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 [Contents] 

 [Index] 

Subsections

• 1. Methods for pricing and hedging of non-standard derivatives
• 2. Robust interest rate (LIBOR) modeling and calibration
• 3. Volatility estimation of financial time series
• 4. Methods for risk management
• 5. Monte Carlo methods in finance

Applied mathematical finance

Collaborator: D. Belomestny, S. Jaschke, A. Kolodko, D. Mercurio, G.N. Milstein, O. Reiß, J. Schoenmakers, V. Spokoiny, J.-H. Zacharias-Langhans

Cooperation with: C. Byrne (Springer, Heidelberg), R. Cont (Ecole Polytechnique, Palaiseau, France), B. Coffey (Merrill Lynch, London, UK), H. Föllmer, W. Härdle, U. Küchler, R. Stehle (Humboldt-Universität (HU) zu Berlin), H. Friebel (Finaris, Frankfurt am Main), P. Glasserman (Columbia University, New York, USA), H. Haaf, U. Wystup (Commerzbank AG, Frankfurt am Main), A.W. Heemink, E.v.d. Berg, D. Spivakovskaya (Technical University Delft, The Netherlands), F. Jamshidian (NIB Capital Den Haag, The Netherlands), J. Kampen (Ruprecht-Karls-Universität, Heidelberg) J. Kienitz (Postbank, Bonn), P. Kloeden (Johann Wolfgang Goethe-Universität Frankfurt am Main), C. März, D. Dunuschat, T. Sauder, S. Wernicke (Bankgesellschaft Berlin AG, Berlin), B. Matzack (Kreditanstalt für Wiederaufbau (KfW), Frankfurt am Main), S. Nair (Chapman & Hall, London, UK), M. Schweizer (Universität München / ETH Zürich), S. Schwalm (Reuters FS, Paris, France), G. Stahl (Bundesaufsichtsamt für das Kreditwesen (BAFin) Bonn), D. Tasche (Deutsche Bundesbank, Frankfurt am Main)

Supported by: BMBF: ``Effiziente Methoden zur Bestimmung von Risikomaßen'' (Efficient methods for valuation of risk measures),
DFG: DFG-Forschungszentrum ``Mathematik für Schlüsseltechnologien'' (Research Center ``Mathematics for Key Technologies''), project E5; SFB 373 ``Quantifikation und Simulation ökonomischer Prozesse'' (Quantification and simulation of economic processes),
Reuters Financial Software, Paris

Description:

The central theme of the project Applied mathematical finance is the quantitative treatment of problems raised by the financial industry, based on innovative methods and algorithms developed in accordance with fundamental principles of mathematical finance. These problems include stochastic modeling of financial data, valuation of complex derivative instruments (options), and risk analysis. The methods and algorithms developed benefit strongly from the synergy with the projects Statistical data analysis and Numerical methods for stochastic models.

1. Methods for pricing and hedging of non-standard derivatives

(D. Belomestny, A. Kolodko, G.N. Milstein, O. Reiß, J. Schoenmakers).

The valuation of financial derivatives based on arbitrage-free asset pricing involves non-trivial mathematical problems in martingale theory, stochastic differential equations, and partial differential equations. While its main principles are established (Harrison, Pliska, 1981), many numerical problems remain such as the numerical valuation of (multidimensional) American equity options and the valuation of Bermudan-style derivatives involving the term structure of interest rates (LIBOR models), [6]. The valuation and optimal exercise of American and Bermudan derivatives is one of the most important problems both in theory and practice, see, e.g., [1]. American options are options contingent on a set of underlyings which can be exercised at any time in some prespecified future time interval, whereas Bermudan options may be exercised at a prespecified discrete set of future exercise dates. In general, the fair price of an American- or Bermudan-style derivative can be represented as the solution of an optimal stopping problem.

