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

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

Introduction Since 70’s: Black-Scholes. ◮ Standar option pricing method. Hypothesis: ⋆ The price follows lognormal distribution. ⋆ The volatility is constant. ◮ Crisis 1987. Model problems. Density Implied volatility Implied volatility Density Strike Strike Asset price Asset price (a) Smile (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

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

SABR model SABR model (Hagan et al. 2002) dF t = α t F β F 0 = ˆ t dW 1 t , f d α t = να t dW 2 t , α 0 = α Forward, F t = S t e ( r − q )( T − t ) , where r is constant interest rate, q constant dividend yield and T maturity date. Volatility, α t . dW 1 t y dW 2 t , correlated geometric brownian motions: dW 1 dW 2 = ρ dt Inicial values: S 0 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

SABR model - Implied volatility α � z � σ B ( K , ˆ � ˆ � ˆ f , T ) = � · · � � � 1 + (1 − β ) 2 + (1 − β ) 4 x ( z ) f f ( K ˆ ln 2 ln 4 f ) (1 − β ) / 2 + · · · 24 K 1920 K � � 1 + (1 − β ) 2 α 2 f ) (1 − β ) / 2 + 2 − 3 ρ 2 f ) 1 − β + 1 ρβνα ν 2 · T + · · · . ( K ˆ ( K ˆ 24 4 24 Note that the previous expression depends on the parameters K , ˆ f and T , also through the functions: � ˆ � z = ν f f ) (1 − β ) / 2 ln α ( K ˆ , K and � � 1 − 2 ρ z + z 2 + z − ρ � x ( z ) = ln . 1 − ρ ´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 6 / 25

SABR model - Obloj correction (2008) � ˆ �  f  ν ln 1 K σ B ( K , ˆ f , T ) = � ˆ � ˆ  · � ·  � � � 1 + (1 − β ) 2 + (1 − β ) 4 x ( z ) f f ln 2 ln 4 + · · · 24 1920 K K � � 1 + (1 − β ) 2 α 2 f ) (1 − β ) / 2 + 2 − 3 ρ 2 f ) 1 − β + 1 ρβνα ν 2 · T + · · · , ( K ˆ ( K ˆ 24 4 24 where the following new expression for z is considered: � f 1 − β − K 1 − β � ˆ ν z = , 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

SABR model - Approx. implied volatility � � K � � K � � f , T ) = 1 σ B ( K , ˆ + A 2 ln 2 1 + A 1 ln + BT , ˆ ˆ ω f f where the coefficients A 1 , A 2 and B are given by − 1 = 2(1 − β − ρνω ) , A 1 1 � (1 − β ) 2 + 3 ν 2 ω 2 � 2 − 3 ρ 2 � � � � = (1 − β ) − ρνω + , A 2 12 (1 − β ) 2 ω + 2 − 3 ρ 2 ω 2 + βρν 1 1 ν 2 , = B 24 4 24 and the value of ω is given by ˆ f 1 − β ω = . α ´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 8 / 25

SABR model - Parameters 0.4 0.8 α = 0.25 β = 1 α = 0.5 β = 0.75 0.7 0.35 α = 0.75 β = 0.5 α = 1 β = 0.25 α = 1.25 0.6 β = 0 0.3 0.5 0.25 0.4 0.2 0.3 0.15 0.2 0.1 0.1 0.05 0 80 85 90 95 100 105 110 115 120 80 85 90 95 100 105 110 115 120 (e) α > 0, the volatility’s reference level. (f) 0 ≤ β ≤ 1, the variance elasticity. 1 0.14 ν = 0.5 ρ = −1 0.9 ν = 1 ρ = −0.5 0.12 ν = 1.5 ρ = 0.0 0.8 ν = 2 ρ = 0.5 0.1 ν = 2.5 ρ = 1 0.7 0.08 0.6 0.06 0.5 0.04 0.4 0.02 0.3 0 0.2 −0.02 0.1 0 −0.04 80 85 90 95 100 105 110 115 120 80 85 90 95 100 105 110 115 120 (g) ν > 0, the volatility of the volatility. (h) − 1 ≤ ρ ≤ 1, the correlation coefficient. SABR parameters ´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 9 / 25

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 V market ( K j , ˆ f , T i ) ≈ V sabr ( K j , ˆ f , T i ) 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

