A new approach for scenario generation in risk management

A new approach for scenario generation in risk management Josef Teichmann TU Wien Vienna, March 2009

Scenario generators Scenarios of risk factors are needed for the daily risk analysis (1D and 10D ahead) due to Basel II legislation.

Scenario generators Scenarios of risk factors are needed for the daily risk analysis (1D and 10D ahead) due to Basel II legislation. Generated scenarios should share the most important stylized facts of the respective time series of risk factor in order to reflect the market s information.

Scenario generators Scenarios of risk factors are needed for the daily risk analysis (1D and 10D ahead) due to Basel II legislation. Generated scenarios should share the most important stylized facts of the respective time series of risk factor in order to reflect the market s information. The actual generation of scenarios must be quick and flexible. Changes of markets and of the risk factors should be easy to implement into the scenario generator.

Standard approaches Monte-Carlo Simulation

Standard approaches Monte-Carlo Simulation Analysis of the marginal distributions of risk factors.

Standard approaches Monte-Carlo Simulation Analysis of the marginal distributions of risk factors. Measurement of covariances from the given time-series.

Standard approaches Monte-Carlo Simulation Analysis of the marginal distributions of risk factors. Measurement of covariances from the given time-series. Construction of multivariate distributions by coupling the marginal distributions through the measured covariance structure.

Standard approaches Historical Simulation

Standard approaches Historical Simulation Draw returns from the set of (rescaled) historical returns.

Standard approaches Historical Simulation Draw returns from the set of (rescaled) historical returns. Do this simultaneously for the whole vector of risk factors.

Problems Problems

Problems Problems The analysis of marginal distributions and the measurement of covariances is ill-defined due to short time series.

Problems Problems The analysis of marginal distributions and the measurement of covariances is ill-defined due to short time series. Coupling of distributions via ill-defined covariances is delicate.

Problems Problems The analysis of marginal distributions and the measurement of covariances is ill-defined due to short time series. Coupling of distributions via ill-defined covariances is delicate. Sampling from the constructed distributions may lead to inconsistent results due to misspecification of covariances and due to undefined relations with historical simulations.

Problems Problems The analysis of marginal distributions and the measurement of covariances is ill-defined due to short time series. Coupling of distributions via ill-defined covariances is delicate. Sampling from the constructed distributions may lead to inconsistent results due to misspecification of covariances and due to undefined relations with historical simulations. Flexibility and Reliability of both methods is limited due to their ad-hoc-nature.

A Synthesis Distributional Aspect

A Synthesis Distributional Aspect Generation of scenarios by SDEs, which describe the local dynamics of the respective risk factors

A Synthesis Distributional Aspect Generation of scenarios by SDEs, which describe the local dynamics of the respective risk factors The SDEs are standard models from mathematical Finance like Black-Merton-Scholes, Cox-Ingersoll-Ross, HJM model etc (adequately chosen for the respective risk factor), in particular they are free of arbitrage.

A Synthesis Aspect of historical simulation

A Synthesis Aspect of historical simulation Calibrate the SDEs by directly constructing the characteristics of the dynamics from the time series of the risk factors.

A Synthesis Aspect of historical simulation Calibrate the SDEs by directly constructing the characteristics of the dynamics from the time series of the risk factors. The longer the time series the closer one recovers the true characteristics.

Interest Rates An example: Risk factors from Interest Rates

Interest Rates An example: Risk factors from Interest Rates Risk factors are the yield curve.

Interest Rates An example: Risk factors from Interest Rates Risk factors are the yield curve. The yield curve evolves according to an HJM equation.

Interest Rates An example: Risk factors from Interest Rates Risk factors are the yield curve. The yield curve evolves according to an HJM equation. Numerics of HJM equations?

Interest Rates An example: Risk factors from Interest Rates Risk factors are the yield curve. The yield curve evolves according to an HJM equation. Numerics of HJM equations? Calibration of the HJM equation to the market?

Numerics of SPDEs On a filtered probability space (Ω, F, (F t ) t 0, P) we consider a d-dimensional standard Brownian motion B. Let H denote a Hilbert space of risk factors, then we consider where dy t = (µ 1 (Y t ) + µ 2 (Y t ))dt + Y 0 H, d σ(y t ) λ i dbt, i (1) i=1 σ(y ) : λ 1,..., λ d λ 1,..., λ d is an invertible, linear map on the span of the set of return directions λ 1,..., λ d depending Lipschitz on Y. (2)

We suppose the usual conditions on the drift µ 1, i.e. µ 1 (Y ) = AY + µ 3 (Y ) for Y dom(a). Here A denotes the generator of a strongly continuous semigroup, which will be in our case always a shift to the right or the identity semigroup, on the Hilbert space of risk factors H.

The volatility factor σ is chosen appropriately for the respective risk factors in order to exclude immediate arbitrages (like, e.g., negative interest rates). It should be interpreted as a priori given factor govering the shape of the support of the risk factors, hence it is a geometric factor. The vector fields µ 1 corresponds to appropriately chosen no-arbitrage conditions and the vector field µ 2 lies in the span of λ 1,..., λ d and corresponds to an appropriate change of measure due to Girsanov s theorem.

