parametric option pricing with chebyshev ... - acmac.uoc… · parametric option pricing = pop we...

Parametric Option Pricingwith Chebyshev Interpolation

Kathrin GlauTechnische Universitat Munchen

joint work with

Maximilian Gaß, Mirco Mahlstedt and Maximilian Mair

Stochastic Methods in Physics and Finance:July 24, 2015, Crete

Parametric Option Pricing = POP

Day-to-day tasks in the financial industry:

calibration, risk assessment, quotation

Respecting credit and liquidity risk results in nonlinear pricingequations, BSDEs and PIDEs. However, today...

...”it’s impossible to implement on large portfolios in an efficient way”...

D. Brigo, Conference Munich 2015

We need reliable complexity reduction techniques.

2


We exploit the recurrent nature of the option pricing problem inan efficient, reliable and general way.

Fast Fourier Transformfor plain vanilla prices — for varying strikes K .

Carr, Madan (1999), Raible (2000),Lee (2004), Lord, Fang, Bervoets and Oosterlee (2008),Feng and Linetsky (2008), Kudryavtsev and Levendorskiy (2009),Boyarchenko and Levendorskiy (2014),. . .

3


BS σ0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

pri

ce

0

0.2

0.4

0.6N = 1

BS price

Cheby approx

BS σ0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

∆ p

rice

-0.02

0

0.02

Figure : Call prices in the BS model4


BS σ0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

pri

ce

0

0.2

0.4

0.6N = 2

BS price

Cheby approx

BS σ0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

∆ p

rice

-0.01

0

0.01



BS σ0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

pri

ce

0

0.2

0.4

0.6N = 3

BS price

Cheby approx

BS σ0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

∆ p

rice

×10-3

-2

0

2



BS σ0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

pri

ce

0

0.2

0.4

0.6N = 4

BS price

Cheby approx

BS σ0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

∆ p

rice

×10-3

-2

0

2



BS σ0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

pri

ce

0

0.2

0.4

0.6N = 5

BS price

Cheby approx

BS σ0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

∆ p

rice

×10-3

-2

0

2



BS σ0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

pri

ce

0

0.2

0.4

0.6N = 6

BS price

Cheby approx

BS σ0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

∆ p

rice

×10-3

-1

0

1



BS σ0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

pri

ce

0

0.2

0.4

0.6N = 7

BS price

Cheby approx

BS σ0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

∆ p

rice

×10-4

-5

0

5



BS σ0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

pri

ce

0

0.2

0.4

0.6N = 8

BS price

Cheby approx

BS σ0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

∆ p

rice

×10-4

-2

0

2



BS σ0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

pri

ce

0

0.2

0.4

0.6N = 9

BS price

Cheby approx

BS σ0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

∆ p

rice

×10-4

-2

0

2



BS σ0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

pri

ce

0

0.2

0.4

0.6N = 10

BS price

Cheby approx

BS σ0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

∆ p

rice

×10-4

-2

0

2



BS σ0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

pri

ce

0

0.2

0.4

0.6N = 11

BS price

Cheby approx

BS σ0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

∆ p

rice

×10-4

-1

0

1



BS σ0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

pri

ce

0

0.2

0.4

0.6N = 12

BS price

Cheby approx

BS σ0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

∆ p

rice

×10-5

-5

0

5



Crucial observations:

• The same pricing problem needs to be solved for a large set ofdifferent parameter values: POP Parametric Option Pricing.

• Model prices typically are continuous or even smooth in theparameter values.

We exploit the functional dependence

x 7→ Pricex , x = parameters

and approximate this function by interpolation.

Related literature: Interpolation of Black Scholes’s call prices inthe volatility, Pistorius, Stolte (2012)

17


0.5

maturity T

1

1.5

2108

6

strike K

42

0

2

3

4

5

0

1

Price

T,K

Figure : Call prices in the CGMY model18

Chebyshev Interpolation

19

Chebyshev Interpolation Points

-1 -0.5 0 0.5 1

0

0.2

0.4

0.6

0.8

1

Chebyshev nodes for D=1

20

Chebyshev Polynomial Interpolation for POP

We approximate

K 7→ PriceK = E[f K (X )

]by Chebyshev polynomial interpolation for K = [−1, 1]

PriceK ≈ 2∑

0≤j≤N

′cj(Price)Tj(K ),

with

nodes Kk := cos(π 2k+1

2N+2

)Chebyshev polynomials Tj(K ) := cos

(j arccos(K )

)coeffients cj(Price) :=

∑′0≤k≤N PriceKk cos

(jπ 2k+1

2N+2

).

