Pricing and risk management of interest rate swaps

European Journal of Operational Research 228 (2013) 102–111

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Stochastics and Statistics

Pricing and risk management of interest rate swaps Sovan Mitra a,⇑, Paresh Date b, Rogemar Mamon c, I-Chieh Wang d a

Glasgow Caledonian University London Campus, London E1 6PX, United Kingdom Brunel University, Uxbridge, UB8 3PH, United Kingdom c University of Western Ontario, London, Ontario N6A 5B7, Canada d Meiho University, Pingtung, Taiwan b

a r t i c l e

i n f o

Article history: Received 29 June 2012 Accepted 19 November 2012 Available online 12 December 2012 Keywords: Swaps Risk management Financial mathematics Numerical analysis Stochastic interest rates

a b s t r a c t This paper reformulates the valuation of interest rate swaps, swap leg payments and swap risk measures, all under stochastic interest rates, as a problem of solving a system of linear equations with random perturbations. A sequence of uniform approximations which solves this system is developed and allows for fast and accurate computation. The proposed method provides a computationally efficient alternative to Monte Carlo based valuations and risk measurement of swaps. This is demonstrated by conducting numerical experiments and so our method provides a potentially important real-time application for analysis and calculation in markets.  2012 Elsevier B.V. All rights reserved.

1. Introduction Financial derivatives pose many significant analytical and computational problems; for instance, see Popovic and Goldsman (2012), Albanese et al. (2012), Bhat and Kumar (2012), amongst others. Swaps are one of the most popular derivatives, which began trading in 1981 and are now a hundred billion dollar market (Toyoshima et al., 2011). A swap is a contract in which two counterparties exchange cashflows at prearranged dates, where the cashflow’s value is derived from some underlying, e.g., interest rates, equities, exchange rates or commodities (Hull, 2000). Swaps have grown in popularity due to their ability to hedge out risks relating to the underlying. They also allow one to speculate upon the underlying over long periods of time (years) on a leveraged basis. Interest rate swaps are one of the most important swaps that are traded (Azad et al., 2012). In addition to the risk management (Kim et al., 2012) and speculative uses, they can be used to manage interest rate borrowing costs, convert fixed borrowing costs to floating rate borrowing costs or vice versa. They have also become increasingly important since the global credit crunch to manage other financing costs (Ashton et al., 2012). Consequently, the necessity to accurately value and manage interest rate swaps is of increasing importance (Hsu and Wu, 2011). Swaps have been researched with respect to general swap pricing and related issues. For instance there have been investigations ⇑ Corresponding author. Tel.: +44 203 369 3000. E-mail address: [email protected] (S. Mitra). 0377-2217/$ - see front matter  2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ejor.2012.11.032

into swap pricing under different asset classes, ranging from equity swaps (Dubil, 2012) to currency swaps (Tamakoshi and Hamori, 2013). Exotic swap pricing has also been investigated, such as variance swaps (Rujivan and Zhu, 2012) and swaptions (Bermin, 2012). Some specific swap types have been researched in more depth since the credit crunch began, for example credit default swaps. On credit default swaps there exists literature on their valuation in portfolios (Lin et al., 2011), the effects of sovereign credit risk upon their value (Ismailescu and Kazemi, 2010), their impact on credit spreads themselves (Chiarella et al., 2011) and pricing them using new techniques (Guarin et al., 2011). In addition to general issues, swaps have been investigated in terms of their impact and interaction with particular factors. For instance, in Bhargava et al. (2012), the transmission of volatility and its impact on swaps in different markets is investigated. Chung and Chan (2010) explore the impact of credit spreads and monetary policy on swap spreads whilst Aizenman and Pasricha (2009) investigate the Federal Reserve’s impact on swaps during the global financial crisis. Feldhutter and Lando (2008) analyse swap spreads and swap rates using a six factor model, and Huang et al. (2008) investigate the swap curve dynamics in US and Hong Kong dollar markets. The research in interest rate modelling itself (such as in Schmidt (2011)) has become a well developed area in itself. Interest rate modelling began with simple stochastic differential equations representing interest rate changes and developing relations with respect to fixed income products. The interest rate modelling has been mainly divided into two areas. The first area focuses on models which there are models which are calibrated

S. Mitra et al. / European Journal of Operational Research 228 (2013) 102–111

by observing the term structure of interest rates, e.g. the Vasicek model (Chen and Hu, 2010). The second area encompasses interest rate models that are derived on a no arbitrage argument, such as a replicating portfolio. Examples of such no arbitrage models include the Hull–White model (Hull and White, 2009) and the Black–Derman–Toy model (Weissensteiner, 2010). Within both the no arbitrage and the term structure calibrated models, a major group of interest rate models are the one factor models (models with one source of randomness). The most significant one factor models include the Ho–Lee model (Ho and Lee, 1986) and the Cox–Ingersoll–Ross model (Zhou and Mamon, 2011). Multifactor models have also been developed and examples include the Brace–Gatarek–Musiela model (Suárez-Taboada and Vázquez, 2010) and the Heath–Jarrow–Morton model (Falini, 2010) to name a few. The main areas researched within interest rate swaps include modelling with additional factors such as credit risk. For instance, Yang et al. (2010) price interest rate swaps taking into account bilateral risk of default, the impact of credit and liquidity risk is investigated by Liu et al. (2006) and Yang et al. (2010). Interest rates swaps have also been investigated with respect to macroeconomic factors such as in Azad et al. (2012) and with respect to different stochastic processes (Hsu and Wu, 2011). There has also been research on modelling different characteristics of interest rate swaps, such as the spreads (Fang et al., 2012; Huang and Chen, 2007; Chan et al., 2009) and the analyses of swap rates (Coleman and Karagedikli, 2012). However, there is no research on generic methods for interest rate swap valuation, nor methods offering fast computation of swaps and swap risk measures, either under deterministic or stochastic interest rates. This is despite that interest rates are known to be an important factor in finance to industry and regulators alike (Saha et al., 2009). In this paper we introduce a new method to evaluate the fair price of swaps, the fixed interest rate and floating interest rate swap payments and interest rate swap risk measures. We introduce a new method that applies to a generic range of stochastic interest rate models and provides a method of fast computation of values and risk. We reduce the valuation and risk measurement (and management) problems under stochastic interest rates into one of solving a system of simultaneous linear equations with random coefficients. A method for solution to problems of this type was developed in Date et al. (2007), which is used here for obtaining accurate approximations. It is worth noting that other authors working in the area of swaps have implemented approximate solutions to improve computation time and enable risk management (e.g., Plat and Pelsser (2009)). However this is not related to any work on interest rate swaps or risk measures. Also, computational methods in general are of interest to finance (e.g., Mitra and Date (2010)) propose a new computational method for calibration. Balbas et al. (2010) address the problem of optimising risk functions in finance. Hieber and Scherer (2010) propose an efficient Monte Carlo method of pricing barrier options and Deelstra et al. (2009) offer an approximation method for pricing basket options in short computation time. The issue of valuation and risk management of interest rate swaps under stochastic interest rates is of importance to finance. The ability to incorporate stochastic processes into one’s model is advantageous in finance but complicates modelling and analysis, such as in Fu and Yang (2012) and Bao et al. (2012). It can be empirically observed that interest rates fluctuate unpredictably over time and most interest rate models include a stochastic component. Furthermore, events such as the global credit crunch and the failure of Long Term Capital Management (Basu, 2011; Griffiths et al., 2010) demonstrate the importance of measuring and managing interest rate risks.

