An Assessment on the Appropriateness of the use of the ...

An Assessment on the Appropriateness of the useof the LFMM in South Africa

Team: The UCT Pigeons

Xi Kleisinger-Yu, Wade Cresswell, Chris Sterley, Aurore DeSavigny

University of Cape Town

05 July 2018

Introduction

I A particular insurance company wishes to use the LognormalForward-LIBOR Market Model (LFMM) in order to price exotic andlonger-dated interest rate derivitives in the South African market

I We are interested in investigating whether there exists aparsimonious instantaneous volatility and correlationparameterisation for long-term modeling with the LFMM after jointcalibration

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Introduction

There exists two popular market models, the LFMM and theLSMM:

I Caps/Floors are priced consistently in LFMM model, and usethe BS 76 formula

I Swaptions are priced consistently in LSMM model, and usethe BS 76 formula

But these two models are not compatible!

Under LFMM model: we price caps using a closed formexpression, price swaptions using approximations, and examine theappropriateness of this model for the South African market.

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Introduction

South African Interbank Interest Rate Market

I Single-curve market

I At-the-money swaption implied volatility term structures

Euro swaption volatility South African swaption volatility

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Outline

I LFMM model

• Structure

• Cap Pricing

• Swaption Pricing

• Approximations

I Data description and manipulation

• South African yield curves

• Caps

• Swaptions

I Bootstrapping

I Instantaneous volatility calibration

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Outline II

I Instantaneous Correlation

• Characteristics

• Parameterisations

• Swaptions

• Swaption Grid

• Black Volatility Approximations

I Callibration Results

I Conclusions

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1. LFMM (Lognormal Forward-LIBOR Market Model)STRUCTURE

I Bond tenor dates Ti ∈ [0.25, 0.5.., 10], τi = Ti+1 − Ti = 0.25

I P(t,Ti

): price at time t of a ZCB paying a unit of currency

at maturity Ti

I Fi(t) : value at time t of the simply-compoundedforward-LIBOR rate from Ti to Ti+1

Fi (t) := F(t;Ti ,Ti+1

)=

1

τi

(P(t,Ti

)P(t,Ti+1

) − 1

)(1)

I Qm+1 : Tm+1-forward probability measure associated with thenumeraire asset P

(t,Tm+1

), 0 ≤ m ≤ M

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LFMM dynamics of Fi (t) under Qi+1:

dFi (t) = σi (t)Fi (t) dW i+1i (t) (2)

I σi(t) : instantaneous volatility of Fi (t)

I Qm+1 : Tm+1-forward probability measure associated with thenumeraire asset P

(t,Tm+1

), 0 ≤ m ≤ M

I Wi+1i (t) : i th component of a M-dimensional Qi+1 standard

Brownian motion W i+1(t) with instantaneous correlationsgiven by:

d⟨W i+1

i ,W i+1j

⟩t

= ρi ,j dt (3)

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LFMM dynamics of Fi (t) under Qm+1:

dFi (t) = µi (t) dt + σi (t)Fi (t) dWm+1i (t), (4)

for t ≤ Ti , where

µi (t) :=

−σi (t)Fi (t)

m∑j=i+1

ρi,jτjσj (t)Fj (t)1+τjFj (t)

for i < m;

0 for i = m;

σi (t)Fi (t)i∑

j=m+1

ρi,jτjσj (t)Fj (t)1+τjFj (t)

for i > m,

(5)

d⟨Wm+1

i ,Wm+1j

⟩t

= ρi ,j dt. (6)

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1. LFMM (Lognormal Forward-LIBOR Market Model)CAP PRICING

Cap(

0,T ,K)

:= E

[β−1∑i=α

τiD(0,Ti+1

)(Fi(Ti

)− K

)+]

=

β−1∑i=α

τiP(0,Ti+1

)Ei+1

[(Fi(Ti

)− K

)+]

=

β−1∑i=α

τiP(0,Ti+1

)Bl(Fi (0), vFi

√Ti ,K , 1

)I D

(0,Ti+1

): stochastic discount factor for the term

[0,Ti+1

]I Ei+1/ Vari+1 : expectation/variance under the measure Qi+1

I v2Fi: average percentage variance of Fi (t) for t ∈

[0,Ti

)as

v2Fi:=

1

TiVari+1

[∫ Ti

0

dFi (t)

Fi (t)

]=

1

Ti

∫ Ti

0σ2i (t) dt (7)

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1. LFMM (Lognormal Forward-LIBOR Market Model)CAP PRICING

Black’s formula

Bl(F , v ,K , ξ

)= ξ

[FΦ(ξd1(F , v ,K

))− KΦ

(ξd2(F , v ,K

))],

d1(F , v√T ,K

)=

log(FK

)+ v2T

2

v√T

(8)

d2(F , v√T ,K

)= d1

(F , v√T ,K

)− v√T , (9)

where Φ(·) is a standard normal cumulative distribution function.