∑′ indicates that the first summand is multiplied with 1/2.

21

Accuracy of Chebyshev Interpolation

By implementing Chebyshev interpolation Platte & Trefethen (2008) aim

. . . “to combine the feel of symbolics with the speed of numerics.”

<

i<

−1 1

ab

a + b = ρ

Theorem (Bernstein (1912), Trefethen (2013))

If g : [−1, 1]→ R is analytic and bounded in the ellipse with foci ±1 andsemiaxis lengths summing to %, then for IN(g)(x) :=

∑′0≤j≤N cj(g)Tj(x),

‖g − IN(g)‖∞ ≤ C%−N

%− 1.

22

Tensorized Chebyshev Interpolation

pk1-1 -0.5 0 0.5 1

pk2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Chebyshev nodes for D=2

Figure : A set of d-variate Chebyshev points x ∈ [−1, 1]d for d = 2 andN = 20 as defined in (??). Comparing the arrangement of points xi ford = 2 to the arrangement for d = 1 as shown in Figure ??, thetensorizing structure is clear to see.

The basis functions Tj , j ∈ Nd0 , possess a tensorizing structure, and

can thus be expressed by their univariate siblings defined in (??),

T(j1,...,jd )(x1, . . . , xd) =d∏

i=0

Tji (xi ). (1)

23


We approximate

x = (K ,T , p) 7→ PriceK ,T ,p := E[f K (X p

T )]

by tensorized Chebyshev polynomial interpolation for x ∈ [−1, 1]D

IN1...ND(Price)(x) :=

N1∑j1=0

. . .

ND∑jD=0

c(j1,...,jD)

D∏i=1

Tji (xi ),

with coefficients

c(j1,...,jD ) =( D∏

i=1

21{ji>0}

(Ni + 1)

) N1∑k1=0

. . .

ND∑kD=0

Pricex(k1,...,kD )

D∏i=1

cos

(jiπ

2ki + 1

2N + 2

).

24


For % = (%1, . . . , %D) ∈ (1,∞)D let the tensorized Bernstein ellipse

B([−1, 1]D , %) = B([−1, 1]D , %1)× . . .× B([−1, 1]D , %D).

Theorem (Gaß, G., Mahlstedt, Mair (2015))

If g has an analytic and bounded extension to B([−1, 1]D , %) forsome parameter vector % ∈ (1,∞)D , then∥∥g − IN(g)

∥∥∞ ≤ C%−N ,

where % = min1≤i≤D

%i and N = min1≤i≤D

Ni .

Proof by a generalization of the univariate case by induction over D.

25


The error decay is of order O(%M

1/D)in the number M of degrees

of freedom.

The result naturally extends to functions

f : P → R

with hyperrectangular domain P. We call B(P, %) theappropriately transformed tensorized Bernstein ellipse.

26

Application to POP

27

Chebyshev Interpolation Applied to POP

Precomputation phase:

Determine the approximate option prices

Pricex=(K ,T ,p)

at the Chebyshev nodes xk1,...,kD = (x1k1, . . . , xDkD ) e.g. with

• Fourier techniques• Partial (integral) differential equation methods• Monte Carlo

Evaluation phase:

Evaluate the approximate polynomial

IN1...ND(Price)(x) :=

N1∑j1=0

. . .

ND∑jD=0

c(j1,...,jD)

D∏i=1

Tji (xi ),

with explicit coefficients.28


0.5

maturity T

1

1.5

2108

6

strike K

42

0

2

3

4

5

0

1

Price

T,K

Figure : Call prices in a CGMY model, N1 = N2 = 15 Chebyshev nodes29


0.5

maturity T

1

1.5

2108

6

strike K

42

0

0

2

-4

-2

4

×10-5

∆ P

rice

T,K

Figure : Error plot for N1 = N2 = 15, call prices in a CGMY model30


strike K

1 2 3 4 5 6 7 8 9 10

Price

K

0

1

2

3

4

5

Option price

Cheby price (N=20)

strike K

1 2 3 4 5 6 7 8 9 10

∆ P

rice

K

×10-6

-5

0

5

Abs. error

Figure : Price and error plot, N=20, call prices in a CGMY model.31

Conditions on Options and Models for

(Sub)Exponential Convergence

32

Convergence of Chebyshev Interpolation for POP

Task: Determine accessible sufficient conditions for

p = (K ,T , π) 7→ PriceK ,T ,π := E[f K (Xπ

T )]

for p = (K ,T , π) ∈ P to have an analytic bounded extension tosome generalized Bernstein ellipse.

Observe in PriceK ,T ,p = E[f K (X p

T )]

typically

K 7→ f K (X pT ) e.g. K 7→ (X p

T − K )+

is not even differentiable.