103

The plan of this paper is as follows. In Section 2 we introduce notation and formulate the valuation of swaps, fixed and floating leg payments in terms of a linear algebraic equation. In Section 3 we provide an approximate solution to these linear algebraic equations for swaps. In Section 4 we derive approximate solutions to swap risk measures under stochastic interest rates, including applying Generalised Extreme Value Theory. Finally we conduct numerical experiments to demonstrate the computational superiority of our method over the standard Monte Carlo simulation approach. We then end with a conclusion. 2. A linear algebraic formulation of swaps In this section we formulate, under stochastic interest rates, the valuation of the swap, its scheduled fixed and floating rate payments in terms of a system of linear equations. The fixed or floating rate swap cash flow at each prearranged date is also known as the leg of the swap. The solution to the system of such equations will be discussed in Section 3. 2.1. Notation and preliminaries In this paper we denote real vectors by boldface and matrices by capitalised letters. Let  fi denote a generic future cash flow at point in time ti;  pi denote the swap running present value at time ti of future cash flows wj, where j P i; ^i denote the running present value at time ti of future cash p ^ j , where j P i (the swap fixed leg’s running present value flows w at time ti);  ri denote the one period interest rate (or short rate) during the time interval [ti1, ti];  yi denote the present value of generic future cash flows fj where j P i;  v() denote variance;  wi denote the swap future cash flow at time ti; ^ i denote the swap fixed leg payment at time ti (future cash w flow);  VFL denote the present value now (time t0) of all future swap floating leg payments;   denote the notional principal of the interest rate swap. We assume that all our processes are defined under the probability space fX; F ; Pg, where X denotes the sample space, the set of all possible events, F denotes a collection of subsets of X or events and P is the probability measure on F or events. The present value at time ti of a $1 cash flow occurring at time tN > ti is given by

1

PN1 j¼1 ð1 þ r jþ1 Þ

:

ð1Þ

It is known that the absence of arbitrage in a financial market is equivalent to the existence of a risk-neutral (martingale) measure; in this paper we assume the dynamics of ri is governed by a riskneutral probability measure. We assume the short rate ri is of the form

ri ¼ hðr i1 Þ þ zi ;

ð2Þ

where r0 > 0, h(): [0.1) ´ [0.1) is a known and deterministic function and Eðzi Þ ¼ 0, where E denotes the expectation operator. Eq. (2) is sufficiently general so that most single-factor short rate models will reduce to this form after discretisation. It is also assumed that zi is defined on a time varying finite support such that Pðzi 2 ðhðzi1 Þ; 1  hðzi1 ÞÞÞ ¼ 1 holds at each time ti. This

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constraint ensures that the one-period interest rate stays within the interval [0, 1).

Proof. This may easily be proved using (3).

2.2. Valuation of swaps, fixed and floating leg payments In the following sections we show that swap valuation and associated leg payments may be reformulated as a system of linear equations with random coefficients.

^ iþ1 ^iþ1 þ w p ; 1 þ r iþ1

^i1 ¼ p

^ iþ1 w ; 1 þ riþ1

i 2 ½0; N  2;

i ¼ N  1:

ð3Þ

ð4Þ

Lemma 3. The net future interest rate swap cash flows and their present values at each time ti are related by

This relationship may be written in a matrix–vector form as follows. Lemma 1. The future fixed leg swap payments and its present value at each time ti may be shown to be related by

^; ^ ¼ ðQ þ hÞp w

h

2.2.3. Valuation of swap Let us assume we receive fixed leg payments and pay out floating leg payments (although the reverse is also just as easy to value). Using the theory and lemmas given in the previous sections it is possible to express the swap valuation problem in matrix–vector form. As both floating and fixed leg payments must be transferred identically in time then we can value the ^ N to swap by firstly adjusting the future fixed payment from w ^ N þ  . Secondly, we make an initial payment wN where wN ¼ w  at t0. This relationship may be written in a matrix–vector form as follows.

2.2.1. Valuation of fixed leg payments Let wi denote the future fixed leg payments of the swap at the end of period (ti1, ti), i = 1, 2, . . . , N. The discounted present values ^i of the future interest payments may be defined by a recursive p relation

^i ¼ p

where eN is a N sized unit vector ½1 0 0 0> .