• American options: For the multidimensional American option in a generalized Black-Scholes framework, this optimal stopping problem can be converted into a free boundary value problem (Stefan problem). In this context we have developed in [22] a pure Monte Carlo algorithm for determining the (free) exercise boundary of an American option. Due to the thus constructed exercise boundary, the option may be priced and hedged by well-known Monte Carlo simulation of a respective Cauchy boundary value problem. As such we have a pure Monte Carlo procedure for the solution of a Stefan problem which may be considered remarkable. For one dimension the main idea is as follows. Let us suppose that the exercise curve x = g'(t) is known from $\bar{t}$ $\leq$ t $\leq$ T. By using the free boundary conditions and taking boundary limits of the Black-Scholes PDE and its time derivative in the continuation region, we have shown that the derivative g'($\bar{t}$) can be expressed in u_xxx($\bar{t}$, g'($\bar{t}$) +) and known quantities u_x($\bar{t}$, g'($\bar{t}$) +), u_xx($\bar{t}$, g'($\bar{t}$) +) which can be expressed in the PDE coefficients and the pay-off function f. As a result we may compute numerically the third derivative u_xxx via a Taylor expansion by an accurate enough computation of u($\bar{t}$,$\bar{x}$ + $\rho$h^q) in a neighborhood point in the continuation region. The latter can be done by standard Monte Carlo simulation using the known exercise curve for $\bar{t}$ $\leq$ t $\leq$ T, see Figure 1. Having g'($\bar{t}$), the exercise curve can be extended one step, g($\bar{t}$ - h) $\approx$ g($\bar{t}$) - g'($\bar{t}$)h, and then we proceed in the same way. In [22] this method is generalized to the multidimensional case.

  Fig. 1: Backward construction of the exercise boundary in one dimension

  
• American options in time-delayed models: In [10] we have started to consider optimal stopping rules for models driven by stochastic delay differential equations. A natural application is the valuation of American options where the volatility depends on the past. For instance the choice
  $$\sigma_{t}^{}$$ = $$\sigma$$$$\Bigl($$$$\Bigl\Vert$$X_t - $$\int_{{-\infty}}^{0}$$X_t+s$$\rho$$e^$\rho$s ds$$\Bigr\Vert$$$$\Bigr)$$
  postulates that the volatility is driven by the deviation of the current price from an exponentially weighted average over the past, which is close in spirit to classical chart techniques. In certain particular cases the models can be described by higher dimensional Markov diffusion processes with singular diffusion matrix. We study the singular free boundary value problem occurring in the analytical description of the solution.

• Bermudan-style products: For Bermudan options we have constructed in [18] an efficient Monte Carlo method for computing a price upper bound. The method is based on a duality approach for Monte Carlo construction of a Bermudan price upper bound via the Doob-Meyer martingale part of an approximation of the respective Snell envelope process, [15], [36]. In our approach we enclose the theoretical upper bound by approximating from above with an ``up-up'' estimator due to Haugh & Kogan and from below by a newly developed lower estimator, called ``up-low'' estimator. Both the ``up-up'' estimator and the ``up-low'' estimator require in general simulations in simulations and, as a consequence, tend to be time-consuming. However, we show that by taking a suitable convex combination of the lower and upper estimator we can obtain a combined estimator which requires only a very low number of inner simulations and, as a result, this estimator has a higher computational efficiency. For an application to the Bermudan swaption, approximated by the maximum of ``still alive'' swaptions, see Figure 2, and for more details, see [18].

  Fig. 2: Bias of different estimators as function of K inner simulations. The dashed line is the combined estimator

  

  Another new method for constructing an upper bound of the Bermudan/American price using some lower bound is currently in development. This approach is based on the fact that an American option is equivalent to a European option with a consumption process involved. The value of the upper bound V(t, x) at a position (t, x) is constructed by the Monte Carlo method. Our attention focuses on constructing new numerical procedures and their practical implementation. The results of numerical experiments confirm efficiency of the algorithms proposed ([3]).

  In an initiated research cooperation with J. Kampen at Heidelberg University, we aim to value Bermudan-style derivatives in the LIBOR market model based on higher order approximation of Greenian kernels. The Greenian kernels are connected with the (high-dimensional) LIBOR process and integration with respect to these kernels will be implemented on sparse grids.

