Interest rates after the credit crunch crisis: single

versus multiple curve approach

Oleksandr Dmytriiev

Yining Geng

Cem Sinan Ozturk

Abstract

For interest rate derivative pricing, 2007 crisis was a turning point. Prior to

the crisis, market interest rates showed consistencies that allowed the use of a

single curve for both forwarding and discounting. After the crisis, the inconsis-

tencies in the market interest rates led to development of a new method of the

pricing interest rate derivatives, which is called Multi-Curve Framework. We

studied the influence of the multi-curve approach on the interest rate deriva-

tive pricing. We calculated and compared the price of a simple swap in both

multi-curve and single curve approaches. We suggested the generalization of the

lattice approach, which is usually used to approximate the short interest rate

models, for milti-curve framework. This is a novel result, which have not been

developed in the scientific literature. As an example, we showed how to use the

Black-Derman-Toy interest rate model on binomial lattice in multi-curve frame-

work and calculated the price of the 2-8 period swaption in a single (LIBOR)

curve and two-curve (OIS+LIBOR) approaches. This technique can be used for

pricing any interest rate instrument.

1 Introduction

The effects of the financial crisis in the private and public sectors spreaded out in the

worldwide economy through the financial markets. Inevitably, companies and banks

1

have struggled to meet the liquidity and credit requirement that are highly related to

their ability of trading financial products. The proper pricing the financial products

includes taking into account liquidity and credit risk premia to reflect their real values.

Other decisions based on incorrect assumptions can have very bad consequences.

Before the financial crisis in 2007, the pricing of interest rate derivatives was a clear

case with a framework that researchers agreed on. The current crisis has affected the

interest rate derivative valuation as it did almost all the patterns in the finance world.

Before the crisis, there was used the idea of the construction of a single risk free yield

curve, reflecting at the present cost of the future cash flows (discounting yield curve) as

well as the level of forward rates (forward yield curve), see Brigo and Mercurio (2006),

Hull (2001). For example, LIBOR or EURIBOR rates were used for both discounting

of the future cash flows and forwarding. This approach is known as a single curve

framework.

Since August 2007 the primary interest rates of the interbank market, such as

LIBOR, EURIBOR, EONIA, and Federal Funds rate display large basis spreads that

have raised up to 200 basis points. Similar divergences are also found among swap

rates with different floating leg tenors (the swap coupon frequency), see Figure 1. The

financial community has thus been forced to start the development of a new theoretical

framework, including a larger set of relevant risk factors into the models used for

derivatives’ pricing and risk analysis. A single yield curve is no longer adequate to

derive discount factors and forward rates as there is now a need for multiple separated

curves in order to account for the different credit/liquidity risks and different dynamics

of instruments of different maturities, see Mercurio (2008) and Bianchetti and Carlicchi

(2011). One of the ideas is using different yield curves for discounting and forwarding.

For example, LIBOR or EURIBOR can be used for forwarding but these rates are

not considered any longer as a proxy for the risk free rate. The best current proxy

of the risk free rate is considered to be Overnight Indexed Swap (OIS) Rate, which is

EONIA in the Euro-zone. This is basically a brief idea of the multi-curve framework.

Multi-curve approaches for pricing derivatives, especially interest rate derivatives are

currently in the process of being implemented by banks in order to adapt to changes

in market practice.

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Figure 1: Money market rates before and after the financial crisis

As a simple example, we calculated and compared the fixed rate of an interest rate

swap in a single (LIBOR) curve and two-curve (OIS+LIBOR) approaches using the

data taken from the Bank of England.

The interest rate derivatives market is the largest derivatives market in the world.

The Bank for International Settlements estimates that the notional amount outstand-

ing in June 2012 (the Semiannual OTC derivative statistics of the Bank for Interna-

tional Settlements) were US$494 trillion for OTC interest rate contracts, and US$342

trillion for OTC interest rate swaps. According to the International Swaps and Deriva-

tives Association, 80% of the world’s top 500 companies as of April 2003 used interest

rate derivatives to control their cashflows. This compares with 75% for foreign ex-

change options, 25% for commodity options and 10% for stock options. The most

popular vanilla interest rate derivatives are interest rate swaps (fixed-for-floating),

cross currency swaps, interest rate caps, swaptions (options on swaps), bond options,

forward rate agreements, interest rate futures.