ð5Þ

where

½Qij ¼ 1 if i ¼ j;

w ¼ ðQ þ hÞp; where

^ 1 w0 T ; ^ N1    w w :¼ ½wN w p :¼ ½pN1 pN2    p0 T ; ^N þ ; wN ¼ w w0 ¼  : Proof. This may easily be proved using (3) and Lemmas 1 and 2.

h

¼ 1 if i ¼ j þ 1; ¼ 0 otherwise; ½hij ¼ r Niþ1

ð6Þ

if i ¼ j;

¼ 0 otherwise;

The main contributions of Lemmas 1–3 is the recasting of the swap valuations as problems of solving systems of linear equations with random coefficients. Specifically, they require solving equations of the form

T

^ :¼ ½w ^ N1    w ^ 1 ; ^N w w ^ :¼ ½p ^ 0 T ; ^N2    p ^N1 p p where T denotes transpose of a matrix or a vector. Proof. This may easily be proved using (3).

h

2.2.2. Valuation of floating leg payments For the floating leg payments, let VFL represent the present value today (t0) of all floating rate payments made during the life of the swap. Then VFL is given by

~0 ; V FL ¼   p

ð7Þ

where y is the random vector with unknown statistics, Q is defined in (6), f is a known vector and h is a diagonal matrix. We outline a method (first proposed in Date et al. (2007)) of constructing a uniformly convergent approximations to the solution of our system of equations. In this section we use the standard definitions of vector induced matrix norm, denoted by kk and the matrix 2-norm, denoted by kk2:

kyk¼1

 N j¼1 ð1

P

kDk2 ¼ sup kDyk2 ¼

þ rj Þ

Lemma 2. The total swap future floating leg payments and its present value now VFL may be shown to be related by

~0 ; V FL ¼   p

maxjr i j < 1

~0 ; eTN  ¼ ðQ þ hÞeTN p ¼

eTN  ðQ

where kyk is a corresponding vector norm for a vector y. Any function which maps the space of matrices to non-negative real lines and satisfies the axioms of a matrix norm, is denoted by jjjjjj. The following theorem, which summarises the relevant results from Date et al. (2007), provides a uniform approximation of the statistics of vector y. Theorem 1. Assume that

~0 is given by where p

~0 eTN p

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eigðDT DÞ;

kyk2 ¼1

and  is the notional principal. This relationship may be written in a matrix–vector form as follows:

)

f ¼ ðQ þ hÞy;

kDk ¼ sup kDyk;

where

~0 ¼ p

3. Approximate solutions to present values of swap valuation, fixed and floating leg payments

i

1

þ hÞ ;

and fi satisfies the condition

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maxjfi j < c for some c < 1 i

and the inverse of (Q + h) exists with probability 1. Also assume that

PðkQ 1 hk2 < 1Þ ¼ 1: Let us define

y ¼ ðQ þ hÞ1 f

ð8Þ

to the moments of the present value may be found using the moments of the future cash flows and the interest rates. The computation involved in finding the moments of y(M) is simpler than it appears. One should observe that, as pointed out in Date et al. (2007), the matrices Q1h, (Q1h)2 and the vector Q1 f have particularly simple forms as illustrated in the following fact: Fact 1.

and

y

ðMÞ

½Q 1 hij ¼ r Njþ1

M X ¼ ðQ 1 hÞi Q 1 f

if i P j;

¼ 0 otherwise;

i¼0

½ðQ 1 hÞ2 ij ¼

with

i X r Nkþ1 r Njþ1

if i P j;

k¼1

ðQ 1 hÞ0 ¼ I:

¼ 0 otherwise;

Then it is proven that:

½Q 1 fj ¼

kQ 1 hkMþ1 1  kQ 1 hk

kQ 1 fk

It is useful to examine these matrices; for N = 3 we have

0

Q

kQ 1 hk2 2

1

ð9Þ

holds for any vector-induced norm kk, provided kQ1hk < 1 with probability 1. (iv) If rmin, rmax are positive constants such that, with probability 1, ri 2 (rmin, rmax) "i, then

rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! Nþ1 NðN þ 1Þ ; r max rmin 2 2

ð10Þ

holds with probability 1. (v) If rmin, rmax are defined as above, then for any  > 0, there exists a matrix norm jjjjjj, s.t. jjjQ1hjjj 2 (rmin, rmax + ).

r3 B h ¼ @ r3 r3 0

B Q 1 f ¼ @

Remark 2. It is reasonable to assume that we have uniformly bounded cash flows and short rate bounded by 100% in a realistic financial market. Also, if r is a deterministic non-negative scalar bounded by unity, then M X f  ðrÞi f 1þr i¼0

follows from a Taylor expansion centred around r = 0. The expression for y involves a direct inversion of a random matrix (Q + h), however, the expression for y(M) is a multivariate polynomial in ri and fi (involving inverse only of a deterministic matrix Q). The y(M) expression is therefore significantly simpler than the expression for y. Moreover, the moments of y(M) are defined in terms of (joint) moments of ri and fi. Thus a uniform approximation

0

1

r2

C 0 A; r1

f3

1

r2

f3 þ f2

C A:

Using Theorem 1 we can relate the present value of swap fixed leg payments to future fixed leg payments as

! M X 1 i 1 ^¼E ^ : ðQ hÞ Q w p

ð12Þ

i¼0

The present value of net swap cash flows can be related to their future net cash flows by

! M X ðQ 1 hÞi Q 1 w :

ð13Þ

i¼0

h

Remark 1. As pointed out in Date et al. (2007), the rmax is a bound on the interest rate per period which is not annualised and is likely to be a small number. Since NðNþ1Þ 6 N 2 , a sufficient condition for 2 the bound kQ1hk2 < 1 to be satisfied is Nrmax < 1. However, the error bound in (10) is still conservative. As may be seen from its proof, this conservatism stems from the use of trace of a positive semi-definite matrix to bound from above its maximum eigenvalue. Whilst this bound is an equality in the worst case, it is very conservative for well-conditioned matrices.

0

f3 þ f2 þ f1

p¼E Proof. See Date et al. (2007).