• Monte Carlo evaluation of Greeks by finite differences: In many cases the price of an option can be represented as the solution of a Cauchy boundary value problem for a parabolic PDE. As a consequence, option prices can be computed by Monte Carlo simulation of a respective system of stochastic differential equations. Derivatives of the solution, in financial terms called ``Greeks'', can in principal be computed by finite differences of solutions obtained by Monte Carlo simulation. In [26] it is shown that this method is effective when in the Monte Carlo simulation the method of dependent realizations is exploited. An error analysis is given, and numerical experiments are in agreement with the developed theory. A related Monte Carlo method for computation of Greeks was previously constructed in [23], see also [35].

2. Robust interest rate (LIBOR) modeling and calibration

(O. Reiß, J. Schoenmakers).

Robust calibration of the LIBOR market model, a popular benchmark model for effective forward interest rates ([5], [17], [28]), to liquidly traded instruments such as caps and swaptions has been a challenging problem for several years. In particular, calibration methods which avoid the use of historical data are very desirable, both from a practical and a more fundamental point of view. The dynamics of the LIBOR model is given by

$$\sum_{{j=i+1}}^{{n-1}} {\frac{{\delta_jL_iL_j \sigma_i\sigma_j\rho_{ij}}}{{% 1+\delta_jL_j}}} \sigma_{i}^{}$$ (1)

where the LIBOR/EurIBOR processes L_i are defined in [t₀, T_i], with $\delta_{i}^{}$ = T_i+1 - T_i being day count fractions and $\sigma_{i}^{}$ being scalar deterministic volatility functions. Further, (W⁽ⁿ⁾_i(t) | t₀ $\leq$ t $\leq$ T_n-1) are correlated Wiener processes under the so-called terminal measure $\mbox{$\rm {I\!P}$}$_n, with deterministic local covariance structure

< dW⁽ⁿ⁾_i, dW⁽ⁿ⁾_j > = $$\rho_{{ij}}^{}$$dt.

In order to construct a stable calibration procedure, we first introduce the following economically motivated parametrization of the scalar volatility and correlation function,

$$\sigma_{i}^{}$$ : = c_ig(T_i - t)

$$\rho_{{ij}}^{} \rho^{{(0)}}_{{i-m(t),j-m(t)}} \geq$$ : =

with a simple parametric function g(s) : = g_$\infty$ + (1 - g_$\infty$ + as)exp(- bs), and, for example, one of the correlation structures developed by Schoenmakers & Coffey in [40], [41], based on some semi-parametric framework, [19], [39].

$$\rho_{{ij}}^{} \left[\vphantom{-\frac{\vert j-i\vert}{m-1} \left(-\ln\rho_{\infty} +\right.}\right. {\frac{{\vert j-i\vert}}{{m-1}}} \left(\vphantom{-\ln\rho_{\infty} +}\right. \rho_{{\infty}}^{}$$ =

$$\left.\vphantom{\left.+\eta\frac{i^2+j^2+ij-3mi-3mj+3i+3j+2m^2-m-4}{(m-2)(m-3)} \right)}\right. \left.\vphantom{+\eta\frac{i^2+j^2+ij-3mi-3mj+3i+3j+2m^2-m-4}{(m-2)(m-3)} }\right. \eta {\frac{{i^2+j^2+ij-3mi-3mj+3i+3j+2m^2-m-4}}{{(m-2)(m-3)}}} \left.\vphantom{+\eta\frac{i^2+j^2+ij-3mi-3mj+3i+3j+2m^2-m-4}{(m-2)(m-3)} }\right) \left.\vphantom{\left.+\eta\frac{i^2+j^2+ij-3mi-3mj+3i+3j+2m^2-m-4}{(m-2)(m-3)} \right)}\right]$$

$$\eta \eta \rho_{{\infty}}^{}$$

The thus designed volatility structure relies on sensible assumptions regarding the behavior of forward rates and implies a kind of time shift invariance when c_i $\equiv$ c. Calibration of this volatility structure to a set of market cap and swaption volatilities comes down to fit the model cap and swaption volatilities to a rather flat surface of market quotes, see the first picture in Figure 3.

