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Volatility Model Risk Measurement and Strategies Against Worst Case

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Volatility Model Risk Measurement and Strategies Against Worst Case Volatility Mo del Risk measurement and strategies against worst case volatilities y Risklab Pro ject in Mo del Risk 1 Mo del Risk: our approach Equilibrium or (absence of ) arbitrage mo dels, but also p ortfolio management applications and risk management pro cedures develop ed in nancial institu- tions, are based on a range of hyp otheses aimed at describing the market setting, the agents risk app etites and the investment opp ortunityset. When it comes to develop or implement a mo del, one always has to make a trade-o between realism and tractability. Thus, practical applications are based on mathematical mo dels and generally involve simplifying assumptions which may cause the mo dels to diverge from reality. Financial mo delling thus in- evitably carries its own risks that are distinct from traditional risk factors such as interest rate, exchange rate, credit or liquidity risks. For instance, supp ose that a French trader is interested in hedging a Swiss franc denominated interest rate book of derivatives. Should he/she rely on an arbitrage or an equilibrium asset pricing mo del to hedge this b o ok? Let us assume that he/she cho oses to rely on an arbitrage-free mo del, he/she then needs to sp ecify the number of factors that drive the Swiss term structure of interest rates, then cho ose the mo delling sto chastic pro cess, and nally estimate the parameters required to use the mo del. The Risklab pro ject gathers: Mireille Bossy, Ra jna Gibson, François-Serge Lhabitant, Nathalie Pistre, Denis Talay and Ziyu Zheng. It is a joint collab oration between the Risklab institute (Zürich), INRIA Sophia Antip olis, and University of Lausanne. Author's current addresses: M. Bossy, D. Talay and Z. Zheng: INRIA Sophia Antip olis, 2004 route des Lucioles, BP 93, 06902, Sophia Antip olis Cedex, France. N. Pistre: ENSAE, 3 avenue Pierre Larousse, 92245 Malako Cedex, Paris, France. R.Gibson: Swiss Banking Institute, 14 Plattenstrasse, 8032, Zurich, Suisse. F-S. Lhabitant: UBS AG, Global Asset Management, Klausstrasse 10, 8034 Zurich. y To app ear in the Journal of the French Statistics So ciety. 1 In order to characterize the random evolution of the term structure of interest rates, mo dels with one-factor, generally chosen as the short term rate, have been developp ed b ecause they are easy to implement (see, e.g., Merton [21] , Vasicek [25], Cox, Ingersoll and Ross [10], Hull and White [17], etc.), even if most empirical studies using a principal comp onent analysis have decomp osed the motion of the interest rate term structure into three indep endent and non-correlated factors, which resp ectively capture the level shift in the term structure, the twist in opp osite direction of short and long term rates, and the buttery factor that captures the fact that the inter- mediate rate moves in the opp osite direction of the short and the long term rates. The multi-factor mo dels do signicantly b etter than single-factor mo d- els in explaining the dynamics and the shap e of the entire term structure, but the latter provide analytical expressions for the prices of simple interest rates contingent claims, whereas a multi-factor mo del generally leads to nu- merically or quasi analytically solve partial dierential equations in a higher dimension to obtain prices and hedge ratios for the interest rate-contingent claims. The factor(s) dynamics sp ecication is another source of Mo del Risk. The dynamics sp ecications cover a large sp ectrum of distributional assumptions such as pure diusion pro cesses or mixed jump-diusion pro cesses. The sto chastic pro cesses can have time-varying or constant drift and/or volatility parameters and they can rely on a linear or a non-linear sp ecication of the drift (see Ait- Sahalia [3]) Once the mo del is xed, one has to estimate its parameters. This step do es not really provide help to verify the adequacy of the mo del on past data, since the theory of parameter estimation generally assumes that the true mo del b elongs to a parametrized family of mo dels. Moreover, a mis- sp ecied mo del do es not necessarily provide abad t to the data. A study by Jacquier and Jarrow [18] prop oses to incorp orate mo del error and parameter uncertainty into a new metho d of contingent claims mo d- els' implementation. Using Markov Chains, Monte Carlo estimators, their conclusion, for a single sto ck option case study, suggests that the pricing p erformance of the "extended" BlackandScholes mo del dominates the sim- ple Black and