There have been developed many different theoretical models to model the evolu-

tion of the interest rate and correspondingly the behavior of the interest rate deriva-

tives. One of the frameworks to describe the future evolution of the interest rates is

describing the future evolution of the short rate (instantaneous spot rate). The other

major framework for interest rate modeling is the Heath–Jarrow–Morton framework

(HJM). Within this framework, the evolution of the instantaneous forward rate is

considered. Unlike the short rate models described above, HJM class of models is

3

generally non-Markovian.

Short rates models can be divided into two groups. One factor short rate models

is one of the groups. In this group, a single stochastic factor – the short rate –

determines the future evolution of all interest rates. The other group of short rate

models is constituted by the multi-factor models, where other stochastic factors are

present along with the short rate. We will consider only one factor short rate models

in this thesis.

The short rate (instantaneous spot rate) is to some extent a mathematical abstrac-

tion, as this is a spot rate between two infinitesimally close time periods. The discrete

approximation of these models on multi-dimensional lattice (tree) is usually used to

apply these models to the market. Despite the extensive research of multi-curve ap-

proach of pricing interest rate derivatives, the generalization of the discrete lattice

models has not been done yet. In this thesis, we show how multi-curve approach

works for the lattice models. As an example we considered muli-curve framework for

the Black-Derman-Toy interest rate model on binomial lattice and calculated the price

of the 2-8 period swaption in a single (LIBOR) curve and two-curve (OIS+LIBOR)

approaches using 5 years historical data taken from the Bank of England. Our gen-

eralization can be used for any type of interest rate derivative pricing which has not

been done before in the literature.

2 Literature Review

In this section, we will briefly discuss the main developments in the theory of pricing

of the interest rate derivatives. We will start from the most used ones, which are

swaps. For the pre-crisis era, the pricing of interest rate swaps was solid and agreed

phenomenon, the literature on this issue solely focused on using the single curve for

the both legs (fixed and floating), and bootstrapping is used for the yield curve con-

struction, see Ron (2000), Boenkost and Schmidt (2004) and Hull (2009). The main

issue has come up with the crises, the increasing spread between different yield curves

pushed the academia to look for a new method to make the pricing of the interest rate

derivatives more reliable and precise. First ideas for the multi curve framework was

the discussions on the affects of changing the discounting curve by Henrard (2007).

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Later on, Henrard (2010) proposes a coherent valuation framework for derivatives

based on different LIBOR tenors still using the traditional bootstrapping technique,

though assuming the discounting curve as given. Amedextran and Bianchetti (2009)

poses a bootstrap market segmentation and swap rates within each tenor separately,

what means that would makes the model subject to arbitrage. An extended version

of this model proposed in Bianchetti (2010), which uses the analogy of currency in

order to avoid arbitrage opportunities. Similar approaches have recently been used

by Chibane and Sheldon (2009) and Kijima, Tanaka and Wong (2009). Mercurio

(2008) constructs a LIBOR market model using the the joint evolution of FRA rates

and implied forward rates. However, this paper lacks a discussion of a multi-currency

situation . Johannes and Sundaresan (2007) and Whittall ( 2010b ) extends the multi-

curve framework by taking the effect of collateralization into account. The works of

Fujii, Shimada and Takahashi (2009a), Fujii, Shimada, and Takahashi (2009b), and

Scavenius, Linderstrom (2010) provide a new framework to build a coherent term

structure in the presence of differential base and provides a multi-currency environ-

ment. Instead of building a yield curve by bootstrapping from different instruments

of the liquid market, the forward rates are restored to incorporate the effect of the

differential base.

