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The IUP Journal of Financial Risk Management
Rational Greeks
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Risk analyses often require the revaluation of complex financial instruments over a large set of risk factor scenarios. In many instances, full revaluation is numerically-intensive and hence too costly. In that case, one can resort to partial revaluations using Greeks: partial derivatives of the instrument with respect to the underlying value drivers. Taylor series approximations to pricing functions are only valid for relatively small changes in the risk factors and show large approximation errors in the tails of the distribution. This paper shows how partial revaluation can be improved by using (modified) Padé rational approximants. Firstly, it is shown how rational approximation enhances the estimation of effective Greeks from perturbated risk factors. Secondly, it is shown that rational approximants outperform conventional Taylor approximants in approximating pricing functions. Rational Greeks signify a substantial improvement in approximation accuracy, even for substantial changes in risk factors.

 
 
 

Risk analyses often require the revaluation of complex financial instruments over a large set of risk factor scenarios. In many instances, especially in historical and Monte-Carlo simulation approaches to Value-at-Risk, full revaluation is numerically-intensive and hence too costly. Lattice models, for example, are flexible and convenient vehicles for pricing complex derivatives, but repricing over many different factor scenarios is prohibitive. For that reason, Taylor series approximations to the pricing function are employed based on partial derivatives of the instrument with respect to the underlying value drivers. Hence, partial simulations of non-linear derivatives are based on estimates of delta, gamma, vega and other relevant Greeks. Likewise, partial revaluation of fixed income positions are based on duration and convexity measures. Drawback is that the Taylor series approximations are only valid for relatively small changes in the risk factors. This results in large approximation errors in the tails of the value distribution—and especially tail information is relevant in risk analyses. Greeks are not only relevant in shortcuts to revaluing derivatives, but also for describing their risk profile. This allows the adequate hedging of derivative and fixed income positions. When the underlying valuation model is complex, effective Greeks are approximated by numerical derivatives.

 
 
 

Financial Risk Management Journal, Rational Greeks, Value-at-Risk, Monte-Carlo, Drawback, Taylor series.