Scholes mo del within but not out-of-sample. Usually, the mo del parameters must be estimated by tting a given set of market data. However, in nance it has b een proved that natural estimators suchasmax- imum likeliho o d and generalized metho d of moments estimators applied to time-series of interest rates may require a very large observation p erio d to converge towards the true parameter values, yet it seems highly unrealis- tic to assume constant parameters over such a long period, see, Fournié & Talay [13]. Another imp ortant problem arises from the time discretization 2 when we numerically compute the statistical estimates of the parameters of a sto chastic mo del. All the sources of Mo del Risk we have just listed, have strong nancial rep ercussions (losses incurred by a bank or a nancial institution due to Mo del Risk are fairly common) on the pricing, hedging, risk management and the denition of regulatory capital adequacy rules. As far as derivatives pricing/hedging is concerned, a large part of the lit- erature relies on the assumption of absence of arbitrage opp ortunities since the seminal articles of Black and Scholes [8] and Merton [21]. The principle is simple: given a distribution on a primitive asset price (for a sto ck, a b ond, or a commo dity), if wemake the simplied hyp othesis that there are no fric- tions and that the risk structure is not to o complex, one can show that the derivative cash-ows can b e replicated by a dynamic trading strategy involv- ing the primitive assets. The Absence of Arbitrage hyp othesis stipulates that the price of the derivative should b e the same as the price of the replicating p ortfolio or there would be arbitrage prots (free lunches) in the market. However, in this framework the pricing or hedging p erformance is not indep endent of the mo del used for the underlying asset price. One imp ortant problem is that the pricing mo dels are often derived under a p erfect and complete market paradigm, whereas markets are actually incomplete and imp erfect. Violations of the p erfect market assumption generally preclude us from observing uniqueness of the asset's theoretical price, and we are often left only with bounds to characterize transaction prices. In an interesting study fo cusing on equity options, Green and Figlewski [15] thus explain why option writers will charge a volatility mark-up to protect their Prot and Loss against Mo del Risk. The presence of Mo del Risk will also aect the p erformance of mo del based hedging strategies. In a continuous-time and frictionless market, a market maker, for ex- ample, the seller of an option, can synthetically create an opp osite p osition (called delta-hedging strategy) which eliminates his/her risk completely. If the hedger uses an alternative (wrong) option pricing mo del, his/her price for the option diers from the true (market) price and provide an incorrect hedge ratio. In the presence of Mo del Risk, even though we assume fric- tionless markets, the self-nancing delta-hedging strategy do es not replicate the nal pay-o of the option p osition. Bossy and al. [9] study the case of b ond option hedging, the Mo del Risk is dened by the Prot and Loss of the seller of an option who b elieves that the true mo del is one of the univariate Markov mo dels nested in the Heath-Jarrow-Morton [16] framework, whereas the true mo del is actually another mo del b elonging to the same class. In a subsequent study, Akgun [4] extends the metho dology develop ed by Bossy 3 et al. [9] to include omitted jumps by the trader who uses the wrong pure diusion univariate term structure mo del to hedge his derivatives exp osure. In his setting, the jumps are driven by a nite state-space comp ound Poisson pro cess. This in turn allows him to show that omitting jump risk can be fairly devastating, as evidenced by simulated forward Mo del Risk Prol and Loss probability distributions, esp ecially when shorting in and at-the-money naked or spread option p ositions. Indeed, it is imp ortant for regulators to measure the trading and the banking books interest-rate risk exp osures correctly. The Basle Committee on Banking Sup ervision [1], [2] issued directives to help nancial institutions evaluate the interest-rate risk exp osures of their exchange traded and over- the-counter derivative activities, as well as for their on and o-balance sheet items. Regulators ask the banks and other nancial institutions to set aside equity in order to cover market risk driven losses. Prop osition 6 of the Basle Committee Prop osal [2] states that banks can calculate their market risk capital requirements as a function of their fore- casted ten-days-ahead value-at-risk.
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