ð11Þ

i¼1

(i) y(M) ? y  with probability  1. (ii) limM!1 E kyðMÞ  yk22 ¼ 0. Furthermore the following statements hold: (iii) With probability 1,

kyðMÞ  yk 6

j X fNiþ1 :

The swap floating rate payments VFL can be calculated as follows. Lemma 4. For an interest rates swap of notional principal , the following relation holds for VFL for the second and third order approximations respectively V 2FL and V 3FL :

0 V 2FL

11

0

B CC B ! N N B X CC B X B CC B r i r j CC; ¼  BE r i  EB B CC B i¼1 @ @i; j ¼ 1 AA 0

i6j 1

C B N C B X C B V 3FL ¼ V 2FL   EB ri r j r k C: C B A @i; j; k ¼ 1 i6j6k We therefore have a method of approximating the calculation of interest rate swap values, fixed and floating leg payments.

4. Approximation of swap risk measures under stochastic interest rates In this paper so far we made two contributions. First, it was shown in Section 2 that swap pricing, swap floating leg and fixed

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S. Mitra et al. / European Journal of Operational Research 228 (2013) 102–111

leg payments can be modelled as a system of linear equations with random coefficients. Second, an approximation to the solution of such systems of equations was provided in Section 3 using the results in Date et al. (2007). In this section we make another contribution; we address the problem of measuring the risk of interest rate swaps. Many risk measures require calculation of statistical functions of interest rates, which in turn can be derived from moments of the distribution. Thus the difficulty in obtaining risk of interest rate  measures  swaps is one of finding the moments E Pij¼1 rj of non-trivial stochastic interest rates processes. Using Theorem 1 we show how to obtain the moments of r, which in turn enables us to measure the swap’s risk under stochastic interest rates, for a range of risk measures. Such a method not only provides swap risk management solutions to intractable interest rate models but also provides a computationally faster method to calculating them. We also note that calculating the moments of interest rates is required to determine the values of swap payments, as they are functions of the interest rate moments. 4.1. Risk management of swaps A key problem in swap dealing is their risk management. For example, Procter and Gamble in 1994 incurred losses in the millions arising from two interest rate swaps (Smith, 1997). Since then, significant effort has been devoted to the risk management of interest rate swaps. Risk management requires that one manages an instrument’s risk quantitatively (Dowd, 2011) and international regulatory requirements (e.g., Basel Accord) now make risk management practises mandatory. In fact poor risk management can result in bankruptcies, threaten collapses of an entire financial sector (Ye et al., 2012). The early work on fixed income product risk measurement was concerned with fixed income specific risk measures, for instance duration, convexity and interest rate gap sensitivity. A review of such measures can be found in Hull (2000). However, such measures do not focus on the loss (or gain) incurred and so do not fully quantify the risk involved. Moreover, the literature and theory on risk measures has considerably moved on since then, where it has moved towards focusing on applying generic risk measures which can be applied to any security rather than fixed income products, particularly since the work of Artzner et al. (1997) on variance, Value at Risk, etc. A review of such risk measures that are typically applied in financial mathematics can be found in Szegö (2005). The trend in risk measurement using generic risk measures has been reflected in industry and research articles. For example, financial regulators use Value at Risk on interest rate risk (Ferreira and Lopez, 2005) whilst in Grundke (2005), Value at Risk is used to measure the impact of interest and credit risks on portfolios. One should also note that risk measures that one may typically associate with shares are applicable to fixed income investments. This has been reflected in the industry and academic literature. For instance, Kahn and Rudd (1995) uses the information ratio and CAPM-based measures to evaluate fixed income investments; see also Arnott et al. (2010) for the use of Sharpe ratio, information ratio and variance to evaluate fixed income investments. In Duarte et al. (2007), interest rate swaps are evaluated using Sharpe ratios. Risk measurement is based on random variables. Definition 1. In the probability space fX; F ; Pg a random variable E is a mapping

E : X ! R; such that E is F -measurable.

A random variable can be seen as a function mapping events (the sample space) to real numbers R. We denote G as the set of linear functions for E : X ! R, that is G is the set of random variables for E Szegö (2005). Following Artzner et al. (1999), a risk measure q is a function mapping G to R , that is

q : G ! R: Risk measures nowadays play a significant role in industry, for instance in Balbás et al. (2011) risk measures are used to optimise reinsurance. For interest rate swaps we are concerned with the risk of interest rates. The difficulty in measuring and managing interest rate risk is that most risk measures are typically statistical functions of the random variable, e.g. quantiles. To obtain analytical solutions to the most simplest statistical functions (e.g., expectation) can be non-trivial for stochastic interest rate models. Hence, risk management becomes non-trivial and in such cases we must resort to computational methods to obtain relevant statistical properties. However, Monte Carlo methods are generally computationally expensive. In the proceeding section, using Theorem 1 we will show how to calculate some popular risk measures under stochastic interest rates. Moreover, we show how we can apply Extreme Value Theory to swaps using Theorem 1, which is an important tool for risk management of extreme losses. 4.2. Swap risk measures under stochastic interest rates Given a parametrised model of the short rate rk, it is easy to obtain the moments. For the benefit of conciseness we consider only two stochastic interest rate models (although many exist (Gauthier and Simonato, 2012)) namely: Cox-Ingersoll-Ross model (CIR) and the Hull–White model, and consider up to the first three moments of them:

pffiffiffiffi CIR model : r kþ1 ¼ ab þ ð1  aÞr k þ r r k zkþ1 ;

ð14Þ

Hull—White model : r kþ1 ¼ ab þ ð1  aÞr k þ rzkþ1 :

ð15Þ

In these models zk are zero mean, unit variance, bounded, symmetric and independent random variables. The parameters a, b, r are either constants or are known functions of k. The intuition behind the terms in the equations can be understood as their discrete time equivalents in their respective original and continuous time formulations of the CIR and Hull–White models. For instance, random variable zk represents the Wiener process one would find in the continuous time models, k represents time, a, b and r (also known as volatility) are their equivalents in their respective continuous time formulations (that is constants or known functions of time). In our model we also assume that

Pðr i 2 ½0; 1ÞÞ ¼ 1:

