatility forecast via GARCH, and HGARCH.
- PDE solutions, comparison with existing trees.
- Interacting agents models and definition of crash precursors.
- MC generator for interest rates.

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.

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].

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.

[2] B. Oskendal,

[3] E. Barucci and M.E. Mancino,

[4] E. Fournie, J.M. Lasry, J. Lebuchoux, P.L. Lions and N. Touzi,

[5] E. Fournie, J.M Lasry, J. Lebuchoux and P.L. Lions,

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

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).

[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.

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).

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.

[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:

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].

- The large scale evolution should obeys a diffusion process (like markets are supposed to behave) which satisfy a global conservation law.
- More generally a feedback mechanism must operate to attract the dynamics to a critical state.

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 analysis’ 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.

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.

[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

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

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