We provide more details about two relevant for us papers. ”Interest Rates and

The Credit Crunch: New Formulas and Market Models” by Mercurio (2008) intro-

duces a new LIBOR market model, which is based on modeling the joint evolution

of FRA rates and forward rates belonging to the discount curve. They started by

analyzing the basic log-normal case and then added stochastic volatility. The dy-

namics of FRA rates under different measures is obtained, closed form formulas for

caplets and swaptions derived in the log-normal and Heston (1993) cases. ”Interest

Rates After The Credit Crunch: Multiple-Curve Vanilla Derivatives and SABR” by

Bianchetti and Carlicchi(2011) focuses on the fixed income market and analyzes the

most relevant empirical evidences regarding the divergence between LIBOR and OIS

rates, the explosion of Basis Swaps spreads, and the diffusion of collateral agreements

and CSA-discounting in terms of credit and liquidity effects.

Term-structure models are essential for the valuation of interest rate dependent

5

claims. Although term-structure experts have produced a variety of useful models,

they involve complex mathematics, which limits their accessibility to investment prac-

titioners who are not engaged in this area of specialization. ”Term-Structure Models

Using Binomial Trees” by Sochacki (2001) and ”Term Structure Lattice Models” by

Haugh (2010) developed general overview for those who need guidelines for implemen-

tation purposes.

The short term interest rate model must incorporate empirical and intuitive char-

acteristics of interest rates. The basic models, which attempt to do so, are the

Ho–Lee (HL; Ho and Lee 1986), Hull–White (HW; Hull and White 1994), Kalotay–

Williams–Fabozzi (KWF; Kalotay, Williams, and Fabozzi 1993), Black–Karasinski

(BK; Black and Karasinski 1991), and Black–Derman–Toy (BDT; Black, Derman

and Toy 1990). Historically, the HL model was introduced first, then the BDT

model, then the BK model, then the KWF model, and then the HW model. The

Ho–Lee Model was the first no-arbitrage term-structure model. It assumes con-

stant and identical volatility for all spot and forward rates and does not incorpo-

rate mean reversion. The Hull–White Model extends the Ho–Lee model to allow

for mean reversion. The Kalotay–Williams–Fabozzi Model assumes a lognormal dis-

tribution and eliminates the problem of negative short rates, which can occur with

the Ho–Lee and Hull–White models. The Black–Karasinski Model is an extension of

the Kalotay–Williams– Fabozzi Model. This model controls the growth in the short

rate. The Black–Derman–Toy Model is the model which permits independent and

time-varying spot-rate volatilities.

3 Theoretical discussion and hypotheses

For the full picture, we start this section with the brief discussion of the different

interest rates, such as spot rate, instantaneous spot rate, forward rate, instantaneous

forward rate and also difference between their simply and continuously compounding.

We will denote the price of zero coupon bond with maturity at T by P (t, T ). The

simply-compounded spot interest rate is defined by the following expression:

L(t, T ) =1− P (t, T )

τ(t, T )P (t, T ), (1)

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where τ(t, T ) is the year fraction between time t and T . A simply compounded

investment of one currency unit at time t will grow to the amount (1 + L(t,T )m

)m in m

periods of time during a year. As m→∞, this quantity converges. This way we can

receive continuously compounded spot rate Lc(t, T ) = − lnP (t,T )τ(t,T )

. Instantaneous sport

rate (or short rate) is defined as

r(t) = limT→t+

Lc(t, T ) (or limT→t+

L(t, T )) (2)

The instantaneous sport rate is a spot rate between two infinitesimally small time

periods, that is why this continuously compounded rate is used in the stochastic

differential equations (SDE) of short interest rate models. For t < T < S, the simply-

compounded forward interest rate with maturity at S and expiry at T is

F (t;T, S) =1

τ(T, S)

(P (t, T )

P (t;T )− 1

). (3)

Continuously compounded forward rate can be defined using the same considerations

as the spot rate. We have Fc(t;T, S) = − lnP (t,S)−lnP (t,T )τ(T,S)

. Instantaneous forward

interest rate, which is the forward rate between two infinitesimally small time periods

in the future, is defined as

f(t, T ) = limS→T+

Fc(t;T, S) = −∂ lnP (t;T )

∂T(or lim

S→T+F (t;T, S)). (4)

This continuously compounded rate is used in SDEs of HJM framework.