ð16Þ

Condition (16) is crucial for the approximation technique to converge, i.e. for Theorem 1 to hold. Euler-discretisation of classical Hull–White and the CIR models will still give models which look similar to the ones above. However, it is worth remembering the crucial constraint above means a non-Gaussian noise on a finite support. For these models, one may obtain expressions for the first three conditional moments as given in the next lemma. Lemma 5. (i) For the CIR model, i1 X Eðr i Þ ¼ ð1  aÞi r0 þ ab ð1  aÞl ;

ð17Þ

l¼0

Eðr i r j Þ ¼ Eðr i ÞEðrj Þ þ r2

j X ð1  aÞiþj2l Eðrl1 Þ;

and

ð18Þ

l¼1

Eðr i r j r k Þ ¼ Eðr i ÞEðr j ÞEðr k Þ:

ð19Þ

S. Mitra et al. / European Journal of Operational Research 228 (2013) 102–111

addresses the question ‘‘How much can one expect to lose, with a given cumulative probability b, for a given time horizon T?’’. VaR is therefore defined for some function X(t) as (Szegö, 2005)

(ii) For the Hull–White model, i1 X

Eðr i Þ ¼ ð1  aÞi r0 þ ab

ð1  aÞl ;

l¼0

Eðr i r j Þ ¼ Eðr i ÞEðrj Þ þ r2

107

FðXðTÞ 6 VaRÞ ¼ b;

j X

iþj2l

ð1  aÞ

;

and

l¼1

Eðr i r j r k Þ ¼ Eðr i ÞEðr j ÞEðrk Þ: In these cases, i P j P k is assumed without loss of generality.

where F() is the cumulative probability distribution function and b is the cumulative probability associated with threshold value VaR. Typically b is chosen to be 90%, 95% or 99%. The one-sided Chebyshev inequality gives a boundary on quantiles for any distribution if X P 0

FðX  E½X P s Proof. See Date et al. (2007). Remark 3. It is worth noting that it is possible to obtain moments for more complicated models. However such models tend not to have analytic expressions for their moments and so verifying the solutions we obtain from our approximations are non-trivial. The ability to calculate statistical moments enables us to calculate many important risk measures, which we shall now demonstrate with some important examples. For conciseness we give examples of risk measures associated with the CIR model:

pffiffiffiffiffiffiffiffiffiffi v ðXÞÞ 6

1 ; 1 þ s2

where s is a constant. Using the Chebyshev inequality we can determine VaR since

pffiffiffiffiffiffiffiffiffiffi FðX P s v ðXÞ þ E½XÞj 6

1 ; 1 þ s2

therefore

FðX 6 VaRÞ 6 1 

1 ; 1 þ s2

where 4.2.1. Markowitz’s variance risk measure The Markowitz variance risk measure (Leung et al., 2012) is a popular risk measure in industry and research. Additionally variance v() has the property that it is a coherent risk measure (Artzner et al., 1999). The variance v(ri) for ri is given by

v ðri Þ ¼ Eðr2i Þ  E2 ðri Þ;

pffiffiffiffiffiffiffiffiffiffi v ðXÞ þ E½X; VaR : s ¼ pffiffiffiffiffiffiffiffiffiffi v ðXÞ þ E½X VaR ¼ s

Therefore

Fðr i 6 VaRÞ 6 1 

so it can be easily calculated using Theorem 1.

1 ; 1 þ s2

where 4.2.2. Sharpe ratio The Sharpe ratio 1i at time ti can be interpreted as the excess return above the risk free rate per unit of risk (Pätäri et al., 2012). The Sharpe ratio provides a measure of the quality of the return for a given level of risk and using Theorem 1 is expressed as: i

1i ¼

r 0 ðð1  aÞ  1Þ þ ab pffiffiffiffiffiffiffiffiffiffiffi v ðri Þ

Pi1

l¼0 ð1

r 0 ðð1  aÞi  1Þ þ ab #

Pi1

l¼0 ð1

 aÞ

:

 aÞl

;

where # is the CAPM sensitivity of ri (or any asset) to the market. 4.2.4. Information ratio The information ratio vi at time ti can be understood as a tracking error metric and used to gauge the skill of investment  denote our benchmark return then the information managers. Let b ratio is given by Theorem 1 as

vi

v ðri Þ þ E½ri 

:

4.3. Generalised Extreme Value Theory

l

4.2.3. CAPM (Capital Asset Pricing Model) The CAPM model (Clark et al., 2011) is a popular model of risk. It postulates that risk should be rewarded in proportion to the expected return above the riskless rate, for some risky asset. By Theorem 1 CAPM risk ki at time ti is expressed as

ki ¼

VaR

s ¼ pffiffiffiffiffiffiffiffiffiffiffi

Pi1  r 0 ð1  aÞi þ ab l¼0 ð1  aÞl  b pffiffiffiffiffiffiffiffiffiffiffi : ¼ v ðri Þ

4.2.5. Bounds on risk measures: Value at Risk (VaR) Statistical moments also enable us to find bounds on certain risk measures, e.g. VaR. VaR is a popular risk measure and

An important area of risk management and measurement is measuring the risk of extreme losses, in fact most risk management is concerned with extreme loss management (e.g. credit crunch). Since it is generally difficult to obtain any reliable data on extreme losses (due to their rareness) this poses an estimation problem. However if we apply Extreme Value Theory (EVT) we can obtain more accurate estimates. One popular area of EVT is Generalised Extreme Value Theory (GEV). By the Fisher–Tippett Theorem if X is from some unknown distribution then the extreme values of X converges to the GEV distribution (Dowd, 2011):

"  1=n # nðx  mÞ for n – 0; FðxÞ ¼ exp  1 þ h    ðx  mÞ ¼ exp  exp  for n ¼ 0; / where x satisfies

1þ

nðx  mÞ > 0: h

The parameters m, / are the location and scale parameters respectively and n is the tail index. The tail index n gives different distributions depending on its value; the GEV becomes the  Frechet distribution if n – 0.  Gumbel distribution if n = 0.