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1. LFMM (Lognormal Forward-LIBOR Market Model)SWAPTION PRICING

I Interest-Rate Swap (IRS) : exchange payment streams(usually fixed IR against floating IR).

At each date Ti ∈ T ={Tα+1, ...,Tβ

}an investor will pay

the amount τiK and receive τiL(Ti ,Ti+1) = τiFi (Ti ).

I Forward swap rate Sα,β(t) at time t for the sets of times T :fair fixed rate of an IRS

Sα,β(t) =

1−β−1∏j=α

11+τjFj (t)

β−1∑i=α

τii∏

j=α

11+τjFj (t)

. (10)

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LSMM dynamics of Sα,β(t) under Qα,β:

dSα,β(t) = σSα,β (t)Sα,β(t) dW α,β(t), (11)

I σSα,β (t) : instantaneous volatility of Sα,β(t)

I Qα,β : ”swap” measure associated to the numeraire

Cα,β(t) =

β−1∑i=α

τiP(t,Ti+1

). (12)

I W α,β(t) is a standard Brownian motion under Qα,β

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PS(

0,T ,K)

:= E[D(0,Tα

)(Sα,β

(Tα)− K

)+Cα,β

(Tα)]

= Cα,β(0)Eα,β[(

Sα,β(Tα)− K

)+]= Cα,β(0)Bl

(Sα,β(0), vSα,β

√Tα,K , 1

)(13)

I Qα,β : ”swap” measure associated to the numeraire Cα,β(t)

I Eα,β/Varα,β : expectation/variance under measure Qα,β

I v2Sα,β : average percentage variance of Sα,β(t) for t ∈[0,Tα

]v2Sα,β :=

1

TαVarα,β

[∫ Tα

0

dSα,β(t)

Sα,β(t)

]=

1

Tα

∫ Tα

0σ2Sα,β (t) dt,

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1. LFMM (Lognormal Forward-LIBOR Market Model)SWAPTION PRICES

LFMM dynamics

dFi (t) = σi (t)Fi (t) dW i+1i (t)

⇓

Approximate v2Sα,βaverage percentage variance of Sα,β(t)

⇓

Price Swaptions using Black’s formula

PS(

0,T ,K)

= Cα,β(0)Bl(Sα,β(0), vSα,β

√Tα,K , 1

)15/56

1. LFMM (Lognormal Forward-LIBOR Market Model)APPROXIMATIONS

Assumption 1

v2Sα,β ≈1

TαEα+1

[∫ Tα

0d⟨log Sα,β

⟩t

]=:(vLNSα,β

)2(14)

Assumption 2

Sα,β(t) =

β−1∑i=α

wi (t)Fi (t), (15)

where

wi (t) =

τii∏

j=α

11+τjFj (t)

β−1∑k=α

τkk∏

j=α

11+τjFj (t)

(16)

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I wi (t) and Fi (t) in (15) are independent

I wi (t) are stable over time.

Sα,β(t) ≈β−1∑i=α

wi (0)Fi (t) (17)

Approximation 1 : The Rebonato formula

(vLNSα,β

)2≈ 1

Tα

β−1∑i ,j=α

wi (0)wj(0)Fi (0)Fj(0)

S2α,β(0)

ρij

∫ Tα

0σi (t)σj(t) dt

(18)

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I wi (t) not independent of Fi (t)

dSα,β(t) =

β−1∑h=α

w̃h(t) dFh(t) =

β−1∑h=α

w̃h(t)σh(t)Fh(t) dWm+1h (t)

(19)where

w̃h(t) = wh(t) +

β−1∑i=α

Fi (t)∂wi (t)

∂Fh(20)

and

∂wi (t)

∂Fh=

wi (t)τh1 + τhFh(t)

β−1∑k=h

τkk∏

j=α

11+τjFj (t)

β−1∑k=α

τkk∏

j=α

11+τjFj (t)

− I{i≥h}

(21)

