Scenario Generation To apply SP models we need to generate scenarios which represent the uncertainty IN A SENSIBLE WAY, taking into account the goal of the model and its structure, the available information, computer facilities and available softare. (Differs from generation of a SINGLE SCENARIO for forecasting performance of a holding strategy, etc.)
Scenarios in Financial Applications I. EXPLOITED IN: stochastic immunization and dedication (Hiller and Schaack 1990) tracking models (Dembo 1991) robust optimization (Mulvey, Vanderbei, Zenios 1995) stochastic programming incl. multiperiod and multistage models evaluation of dynamic investment strategies (Mulvey 1996), etc.
Scenarios in Financial Applications II. For financial applications many suitable models for financial time series and stochastic models with continuous time historical data various specific problems, e.g. various levels of information on various sources of uncertainty e.g. demographic data, economic factors correlations of stock market and bond returns problem specific additional random factors (prepayment rates, defaults...) difficulties to model illiquid assets prices, derivatives,... low information level under new conditions (emerging market economies, etc.)
How to Build Sensible Scenarios? THE GOAL of scenario generation procedure is NOT to get a good approximation of the probability distribution but of the optimal value and of the optimal decisions. GOOD APPROXIMATION may mean precision (evaluate errors), robustness, etc. Connection with output analysis is evident. No general recipe exists; it is necessary to exploit past data, experience, models, theory... HARD JOB
Origin of Scenarios can be very diverse. Scenarios atoms of a known genuine discrete probability distribution, obtained in the course of a discretization / approximation scheme, result from simulation, come from limited sample information, follow recognized regulations result from a preliminary analysis of the problem with probabilities of their occurrence that may reflect an ad hoc belief or a subjective opinion of an expert, etc. Generation of scenarios is problem specific and it should reflect both the problem structure and the available information about the underlying probability distribution.
Scenarios and their Generation Aim of scenario generation is to build a manageable problem which will provide good decisions for the true underlying real-life problem. Compromise is needed between precision of the approximation of the probability distribution P and the size and goal of the approximate problem. SCHEME: Use historical data (if any) in conjunction with an assumed background model, apply suitable estimation, simulation and sampling procedures take into account opinion of experts based on their experience, heuristics, etc generation of scenarios cannot be reduced solely to forecasting the future development of the complex system under consideration.
Level of Available Information Full Knowledge of Probability Distribution Probability distribution P is fully specified, scenarios can be obtained by sampling from P or by application of a discretization or simulation scheme. Appears mostly when testing designed models and/or the performance of newly developed solvers using normal, uniform or discrete P. Known Parametric Family Parametric family of probability distributions based on a theoretical model is specified, parameters of probability distribution P are estimated from available data. Choice of parametric form of probability distribution corresponds to choice of model, estimation of parameters to calibration of model and a subsequent simulation, sampling or discretization procedure follow similarly as above. Appears frequently in finance: the relevant stochastic models of interest rates and assets prices appeared relatively early and both discrete and continuous time models have been well developed and supported by time series of historical data.
Examples 1. Discrete time stochastic models Vector Autoregressive Model of 1st order ω t = µ + H(ω t 1 µ) + ε t, ε t N(0,Σ) where eigenvalues of H fulfil λ(h) < 1 and ε t s are jointly independent. Parameters µ,h,σ are estimated from historical data and possibly further adapted to distinct sources of information (e.g., experts forecasts or values of related global parameters); ˆµ, Ĥ, ˆΣ estimates. Starting with a known vector ω 0 and using the calibrated model, scenarios are constructed as ω s t = ˆµ + Ĥ(ωs t 1 ˆµ) + ˆε s t, where ˆε s t is obtained as an observation from N(0, ˆΣ) by a suitable discretization or simulation technique. Similarly, also autoregressive and ARMA models of higher orders or econometric models with lagged variables may be applied. Factor analysis can be used to get a small number, say M, of one-dimensional independent factors.
