Greenwood (LPI Swaps - Pricing and Trading - Uk

6/9/2010

Risk and Investment Conference 2010 Mark Greenwood and Simona Svoboda

LPI swaps Pricing and Trading 13-15 June 2010 © 2010 The Actuarial Profession  www.actuaries.org.uk

Agenda 1. What is LPI? 2. The inflation options p market 3. The inflation volatility smile 4. Modelling the inflation volatility smile 5. SABR 6. A simple LPI model 7. Conclusion and discussion

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Types of LPI LPI, Limited Price Indexation, is the RPI link in pensions Following Wilkie(1988), Wilkie(1988) we define various types of LPI indexation: – Type 1: LPIt = RPIt – Type 2: LPIt = max[RPI0*(1+floor%)^t , RPIt ] – Type 3: LPIt = max[RPI0, RPI1…,RPIt ] – Type 4: LPIt = LPIt-1* min[max[1+floor%, RPIt /RPIt-1], 1+cap%] – Type 5: LPIt = LPIt-1 * [1+participation% * ( RPIt/RPIt-1-1 )] For 0% floors, Type 1 < Type 2 < Type 3 < Type 4 2

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Illustration of types of LPI RPIt

0% floor

(Type 1)

y/y%

indext

pension

Type 2 floor at 0%

0

100.00

100.00

100.00

1

101 00 101.00

1 0% 1.0%

100 00 100.00

101 00 101.00

1 0% 1.0%

101 00 101.00

1 0% 1.0%

101 00 101.00

2

103.53

2.5%

100.00

103.53

2.5%

103.53

2.5%

103.53

2.5%

3

99.90

-3.5%

100.00

100.00

-3.4%

103.53

0.0%

103.53

0.0%

4

98.90

-1.0%

100.00

103.53

0.0%

103.53

0.0%

104.84

6.0%

100.00

100.00 115 104.84

0.0%

5

4.8%

104.84

1.3%

109.74

6.0%

indexed pension w with floor

RPIt t

increase

Type 3 floor at 0% pension

increase

100.00

Type 4 floor at 0% pension

increase

100.00 1 0% 1.0%

110

105

100 Type 4 0% floor, no cap Type 3 floor Type 2 0% floor Type 1 (no floor)

95

90 © 2010 The Actuarial Profession  www.actuaries.org.uk

0

1

2

year

3

4

5

3

2

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The UK inflation options market •

The building blocks of the RPI derivatives market are zero coupon inflation swaps. These are a hedge for type-1 LPI.

•

Three forms of “vanilla” RPI inflation options trade: 1. ‘Year-on-year (y/y) RPI caps and floors. The cap has T caplets with payoffs max[ 0, (RPIt/RPIt-1-1) – K% ] at times t=1,2, …,T. 2. RPI index caps and floors with a single payoff at maturity. The cap payoff is max[ 0, RPIt/RPI0 -(1+K%)T ]. Hedge for LPI type2 liabilities liabilities. 3. LPI swaps hedge LPI type-4 liabilities. The most common collar strikes traded are [0%,5%], [0%,3%] and [0%,∞] (i.e. no cap).

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The UK inflation options market • Participants • Maturities • Liquidity • Equilibrium • Execution costs • Price transparency

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Inflation options screens

Source: Bloomberg 6 June 2010 © 2010 The Actuarial Profession  www.actuaries.org.uk

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The implied inflation volatility smile • Inflation options trade on price, not volatility • Prices can be inverted to implied volatilities using the conventional options pricing formulae for each market • For RPI index options, it is natural to use the Black Scholes (lognormal) model to price since the index is always positive and is expected to grow exponentially:

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Implied RPI index volatility smile Type-2 LPI liabilities can be priced directly from the implied volatility given the strike and maturity Black Scholes s implied index volatility

17.5%

30y RPI index option vol smile 20y RPI index option vol smile 10y RPI index option vol smile 5y RPI index option vol smile

15.0% 12.5% 10.0% 7.5%

Source: Bloomberg composite 6 June 2010

5.0% 2.5% 0.0% -1.0%

0.0%

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1.0%

2.0%

3.0%

4.0%

5.0%

6.0%

index option strike (rate)

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The inflation implied volatility smile: y/y options • For RPI y/y options, the y/y rate may be negative so the Black Scholes lognormal model is not appropriate • The market convention is to assume the underlying y/y rate has a normal distribution and use the Bachelier(1900) model. The resulting vol  is called the normal vol or basis point vol:

See VBA code in the spreadsheet on the conference website. © 2010 The Actuarial Profession  www.actuaries.org.uk

