Sensitivity
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Risk Sensitivity (Greek) Introduction
Risk sensitivities, also referred to as Greeks, are the measure of a financial instrument’s value reaction to changes in underlying factors. The value of a financial instrument is impacted by many factors, such as interest rate, stock price, implied volatility, time, etc. Sensitivities are risk measures that are more important than fair values.
Financial sensitivities of a derivative product are just the partial derivatives of the price function with respect to market input variables. Since the derivatives is a function of the set of market variables, the sensitivtiy function, in most cases, may be a complicated compounding function. Therefore, repeatly applying the chain-rule is the major procedure of calculating these partial derivatives.
Greeks are vital for risk management. They can help financial market participants isolating risk, hedging risk and explaining profit & loss. This presentation gives certain practical insights onto this topic.
Without loss of generality, we discuss the sensitivities of interest rate derivatives. For example, Delta is for the sensitivity related to zero curve rates and Vega is for the sensitivity with respect to implied volatilities. For some simple derivative product, such as standard IR swaps and IR European swaptions, these sensitivities can be calculated analytically.
However, for those products whose pricing involves an IR term structure model, the calculation of the sensitivities is not a trivial task. It is because not only the pricing may be complicated but also the calibration of the term structure model has to be taken.
There is a naive approach to the sensitivity calculations which includes individually perturbing a market input parameter, re-generating the IR curve and/or re-calibrating the IR term structure model, and then evaluating the product. This works well for vanilla products. For those models combined with calibrations, the above method may not be feasible.
We assume that the value of a given derivative product is a function of a set of input data. Naturally, the sensitivities of the product are defined as the vector of first order partial derivatives of the value with respect to those input data.
Suppose that, to price a given product, we need a set of model. In the model set, there may be a single model or multiple models. Those input data are used to calibrate model parameters.
The Delta of a derivatives is defined as
where · P is the option price defined above, · S is the stock price on the valuation date.
The Gamma is similarly defined as
The Theta is defined as the derivative of the option price with respect to the valuation time:
where time t is expressed in days.
The Rho is defined as the derivative of the option price with respect to the parallel shift of the risk free yield curve (with the spread curve unchanged):
The Vega is defined as the derivative of the option price with respect to a parallel shift to the volatility curve:
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