SABR model Alvaro Leitao Lecture group, CWI November 18, 2013 - - PowerPoint PPT Presentation

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SABR model Alvaro Leitao Lecture group, CWI November 18, 2013 - - PowerPoint PPT Presentation

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SABR model Alvaro Leitao Lecture group, CWI November 18, 2013 Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 1 / 25 Outline Introduction 1 SABR model 2 dynamic SABR model 3 SABR model applications 4

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SABR model

´ Alvaro Leitao

Lecture group, CWI

November 18, 2013

´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 1 / 25

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SLIDE 2

Outline

1

Introduction

2

SABR model

3

dynamic SABR model

4

SABR model applications

´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 2 / 25

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Introduction

Since 70’s: Black-Scholes.

◮ Standar option pricing method. Hypothesis: ⋆ The price follows lognormal distribution. ⋆ The volatility is constant. ◮ Crisis 1987. Model problems.

Implied volatility Strike Density Asset price

(a) Smile

Implied volatility Density Strike Asset price

(b) Skew

Models which modify the price distribution. Models which allow non-constant volatility.

◮ Local volatility models: Dupire. ◮ Stochastic volatility models: Heston or SABR. ´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 3 / 25

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Local vs. Stochastic volatility models

LVM volatility is a function. Both capture smile well. Both can be used for pricing. LVM show an opposite dynamic. LVM problems with risk measures. SVM solve it. Volatility also follows a stochastic process.

´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 4 / 25

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SABR model

SABR model (Hagan et al. 2002)

dFt = αtF β

t dW 1 t ,

F0 = ˆ f dαt = ναtdW 2

t ,

α0 = α Forward, Ft = Ste(r−q)(T−t), where r is constant interest rate, q constant dividend yield and T maturity date. Volatility, αt. dW 1

t y dW 2 t , correlated geometric brownian motions:

dW1dW 2 = ρdt Inicial values: S0 y α. Model parameters: α, β, ν and ρ. S-tochastic A-lpha B-eta R-ho model.

´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 5 / 25

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SABR model - Implied volatility

σB(K, ˆ f , T) = α (K ˆ f )(1−β)/2

  • 1 + (1 − β)2

24 ln2 ˆ f K

  • + (1 − β)4

1920 ln4 ˆ f K

  • + · · ·

·

  • z

x(z)

  • ·
  • 1 + (1 − β)2

24 α2 (K ˆ f )1−β + 1 4 ρβνα (K ˆ f )(1−β)/2 + 2 − 3ρ2 24 ν2

  • · T + · · · .

Note that the previous expression depends on the parameters K, ˆ f and T, also through the functions:

z = ν α (K ˆ f )(1−β)/2 ln ˆ f K

  • ,

and

x(z) = ln 1 − 2ρz + z2 + z − ρ 1 − ρ

  • .

´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 6 / 25

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SABR model - Obloj correction (2008)

σB(K, ˆ f , T) = 1

  • 1 + (1 − β)2

24 ln2 ˆ f K

  • + (1 − β)4

1920 ln4 ˆ f K

  • + · · ·

·   ν ln ˆ

f K

  • x(z)

  ·

  • 1 + (1 − β)2

24 α2 (K ˆ f )1−β + 1 4 ρβνα (K ˆ f )(1−β)/2 + 2 − 3ρ2 24 ν2

  • · T + · · · ,

where the following new expression for z is considered:

z = ν

  • ˆ

f 1−β − K 1−β α(1 − β) ,

and x(z) is given by the same previous expression. The omitted terms can be neglected.

´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 7 / 25

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SABR model - Approx. implied volatility

σB(K, ˆ f , T) = 1 ω

  • 1 + A1 ln

K ˆ f

  • + A2 ln2

K ˆ f

  • + BT
  • ,

where the coefficients A1, A2 and B are given by A1 = −1 2(1 − β − ρνω), A2 = 1 12

  • (1 − β)2 + 3
  • (1 − β) − ρνω
  • +
  • 2 − 3ρ2

ν2ω2 , B = (1 − β)2 24 1 ω2 + βρν 4 1 ω + 2 − 3ρ2 24 ν2, and the value of ω is given by ω = ˆ f 1−β α .

´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 8 / 25

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SABR model - Parameters

80 85 90 95 100 105 110 115 120 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 α = 0.25 α = 0.5 α = 0.75 α = 1 α = 1.25

(e) α > 0, the volatility’s reference level.

80 85 90 95 100 105 110 115 120 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 β = 1 β = 0.75 β = 0.5 β = 0.25 β = 0

(f) 0 ≤ β ≤ 1, the variance elasticity.

80 85 90 95 100 105 110 115 120 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ν = 0.5 ν = 1 ν = 1.5 ν = 2 ν = 2.5

(g) ν > 0, the volatility of the volatility.

80 85 90 95 100 105 110 115 120 −0.04 −0.02 0.02 0.04 0.06 0.08 0.1 0.12 0.14 ρ = −1 ρ = −0.5 ρ = 0.0 ρ = 0.5 ρ = 1

(h) −1 ≤ ρ ≤ 1, the correlation coefficient.