Now, we are ready to discuss the pricing of the interest rate derivatives. We will

start from the swaps. Interest rate swap (IRS) is a contract in which two counter-

parties agree to exchange interest payments of different character based on an under-

lying notional principle amount that is not exchanged. There are several types of the

swaps, which are coupon swaps (exchange of fixed rate for floating rate instruments in

the same currency), basis swaps (exchange of floating rate for floating rate instruments

in the same currency) and cross currency interest rate swaps (exchange of fixed rate

instruments in one currency for floating rate in another). We will consider coupon

swaps only. For coupon swaps, the cash flows are made up of two legs. The fixed leg

(fixed rate interest rate payments) is the stream of cash flows made by a swap buyer;

the floating leg (floating rate interest rate payments) is the stream of cash flows made

by a swap seller. There is no any upfront payment (price) to enter a swap agreement.

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This way, the fixed rate of the swap can be found using no-arbitrage consideration and

a condition that the present value of the floating leg should equal the present value of

the the fixed leg. Present value (PV) of interest payments on the fixed leg is

PVfixed = Nm∑j=1

τ(TKj−1, T

Kj

)·K · P

(0, TKj

),

PV of the interest payments on the floating leg is

PVfloating = Nn∑j=1

τ(T Fj−1, T

Fj

)· F(T Fj−1, T

Fj

)· P(0, T Fj

),

where N is a notional principle; K is a fixed interest rate; F (Tj−1, Tj) = F (0;Tj−1, Tj)

is a forward interest rate defined by Eq. (3); the swap fixed leg pays at each time

{TK1 , ..., TKm } and the swap floating leg pays at each time {T F1 , ..., T Fn }. Both fixed

and floating legs terminate simultaneously, so TKm = T Fn . We immediately receive the

non-arbitrage expression of the fixed rate:

K =

∑nj=1 τ

(T Fj−1, T

Fj

)· F(T Fj−1, T

Fj

)· P(0, T Fj

)∑mj=1 τ

(TKj−1, T

Kj

)· P(0, TKj

) (5)

Which forward curve and discount curve should we use? In multi-curve framework,

generally, we should use different yield curves for discounting and forwarding. For

example, we can use LIBOR/EURIBOR for forwarding and EONIA for discounting.

Both discounting and forwarding yield curves should be obtained from vanilla interest

rate instruments with homogeneous corresponding underlying rate tenors (in our case

TK for discounting curve and T F for forward curve), see Mercurio (2009), Bianchetti

and Carlicchi (2012) In a single-curve approach, when only one yield curve is used (for

example LIBOR or EURIBOR) for both discounting and forwarding, Eq. (5) can be

significantly simplified and looks as follows:

Ksingle =1− P (0, TKm )∑m

j=1 τ(TKj−1, T

Kj

)· P(0, TKj

) . (6)

Now, we are moving to pricing more complicated interest rate derivatives. One of

the plain-vanilla option in the interest market is the European swaption. A swaption

gives the right for a buyer to enter at time T Fa = TKc an IRS with payment times for

the floating and fixed legs given, respectively, by {T Fa+1, ..., TFn } and {TKc+1, ..., T

Km },

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with T Fn = TKm , fixed rate is K. Below, we will follow the Mercurio (2009) approach.

In a multi-curve framework, the swaption payoff at the time T Fa = TKc is

Payoff =(Ka,n,c,m(T Fa )−K

)+ ·m∑

j=c+1

τ(TKc , T

Kj

)· P(TKc , T

Kj

), (7)

where Ka,n,c,m(T Fa ) is the forward swap rate, derived from Eq. (5):

Ka,n,c,m(t) =

∑nj=a+1 τ

(T Fj−1, T

Fj

)· F(t;T Fj−1, T

Fj

)· P(t, T Fj

)∑mj=c+1 τ

(TKj−1, T

Kj

)· P(t, TKj

) . (8)

The Payoff (7) can be priced under the risk neutral swap measure Qc,m, whose asso-

ciated numeraire is the annuity∑m

j=c+1 τ(TKj−1, T

Kj

)· P(t, TKj

). We get:

PS(t,K;T Fa+1, ..., TFn , T

Kc+1, ..., T

Km )

=m∑

j=c+1

τ(t, TKj

)· P(t, TKj

)· EQc,m

[(Ka,n,c,m(T Fa )−K

)+ | Ft], (9)