33


Our approach: Fourier representation of option prices,

PriceK ,T ,p =1

(2π)d

∫Rd+iη

f K (−z)ϕT ,p(z) dz

with the characteristic functions f K of f K and ϕT ,p of X pT .

34

Conditions

Let the parameter set P = P1 × P2 ⊂ RD be a hyperrectangle.Let % ∈ (1,∞)D with %1 := (%1, . . . , %m) and %2 := (%m+1, . . . , %D)and let weight η ∈ Rd .

(A1) Integrability of x 7→ e〈η,x〉 f K (x) for all K ∈ P1.

(A2) Analyticity of K 7→ f K (z − iη) in B(P1, %1) for all z ∈ Rd .

(A3) Exponential moment E(

e−〈η,XpT 〉)<∞ for all (T , p) ∈ P2.

(A4) Analyticity of (T , p) 7→ ϕT ,p(z + iη) in B(P2, %2) for all z ∈ Rd .

(A5) Uniform bound: There exists h ∈ L1(Rd) such that

max(K ,T ,p)∈B(P,%)

∣∣f K (−z − iη)ϕT ,p(z + iη)∣∣ ≤ h(z) for all z ∈ Rd .

35


PriceK ,T ,p = E[f K (X p

T )]

Proposition (Eberlein, G., Papapantoleon (2010))

Assume (A1), (A3) and (A5). Then each PriceK ,T ,p = E[f K (X p

T )]

has the Fourier representation

Price(K ,T ,p) =1

(2π)d

∫Rd+iη

f K (−z)ϕT ,p(z) dz .

36


Theorem (Gaß, G., Mahlstedt, Mair (2015))

Let % ∈ (1,∞)D and weight η ∈ Rd . Conditions (A1)–(A5) imply

max(K ,T ,p)∈P

|Price(K ,T ,p) − IN(Price(·))(K ,T , p)| ≤ C%−N ,

where % = min1≤i≤D

%i and N = min1≤i≤D

Ni .

37

Examples investigated in more detail

Models

• Black Scholes, Merton, NIG, CGMY,. . .

• time-inhomogeneous Levy

• Heston, affine (jump) models

Payoff profiles

• European: call, put, digital down&out, power,. . .

• Baskets: call on basket, call on minimum,. . .

38

Detailed analysis in Levy and affine models

39

European options in Levy models

Spt = S0 eL

pt with Lp a Levy process and special semimartingale for

varying parameters p = (b, σ)

E(

eizLpt)

= etψp(z),

ψp(z) =σ2z

2+ ibz +

∫R

(eizx −1− izx

)F (dx)︸︷︷︸

=:ψ(z)

.

Martingale condition

b = b(r , σ) = r − σ2

2−∫R

(ex −1− x

)F (dx).

Fair value of a European option

Price(r ,K ,S0,T ,p) = e−rT E(f K (S0 eL

pT )).

40

European options in Levy models

Let P = [K ,K ]× [T ,T ]× [b, b]× [σ, σ] with T > 0.

Corollary (Gaß, G., Mahlstedt, Mair (2015))

Let η ∈ R, ρ1 > 1. Assume (A2), that∫|x|>1

(e−ηx ∨ ex)F (dx) <∞ and

(A1’) supK∈B([K ,K ],%1)

∥∥ e〈·,η〉 f K∥∥L1(Rd )

<∞ and∣∣∣f p1 (−z − iη)

∣∣∣ ≤ c1ec2|z|

for some c1, c2 > 0.

If additionally

(i) σ > 0, or

(ii) there exist α ∈ (1, 2] and C1,C2 > 0 such that

<(ψ)(z + iη) ≤ C1 − C2|z |α for all z ∈ R,

then there exist constants C > 0 and % > 1 such that for N = min1≤i≤4

Ni ,

max(K ,T ,b,σ)∈P

∣∣Price(K ,T ,b,σ) − IN(Price(·))(K ,T , b, σ)∣∣ ≤ C%−N .

41

Basket Options in Affine Models

Parametrized affine process X π′with state space D ⊂ Rd , π′ ∈ Π′

and with C-valued ϕπ′

and Cd -valued φπ′

solutions of generalizedRicatti equations satisfying

ϕ(t,x ,π′)(z) = E(

ei〈z,Xπ′t 〉∣∣Xπ′

0 = x)

= eϕπ′ (t,iz)+〈φπ′ (t,iz),x〉, (2)

for every t ≥ 0, z ∈ Rd and x ∈ D.