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∫ Tα

0d⟨log Sα,β

⟩t

=

∫ Tα

0

β−1∑h,j=α

Gh,j(t)σh(t)σj(t)ρh,j dt, (22)

where Gh,j(t) =w̃h(t)w̃j (t)Fh(t)Fj (t)

S2α,β(t)

and Gh,j(t) ' Gh,j(0)

Approximation 2 : The Hull-White formula

(vLNSα,β

)2≈ 1

Tα

β−1∑h,j=α

Gh,j(0)ρhj

∫ Tα

0σh(t)σj(t) dt. (23)

Approximation 3 : Kawai expansion

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Goal

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South African Market Data Description

I Historical nominal swap curves (NACC zero-coupon swapyields)

• from the JSE

• yield curves : Jan 2004 - March 2018 (= 3559 business days)

• data included up to 30 years (165 points per yield curve)

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South African Yield Curves

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South African Market Data Description

I Market cap volatilities

• from Bloomberg

• Jan 2012 - March 2018

• Annual terms and tenor of 0.25

Keep data only on the common dates (Jan 2012 - March 2018i.e. 1543 business days, up to 10 years)

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South African Cap Volatilities

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South African Market Description

I Market swaption volatilites

• from Bloomberg

• November 2010 - March 2018

• range of tenors and terms

Data between 24 November 2010 till 24 December 2010removed, and several other points estimated

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South African Swaption Volatilities: 1 Year Term

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From the NACC Spot Rates to the Forward rates

For each date from Jan 2012 - March 2018, for allTi ∈ [0.25, 0.5, 0.75, 1, ..10] :

I Apply linear interpolation to estimate the exact spot ratesr(0,Ti )

I Compute the zero-coupon bond prices P(0,Ti ) as :

P(0,Ti ) = e−r(0,Ti )Ti (24)

I Compute the forward-LIBOR rates Fi (0) from Ti to Ti+1,with τi = Ti+1 − Ti = 0.25:

Fi (0) := F(0;Ti ,Ti+1

)=

1

τi

(P(0,Ti

)P(0,Ti+1

) − 1

). (25)

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NACC Spot Rates

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Simple Forward Rates

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From the Caps to the Caplets volatilities

I Market data includes cap volatilities but not caplet volatilities

I 10 cap volatilities quoted with annual term to maturities, i.e.4 caplets between cap periods

I Relative and absolute strike caps

I So how can we calibrate these important caplet volatilities?

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Bootstrapping Caplet Volatilities

I Strengths of bootstrapping

• simplicity

• efficiency

• guaranteed result (if one exists)

I Weaknesses of bootstrapping

• too simple?

• rigid assumptions

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The Bootstrapping algorithm

I Sort the caps into increasing order of maturity

I Price of the caps

I Generate a series of price differences between adjacent caps

Diffk = Cap(0,Tk+4,K

)− Cap

(0,Tk ,K

)=

β+3∑i=β

τiP(0,Ti+1

)Bl(Fi (0), vCap

√Ti ,K , 1

)I Assign caplets to the relevant periods

I Assign a common caplet volatility to each period and solve:

Cap(0,Tk+4,K

)− Cap

(0,Tk ,K

)=

β+3∑i=β

τiP(0,Ti+1

)Bl(Fi (0), vCaplet

√Ti ,K , 1

)34/56

Bootstrapped Caplet Volatilities

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Calibration of Instantaneous Volatilities

NB: Fixed instantaneous correlation (using historical data)

σi (t) =[a + b

(Ti − t

)]e−c(Ti−t) + d

I Calibrate instantaneous volatilities to best recover Swaptionprices

OR

I Calibrate instantaneous volatilities to best recover bothSwaption and Cap prices

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Instantaneous Correlation Characteristics

Necessary:

I |ρi ,j | ≤ 1 for all i , j

I ρ positive semidefinite

I ρi ,i = 1 for all i

Desired for Forward Rates:

I i → ρi ,j increasing for all i ≥ j ”decreasing along row”

I i → ρi+p,i increasing for fixed ρ ”increasing alongsub-diagonals”

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Instantaneous Correlation Parametarisations

I Single-Parameter

ρi ,j = exp(−β|Ti − Tj |)

I Two-Parameter

ρi ,j = exp[− |i − j |

M − 1

(log ρ∞ + η

M − 1− i − j

M − 2

)]I Two-Parameter Improved

ρi ,j = exp[− |i − j |

M − 1

(− log ρ∞

+ ηi2 + j2 + ij − 3Mi − 3Mj + 3i + 3j + 3M2 −M − 4

(M − 2)(M − 3)