Examples 2. Black-Derman-Toy Model (will be detailed later on) Continuous time stochastic models are mostly represented by stochastic differential equations of the type dω(t) = b(t,ω(t))dt + σ(t,ω(t))dw(t), ω(0) = ω 0, t 0, (1) where W is the Wiener process and coefficients b(t,ω(t)),σ(t,ω(t)) fulfil some assumptions. For instance, solutions of (1) with b,σ independent of t and satisfying certain integrability conditions are known as diffusion processes. Recall that Wiener process W(t) is stochastic process with independent increments and continuous trajectories such that W(0) = 0 a.s., for positive s, t, the distribution of W(t) W(s) is normal, N(0, t s ).
Example Vašíček s Model for Spot Rates dω(t) = b(k ω(t))dt + σdw(t), t 0 with b > 0, constant, corresponds to choice b(t,ω(t)) = b(k ω(t)), σ(t,ω(t)) = σ in (1). It is so called Ornstein-Uhlenbeck process, Markov process with normally distributed increments. In contrast to Wiener process, it has stationary distribution. Instantaneous drift b(k ω(t)) forces the process towards its long-term mean k, the so called mean reversion property, and the constant instantaneous variance σ 2 makes it to fluctuate around k in continuous but erratic way. This means that negative values of ω(t) cannot be excluded. To apply the model for generation of scenarios of interest rates means to estimate parameters and to choose suitable time discretization < 1/b. Then ω (n+1) = bk + ω n (1 b ) + σ ε, n = 0,1,... with an arbitrarily given initial value ω 0 and with ε independent, N(0,1).
Sample information Sample information about the true probability distribution is mostly based on observed past data. If the data are homogeneous enough, they can be viewed as independent, identically distributed random variables (vectors). Use of empirical distribution is straightforward. Otherwise, one could think of a preprocessing procedure to treat missing data, smoothing, etc., or of an adjustment for to fit specific values of (sample) moments. Simplest idea use as scenarios the past observations obtained under comparable circumstances and assign them equal probabilities. For example one may construct scenarios of joint assets returns for a T-period model as n T distinct T-tuples of their subsequent observations from n previous periods and to assign them equal probabilities, (n T) 1.
Low Information Level The procedures fail if there are no reliable data. Then, scenarios and their probabilities are mostly based on experts forecasts or even on governmental regulations. For instance, to test the surplus adequacy of an insurer, New York State Regulation 126 suggests seven interest rate scenarios to simulate the performance of the surplus. Under the heading of low information level one may also include cases when true probability distribution P is described only by several moment values or/and some simple qualitative properties.
Miscelaneous Sources of Uncertainties In most applications one can trace interactions of various information levels and to use all available information is the best thing to do. Past information is often combined with experts oppinion probably one possible origin of postulated form of true probability distribution. Different information levels and different time-scales of collecting and recording data may apply separately to distinct parameters of model. In portfolio management, different classes of securities require different treatment, deposits and liabilities can be driven by conceptually different external factors such as mortality rates. For management of new pension plan, the random factors which run the bond market can be well described by an interest rates model, basic requirements on premium and pension payments are partly known thanks to extensive demographic data whereas the uncertainty relates to future preferences of clients. Scenarios and their probabilities are then based on experts forecasts. For independent sources of uncertainties, number of scenarios needed to represent their mutual influence on results is product of the number of scenarios used to represent the impact of each source separately.