Fair value of a European option

Price(K ,T ,x ,π′) = E(f K (X π′

T )|Xπ′0 = x

)(3)

where f K is a parametrized family of measurable payoff functionsf K : Rd → R+ for K ∈ P1.

42

Basket Options in Affine Models

Corollary (Gaß, G., Mahlstedt, Mair (2015))

Assume (A1’), (A2), (A3’) for η ∈ Rd and hyperrectangular P ⊂ RD. Let(i) for every p2 = (t, x , π′) ∈ P2 ⊂ RD−m and every z ∈ R+ iη,

ϕp2=(t,x,π′)(z) = E(

ei〈z,Xπ′t 〉∣∣Xπ′

0 = x)

= eφπ′ (t,iz)+〈ψπ

′(t,iz),x〉,

(ii) (t, π′) 7→(φπ′(t, iz − η), ψπ

′(t, iz − η)

)has an analytic extension to

a Bernstein ellipse B(Π′, %′) with %′ ∈ (1,∞)D−m for every z ∈ Rd ,

(iii) there exist α ∈ (0, 2] and C1,C2 > 0 such that for everyp2 = (t, x , π′) ∈ P2 and every z ∈ R,

<(φπ′(t, iz) + 〈ψπ

′(t, iz), x〉

)(z + iη) ≤ C1 − C2|z |α.

Then there exist constants C , % > 0 such that

maxp∈P1×P2

∣∣Pricep − IN(Price(·))(p)∣∣ ≤ C%−min1≤i≤D Ni .

The existence of exponential moments of affine processes is investigatedin [?] where moreover criteria are provided under which formula (2) andthe related generalized Riccati system can be extended to complexexponential moments z ∈ Cd . The question has already been treated forimportant special cases where more explicit conditions are available [?]treat affine diffusions and [?] investigate affine processes with killingwhen the jump measures possess exponential moments of all orders.

43

Numerical Results

44

Numerical Experiments: European Call

Fixed strike, K = 1

p2 K ∈ T ∈

BS σ = 0.2 [1, 10] [0.5, 2]

Merton σ = 0.15, [1, 10] [0.5, 2]α = −0.04,β = 0.02,λ = 3

CGMY C = 0.6, [1, 10] [0.5, 2]G = 10,M = 28,Y = 1.1

Heston T = 2, [1, 10] v0 ∈ [0.12, 0.42]κ = 1.5,θ = 0.22,σ = 0.25,ρ = 0.1

45


2

T

1.5

Chebyshev price error, BS

10.50

5K

-1

0

1

10

×10-4

∆Price

2

T

1.5

Chebyshev price error, Merton

10.50

5K

-5

0

5

10

×10-4

∆Price

2

T

1.5

Chebyshev price error, CGMY

10.50

5K

1

-1

0

10

×10-6

∆Price

10

K

Chebyshev price error, Heston

50

0.050.1

v0

0.15-5

0

5

×10-5

∆Price

Figure : Absolute pricing error. We compare the Chebyshev interpolationwith N1 = N2 = 20 to classic Fourier pricing by numerical integration.

46


Chebyshev N

0 5 10 15 20 25 30 35 40 45 50

error

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

Chebyshev error decay

εL∞ (BS)

εL2 (BS)

εL∞ (Merton)

εL2 (Merton)

εL∞ (CGMY)

εL2 (CGMY)

εL∞ (Heston)

εL2 (Heston)

Figure : Convergence study for the BS, Merton, CGMY, Heston model

47

Numerical Experiments: Digital Down&Out

Chebyshev N

0 5 10 15 20 25 30 35 40 45 50

error

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

Chebyshev error decay

εL∞ (BS)

εL2 (BS)

εL∞ (Merton)

εL2 (Merton)

εL∞ (CGMY)

εL2 (CGMY)

εL∞ (Heston)

εL2 (Heston)

Figure : Convergence study for prices for the BS, Merton, CGMY, Heston

48

Numerical Experiments:Basket, Lookback, Barrier

Basket option for d underlyings

f (S1(T ), . . . ,Sd(T )) =

(( 1

d

d∑j=1

Sj(T ))− K

)+

.

Lookback option for d underlyings, S j(T ) := maxt≤T Sj(t),

f(S(t)0≤t≤T

)=

(( 1

d

d∑j=1

S j(T ))− K

)+

.

Barrier option for d underlyings, down&out, S j(T ) := mint≤T

Sj(t),

f(S(t)0≤t≤T

)=

(( 1

d

d∑j=1

S j(T ))− K

)+

· 1{S j (T )≥80, j=1,...,d}.