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To obtain VaR we use the log quantile function

 1=n ðx  mÞ logðFðxÞÞ ¼  1 þ n for n – 0; /   ðx  mÞ for n ¼ 0: ¼  exp / As can be seen from the log quantile equations, to determine VaR we require parameters m and /. Normally these parameters would be obtained by estimating the moments of the distribution (see Diebolt et al. (2008), for example), which is computationally expensive. However, using Theorem 1 we have expressions for the moments and so can calculate GEV values computationally very efficiently. We can apply this method to any n but for simplicity we set n = 0, then

rffiffiffiffiffiffiffiffiffiffiffiffiffi 6v ðr i Þ ; /¼ 2

p

m ¼ Eðr i Þ  /j; where j is Euler’s constant. 5. Numerical experiments In this section we conduct numerical experiments to compare our method against the standard computational method (Monte Carlo simulation) for interest rate swaps. As Date et al. (2007) provides a detailed study on the computational accuracy of the method we have chosen to omit computational values for the benefit of brevity; the reader should refer to Date et al. (2007) for a detailed discussion. However, there has been no analysis on the computation time of our method compared to Monte Carlo simulation so this will be the focus of our experiments. Specifically, we conduct numerical experiments to compare the computational time required to calculate two important risk measures: VaR under EVT and the Sharpe ratio. 5.1. Method In this section we conducted numerical experiments using a 1.61 gigahertz computer, with 992 megabyte RAM, running Matlab version 6.5. We determined the time required to calculate the risk measures VaR (under EVT) and the Sharpe ratio using the CIR interest rate model. To use the CIR model we used the following constant parameters for Eq. (14) taken from the example in (Date et al., 2007):

a ¼ 0:7366;

b ¼ 0:0037;

r ¼ 0:0049; r0 ¼ 0:0041:

In Date et al. (2007), the parameters were estimated using empirical UK interest rate data and therefore the parameter values reflect realistic values that they may take. For each computation method we determined the time required to calculate the risk measures under different time steps g, i.e. i = g. This was to demonstrate the difference in computation time between the two methods as simulation time (g) is increased. The VaR was calculated for a b of 0.95%, using the EVT-VAR equation as discussed in the previous section. To calculate the Sharpe ratio requires the riskless rate. We assumed the riskless rate was the spot rate for today, which is consistent with literature on interest rate theory. To record the time required to calculate the two risk measures under Monte Carlo simulation we required a distribution. The CIR process was simulated for a given period of time steps (g) and this was repeated 10,000 times so that one could obtain an accurate distribution for risk measurement. The approximation method calculated the risk measures using the equations discussed in the

previous sections. Both the computation times of the Monte Carlo simulation and the approximation methods were obtained using Matlab. 5.2. Results We now present the numerical experiment’s results in tabular and graph form. The results refer to the computational time recorded to calculate each risk measure for each computational method. The ‘linear method’ refers to our computational approach (and Monte Carlo to the Monte Carlo simulation based approach). 5.3. Discussion As one can see from Tables 1, 2 and Figs. 1–4, both computation methods result in monotonically increasing computation times as g increases. This is expected for both risk measures as increasing g must always increase the number of computational operations and so the computation time. However the absolute values and the rate at which either method increases computation time significantly differs. Firstly, in absolute times we can see that our method is significantly faster than the Monte Carlo method. The Monte Carlo times are of the order of 3–550 seconds, however the linear method is of the order of 0.1–0.2 seconds and this is a difference of a factor of 10–1000 multiple increase. The key reason behind this is the difference in computational procedure between the two methods. Our method involves computing a series of linear sums, which can be computationally executed in a short time. In fact linear operations in general are computationally inexpensive. The Monte Carlo based method is computationally more expensive. It involves taking random samples from a Wiener process to simulate the sample path for g time steps, hence as g increases sampling also increases. Secondly, the Monte Carlo method must take a set of sample paths (in our case 10,000) to generate a sufficiently accurate distribution from which to obtain accurate risk measurements. If we reduce the number of sample paths then we reduce our accuracy. Hence an increase in g leads to a significant increase in computation time. Secondly, the rate at which the computation time of the either method increases with g differs significantly. One can observe that our method’s computation time hardly increases with g, in fact it remains fairly constant. This is because our method consists of a linear sum and so is computationally quicker to calculate. The increase in computation by increasing g is not significant. Table 1 Sharpe ratio computation times (seconds). Time steps (g)

Monte Carlo

Linear method

5 10 20 40 60 80 100 150 200 250 300 350 400 450 500 550 600 650 700

3.9 7.7 15.2 30.3 45.4 60.4 75.5 113.5 152.1 191.7 230.6 265.0 302.6 340.0 380.0 416.6 455.0 495.4 531.1

0.14 0.14 0.14 0.14 0.17 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19

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For the Monte Carlo method, we observe a linear relation between g and computation time. This is undesirable because large g computations will be infeasible and long term forecasts are important calculations in risk management. We expect a linear relation between computation time and g because the Monte Carlo method consists of taking many samples paths to generate an accurate distribution. The time required to generate each sample will monotonically increase with g. The rate of increase in computation time is generally an important characteristic of computational processes. This is because the rate enables us to determine whether large computations (or in our case g) can be achieved in feasible times. Additionally, for financial applications the computation time is even more important because long times would not be useful to real time market applications. This is because parameter values, market conditions, etc. can change very quickly (in the order of seconds to minutes). This implies the Monte Carlo method would not be suitable for real time market applications and so our method provides a viable alternative.

Table 2 EVT VaR computation times (seconds). Time steps (g)

Monte Carlo

Linear method

5 10 20 40 60 80 100 150 200 250 300 350 400 450 500 550 600 650 700

3.9 7.7 15.2 30.2 45.2 60.3 75.3 113.0 150.6 188.5 225.7 263.7 300.8 339.3 377.0 413.4 451.5 489.0 530.0

0.11 0.11 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.14

Sharpe Ratio Computation Time Under the Linear Method 0.19 0.185 0.18

Time (sec)

0.175 0.17 0.165 0.16 0.155 0.15 0.145 0.14

0

100

200

300

400

500

600

700

Time Steps Fig. 1. Sharpe ratio computation time under the linear method.