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The RPI y/y inflation options market RPI y/y atm inflation vol > GBP LIBOR atm caplet floorlet vol out to 5 years

Implied at-the--money normal volatility

2 00% 2.00% GBP RPI y/y normal vol 1.75% GBP LIBOR6m normal vol 1.50% 1.25% 1.00% 0 75% 0.75%

Source: RBS, 6 June 2010

0.50% 0.25% 0.00% 0

5

10

15

20

25

30

maturity

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The RPI y/y inflation options market RPI y/y inflation vol smile compared with LIBOR cap floor vol smile 2 00% 2.00% 20 year atm LIBOR atm flat vol 1.75%

20 year atm RPI y/y flat vol

no ormal volatility

1.50% 1.25% 1.00% 0 75% 0.75%

Source: Bloomberg composite 6 June 2010

0.50% 0.25% 0.00% -2.0% © 2010 The Actuarial Profession  www.actuaries.org.uk

0.0%

2.0%

4.0% strike

6.0%

8.0%

10.0% 11

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Modelling the implied volatility smile • Historically, the implied volatility smile first appeared in equity option markets • Techniques developed for equities have subsequently been used in interest rate and inflation markets – Alternate stochastic process – Local volatility models – Stochastic volatility models – Lévy processes

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Volatility smiles - alternate stochastic process • The presence of a smile may interpreted as a deviation from lognormality g y in the dynamics y of the underlying y g • Popular processes include the CEV (constant elasticity of variance) process dS   dt   S  dW

where 0    1

• And the shifted-lognormal (displaced-diffusion) process d S      dt  S    dW

• Both processes allow a monotonically downward sloping skew, however alternate techniques are needed to introduce curvature. © 2010 The Actuarial Profession  www.actuaries.org.uk

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Volatility smiles – local volatility models • Volatility is a function of the underlying itself, hence dS   Sdt   t , S dW

• First introduced by Dupire(1994) and Derman and Kani(1994).

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Stochastic volatility models • The underlying and its volatility are driven by correlated Brownian motions • A well known stochastic volatility model is due to Heston(1993) dS t   S t dt 

v t S t dW t1

dv t     v t dt  

v t dW t 2 where

dW t1 dW t 2   dt

• Semi-analytical option prices may be found via Fourier transform techniques as C S , v , t   SP1  KP t , T P2 where P(t,T) is the T-maturity discount bond and P1 and P2 are the associated probabilities evaluated via numerical integration. © 2010 The Actuarial Profession  www.actuaries.org.uk

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Lévy processes • General class of processes with independent and stationary increments • This includes familiar processes such as Brownian motion and Poisson process which introduces random jumps into the dynamics • All other Lévy processes are generalisations of a Brownian motion and a possibly infinite number of Poisson processes.

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Lévy processes Lévy processes that have become popular in finance include: – Jump-diffusion p – dynamics y are driven by y a diffusion and a finite number of Poisson processes, – Variance-gamma – may be interpreted as a Brownian motion evaluated at a time given by a gamma process, see Madan et. al. (1990,1991,1998), – Normal-Inverse-Gaussian – may be interpreted as a Brownian motion evaluated at a time given by an InverseInverse Gaussian process, see Barndorff-Nielsen (1997,1998).

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SABR (Stochastic Alpha Beta Rho) model • This stochastic volatility model is the market standard for fitting an implied volatility smile, dF t   t F t  dW t1 ,

F0  f

d  t   t dW t ,

0  

2

where dW t1 dW t 2   dt and ,  are constants such that 0    1 and   0 . • It is not a true stochastic volatility model since a separate set of parameters ,  and  are associated with each maturity. • The great advantage is that the equivalent Black-Scholes (lognormal) implied volatility may be approximated analytically by B(K,f) defined as follows: 18

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SABR (Stochastic Alpha Beta Rho) model σ B K,f





 fK 



  1  

  log 2 1  24  2

2

 2   24

2

 fK 



f

K





4

1920

log 2

f

K

 z        z      

 1  2  3 2 2    T     4  fK  2 24  

where   1   and

z

  fK  2 log l 

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f

K

 1  2 z  z 2  z       1    

  z   log 

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SABR Normal model • For options on the RPI y/y rate, we use a normal model of the underlying and the normal option pricing formula of Bachelier. Hence: 1 dF t   t dW t ,

F0  f

d  t   t dW t , 2

0  

where dW t1 dW t 2   dt and  is constant such that   0 . • The equivalent normal implied volatility may be approximated analytically as   z   2  3 2 2   1   T   σ N K,f      24   z     where  1  2 z  z 2  z        z   log  z   f  K ,  