)]I Rebonato’s Angles (M-Parameter)

ρi ,j = cos(θi − θj)38/56

Swaption Grid

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Swaption Black Volatility ApproximationsThe Hull-White formula: The squared Black swaption volatility is

(vSα,β

)2 ≈ 1

Tα

β−1∑h,j=α

Gh,j(0)ρhj

∫ Tα

0σh(t)σj(t) dt.

where

Gh,j(t) =w̃h(t)w̃j(t)Fh(t)Fj(t)

S2α,β(t)

.

w̃h(t) = wh(t) +

β−1∑i=α

Fi (t)∂wi (t)

∂Fh

and

∂wi (t)

∂Fh=

wi (t)τh1 + τhFh(t)

β−1∑k=h

τkk∏

j=α

11+τjFj (t)

β−1∑k=α

τkk∏

j=α

11+τjFj (t)

− I{i≥h}

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Two Approaches of Calibration

I Fix instantaneous volatility of the forward rates (to thatimplied by the Caps), and calibrate instantaneous correlationsto Swaptions in the market

OR

I Fix instantaneous correlation and calibrate the instantaneousvolatilies to either exclusively Swaptions, or a weightedcombination of the Swaptions and Caps

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Calibration of Instantaneous Correlation

NB: Fixed instantaneous volatilities (using Cap data)

I Single-Parameter and both Two-Parameter parametarisationsare found by minimising the average absolute error of theswaption prices

I Rebonato’s Angles calibration however, requires a hybridbootstrapping approach

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Swaption Grid, Again

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Hybrid Bootstrapping Calibration of Rebonato’s Angles

ρi ,j = cos(θi − θj), 0 ≤ θ ≤ π

2

σ matrix ρ matrix

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Calibration ResultsSwaption vol calibration with fixed correlation

Fix instantaneous correlation and calibrate the instantaneous volatilitiesof the forward rate only to Swaptions prices.

I absolute relative error εα,β :=|σmarketα,β −σ

modelα,β |

σmarketα,β

I average relative error = 20.85398%

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Calibration ResultsSwaption vol calibration with fixed correlation

Fix instantaneous correlation and calibrate the instantaneous volatilitiesof the forward rate to a weighted combination of Caps and Swaptionsprices.

I we assume equal weights

I average relative error = 25.09454%

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Calibration Results

Swaption vol calibration with fixed caplet volatilities

Fix instantaneous volatility of the forward rates (to that implied by theCaps) and calibrate instantaneous correlations to Swaptions in themarket.

Single-Parameter Two-Parameter improv. Two-Parameter Rebonato’s

β ρ∞, η ρ̃∞, η̃ angle θaverage

relative error 36.14376% 35.28341% 34.37879% 5.46626%

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Calibration Results


I Single-Parameter parametrisation

I average relative error: 36.14376%

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Results


I Two-Parameter parametrisation


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Results


I improved Two-Parameter parameterisation


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Results


I Rebonato’s angle parameterisation


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Summary

I The South African inter-bank interest rate market has a peculiarswaption volatility structure

There are two approaches to calibrating the LFMM:

I Using fixed instantaneous volatilities (calibrated to caps in themarket), we calibrate instantaneous correlations to swaption prices

I Using fixed instantaneous correlations (historical correlations), wecalibrate the instantaneous volatilities to the cap and swaption prices

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Conclusion I

Using fixed instantaneous volatilities:

I The correlation parameterisations that maintained the desirablecharacteristics resulted in a low quality of fit to the swaption prices.

I Using a more flexible parameterisation (Rebonato’s Angles) a higherquality of fit is achieved, but the correlation matrix oscillates beyondthe point of interpretation

I This corroborates the findings of Brigo & Mercurio in developedmarkets.

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Conclusion II

Using fixed instantaneous correlations:

I Calibrating the instantaneous volatilities to cap and swaption dataresults in vastly different volatilities in comparison to those impliedby the market

I Further, the resultant swaption prices deviate significantly fromthose in the market

I Brigo & Mercurio show ’exact’ calibration to Swaption data througha cascading calibration algorithm. We have shown that these resultscannot be replicated with parametric volatility structures within theSouth African market

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Conclusion III

When using parsimonious instantaneous volatility and correlationparameterisations (which would be used for long-term modeling),the resultant fit is of poor quality.

Only once more flexible structures are introduced can the modelproduce an adequate fit.

We conclude that the LFMM is not appropriate for thispurpose within the context of the South African inter-bankinterest rate market.

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Thank you!

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