Generally, in the multi-curve approach, the forward swap rate (8) depends on all

yield curves and correspondingly has very complicated dynamics. If we assume that

we know its volatility function and it evolves , under Qc,m, according to a driftless

geometric Brownian motion:

dKa,n,c,m(t) = σa,n,c,mKa,n,c,m(t) dW (t),

the payoff (7) can be explicitly calculated, leading to the generalized the Black–Scholes

formula:

PS(t,K;T Fa+1, ..., TFn , T

Kc+1, ..., T

Km )

=m∑

j=c+1

τ(t, TKj

)· P(t, TKj

)· Bl(K,Ka,n,c,m, σa,n,c,m

√T Fa − t, w), (10)

where the Black-Scholes function is defined as following:

Bl(K,S, σ, w) = wSΦ(wd1)−KwΦ(wd2) and d1 = 0.5σ2−ln(K/S)σ

; d2 = −0.5σ2−ln(K/S)σ

.

The problem with the Eq (10) is deriving and justifying the yield curve

for the forward swap rate (8) what is not always feasible, as it depends on

both discount and forward yield curves. In this thesis we develop alter-

native approach. The idea of this approach is the following. We propose

9

to describe separately the discount and forward yield curves in the frame-

work of one of the short rate models. Generally, we can chose different

short rate models for the discount and forward interest rates. Then, we

approximate continuous time models on discrete lattice, calculate interest

rate trees (binomial or trinomial) for both discount and forward rates, and

calculate the cash flows tree using both forward and discount interest rate

trees. This approach has not been considered in the literature and can be

used for pricing any interest rate derivative.

For the full picture, we provide the brief information about the short interest rate

models. It is very well known that the interest rates change through time, so we need

to use a mathematical process to model them. There are also broad evidences that

the interest rates are stochastic (for example, see Tse (1995), Ball and Torous (1998)),

so stochastic differential equations (SDE) should be used to capture the dynamics of

the short-term interest rates. The main demand for the short rate models is that they

must incorporate empirical and intuitive characteristics of interest rates. The most

general form of SDE for one factor short rate model is the following:

df [r(t)] = {θ(t) + ρ(t)g[r(t)]}dt+ σ[r(t), t]dWt, (11)

where f and g are suitably chosen functions; θ, which is the drift of the short rate,

is determined by the market; ρ, which is the ’tendency to an equilibrium short rate’

(mean reversal), can be chosen by the user of the model or dictated by the market;

the term σ is the local volatility of the short rate.

There are more than a dozen of different models and we have already briefly

discussed 5 of them in the literature review section, which are the Ho–Lee Model,

the Hull–White Model, the Kalotay–Williams–Fabozzi Model, the Black-Karasinski

Model, and the Black–Derman–Toy Model.

We will consider the Black–Derman–Toy Model in more details. The equation

describing the interest rate dynamics in the BDT model has f(r) = ln(r) and g(r) =

ln(r) and ρ(t) = d ln[σ(t)]dt

= σ′(t)σ(t)

in Eq. (11). The short rate in the BDT model follows

the lognormal process:

d ln r(t) = {θ(t) +σ′(t)

σ(t)ln r(t)}dt+ σ(t)dWt. (12)

10

It is interesting that it was first developed for in-house use by Goldman Sachs in the

1980s and was published only in 1990. To use this mordel for the financial markets it

can be approximated to the discrete space of binomial tree. Clewlow and Strickland

(1998) show that for each step i in the binomial tree, the expression in the above

equation can be operationalized to give an expression for the short rate at each node

in the tree. That is,

ri,j = aieσij√

∆t, (13)

where j numerate different possible states for every step i; ln(ai) is a drift parameter

and σi is a volatility parameter of the forward rate at the step i. Parameters ai can

be found from the calibration to the observed term-structure of corresponding market

spot rates. Volatility parameter σi is a volatility of the forward rate with the tree

period tenor and and expiration at time period i. This volatility can be found using

historical data (historical volatility) or can be estimated from the current security’s

prices (implied volatility).