Application Bond Portfolio Management Problem Kahn (1991): Many years ago, bonds were boring. Returns were small and steady. Fixed income risk monitoring consisted in watching duration and avoiding low qualities. But as interest-rate volatility has increased and the variety of fixed income instruments has grown, both opportunities and dangers have flourished... TRADITIONAL APPROACHES FOR BOND PORTFOLIO MANAGEMENT (e.g. immunization or deterministic dedication models) BASED ON NON REALISTIC ASSUMPTIONS SHOULD BE REVISED INSERT: PROBLEM FORMULATION
Scenario Generation via BDT Model I. Use interest rate scenarios obtained from the binomial lattice constructed according to the Black-Derman-Toy model. The obtained one-period short term rates valid for scenario s and for the time interval (l,l + 1] are They are determined by base rates r l0 lattice volatilities k l ρ s l = r l0 k i l(s) l, i l (s) = l ωτ. s (2) and by chosen sampling strategy which determines the exponent l τ=1 ωs τ, i.e., the number of up moves in the time discretization points τ = 1,...,l. Theoretical binomial lattice consists of 2 T 1 different paths or vectors of interest rates; their components are given by (2). INSERT FIGURE 51n τ=1
Scenario Generation via BDT Model II. To calibrate Black-Derman-Toy model, i.e., to get base rates r l0 and lattice volatilities k l l, means to use yield and volatility curve related to yields to maturity of zero coupon government bonds of all maturities corresponding to the chosen time steps of the lattice. Such bonds are rare in the market and have to be replaced by synthetic zero coupon bonds whose yields correspond to yields of fixed coupon government bonds that do not contain any special provision such as call or put options. To estimate yield curve from existing market data on yields of fixed coupon government bonds at the given day we applied regression analysis; we chose simple form of yield curve applied by Bradley and Crane g(t;θ,β,γ) = θt β e γt and fitted its linearized form to the logarithms of yields: For market information consisting of yields u i,i = 1,...,N of various fixed coupon government bonds (without option) characterized by their maturities T i, the postulated model is ln u i = ln θ + β ln T i + γt i + ε i, i = 1,...,N (3) where random errors ε i,i = 1,...,N are independent, N(0,σ 2 ).
Scenario Generation via BDT Model IV. COMMENT: There is a good reason to accept approximately normal errors; this is in line with the assumed log-normal process of short rates approximated by the BDT lattice. INSERT: Data yields for government bonds traded on September 1, 1994. Plot of yield curve for September 1, 1994, FIGURE fign The fit is sensitive to data. Compare the fit with and without the long bond BTP36665 maturing in 2023. (Maturities of all remaining bonds are less than 10 years.)
Scenario Generation via BDT Model V. Least squares estimates ln θ, ˆβ, ˆγ of parameters lnθ,β,γ are approximately normal, with the mean values equal the true parameter values and the covariance matrix σ 2 Σ 1, Σ = G G where σ 2 is estimated by s 2 = 1 N 3 min ln θ,β,γ N (ln u i ln θ β ln T i γt i ) 2 (4) i=1 and the N 3 matrix G consists of rows (1,ln T i,t i ), i = 1,...,N. To estimate the yields of zero coupon bonds of all required maturities which are not directly observable, t T i, we replace the unobservable logarithm of yield by the corresponding value on the already estimated log-yield curve. Such estimates are subject to an additional error.
Scenario Generation via BDT Model VI. Techniques for obtaining volatilities of yields are less obvious. There is not enough data for fitting the volatility curve by a regression model. Most of the authors work with an ad hoc fixed constant volatility; the volatility curve may be estimated from historical data or based on the Risk Metrics datasets which provide historical volatilities computed daily for several main maturities, 1 year, 2, 3, 4, 5, 7, 9, 10, 15, 20 and 30 years. It is suggested to estimate the missing yields by linear interpolation and to use the volatilities and correlations of the reported yields to compute the approximate values of yield volatilities for these nonincluded maturities. Approximate standard deviations of lnu may be obtained from the parametric model provided that the errors in regression are normally distributed. Based on the obtained yield and volatility curves, calibration of binomial lattice in agreement with no-arbitrage valuation principles follows by a numerical procedure suitable for solving the large system of nonlinear equations for the base rates r l0 and lattice volatilities k l, l = 1,...,T.
Scenario Generation via BDT Model VI. According to (2), the fitted binomial lattice provides different 2 T 1 scenarios of interest rates identified by the binary fractions of T 1 ones or zeroes. Smaller number of scenarios has to be selected. The components of selected S scenarios are computed according to (2) using scenario independent base rates r t0, t = 1,...,T 1, volatilities k t, t = 1,...,T 1, and scenario dependent position on the lattice given by the exponent i t (s) in (2) number of up moves needed to reach the position on the lattice within t periods. The prices ξjt s,ζs s jt and cash flows fjt are evaluated along each of these scenarios and the problem (2) (7) is solved. The main output is the optimal value - the maximal attainable expected utility of final wealth at time T 0 and an optimal first-stage solution, say x,y,z 0,y+ 0. BACK TO APPLICATION NUMERICAL EXPERIMENT WHAT DECISION TO ACCEPT? VALIDATION OF RESULTS or OUTPUT ANALYSIS NEEDED!