The LIBOR model is in a sense designed to price cap(let)s in closed form. Indeed, for any function g and correlation structure $\rho^{{(0)}}_{}$, caps can be matched perfectly by appropriate choice of the coefficients c_i. However, since caps are in fact one period swaptions, the model swaption volatility surface intersects with the market volatility surface at the 1 period swaption line, regardless of the choice of g and $\rho^{{(0)}}_{}$. See the second picture in Figure 3 for a model swapvol surface for some rather arbitrary choice of g and $\rho^{{(0)}}_{}$. This is in fact an intrinsic stability problem (see also [37]), since basically two explaining powers are available determining one rotation angle. We have resolved this problem by introducing economically motivated regularizations of the least squares object function and implemented the so developed stable procedures for testing against market data. Meanwhile our calibration methods are gaining interest, appearing from consulting requests (Reuters FS, Bankgesellschaft Berlin AG) and a currently developing book project ([38]).

Fig. 3: Market swaption vols and model vols for some typical but arbitrary choice of g,$\rho^{{(0)}}_{}$

3. Volatility estimation of financial time series

(D. Mercurio, V. Spokoiny).

In cooperation with the project Statistical data analysis, new techniques for the volatility estimation for financial time series have been developed, [12], [21], [30].

• Locally time-homogeneous volatility estimation: The usual approach to volatility estimation starts from the definition of a parametric model, such as GARCH, and then fits this model to the data, globally, for example, by maximum likelihood estimation. On the other hand, practitioners usually prefer to focus on the most recent data, which are perceived as more informative. Therefore, exponentially weighted moving average and window estimators are often applied in practice. However, the results of these techniques strongly depend on the choice of the smoothing parameters.
  The attempt of the technique proposed in [21] is precisely to put the window estimator on a sound statistical basis. The most recent window of data on which the estimation is performed is selected through a multiple testing procedure, where the probability of rejecting a homogeneous interval is specified. Applications to exchange rate data show good results. The method is successfully applied to problems of volatility forecasting and Value at Risk evaluation.

  Fig. 4: Returns and estimated volatility for the JPY/USD exchange rate

  
• Volatility estimation from discrete observations: Since many artifacts are observed in high-frequency or tick data during the day, one often uses daily data in historical financial time series to draw inference on the underlying model parameters. In mathematical finance the underlying models are, however, usually formulated in continuous time. Nevertheless, classical methods for estimating the volatility in diffusion-type models are only asymptotically consistent for observation distances tending to zero. In [12] and [30], we studied the prototype problem of estimating the drift and diffusion (volatility) coefficient in a scalar diffusion X from the observation (X_n$\Delta$)_{0$\le$n$\le$N} for arbitrary $\Delta$ > 0. We prove optimality of our nonparametric estimation method in a minimax sense for fixed $\Delta$ > 0 and the asymptotics N$\to$$\infty$. First simulation results indicate that already for relatively small observation distances $\Delta$ our method outperforms classical procedures based on quadratic variation estimation.

4. Methods for risk management

(S. Jaschke, D. Mercurio, O. Reiß, J. Schoenmakers, V. Spokoiny, J.-H. Zacharias-Langhans).

Since the Basel Committee's proposal for ``An internal model-based approach to market risk capital requirements'' (1995) was implemented in national laws, banks have been allowed to use internal models for estimating their market risk and have been able to compete in the innovation of risk management methodology. Since all banks are required to hold adequate capital reserves with regard to their outstanding risks, there has been a tremendous demand for risk management solutions. A similar ``internal ratings-based approach'' is planned for the controlling of credit risk in the ``Basel II'' process, which is due to be implemented in national laws by 2006. Meanwhile, credit derivatives play an important role as vehicle for banks to transform credit risk into de jure market risk and to potentially lower the required reserves. Such problems of risk measurement and risk modeling are the subject of the research on ``Mathematical methods for risk management''. This research is supported by the BMBF project ``Efficient methods for valuation of risk measures'', which continued in 2003 in cooperation with the Bankgesellschaft Berlin AG. Problems of both market and credit risk from the viewpoint of supervisory authorities are being worked on in cooperation with the BAFin.

Although the basic principles of the evaluation of market risks are now more or less settled, e.g., [2], [8], [9], [29], in practice many thorny statistical and numerical issues remain to be solved. Specifically the industry standard, the approximation of portfolio risk by the so-called ``delta-gamma normal'' approach, can be criticized because of the quadratic loss approximation and the Gaussian assumptions. Further, in the context of the ``Basel II'' consultations fundamental questions arise in the area of Credit Risk Modeling.