SABR parameters ´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 9 / 25

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SABR model - Calibration

The calibration process tries to obtain a set of model parameters that makes model values as close as possible to market ones, i.e Vmarket(Kj, ˆ f , Ti) ≈ Vsabr(Kj, ˆ f , Ti) In order to achieve this target we must follow several steps: Prices or volatilities. Representative market data. Error measure. Cost function. Optimization algorithm. Fix parameters on beforehand. Calibrate and compare the obtained results.

´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 10 / 25

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SABR model - Calibration example

80 85 90 95 100 105 110 115 120 16 16.5 17 17.5 18 18.5 19 19.5 20 20.5 21 strike (% of spot) σimpl σmodel σmarket

(i) 3 months maturity.

80 85 90 95 100 105 110 115 120 16 16.5 17 17.5 18 18.5 19 19.5 20 20.5 21 strike (% of spot) σimpl σmodel σmarket

(j) 6 months maturity.

80 85 90 95 100 105 110 115 120 16 16.5 17 17.5 18 18.5 19 19.5 20 20.5 21 strike (% of spot) σimpl σmodel σmarket

(k) 12 months maturity.

80 85 90 95 100 105 110 115 120 16 16.5 17 17.5 18 18.5 19 19.5 20 20.5 21 strike (% of spot) σimpl σmodel σmarket

(l) 24 months maturity.

´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 11 / 25

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SABR model - Drawback

3 m 6 m 12 m 24 m 90% 95% 100% 105% 110% 0.16 0.17 0.18 0.19 0.2 0.21 0.22 Maturities Strikes (% spot) σmarket

Figure: Market volatility surface.

´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 12 / 25

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SABR model - Drawback

80 85 90 95 100 105 110 115 120 16 16.5 17 17.5 18 18.5 19 19.5 20 20.5 21 strike (% of spot) σimpl σmodel σmarket

(a) 3 months maturity.

80 85 90 95 100 105 110 115 120 16 16.5 17 17.5 18 18.5 19 19.5 20 20.5 21 strike (% of spot) σimpl σmodel σmarket

(b) 6 months maturity.

80 85 90 95 100 105 110 115 120 16 16.5 17 17.5 18 18.5 19 19.5 20 20.5 21 strike (% of spot) σimpl σmodel σmarket

(c) 12 months maturity.

80 85 90 95 100 105 110 115 120 16 16.5 17 17.5 18 18.5 19 19.5 20 20.5 21 strike (% of spot) σimpl σmodel σmarket

(d) 24 months maturity.

´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 13 / 25

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Dynamic SABR model

dynamic SABR model

dFt = αtF β

t dW 1 t ,

F0 = ˆ f dαt = ν(t)αtdW 2

t ,

α0 = α Forward, Ft. Volatility, αt. dW 1

t y dW 2 t , correlated geometric brownian motions:

dW1dW 2 = ρ(t)dt Inicial values: S0 y α. Model parameters: α, β and ones that ν(t) and ρ(t) can provide. Approximation of implied volatility provided by Osajima (2007).

´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 14 / 25

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SLIDE 15

Dynamic SABR model

dynamic SABR model

dFt = αtF β

t dW 1 t ,

F0 = ˆ f dαt = ν(t)αtdW 2

t ,

α0 = α Forward, Ft. Volatility, αt. dW 1

t y dW 2 t , correlated geometric brownian motions:

dW1dW 2 = ρ(t)dt Inicial values: S0 y α. Model parameters: α, β and ones that ν(t) and ρ(t) can provide. Approximation of implied volatility provided by Osajima (2007).

´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 14 / 25

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Dynamic SABR model - Approx. implied volatility

σB(K, ˆ f , T) = 1 ω

  • 1 + A1(T) ln

K ˆ f

  • + A2(T) ln2

K ˆ f

  • + B(T)T
  • ,

where

A1(T) = β − 1 2 + η1(T)ω 2 , A2(T) = (1 − β)2 12 + 1 − β − η1(T)ω 4 + 4ν2

1(T) + 3

  • η2

2(T) − 3η2 1(T)

  • 24

ω2, B(T) = 1 ω2 (1 − β)2 24 + ωβη1(T) 4 + 2ν2

2(T) − 3η2 2(T)

24 ω2

  • ,

with

ν2

1(T) = 3

T 3 T (T − t)2ν2(t)dt, ν2

2(T) = 6

T 3 T (T − t)tν2(t)dt, η1(T) = 2 T 2 T (T − t)ν(t)ρ(t)dt, η2

2(T) = 12

T 4 T t s ν(u)ρ(u)du 2 dsdt.

´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 15 / 25

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Dynamic SABR model - ρ(t) and ν(t) functions

ρ(t) and ν(t) have to be smaller for long terms (t large) rather than for short terms (t small).

Constant

◮ ρ(t) = ρ0 ◮ ν(t) = ν0 ◮ α, β, ρ0, ν0, SABR model.