Sharpe Ratio Computation Time under Monte Carlo

600

500

Time (sec)

400

300

200

100

0 0

100

200

300

400

500

Time Steps Fig. 2. Sharpe ratio computation time under Monte Carlo.

600

700

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VaR Computation Time Under Linear Method 0.15

0.145

0.14

Time (sec)

0.135

0.13

0.125

0.12

0.115

0.11

0

100

200

300

400

500

600

700

Time Steps Fig. 3. VaR computation time under linear method.

VaR Computation Time Under Monte Carlo 600

500

Time (sec)

400

300

200

100

0 0

100

200

300

400

500

600

700

Time Steps Fig. 4. VaR computation time under Monte Carlo.

6. Conclusion In this paper, we have formulated a new way of valuing swaps, fixed and floating rate swap payments and swap risk measures under stochastic interest rates. We formulated them as a problem of solving a system of linear equations with random perturbations. We solved them using a sequence of uniform approximations, which allowed fast computation of results. We conducted numerical experiments which showed that our approach provides significant savings in computation time compared to Monte Carlo simulation methods. Our paper offers potential avenues for exploring accelerations of Monte Carlo techniques for swap valuation and risk measurement. This may be achieved by combining the methods with variance reduction and importance sampling techniques for Monte Carlo simulations. The linear formulation of equations may offer significant potential benefits for computational optimisation of portfolios, whereby powerful optimisation techniques can be applied from stochastic linear programming methods. Furthermore,

we believe our method can be adapted to investigate exotic derivatives, which pose many non-trivial analytical and computational challenges. We believe our paper offers computational advantages that will be of significant benefit to academic research as well as industry, where it is important to calculate swap and risk measures in short time periods. This requirement is becoming increasingly important with the growing trend of computerised and high frequency trading in industry. References Aizenman, J., Pasricha, G., 2009. Selective swap arrangements and the global financial crisis: Analysis and interpretation. International Review of Economics and Finance 19, 353–365. Albanese, C., Lo, H., Tompaidis, S., 2012. A numerical algorithm for pricing electricity derivatives for jump-diffusion processes based on continuous time lattices. European Journal of Operational Research 222, 361–368. Arnott, R., Hsu, J., Li, F., Shepherd, S., 2010. Valuation-indifferent weighting for bonds. Journal of Portfolio Management 36, 117–130. Artzner, P., Delbaen, F., Eber, J., Heath, D., 1997. Thinking coherently. Risk 10, 68–71.

S. Mitra et al. / European Journal of Operational Research 228 (2013) 102–111 Artzner, P., Delbaen, F., Eber, J., Heath, D., 1999. Coherent measures of risk. Mathematical Finance 9, 203–228. Ashton, P., Doussard, M., Weber, R., 2012. The financial engineering of infrastructure privatization. Journal of the American Planning Association 78, 300–312. Azad, A., Fang, V., Hung, C., 2012. Linking the interest rate swap markets to the macroeconomic risk: the UK and US evidence. International Review of Financial Analysis 22, 38–47. Balbas, A., Balbás, B., Balbás, R., 2010. Minimizing measures of risk by saddle point conditions. Journal of Computational and Applied Mathematics 234, 2924– 2931. Balbás, A., Balbás, B., Heras, A., 2011. Stable solutions for optimal reinsurance problems involving risk measures. European Journal of Operational Research 214, 796–804. Bao, Q., Li, S., Gong, D., 2012. Pricing VXX option with default risk and positive volatility skew. European Journal of Operational Research 223, 246–255. Basu, S., 2011. Comparing simulation models for market risk stress testing. European Journal of Operational Research 213, 329–339. Bermin, H., 2012. Bonds and options in exponentially affine bond models. Applied Mathematical Finance 19, 513–534. Bhargava, V., Malhotra, D., Russel, P., Singh, R., 2012. An empirical examination of volatility spillover between the indian and us swap markets. International Journal of Emerging Markets 7, 289–304. Bhat, H., Kumar, N., 2012. Option pricing under a normal mixture distribution derived from the Markov tree model. European Journal of Operational Research 223, 762–774. Chan, W., Wong, C., Chung, A., 2009. Modelling Australian interest rate swap spreads by mixture autoregressive conditional heteroscedastic processes. Mathematics and Computers in Simulation 79, 2779–2786. Chen, H., Hu, C., 2010. A relaxed cutting plane algorithm for solving the Vasicektype forward interest rate model. European Journal of Operational Research 204, 343–354. Chiarella, C., Fanelli, V., Musti, S., 2011. Modelling the evolution of credit spreads using the Cox process within the HJM framework: a CDS option pricing model. European Journal of Operational Research 208, 95–108. Chung, H., Chan, W., 2010. Impact of credit spreads, monetary policy and convergence trading on swap spreads. International Review of Financial Analysis 19, 118–126. Clark, E., Jokung, O., Kassimatis, K., 2011. Making inefficient market indices efficient. European Journal of Operational Research 209, 83–93. Coleman, A., Karagedikli, Ö., 2012. The relative size of exchange rate and interest rate responses to news: an empirical investigation. The North American Journal of Economics and Finance 23, 1–19. Date, P., Mamon, R., Wang, I., 2007. Valuation of cash flows under random rates of interest: a linear algebraic approach. Insurance: Mathematics and Economics 41, 84–95. Deelstra, G., Petkovic, A., Vanmaele, M., 2009. Pricing and hedging Asian basket spread options. Journal of Computational and Applied Mathematics 233, 2814– 2830. Diebolt, J., Guillou, A., Naveau, P., Ribereau, P., 2008. Improving probabilityweighted moment methods for the generalized extreme value distribution. REVSTAT – Statistical Journal 6, 33–50. Dowd, K., 2011. An Introduction to Market Risk Measurement. Wiley Finance. Duarte, J., Longstaff, F., Yu, F., 2007. Risk and return in fixed-income arbitrage: Nickels in front of a steamroller? Review of Financial Studies 20, 769–811. Dubil, R., 2012. Swap markets. Financial Engineering and Arbitrage in the Financial Markets, 95–117. Falini, J., 2010. Pricing caps with HJM models: the benefits of humped volatility. European Journal of Operational Research 207, 1358–1367. Fang, V., Azad, A., Batten, J., Lin, C., 2012. Business Cycles and the Impact of Macroeconomic Surprises on Interest Rate Swap Spreads: Australian Evidence. Emerald Group Publishing Limited. Feldhutter, P., Lando, D., 2008. Decomposing swap spreads. Journal of Financial Economics 88, 375–405. Ferreira, M., Lopez, J., 2005. Evaluating interest rate covariance models within a value-at-risk framework. Journal of Financial Econometrics 3, 126–168. Fu, J., Yang, H., 2012. Equilibruim approach of asset pricing under lévy process. European Journal of Operational Research 223, 701–708. Gauthier, G., Simonato, J., 2012. Linearized Nelson–Siegel and Svensson models for the estimation of spot interest rates. European Journal of Operational Research 219, 442–451. Griffiths, M., Lindley, J., Winters, D., 2010. Market-making costs in treasury bills: a benchmark for the cost of liquidity. Journal of Banking and Finance 34, 2146– 2157. Grundke, P., 2005. Risk measurement with integrated market and credit portfolio models. Journal of Risk, 763–794.