In order to show how the callibration process works we will use Arrow-Debreu

prices. We say that a security is an elementary one if it pays 1 at time i, state j and 0

at every other time and state, its price at time 0 is Pe(i, j). The elementary security

prices satisfy the forward equations:

Pe(k + 1, s) = Pe(k,s−1)2(1+rk,s−1)

+ Pe(k,s)2(1+rk,s)

, 0 < s < k + 1,

Pe(k + 1, 0) = Pe(k,0)2(1+rk,0)

,

Pe(k + 1, k + 1) = Pe(k,k)2(1+rk,k)

.

Let us return to the calibration of the BDT model to the observed term-structure.

Consider an n-period lattice and let (s1, ..., sn) be the term-structure observed in the

market assuming that spot rates are compounded per period. We have:

1

(1 + si)i=

i∑j=0

Pe(i, j). (14)

This equation simply says that today (time 0) market price of a zero coupon bond,

which pays 1 at period i should be equal to its calculated price by using binomial tree.

11

We can rewrite the right-hand side of Eq. (14) using the forward equations, we have:

1

(1 + si)i=Pe(i− 1, 0)

2(1 + ai−1)+

i−1∑j=1

(Pe(i− 1, j)

2(1 + ai−1ebi−1j)+

Pe(i− 1, j − 1)

2(1 + ai−1ebi−1(j−1))

)+

Pe(i− 1, i− 1)

2(1 + ai−1ebi−1(j−1)). (15)

Eq. (15) is used to solve iteratively for the ai’s.

4 Methodology and data

4.1 Single Curve Approach

In the single-curve framework, a single yield curve is used for both discounting and

forwarding. By single-curve, we mean that the same instruments are used to derive

all the curves: the discount curve, the spot curve, the forward curve.

Ametrano and Bianchetti (2009) summarize the traditional single-curve pricing

and hedging framework in the following steps:

1. Select one finite set of vanilla (i.e. basic) interest rate instruments with increasing

maturities.

2. Build one spot curve by using the selected instruments and bootstrapping (a

method for building the curve incrementally in increasing maturity order).

3. Compute on the same curve, forward rates and discount factors and work out

the prices by summing up the discounted cash flows.

4. If necessary, compute the delta sensitivity, i.e. how the prices react to changes

in interest rates, and hedge the resulting delta risk using the suggested amounts

(hedge ratios) of the same set of vanillas.

4.2 Multi-Curve Approach

Multi-curve models for pricing and valuation represent a general term for the descrip-

tion of circumstances in which discount curves and forward curves differ. There are

several reasons for this. First, price discovery in financial markets in the course of the

12

financial crisis takes into account differences in tenors which result in different tenor-

specific interest rates. This difference is also known as “tenor basis spread”. Second,

changes in market conventions and institutional changes within financial markets drive

the implementation of so called “OIS-discounting” for collateralized trades.

Why does crisis make multi-curve methods so necessary? Since, during the financial

crisis the spreads between interest rates of different tenors widened and have since

that time remained on a significant level. Consequently the dependence of interest

rates on tenors for the valuation of derivatives became significant. Also, Multiple

Curve Approach tries to eliminate the problem of pricing and hedging plain vanilla

single-currency interest rate derivatives using multiple distinct yield curves for market

coherent estimation of discount factors and forward rates with different underlying

rate tenors.

Tenor basis spread: different tenor-specific interest rates; longer tenors are riskier

and forwards related to longer tenors should be priced higher.

Portfolio of interest rate derivatives with different underlying tenors requires sep-

arate forward curves.

Separation of forward curve and discounting curve: a unique discounting curve

for all tenor forward curves is used; the Overnight Indexed Swap (OIS) curve is now

commonly used to discount for collateralized derivative deals.

Ametrano and Bianchetti (2009) give the steps for multiple curve approach:

1. Build the discounting curve using a bootstrapping technique. The typical in-

struments are OIS swaps.

2. Select vanilla instruments linked to LIBOR/EURIBOR for each tenor curve

homogeneous in the underlying rate (typically with 3M, 6M, 12M tenors). The

typical instruments are FRA contracts, Futures, Swaps and Basis swaps.

3. Build the forward curves using the selected instruments by means of bootstrap-

ping technique, use these forward rates to find corresponding cashflows.

4. The discounting curve is used to discount the cashflows and calculate the present

value of the derivative.

