Author: Ing. Michal Šára and RNDr. Martin Janeček
Abstract
The purpose of this paper is to show one possible approach how to construct a proxy for a term structure of volatilities in currencies (in particular in the Czech crown), where there is a limited interest rate derivatives market (swaption market, cap/poor market, etc.). The usual way how to get a particular term structure of volatility is from the market prices of interest rate derivatives by inverse engineering, i.e. extracting volatilities by equating the market price with the particular Black-Scholes formula for the derivative.This is a very common procedure in countries where such interest rate markets exist. However, in many countries (including the Czech Republic) there are no such markets and yet there is a great need (by banks, investment companies, insurance companies ) to have such interest rate derivatives priced in its own currency.
The approach derived below describes the decomposition of the known (swaption or caplet) volatility into a semi-historical part and an implied part.While thesemi-historical part can be computed for any unknown (swaption or caplet) volatilityfrom a term stucture of interest rate, the implied part cannot be determined without a particular market. Therefore, there is no possible way to arrive at the precise implied volatility for currencies where there are no such markets.The question remains whether we can at least estimate or approximate these volatilities. In this paper we call the resulting quantities pseudo-implied volatilities.
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