1 



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Weaknesses of the SABR model • These analytic approximations are derived via singular perturbation techniques p q relying y g on a ‘small volatility’ y expansion, p , hence assuming both volatility  and volatility-of-volatility  are small. • For extreme parameter values and strikes far away-from-themoney, these approximations break down. • This is well known by the market and each market participant has their own set of proprietary ‘fixes’ fixes . • The breakdown of these approximations is most clearly visible if one examines the probability density function derived from call spreads using implied volatilities, which becomes negative at extreme parameter values and away-from-the-money. © 2010 The Actuarial Profession  www.actuaries.org.uk

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SABR model: negative probabilities SABR model implied density for F=3.63%, =1.25%, =50%, =15%, =22% SABR implied p density y for 30y y 6-month LIBOR rate 5.0%

pro obability density

4.0% 3.0% 2.0% 1.0% 0.0% -1.0% 0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

8.00%

rate (forward = 3.63%) 22

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Simple LPI model: RPI y/y history and forward rates market forward rate

historical rate 10% 9% 8% 7% RPI y/y rate

6%

5%

5% 4%

3%

3% 2% 1%

0%

0% -1% -2% -3% 1980

1990

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2000

2010

2020

calendar year

2030

2040

2050 23

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SABR normal volatility smiles: effect of parameters 1.75%

base: atmvol=0.60%, =0%, =40% higher atmvol: atmvol=0.80%, =0%, =40% negative skew: atmvol=0.60%, =-40%, =40% less smile: atmvol=0.60%, =0%, =20%

1.50%

normal vol

1.25%

1.00%

0.75%

0.50%

0.25% -1.0%

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

6.0%

7.0%

8.0%

strike 24

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Simple LPI model • Type 4: LPIt = LPIt-1* min[max[1+floor%, RPIt /RPIt-1], 1+cap%] • A 30y LPI swap has 60 RPI y/y options embedded in the swap: LPI30 = LPI0* (RPI1 /RPI0 -1 + floor1 - cap1) * (RPI2 /RPI1 -1 + floor2 - cap2) :

:

:

* (RPI30 /RPI29 -1 + floor30 - cap30) • Most LPI swap trades use 3 strikes: [0,5], [0,3] and [0,]. SABR RPI y/y normal model has 3 parameters: ,  and (in our example spreadsheet we reparameterise as: atm vol,  and ) • In practice a unique fit can be usually identified © 2010 The Actuarial Profession  www.actuaries.org.uk

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Slide 26 m2

mark greenwood, 08/06/2010

6/9/2010

Simple LPI model 1. FIND SABR PARAMETERS 2. TO GIVE BEST FIT

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Simple LPI model: LPI[0,5] fit good 0.15%

bp p spread to RPI swap

[[0,5] , ] SABR simple p LPI 0.10%

[0,5] market

0.05%

0.00% 0

10

20

30

40

50

-0.05%

-0.10% maturity

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Slide 28 m4

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Simple LPI model: LPI[0,] fit good

bp spread to RPI swap b

0.50%

0.40%

0.30%

0.20%

[0,inf] SABR simple LPI

0.10%

[0,inf] market 0.00% 0

10

20

30

40

50

maturity 28

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Simple LPI model: LPI[0,3] fit good 0.00% 0

10

20

30

bp p spread to RPI swap

-0.20%

40

50

[0,3] SABR simple LPI [0,3] market

-0.40%

-0.60%

-0.80%

-1.00%

-1.20% maturity © 2010 The Actuarial Profession  www.actuaries.org.uk

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Simple LPI model: LPI[0,2.5] fit good 0.00% 0

10

20

30

40

50

bp spread to RPI swap

-0.20% 0.20% -0.40%

[0,2.5] SABR simple LPI

-0.60%

[0,2.5] market

-0.80%

model -1.317% vs market -1.323% 1 323%

-1.00% -1.20% -1.40% -1.60%

maturity 30

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Simple LPI model: LPI[3,5] fit poor in 50y

model -0.442% vs market -0.409%

0.70%

bp p spread to RPI swap

0.60% 0.50% 0.40% 0.30% 0.20%

[3,5] SABR simple LPI 0.10%

[3,5] market

0.00% 0

10

20

30

40

50

maturity © 2010 The Actuarial Profession  www.actuaries.org.uk

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Slide 32 m8

mark greenwood, 08/06/2010

6/9/2010

Simple LPI model: pros and cons + recovers RPI and main LPI swap rates and allows alternative strikes, maturities and RPI reference dates to be priced quickly + risk to the model parameters (the greeks) is quick and simple