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4.3 Data

The Macro Financial Analysis Division of the Bank of England estimates three kinds

of yield curves for the UK on a daily basis. First one is based on yields on UK

government bonds (gilts), and the second on is based on sterling interbank rates

(LIBOR) and the last one is based on yields on instruments linked to LIBOR and

based on sterling overnight index swap (OIS) rates, which are instruments that settle

on overnight unsecured interest rates (the SONIA rate in the UK).

4.4 Calibration of the BDT model

Here we bring the main steps to calibrate the Black–Derman–Toy Model used to price

the interest rate derivatives. Eq. (15) is used to solve iteratively for the ai’s as follows:

1. Set i = 1 in (15) and note that Pe(0; 0) = 1 to see that a0 = s1.

2. Use the forward equations to find Pe(1; 0) and Pe(1; 1).

3. Set i = 2 in (15) and solve for ai.

4. Continue to iterate forward until all ai’s have been found.

By construction, this algorithm will match the observed term structure to the term

structure on the lattice. By this mean we will have binomial trees for the instrument

needed to be calibrated. You can see our results as an example for it.

5 Results and Discussion

5.1 Single vs. Multi-curve frameworks: Swap fixed rate

We used the data available on the Bank of England website for continuously com-

pounded sterling overnight index swap (OIS) spot rates, which is available out to 5

years horizon, and sterling interbank LIBOR spot rates, which available out to 25

years horizon. Using this data we found the prices of zero coupon bonds (discount-

ing curves), constructed simply compounded spot curves (using Eq. (1)), and simply

compounded forward curves (using Eq. (3)) with 6 month tenor for both OIS rate

14

Figure 2: Simply compounded spot and forward rates, zero coupon bond prices

and LIBOR. As we have already said, the OIS rate is currently considered as a proxy

for a risk free rate and as one can see from Fig. (2), both sport and forward OIS yield

curves lies bellow corresponding LIBOR yield curves, what means that LIBOR rate is

higher and correspondingly riskier. This fact is reflected in the price of corresponding

zero coupon bonds, the ones that correspond to LIBOR have lower price, ensuring

higher rate of return. One can also see that the spread between LIBOR and OIS rate

increases for both spot and forward rates, what means the increase of the risk, when

maturity increases. For a simply compounded sport rate, this spread increases from

0.12% (12 bp) for 0.5 year maturity to 0.37% (37 bp) for 5 year maturity; for a sim-

ply compounded forward rate with 6 month tenor, this spread increases from 0.12%

(12 bp) for 0.5 year maturity to 0.48% (48 bp) for 5 year maturity. It is also very

interesting to bring attention to the form of the forward rate on 25 year horizon. One

can see that the forward rate increases up to some point (12-13 year maturity) and

then starts decreasing. One of the possible economic explanation is that the demand

for long term maturity securities is high in the market, what leads to increase of their

price and correspondingly decrease of the interest rate. It may mean that investors

try to invest into long term securities possibly expecting some bad news in short and

medium terms.

Let us consider a simple example of the swap with the maturity of 5 years and

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both fixed and floating rate (£ LIBOR) payments are made semiannually (6M tenor).

We found the swap fixed rate in a single-curve approach (we used LIBOR for both dis-

counting and forwarding) and multi-curve approach (we used OIS rate for discounting

and LIBOR for forwarding). Fixed rate is

Single curve approach (LIBOR is used for both discounting and forwarding):

1.97%;

Multi-curve approach (OIS rate is used for discounting and LIBOR is used for

forwarding): 1.98%.

We see that two approaches give a small difference in 0.01 % (1 bp). However,

for a notional principle, for example, of £100 million this difference would result in

£10,000 difference in fixed payment every period.

5.2 Generalization of the lattice approach to multi-curve frame-

work

As we have already discussed it briefly in theoretical section, we suggest generalization

of the lattice approach, which is used to approximate the short interest rate models,

for milti-curve framework. The main idea of this generalization is that we suggest

using different binomial/trinomial interest rate trees for forwarding and discounting.

