Scenario-Based Evaluation and Uncertainty
The following problems arise in practice: A concrete instance of the selected equity, FX or interest rate model must be chosen, by instantiating its volatility and other coefficients with plausi- ble values. For example, the Black-Scholes model dS=μS t +∂
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The following problems arise in pr act ice: - A concrete inst an ce of t he selected equity, FX or int erest rat e mod el must be chosen, by instanti at ing it s volatility and ot her coefficients with plausible valu es. For example, the Black-Scholes mod el dS = J.1 St + (J dW might be instant iate d t o dS = 0.05 St + 0.3 dW. - On ce inst antiated , models often prove to o weak to represent the market dyn amics adequately; in t he case of Black-Scholes, t his deficiency shows itse lf in t he oft en cited implied volatility smile. The second problem can be approached with t ime- and space-de pe ndency in t he volatility and ot her coefficients. If this impli es randomness in t he evolut ion of t he volat ility, one has created a st ochastic volatility mod el. The first problem does not disappear , however, and some sort of param et er calibration is necessar y before t he sto chastic volatili ty model can be applied. Unce rtain volatility takes a different approach. Inst ead of choosing a fixed set of a pri ori mod el coefficient s, users specify priorit ies which t hey would like t o see applied when a given portfolio is evaluate d under t he mod el. T hese priorit ies are initi ally stated "in pros e" and have some economic function. T hey usu ally corres pond to stochastic cont rol problems and require dyn ami c programming methods for t heir solution.
4.1 Preliminaries Definition 1 (Scenario). We call a set of (declarative) agent prio rit ies and th e (im perative) evaluation rules th ey imply a scenario. D efinition 2 (Uncertain coefficients) . Model coeffici ents which are variable un der a given scenario are called uncert ain. The evaluation rules of the scenario control th e instanti ation of un certain coeffi cient s, locally or globally. T hese definiti ons are not strict ly forma l. The sou nd ness of t he concept needs to be established for each concrete scena rio. In t his book, we restrict ou rselves to two scenarios : - t he worst -case volatility scenario; R. Buff, Uncertain Volatility Models - Theory and Application © Springer-Verlag Berlin Heidelberg 2002
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4 Scenario-Based Evalu ati on and Uncertainty Patterns for mod el coefficients
Scena rio -
1
0
-+--
Po rtfolio
1
Instantiated model coefficients Fig. 4.1. Both scenari o a nd port folio are required compo nents when mod el coefficients ar e instan ti ated. Mode l coefficients ca n, bu t mu st not , b e restrict ed by pat terns
- the volatility-shock scena rio. We review the worst-case volatility scenario in this cha pte r. It was first develop ed by Avellaneda and Par as as t he A-Uncertain Volatility Mod el or AUVM. Algorithmic issues of worst-case scena rios are moved as original work t o P ar t II . The volatility shock scena rio is an exte nsion of t he worst- case scena rio and is discussed , also as original work , in Chapter 9 of P ar t II. T he benefit of the scena rio approach is clear: no definite a-priori choice of model coefficients has to be made. Fur therm ore, once evaluation rul es have been applied t o inst anti a
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