49

Numerical Experiments:Basket, Lookback, Barrier, American Put

r = 0, Free parameters K ∈ [83.33, 125] and T ∈ [0.5, 2]

Model fixed parameters:p1 p2

BS Sj,0 = 100, σj = 0.2

Heston Sj,0 = 100, κj = 2,θj = 0.22,σj = 0.3,ρj = −0.5,vj,0 = 0.22

Merton Sj,0 = 100, σj = 0.2,αj = −0.1,βj = 0.45,λj = 0.1

Monte Carlo simulation of 1 Mio sample paths, antithetic variates, 400time steps per year. 95% confidence bound for a price level of 10.

50

Numerical Experiments:Basket, Lookback, Barrier, American Put

N = 30 Chebyshev nodes in each varying parameter

Model Option d εL∞ MC price CB price

BS Basket 5 2.0 · 10−4 10.2988 10.2986Heston 1.4 · 10−3 2.8189 2.8203Merton 2.4 · 10−3 4.734 4.7364

BS Lookback 5 5.3 · 10−3 18.8028 18.7975Heston 3.1 · 10−3 24.1618 24.1587Merton 5 · 10−3 21.275 21.28

BS Barrier 5 7.4 · 10−3 10.0076 10.0149Heston 4.5 · 10−3 5.4898 5.4942Merton 7.0 · 10−3 9.2406 9.2476

BS American 1 8.6 · 10−3 9.5515 9.5602

51

Numerical Experiments: Digital Down&Out

2

T

1.5

Chebyshev price error, BS

10.50

5K

-1

1

0

10

×10-3

∆Price

2

T

1.5

Chebyshev price error, Merton

10.50

5K

2

-2

0

10

×10-3

∆Price

2

T

1.5

Chebyshev price error, CGMY

10.50

5K

0

2

-210

×10-6

∆Price

10

K

Chebyshev price error, Heston

50

0.050.1

v0

0.15-5

0

5

×10-4

∆Price

Figure : Absolute pricing error for a European digital down&out option

52

Applications and further developments

53

Benefits

Accurate Acceleration in models with low number of parameter

• PricingPrecomputation of prices once with a fine resolution, thenalways a fast realization available.

• CalibrationInterpolation of the objective function for fast optimization.

• Hedging, risk assessment, parameter uncertainty,...

Gain of freedom in modeling with a low number of parameters:

From ”For liquid options a fast pricer (Fourier-representation) isnecessary.”To ”An accurate pricer is available and the parameterdependence is analytic.”Thus one may include more realistic features, e.g. dividends.

54

Contribution

• Propose Chebyshev interoplation for parametric option pricing

• Show that the method is applicable for a variety of modelsand option types.(a) Theoretically:

- Exponential convergence for analytic parameter dependence.- Explicit conditions on the Fourier transform of payoff and

distribution.- Examples

(b) Numerically:

- Experimental order of convergence for Call and digital in BS.Heston, Merton, CGMY model. (CI combined with Fouriermethod)

- (Path-dependent) multivariate options: CI combined withMonte-Carlo and Finite Differences.

55

Outlook

56

Limitations of Chebyshev Interpolation

Methodological constraints

• Curse of dimensionality: degree of freedom of order ND forD-dimensional parameter space

• Requirement: Tensor structure of parameter space!

Realistic requirements

• Dimension of the parameter space is typically large.

• Parameter constraints often do not result inhyperrectangulars, e.g. entries covariance matrices.

How to find a generic interpolation that is flexible in theparameter space?

57

Literature

S. Bernstein. Sur l’ordre de la meilleure approximation des fonctionscontinues par des polynomes de degre donne. Acad. Royale de Belgique.Classe d. sc. Memoires. Coll. in-4”. Hayez, imprimeur des academiesroyales, 1912.

S. Boyarchenko and S. Levendorskiy. Efficient variations of the Fouriertrans- form in applications to option pricing. Journal of ComputationalFinance, 18(2), 2014.

P. Carr and D. B. Madan. Option valuation and the fast Fouriertransform. Journal of Computional Finance, 2(4): 61-73, 1999.

E. Eberlein, K. Glau, and A. Papapantoleon. Analysis of Fouriertransform valuation formulas and applications. Applied MathematicalFinance, 17(3): 211-240, 2010.

L. Feng and V. Linetsky. Pricing Discretely Monitored Barrier Optionsand Defaultable Bonds in Levy Process Models: A Fast Hilbert TransformApproach. Mathematical Finance, 18(3): 337-384, 2008.

M. Gaß, K. Glau, M. Mahlstedt, M. Mair. Chebyshev Interpolation forParametric Option Pricing. Preprint on Arxiv, 2015

58

Literature

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