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Simple LPI model: the greeks DELTA AND SABR VEGA LADDERS per £1m notional LPI swap

5y 10y 15y 20y 25y 30y 40y 50y TOTAL

DELTA notional('£m) RPI zc 30y 498 944 1339 1756 2186 2620 3436 4204

DELTA 1bp LPI[0,5] 30y -11 -12 45 -40 -72 2094 0 0 2003

DELTA notional('£m) LPI[0,5] 30y -0.02 -0.01 0.03 -0.02 -0.03 0.80 0.00 0.00

VEGA atmvol 1bp LPI[0,5] 30y 63 207 170 237 48 -56 0 0 669

VEGA 1% LPI[0,5] 30y 251 214 230 243 270 169 0 0 1376

RPI zc swap notionals (in £’m) to delta hedge £1million LPI[0,5] liability © 2010 The Actuarial Profession  www.actuaries.org.uk

VEGA 1% LPI[0,5] 30y -251 -309 -467 -625 -667 -432 0 0 -2752

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mark greenwood, 08/06/2010

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Simple LPI model: pros and cons + recovers RPI and main LPI swap rates and allows alternative strikes, maturities and RPI reference dates to be priced quickly + risk to the model parameters (the greeks) is quick and simple + effects on LPI swap rates and greeks of RPI swap scenarios or curve moves are readily explored - is not a true model, recovers RPI zc swap rates but does not recover market prices of y/y and index options - does not price other types of LPI swaps or other inflation derivatives

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RPI vs LPI[0,5] swap market rates

zc RPII swap rate / LPI spread

4.50% 4.25%

0.15% 30y RPI zc swap rate (LHS) 30y LPI zc spread to RPI swap (RHS)

0.10%

4.00%

0.05%

3.75%

0.00%

3.50%

-0.05%

3.25%

-0.10%

3.00%

Source: -0.15% RBS

Oct 09

Nov 09

Jan 10

Mar 10

May 10

trade date © 2010 The Actuarial Profession  www.actuaries.org.uk

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Slide 35 m10

mark greenwood, 08/06/2010

Slide 36 m11

mark greenwood, 08/06/2010

6/9/2010

Conclusion and discussion • The implied RPI volatility smile is an important feature of the inflation options market. The skew towards expensive fl floors/cheaper / h caps iis extreme t as a resultlt off lack l k off natural t l supply of floors • LPI models proposed in literature have had far more general applicability, but have not emphasised the effect of the smile • The simple type-4 LPI model presented may assist with “interpolating” p g values for LPI liabilities, calculation delta and vega risks and understanding dealer execution charges for these greeks.

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Questions or comments?

Expressions of individual views by members of The Actuarial Profession and its staff are encouraged. The views expressed in this presentation are those of the presenter.

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References LPI WlLKIE, A.D. (1988). The use of option pricing theory for valuing benefits with cap and collar guarantees. Transactions of the 23rd International Congress of Actuaries. BEZOOYEN, J.T.S., EXLEY, C.J. and SMITH, A.D. (1997). A market-based approach to valuing LPI liabilities. Paper presented to the Joint Institute and Faculty of Actuaries Investment Conference.

OPTION PRICING MODELS BLACK, F. and SCHOLES, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy 81, 637-659. BACHELIER, L. (1900). Théorie de la spéculation, Annales Scientifiques de l’École Normale Supérieure.

VOLATILTY MODELS DUPIRE, B. (1994). Pricing With a Smile. Risk 7(1), 18-20. DERMAN, E. and KANI I. (1994). Riding on a Smile. Risk 7(2), 32-39. HESTON, S. (1993). A Closed-Form Solution for Options with Stochastic Volatility with Application to Bond and Currency Options. Review of Financial Studies 6(2), 327-343.

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References

VOLATILTY MODELS cont. MADAN, D., MADAN D CARR, CARR P. P and CHANG, CHANG E. E (1998). (1998) The Variance Gamma Process and Option Pricing Pricing, European Finance Review 2, 79-105. MADAN, D. and MILNE, F. (1991). Option Pricing with V.G. Martingale Components, Mathematical Finance 1 (4), 39-55. MADAN, D. and SENETA, E. (1990). The Variance Gamma (V.G.) Model for Share Market Returns, Journal of Business 63(4), 511-524. BARNDORFF-NIELSEN, O. (1997). Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling, Scandinavian Journal of Statistics 24(1), 1-13. HAGAN, P., KUMAR D., LESNIEWSKI A. and WOODWARD R. (2002). Managing Smile Risk, Wilmott September 84-108. September, 84 108

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