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Convexity
Convexity ins and outs
Most interest-rate instruments have convexity, which produces pricing bias when it is not taken into account. This is particularly obvious with CMS/CMT and Swap-in-arrears.
We revisit convexity, describe determinants and especially volatility, which prevails, and explain how.
Acumen calculates the convexity bias using Taylor series expansion or Hull & White model.
Convexity bias in Futures
This document is building on the difference between future and forward rates to exhibit convexity, and explains how to adjust implied future rates before they can be incorporated in a yield curve, for pricing purpose.
We use a case study as a support to the explanation, which consists in stripping a FRA portfolio into future contracts. This outlines the systematic advantage of hedging sensitivity, and is an intuitive approach to the positive convexity bias induced by a positive correlation between term and tenor rates. Since this correlation effect is quantified by Hull and White, we use this model to quantify the convexity bias deductible from implied future rates.
Time adjustment
A rate is usually fixed at beginning and paid at the end of the period.
When it is not, that is fixed at beginning and paid at beginning, it does not benefit from the convexity advantage.
That is why a positive convexity adjustment is added in case the payment is done at the beginning of the period without discounting effect.
Quantos
A Quanto Derivative is jointly approached as :
- A contract where the underlying is observed in one currency and the payoff is made in other currency;
- A contract where the currency risk is removed from the underlying foreign forward index rate;
- A foreign forward contract converted into domestic currency at a fixed exchange rate.
This work provides an insight into two models to price floating legs indexed on a quanto index: the ‘Constant Covariance’ and the ‘Modified Wei’ model. Both approaches are making use of the correlation between the forex level and the domestic / foreign interest rates.
That is why it is necessary to refer to the volatility and correlation levels published by RiskMetrics. Volatility measures can be fine-tuned by unrolling Hull and White term structure model.
The two approaches are also widely using the relation between forex and interest rate markets, exhibited into forward foreign exchange rates, in order to choose proxies in the most efficient way, when variables cannot be modeled directly.