Piecewise

◮ ρ(t) = ρ0, t ≤ T0

ρ(t) = ρ1, t > T0

◮ ν(t) = ν0, t ≤ T0

ν(t) = ν1, t > T0

◮ α, β, ρ0, ν0, ρ1, ν1 and T0

Classical

◮ ρ(t) = ρ0e−at ◮ ν(t) = ν0e−bt ◮ α, β, ρ0, ν0, a and b

General

◮ ρ(t) = (ρ0 + qρt)e−at + dρ ◮ ν(t) = (ν0 + qνt)e−bt + dν ◮ α, β, ρ0, ν0, a, b, dρ, dν, qρ and qν. ´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 16 / 25

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Dynamic SABR model - Classical choice

ν2

1(T)

= 6ν2 (2bT)3

  • (2bT)2/2 − 2bT + 1
  • − e−2bT

, ν2

2(T)

= 6ν2 (2bT)3

  • 2(e−2bT − 1) + 2bT(e−2bT + 1)
  • ,

η1(T) = 2ν0ρ0 T 2(a + b)2

  • e−(a+b)T −
  • 1 − (a + b)T
  • ,

η2

2(T)

= 3ν2

0ρ2

T 4(a + b)4

  • 1 − 8e−(a+b)T +
  • 7 + 2(a + b)T
  • − 3 + (a + b)T
  • .

´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 17 / 25

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Dynamic SABR model - Calibration example

80 85 90 95 100 105 110 115 120 16 16.5 17 17.5 18 18.5 19 19.5 20 20.5 21 strike (% of spot) σimpl σmodel σmarket

(e) 3 months maturity.

80 85 90 95 100 105 110 115 120 16 16.5 17 17.5 18 18.5 19 19.5 20 20.5 21 strike (% of spot) σimpl σmodel σmarket

(f) 6 months maturity.

80 85 90 95 100 105 110 115 120 16 16.5 17 17.5 18 18.5 19 19.5 20 20.5 21 strike (% of spot) σimpl σmodel σmarket

(g) 12 months maturity.

80 85 90 95 100 105 110 115 120 16 16.5 17 17.5 18 18.5 19 19.5 20 20.5 21 strike (% of spot) σimpl σmodel σmarket

(h) 24 months maturity.

´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 18 / 25

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SABR pricing

Monte Carlo:

◮ huge number of forward and volatility paths ◮ V (S0, K) = D(T)E

  • V (ST, K)
  • S1

S2 S3 S4 So

Discretization schemes.

◮ Euler. ◮ Milstein. ◮ log-Euler. ◮ low-bias.

Time step(∆t) or number of time steps.

´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 19 / 25

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SABR pricing - European

1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 0.05 0.1 0.15 0.2 0.25 0.3 strike K Price EURUSD − Pricing European Call Option 3 m Market 3 m Monte Carlo 3 m σmodel 6m Market 6m Monte Carlo 6m σmodel 12m Market 12m Monte Carlo 12m σmodel 24m Market 24m Monte Carlo 24m σmodel

´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 20 / 25

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SABR pricing - Barrier

1.1 1.2 1.3 1.4 1.5 0.05 0.1 0.15 0.2 0.25 strike K Price EURUSD − Pricing Barrier Call Option 3 months 6 months 12 months 24 months

´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 21 / 25

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SABR pricing - Asian

1.1 1.2 1.3 1.4 1.5 0.05 0.1 0.15 0.2 0.25 0.3 strike K Price EURUSD − Pricing Asian Call Option 3 months 6 months 12 months 24 months

´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 22 / 25

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SABR Risk mesures

∆ risk ∂V ∂ ˆ f = ∂BS ∂ ˆ f + ∂BS ∂σB ∂σB ∂ ˆ f Vega risk ∂V ∂σB = ∂BS ∂σB ∂σB ∂α Vanna risk ∂V ∂ρ = ∂BS ∂σB ∂σB ∂ρ Volga risk ∂V ∂ν = ∂BS ∂σB ∂σB ∂ν

´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 23 / 25

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References

Jos´ e Luis Fern´ andez, Ana Mar´ ıa Ferreiro, Jos´ e Antonio Garc´ ıa-Rodr´ ıguez, ´ Alvaro Leitao, Jos´ e Germ´ an L´

  • pez-Salas, and Carlos

V´ azquez. Static an dynamic SABR stochastic volatility models: Calibration and

  • ption pricing using GPUs.

Mathematics and Computers in Simulation, 2013. Patrick S. Hagan, Deep Kumar, Andrew S. Lesniewski, and Diana E. Woodward. Managing smile risk. Wilmott Magazine, 2002. Jan Obloj. Fine-tune your smile: Correction to Hagan et al, 2013. Yasufumi Osajima. The asymptotic expansion formula of implied volatility for dynamic SABR model and FX Hybrid model, 2007.

´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 24 / 25

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Questions

´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 25 / 25

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Thank you

Thanks Gracias

´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 25 / 25