SABR model - Calibration example 21 21 σ model σ model 20.5 σ market 20.5 σ market 20 20 19.5 19.5 19 19 σ impl σ impl 18.5 18.5 18 18 17.5 17.5 17 17 16.5 16.5 16 16 80 85 90 95 100 105 110 115 120 80 85 90 95 100 105 110 115 120 strike (% of spot) strike (% of spot) (i) 3 months maturity. (j) 6 months maturity. 21 21 σ model σ model 20.5 σ market 20.5 σ market 20 20 19.5 19.5 19 19 σ impl σ impl 18.5 18.5 18 18 17.5 17.5 17 17 16.5 16.5 16 16 80 85 90 95 100 105 110 115 120 80 85 90 95 100 105 110 115 120 strike (% of spot) strike (% of spot) (k) 12 months maturity. (l) 24 months maturity. ´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 11 / 25

SABR model - Drawback 0.22 0.21 0.2 σ market 0.19 0.18 0.17 0.16 90% 95% 3 m 100% 6 m 105% 12 m 110% 24 m Maturities Strikes (% spot) Figure: Market volatility surface. ´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 12 / 25

SABR model - Drawback 21 21 σ model σ model 20.5 σ market 20.5 σ market 20 20 19.5 19.5 19 19 σ impl σ impl 18.5 18.5 18 18 17.5 17.5 17 17 16.5 16.5 16 16 80 85 90 95 100 105 110 115 120 80 85 90 95 100 105 110 115 120 strike (% of spot) strike (% of spot) (a) 3 months maturity. (b) 6 months maturity. 21 21 σ model σ model 20.5 σ market 20.5 σ market 20 20 19.5 19.5 19 19 σ impl σ impl 18.5 18.5 18 18 17.5 17.5 17 17 16.5 16.5 16 16 80 85 90 95 100 105 110 115 120 80 85 90 95 100 105 110 115 120 strike (% of spot) strike (% of spot) (c) 12 months maturity. (d) 24 months maturity. ´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 13 / 25

Dynamic SABR model dynamic SABR model dF t = α t F β F 0 = ˆ t dW 1 t , f d α t = ν ( t ) α t dW 2 t , α 0 = α Forward, F t . Volatility, α t . dW 1 t y dW 2 t , correlated geometric brownian motions: dW 1 dW 2 = ρ ( t ) dt Inicial values: S 0 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

Dynamic SABR model dynamic SABR model dF t = α t F β F 0 = ˆ t dW 1 t , f d α t = ν ( t ) α t dW 2 t , α 0 = α Forward, F t . Volatility, α t . dW 1 t y dW 2 t , correlated geometric brownian motions: dW 1 dW 2 = ρ ( t ) dt Inicial values: S 0 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

Dynamic SABR model - Approx. implied volatility � � K � � K � � f , T ) = 1 σ B ( K , ˆ + A 2 ( T ) ln 2 1 + A 1 ( T ) ln + B ( T ) T , ˆ ˆ ω f f where β − 1 + η 1 ( T ) ω A 1 ( T ) = , 2 2 (1 − β ) 2 + 4 ν 2 � η 2 2 ( T ) − 3 η 2 � 1 ( T ) + 3 1 ( T ) + 1 − β − η 1 ( T ) ω ω 2 , A 2 ( T ) = 12 4 24 � (1 − β ) 2 + 2 ν 2 2 ( T ) − 3 η 2 � 1 + ωβη 1 ( T ) 2 ( T ) ω 2 B ( T ) = , ω 2 24 4 24 with � T � T 1 ( T ) = 3 2 ( T ) = 6 ν 2 ( T − t ) 2 ν 2 ( t ) dt , ν 2 ( T − t ) t ν 2 ( t ) dt , T 3 T 3 0 0 � T � T � t �� s � 2 η 1 ( T ) = 2 2 ( T ) = 12 η 2 ( T − t ) ν ( t ) ρ ( t ) dt , ν ( u ) ρ ( u ) du dsdt . T 2 T 4 0 0 0 0 ´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 15 / 25

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 Piecewise ◮ ρ ( t ) = ρ 0 ◮ ρ ( t ) = ρ 0 , t ≤ T 0 ρ ( t ) = ρ 1 , t > T 0 ◮ ν ( t ) = ν 0 ◮ ν ( t ) = ν 0 , t ≤ T 0 ν ( t ) = ν 1 , t > T 0 ◮ α , β , ρ 0 , ν 0 , SABR model. ◮ α , β , ρ 0 , ν 0 , ρ 1 , ν 1 and T 0 Classical General ◮ ρ ( t ) = ( ρ 0 + q ρ t ) e − at + d ρ ◮ ρ ( t ) = ρ 0 e − at ◮ ν ( t ) = ( ν 0 + q ν t ) e − bt + d ν ◮ ν ( t ) = ν 0 e − bt ◮ α , β , ρ 0 , ν 0 , a and b ◮ α , β , ρ 0 , ν 0 , a , b , d ρ , d ν , q ρ and q ν . ´ Alvaro Leitao (Lecture group, CWI) SABR model November 18, 2013 16 / 25