Numerics of SPDEs Assume that A = d dx, the generator of a strongly continuous shift semigroup and that all other vector fields µ 2, µ 3, σ restrict to Cb -vector field on dom(a) with respect to the operator topology (this means in particular that we have a strong solution Y t ). Then the strong Euler scheme yields a strong short time asymptotics of order 0.5, i.e. E t (Y 0 ) =S t Y 0 + µ 2 (Y 0 )t + µ 3 (Y 0 )t + d σ(y 0 )λ i Bt, i i=1 E( Y T E T /N... E T /N (Y 0 ) ) C 1 N as N for Y 0 dom(a).

Layman s calibration We assume a time series, i.e. an observation of equation (1), on equidistant grid points of distance, denoted by Y 1,..., Y K and we ask the simple question if we can estimate the volatility directions λ 1,..., λ d out of the observations Y 1,..., Y K in a simple way? We announce now an equation of type (1) calibrated to the time series dx (K) t =(µ (K) 1 (X (K) t ) + µ 2 (X (K) t ))dt+ (3) K 1 1 + σ(x (K) t ) (σ(y i ) 1 (Y i+1 Y i )) dwt i, (K 1) i=1 where σ is a the known, non-vanishing geometric factor on the risk factors describing the local dynamics.

Furthermore we have to assume the following technical assumptions, which are necessary from the point of view of SPDEs: 1. The constant volatility directions λ 1,..., λ d are elements of dom(a ).

Theorem Let equation (1) be given in the sense that σ and Y 0 dom(a) are given maps, but λ 1,..., λ d are unknown. We collect a time series of observations Y 1,..., Y K on an equi-distant grid of time distance on an interval of length T = K. Refining the observations through = T K leads to the following limit theorem lim X (K) t = Y t K in distribution for any t 0 if X 0 = Y 0.

Theorem The underlying limit theorem is the following Gaussian one, lim t K 0 t 0 = lim K = t 0 σ(x (K) t ) 1 dx (K) t σ(x s (K) ) 1 (µ (K) 1 (X s (K) ) + µ 2 (X (K) )ds K 1 1 (σ(y i ) 1 (Y i+1 Y i ))W i (K 1) σ(y t ) 1 dy t i=1 t 0 σ(y s ) 1 (µ 1 (Y s ) + µ 2 (Y s )ds. s t

IR-markets Calibration to a IR-market

IR-markets Calibration to a IR-market The implemented dynamics calibrated to the time series is of the following type dr t =( d dx r t + α(r t) 2 K 1 1 2(K 1) α(r i=1 i ) 2 (r i+1 r i ) + α(r K 1 t) K 1 i=1 (r i+1 r i ) dwt i, α(r i ). where α(r) is a non-vanishing field on yield rates. 0 (r i+1 r i ) + µ)dt+

Real world Implementation Three steps

Real world Implementation Three steps Discretization of the previous equation: one step is given by ( corresponds to one timestep) r = shift( )r 0 + drift + volas N, where N is a vector of standard normals. volas and drift are constructed directly from the time series as shown.

Real world Implementation Three steps Discretization of the previous equation: one step is given by ( corresponds to one timestep) r = shift( )r 0 + drift + volas N, where N is a vector of standard normals. volas and drift are constructed directly from the time series as shown. Randomization of simulation time (business time versus running calendar time).

Real world Implementation Three steps Discretization of the previous equation: one step is given by ( corresponds to one timestep) r = shift( )r 0 + drift + volas N, where N is a vector of standard normals. volas and drift are constructed directly from the time series as shown. Randomization of simulation time (business time versus running calendar time). Inclusion of events is easy

Real world Implementation Three steps Discretization of the previous equation: one step is given by ( corresponds to one timestep) r = shift( )r 0 + drift + volas N, where N is a vector of standard normals. volas and drift are constructed directly from the time series as shown. Randomization of simulation time (business time versus running calendar time). Inclusion of events is easy, but unnecessary those days.

Features

Features The longer the time series (large K) the better the stylized facts of the simulations.

Features The longer the time series (large K) the better the stylized facts of the simulations. All cross-correlations match perfectly.

Features The longer the time series (large K) the better the stylized facts of the simulations. All cross-correlations match perfectly. The non-shift -drifts can indeed be neglected, since volas shortrate for real values and less than one day.

An example

An example Four FX rates, HJM dynamics for one domestic and four foreign yield curves, 500 BD of data.

An example Four FX rates, HJM dynamics for one domestic and four foreign yield curves, 500 BD of data. Business time versus running time.

Bayer, C., Teichmann, J., Cubature on Wiener space in infinite dimension, Proceedings of the Royal Society London A, to appear, 2008. Ortega, J.-P., Pullirsch, R., Teichmann, J. and Wergieluk, J., A new approach for scenario generation in risk management, preprint, 2009. Teichmann, J.,Another approach to some rough and stochastic partial differential equations, preprint, 2008.