• Evaluation of market portfolio risk: A procedure is constructed, where a small set of scenarios is carefully generated according to a target distribution, which approximates the optimal importance sampling measure adapted to this portfolio, but still reflects small variations in the components accurately, and is therefore perfectly suited to calculate the derivatives on a small sample of scenarios. The sample can be generated by a simple rejection algorithm on a previously generated large sample set, but also by a Markov chain, similar to the annealed importance sampling scheme with Metropolis-Hastings kernel, which is also used in the alternative approach. Here, the target distribution is determined by a weight of the form exp(- $\theta$(L - VaR_$\alpha$)²), approximating for large $\theta$ the condition L = VaR_$\alpha$, see Figure 5.

  
  

  Fig. 5: Sampling in the area of certain losses

  

• Evaluation of credit portfolios: The results obtained in cooperation with the BMBF project ``Efficient methods for evaluation of risk measures'' for the problem of evaluating market portfolio risk were our starting point for the study of credit risk portfolios. In particular, the enhanced Fourier inversion method is adapted for application to the popular credit risk portfolio model CreditRisk⁺, proposed by Credit Suisse First Boston (1997) ([7]). CreditRisk⁺ is a default-mode model which distinguishes between two states, default or survival of an obligor within a one-year period. The popularity of CreditRisk⁺ is due to the following features. The input data and parameters are readily available. For instance, default probabilities and recovery rates are required in the context of the Internal Ratings-Based Approach of the Basel II framework on the regulatory treatment of credit risk. Furthermore, CreditRisk⁺ is very appealing from a computational point of view due to its analytical tractability. In particular, the loss distribution is discrete and its probability generating function is explicitly known,
  

  $$\sum_{{n=0}}^{\infty} \tilde{{X}}$$ G(z) = (2)

  $$\left(\vphantom{ \sum_{i=1}^N w_{0,i} p_i (z^{\nu_i}-1) - \s... ... \left[ 1-\sigma_k^2 \sum_{i=1}^N w_{k,i} p_i (z^{\nu_i}-1) \right] }\right. \sum_{{i=1}}^{N} \nu_{i} \sum_{{k=1}}^{K} {\frac{{1}}{{\sigma_k^2}}} \left[\vphantom{ 1-\sigma_k^2 \sum_{i=1}^N w_{k,i} p_i (z^{\nu_i}-1) }\right. \sigma_{k}^{2} \sum_{{i=1}}^{N} \nu_{i} \left.\vphantom{ 1-\sigma_k^2 \sum_{i=1}^N w_{k,i} p_i (z^{\nu_i}-1) }\right] \left.\vphantom{ \sum_{i=1}^N w_{0,i} p_i (z^{\nu_i}-1) - \s... ... \left[ 1-\sigma_k^2 \sum_{i=1}^N w_{k,i} p_i (z^{\nu_i}-1) \right] }\right)$$ =

  
  where $\nu_{i}^{}$ is the loss exposure of obligor i with average loss probability p_i, and w_{k, i} and $\sigma_{k}^{}$ are economic sector weights and volatilities, respectively. In [32], [33], Fourier inversion techniques are applied successfully to CreditRisk⁺ and generalizations of it. Moreover, in [31], [34], a unified model is proposed which incorporates CreditRisk⁺ and Delta-normal as special cases. Further, in [13] we present an alternative numerical recursion scheme for computing the loss probabilities in (2) of the standard CreditRisk⁺ model, based on well-known expansions of the logarithm and the exponential of a power series. We show that it is advantageous to the Panjer recursion advocated in the original CreditRisk⁺ document, in that it is numerically stable whereas the Panjer algorithm is known to suffer from stability problems. Also we show that this stable recursion method can be extended to a model which incorporates stochastic exposures as proposed by Tasche ([42]).

  Based on knowledge of the complete loss distribution one can easily compute different risk measures such as Value at Risk and Expected shortfall. These risk measures are also important in the context of stochastic optimization ([14]).

5. Monte Carlo methods in finance

(G.N. Milstein, J. Schoenmakers, V. Spokoiny).