111

Guarin, A., Liu, X., Ng, W., 2011. Enhancing credit default swap valuation with meshfree methods. European Journal of Operational Research 214, 805–813. Hieber, P., Scherer, M., 2010. Efficiently pricing barrier options in a Markovswitching framework. Journal of Computational and Applied Mathematics 235, 679–685. Ho, T., Lee, S., 1986. Term structure movements and pricing interest rate contingent claims. Journal of Finance 41, 1011–1029. Hsu, C., Wu, T., 2011. Pricing Asian-style interest rate swaps within a multi-factor Gaussian HJM framework. International Journal of Information and Management Sciences 22, 357–375. Huang, Y., Chen, C., 2007. The effect of Fed monetary policy regimes on the US interest rate swap spreads. Review of Financial Economics 16, 375–399. Huang, Y., Neftci, S., Guo, F., 2008. Swap curve dynamics across markets: case of US dollar versus HK dollar. Journal of International Financial Markets, Institutions and Money 18, 79–93. Hull, J., 2000. Options, Futures and Other Derivatives. Prentice Hall, New Jersey. Hull, J., White, A., 2009. One-factor interest-rate models and the valuation of interest-rate derivative securities. Journal of Financial and Quantitative Analysis 28, 235–254. Ismailescu, I., Kazemi, H., 2010. The reaction of emerging market credit default swap spreads to sovereign credit rating changes. Journal of Banking and Finance 34, 2861–2873. Kahn, R., Rudd, A., 1995. Does historical performance predict future performance? Financial Analysts Journal 51, 43–52. Kim, S., Anderson, S., Zitzler, M., 2012. Hedging instrument in post liquidity crisis: a case of interest rate swaps. Managerial Finance 39, 3. Leung, P., Ng, H., Wong, W., 2012. An improved estimation to make Markowitzs portfolio optimization theory users friendly and estimation accurate with application on the us stock market investment. European Journal of Operational Research 222, 85–95. Lin, J., Liang, G., SEN, W., Zheng, H., 2011. The valuation of the basket CDS in a primary–subsidiary model. Asia-Pacific Journal of Operational Research 28, 213–238. Liu, J., Longstaff, F., Mandell, R., 2006. The market price of risk in interest rate swaps: the roles of default and liquidity risks*. The Journal of Business 79, 2337–2359. Mitra, S., Date, P., 2010. Regime switching volatility calibration by the Baum–Welch method. Journal of Computational and Applied Mathematics 234, 2367–2616. Pätäri, E., Leivo, T., Honkapuro, S., 2012. Enhancement of equity portfolio performance using data envelopment analysis. European Journal of Operational Research 220, 786–797. Plat, R., Pelsser, A., 2009. Analytical approximations for prices of swap rate dependent embedded options in insurance products. Insurance: Mathematics and Economics 44, 124–134. Popovic, R., Goldsman, D., 2012. On valuing and hedging European options when volatility is estimated directly. European Journal of Operational Research 218, 124–131. Rujivan, S., Zhu, S., 2012. A simplified analytical approach for pricing discretelysampled variance swaps with stochastic volatility. Applied Mathematics Letters 25, 1644–1650. Saha, A., Subramanian, V., Basu, S., Mishra, A., 2009. Networth exposure to interest rate risk: an empirical analysis of Indian commercial banks. European Journal of Operational Research 193, 581–590. Schmidt, W., 2011. Interest rate term structure modelling. European Journal of Operational Research 214, 1–14. Smith, D., 1997. Aggressive corporate finance. The Journal of Derivatives 4, 67–79. Suárez-Taboada, M., Vázquez, C., 2010. A numerical method for pricing spread options on LIBOR rates with a PDE model. Mathematical and Computer Modelling 52, 1074–1080. Szegö, G., 2005. Measures of risk. European Journal of Operational Research 163, 5– 19. Tamakoshi, G., Hamori, S., 2013. Dynamic linkages among cross-currency swap markets under stress. Applied Economics Letters 20, 404–409. Toyoshima, Y., Tamakoshi, G., Hamori, S., 2011. Asymmetric dynamics in correlations of treasury and swap markets: evidence from the us market. Journal of International Financial Markets, Institutions and Money 22, 381–394. Weissensteiner, A., 2010. Using the Black–Derman–Toy interest rate model for portfolio optimization. European Journal of Operational Research 202, 175–181. Yang, X., Yu, J., Li, S., Cristoforo, A., Yang, X., 2010. Pricing model of interest rate swap with a bilateral default risk. Journal of Computational and Applied Mathematics 234, 512–517. Ye, W., Liu, X., Miao, B., 2012. Measuring the subprime crisis contagion: evidence of change point analysis of copula functions. European Journal of Operational Research 222, 96–103. Zhou, N., Mamon, R., 2011. An accessible implementation of interest rate models with Markov-switching. Expert Systems with Applications 39, 4679–4689.