# Interest Rate & Fx Models

7. Risk Management

Andrew Lesniewski

Courant Institute of Mathematical Sciences

New York University

New York

March 8, 2012

Interest Rates & FX Models

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Contents

1 Introduction

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2 Delta risk management

2.1 Input perturbation sensitivities . . . . . . . . . . . . . . . . . . . 2.2 Regression based sensitivities . . . . . . . . . . . . . . . . . . . .

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3 Gamma risk management

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4 Vega risk management

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5 Risk management under SABR

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1 Introduction

One of the most important tasks faced by a portfolio manager, a trader, or a risk manager is to manage the interest rate exposure of a portfolio of ﬁxed income securities such as government bonds, corporate bonds, mortgage backed securities, structured interest rate products, etc. This interest rate risk may manifest itself in various ways: (i) risk to the level of rates, (ii) risk to the convexity of instruments, and (iii) risk to the volatility of rates.

Traditional risk measures of options are the greeks: delta, gamma, vega, theta, etc.1 , see for example [3]. Recall, for example, that the delta of an option is the derivative of the premium with respect the underlying. This poses a bit of a problem in the world of interest rate derivatives, as the interest rates play a dual role in the option valuation formulas: (a) as the underlyings, and (b) as the discount rates. One has thus to differentiate both the underlying and the discount factor when calculating the delta of a swaption!

The key issue is to quantify this exposure and, if required, offset aspects it by taking positions in liquid vanilla instruments such as Eurodollar futures, swaps, swaptions, caps/ﬂoors, etc.

In addition to the various facets of interest rate risk, ﬁxed income portfolios carry other kinds of risk. Government bonds carry foreign exchange risk and sovereign credit risk, corporate bonds are exposed to credit and liquidity risk, and mortgage backed securities have prepayment and credit risk. CAT bonds carry 1

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Risk Management

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risks to natural catastrophes such as earth quakes and hurricanes. These other types of risk are the deﬁning characteristics of the relevant instruments and are, in fact, their raison d’ˆ tre. Discussion of these risks is beyond the scope of this e

course, and we refer the reader to specialized literature.

There are other widely used risk metrics such as the value at risk (VaR), which we do not discuss in this course. The focus by VaR is less myopic than the sensitivity measure given by the greeks; instead it is a measure of the global tail risk. Under a given level of risk tolerance, what is the maximum loss that a portfolio can sustain? VaR is a tool used by banks to set aside their economic capital, clearing houses to set the margin levels, and hedge funds to set the risk capital to their traders.

2 Delta risk management

We begin with the dominant portion of the interest rate risk, namely the delta risk. Traditionally, this risk has been designated to as the duration risk. We let Π denote this portfolio, whose detailed composition is not important for our discussion. We will discuss two commonly used approaches to measure the interest rate risk of Π. Two methods of computing the delta are commonly used in the industry.

2.1 Input perturbation sensitivities

In this approach we compute the sensitivities of the portfolio to the benchmark instruments used in the curve construction, and replicate the risk of the portfolio by means of a portfolio consisting of the suitably weighted benchmark instruments. (a) Compute the partial DVO1s of the portfolio Π to each of the benchmark instruments Bi : We shift each of the benchmark rates down 1 bp and calculate the corresponding changes δi Π in the present value of the portfolio. (b) Compute the DVO1s δi Bi of the present values of the benchmark instruments under these shifts. (c) The hedge ratios ∆i of the...

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