Monte Carlo methods are very important in the field of applied mathematical finance, and we present here some interesting applications.

• Forward-reverse simulation of worse case probabilities: Previously, Milstein, Schoenmakers and Spokoiny ([25]) developed in cooperation with the project Statistical data analysis a new transition density estimator for diffusion processes which is basically root-N consistent for any dimension of the diffusion process. This estimator,
  

  $$\hat{{p}} {\frac{{1}}{{MN\delta^{d}}}} \sum_{{m=1}}^{{M}% } \sum_{{n=1}}^{{N}} \left(\vphantom{ \frac{{X}_{t,x}^{(n)}(t_{1})-{Y}% _{t_{1},y}^{(m)}(T)}{\delta}}\right. {\frac{{{X}_{t,x}^{(n)}(t_{1})-{Y}% _{t_{1},y}^{(m)}(T)}}{{\delta}}} \left.\vphantom{ \frac{{X}_{t,x}^{(n)}(t_{1})-{Y}% _{t_{1},y}^{(m)}(T)}{\delta}}\right) \cal {Y}$$ (3)

  
  which does not suffer from the ``curse of dimensionality'', is based on forward simulation of the given process X and reverse simulation of an adjoint process (Y,$\cal {Y}$) which can be constructed via the (formal) adjoint of the generator of the original process. For estimating worst-case scenario probability densities in financial applications it is desirable to have a variation of the method in [25] for discrete time models, which have basically more potential for modeling heavy tails. To this end we have constructed in [24] the discrete adjoint process for a large class of discrete time Markov processes such that the forward-reverse density estimator (3) goes through for such processes as well. Several financial applications are currently studied in cooperation with the project Statistical data analysis.

• Variance reduction by Stratification: In cooperation with the project Numerical methods for stochastic models we have been considering the following problem in [20]. Many financial portfolios are influenced by a variety of underlyings, but which are only locally correlated. The valuation of such products leads to high-dimensional integration, but the integrands possess a small effective dimension.

  Prototypically we exhibit the following situation, describing a portfolio of a set {I_t¹, I_t²,..., I_t^m}, of m underlyings with respective shares w₁, w₂,..., w_m, determining the present value V_t = $\sum_{{j=1}}^{m}$w_jI_t^j. There are many numerical schemes for the valuation of such financial products. Often a reasonable approximation is obtained using the $\Delta$ - $\Gamma$-normal-method for the forecast of V_t+1 at a future time knowing V_t, given by

  $$\approx \theta \Delta \langle \Gamma \rangle$$ (4)

  with $\theta$,$\Delta$, and $\Gamma$ completely determined by the structure of the portfolio, and y Gaussian innovations, see [11]. In this case we obtain a quadratic functional in the underlyings. This remains quadratic when turning to the independent risk factors. Normally, $\Gamma$ is sparse.

  In this situation, the ANOVA decomposition of the integrand in (4) admits only up to bivariate contributions, such that it makes sense to apply Monte Carlo methods, which intrinsically use this information. For the situation at hand, we propose the use of randomized orthogonal arrays of strength 2 ([4]), which are known to show super-convergence, i.e. converge faster than usual Monte Carlo. Test calculations based on real-world data provided by the Bankgesellschaft Berlin AG actually show superiority above conventional simulations, see Figure 6.

  Fig. 6: Convergence Monte-Carlo (1) vs. orthogonal arrays (2)

  

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38.          , Robust Libor Modelling and Pricing of Derivatives Products, in preparation.

39. J.G.M. SCHOENMAKERS, B. COFFEY, LIBOR rate models, related derivatives and model calibration, WIAS Preprint no. 480, 1999.

40.          , Stable implied calibration of a multi-factor LIBOR model via a semi-parametric correlation structure, WIAS Preprint no. 611, 2000.

41.          , Systematic generation of parametric correlation structures for the LIBOR market model, Int. J. Theor. Appl. Finance, 6 (2003), pp. 507-519.

42. D. TASCHE, Capital allocation with CreditRisk⁺, to appear in: CreditRisk⁺ in the Banking Industry, F. Lehrbass, M. Gundlach, eds., Springer, Berlin Heidelberg, 2004.

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