The procedure is the following. First, we need to chose the short rate models, which

we use do describe the dynamics of the forward and discount interest rates (generally,

it may be different models). Then, using observed market term-structure, we use

calibration to find the model parameters and construct the binomial/trinomial trees

separately for discounting and forwarding interest rates. Finally, using separate trees

we calculate the cash flows to finish the task of the interest rate derivative valuation.

We will show how it works calculating as an example the price of the 2-8 payer

swaption. Specifically, the 2-8 swaption is an option that expires in 2 periods (1 year

in our case) to enter an 8-period swap with both fixed and floating leggs paying semi-

annually in period 3 through period 10. Floating payment are based on the prevailing

LIBOR rate of the previous months. For this particular problem, we will consider the

semi-anuyal fixed rate is set at 1% (correspond to annyal fixed rate of 2 %), which

is close to what we found in the privious section. As a model for short rate, we will

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choose the BDT model as this model garantees nonnegative interest rtes and allows

short rate volatility variation through the time.

5.3 Binomial trees for the LIBOR and OIS rates

To construct the binomial trees, we converted continuously compounded LIBOR and

OIS rate, which are taken from the Bank of England, to semi-annual simply com-

pounded interest rates. Following the instructions in the section 4.4, we calibrated

the Black–Derman–Toy (BDT) models to match the market interest rates. We used

past 5 years historical volatility from the data provided by the bank of England in the

calibration of binomial tree.

Specifically, taking LIBOR as an example:

1. Set a0 = s1= 0.0028

2. Use the forward equations to find Pe(1; 0) = Pe(1; 1) = 0.4986

3. Set i = 2 in Eq. (15) to solve for a2.

4. Continue to iterate forward until all ai’s have been found.

Using Eq. (13) we find the binomial trees for both LIBOR and OIS rate.

Calibrated ai’s for LIBOR:

a0 a1 a2 a3 a4 a5 a6 a7 a8 a9

0.0028 0.0040 0.0065 0.0087 0.0107 0.0122 0.0134 0.0143 0.0150 0.0155

Calibrated ai’s for OIS:

a0 a1 a2 a3 a4 a5 a6 a7 a8 a9

0.0022 0.0020 0.0059 0.0070 0.0089 0.0103 0.0113 0.0121 0.0127 0.0131

5.4 Cash flow trees in single curve and multi-curve approaches

To construct the cash flow tree for the swaption in a multi-curve approach we need

both LIBOR and OIS interest rate trees constructed above. We use LIBOR interest

rate tree to compare fixed rate with the floating rate of this tree for every period. To

find the present value of the cash flow due to the difference in fixed and floating rates

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Figure 3: LIBOR Binomial Tree

Figure 4: OIS rate Binomial Tree

Figure 5: Cash flow tree in a single-curve framework

Figure 6: Cash flow tree in a multiple-curve framework

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we need to discount it and for this purposes we use OIS interest rate tree. We also

use the fact, that interest rate of every node, during one period, can develop into one

of two possible (binomial) states, which have equal 1/2 probability.

The final current price of the swaption at t = 0 is the sum of all possible, proper

discounted and proper weighted future cash-flows. We found it in both single curve (we

used only LIBOR tree for both discounting and forwarding) and multi-curve frame-

works for a notional swaption amount of a one unit. We see from Fig. (5) and (6),

the swaption price in a multiple curve method, which is 0.0027, is higher than the

one in the single curve framework, which is 0.0025. It can be explained that OIS

rate is lower comparing to LIBOR what leads to lower discounting. If we consider a

swaption with notional amount of £100 million we obtain £20,000 different in price

in two approaches.

6 Conclusion

We studied the influence of the modern, after crisis multi-curve approach on the in-

terest rate derivatives. We calculated and compared the price of a simple swap in

both multi-curve and single curve approaches. We suggested the generalization of the

lattice approach, which is usually used to approximate the short interest rate models,

for milti-curve framework. This is a novel result, which have not been developed in the

scientific literature. As an example, we showed how to use the Black-Derman-Toy in-

terest rate model on binomial lattice in multi-curve framework and calculated the price

of the 2-8 period swaption in a single (LIBOR) curve and two-curve (OIS+LIBOR)

approaches. This technique can be used for pricing any interest rate instruments.

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