Pricing and calibration in local volatility models via fast quantization

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Pricing and calibration in local volatility models via fast quantization

Lucio Fiorin

Universit`a degli studi di Padova

Parma, 29

^th

January 2015.

Joint work with Giorgia Callegaro and Martino Grasselli

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Quantization: a brief history

Birth: back to the 50’s, due to the necessity to optimize signal transmission, by appropriate discretization procedures;

Applications: information theory, cluster analysis, pattern and speech recognition, numerical integration and numerical probability (90’s);

Idea: approximating a signal admitting a continuum of possible values, by a signal that takes values in a discrete set;

Two types (probability):

Vector quantization random variables;

Functional quantization stochastic processes;

How: numerical procedures mostly based on stochastic optimization algorithms very time consuming.

Lucio Fiorin Universit`a degli studi di Padova

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Quantization: technical introduction

Given an R

^d

-valued random variable X on (Ω, A, P), X ∈ L

^r

, quantizing X on a grid Γ = (x

1

, . . . , x

N

) consists in projecting X on Γ, following the closest neighbor rule.

A grid Γ

^?

minimizing the L

^r

−mean quantization error

||X − Proj

_Γ

(X )||

_r

= || min

_{1≤i ≤N}

|X − x

ⁱ

| ||

_r

over all the grids with size at most N is the L

^r

-optimal quantizer.

The projection of X on Γ

^?

, Proj

_Γ?

(X ) is called the quantization of X :

Proj

_Γ^?

(X ) =

N

X

i =1

x

i

11

_C_i_(Γ^?₎

(X )

where C

_i

(Γ

^?

) ⊂ {ξ ∈ R

^d

: ||ξ − x

_i

|| = min

_{1≤j ≤N}

||ξ − x

_j

||}, is

called the Voronoi partition, or tessellation induced by Γ

^?

.

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Figure: Voronoi diagram for the optimal grid of N (0, I

2

).

The L

^r

−mean quantization error goes to zero as the grid size N → +∞ and the convergence rate is rules by the so-called Zador Theorem.

From a numerical point of view, finding an optimal quantizer may be a very challenging and time consuming task. This motivates the introduction of sub-optimal criteria: stationary quantizers.

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Stationary quantizers

Definition: An N-quantizer Γ = {x

1

, · · · , x

_N

} inducing the quantization Proj

_Γ

(X ) of X is said to be stationary if

E [X |Proj

Γ

(X )] = Proj

_Γ

(X ).

Optimal quantizers are stationary;

Stationary quantizers ˜ Γ are critical points of the distortion function associated with Γ:

D(Γ) :=

N

X

i =1

Z

Ci(Γ)

|z − x

_i

|

²

d P

X

(z).

Stationary quantizers are interesting insofar they can be found through zero search recursive procedures like Newton’s

algorithm or fixed point procedures.

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“Step by step marginal quantization”: ideas

A new quantization approach recently introduced by Pag` es and Sagna, 2014.

Consider a continuous-time Markov process Y

dY

t

= b(t, Y

t

)dt + a(t, Y

t

)dW

t

, Y

0

= y

0

> 0, where W is a standard Brownian motion and a and b satisfy the usual conditions ensuring the existence of a (strong) solution to the SDE;

Given T > 0 and {t

0

, t

1

, . . . , t

M

}, with t

₀

= 0 and t

M

= T ,

∆ = t

_k

− t

_k−1

, k ≥ 1, the Euler scheme for the process Y is Y e

_t_k

= e Y

_t_k−1

+ b(t

_k−1

, e Y

_t_k−1

)∆ + a(t

_k−1

, e Y

_t_k−1

)∆W Y e

_t₀

= e Y

₀

= y

₀

where t

_k

= k∆ and ∆W := (W

tk

− W

_t_k−1

) ∼ N (0, ∆);

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KEY REMARK: for every k = 1, . . . , M L

Y e

tk

| Y e

tk−1

= x

∼ N m

_k−1

(x), σ

_k−1²

(x)

(1)

where

m

_k−1

(x ) = x + b(t

_k−1

, x )∆

σ

²_k−1

(x ) = [a(t

k−1

, x )]

²

∆.

IDEA: quantize recursively, using vector quantization, every marginal random variable e Y

tk

, given its (Gaussian) conditional distribution given e Y

tk−1

;

It can be seen (see Pag` es and Sagna, 2014) that the error

made by quantizing the Euler scheme can be controlled, under

some mild regularity assumptions on the process.

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“Step by step marginal quantization”: stationary quantizers The distortion function at time t

_k+1

, relative to e Y

t_k+1

, is

D

k+1

(x) =

N

X

i =1

Z

Ci(x)

(y

k+1

− x

_i

)

²

P

Y e

tk+1

∈ dy

_k+1

(2)

where N is the (fixed) size of the quantizer x = {x

₁

, x

₂

, . . . , x

_N

}.

We are looking for x ∈ R

^N

such that

∇D

_k+1

(x) = 0.

QUESTION: Is it possible to apply Newton-Raphson now?

ANSWER: NO! We do NOT know the distribution of e Y

_t_k+1

!

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Using the conditional distribution in (1) we have

P( e Y

tk+1

∈ dy

_k+1

) = Z

R

φ

_m_k_(y_k_),σ_k_(y_k₎

(y

k+1

) P( e Y

tk

∈ dy

_k

)dy

k+1

where φ

_m,σ

is the density function of a N (m, σ

²

).

Due to the discrete nature of the quantizer, the integral in (2) becomes a finite sum;

We deduce a recursive procedure to obtain the stationary (here optimal) quantizer at time t

_k+1

, based on the quantizer at time t

k

, k ∈ {1, . . . , M} (Y

0

= y

0

is not random);

The distorsion is continuously differentiable: it is possible to

compute the gradient and the Hessian matrix of the distortion

function Newton-Raphson faster computations wrt

stochastic algorithms.

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The Quadratic Normal Volatility model

Much attention in the financial industry due to its analytic tractability and flexibility (Blacher, 2001; Ingersoll, 1997; Lipton, 2002; Andersen, 2011):

dY

t

= (e

1

Y

_t²

+ e

2

Y

t

+ e

3

)dW

t

, Y

0

= y

0

> 0, (3) for some e

₁

, e

₂

, e

₃

∈ R, where W is taken under

the risk neutral measure.

Includes, as special cases, the Brownian motion, the geometric Brownian motion and the inverse of a three-dimensional Bessel process. We refer to (Andersen, 2011) and (Carr, Fisher and Ruf, 2013) for technical properties of the model.

Mimicking a quadratic spot volatility gives some chances to get an implied volatility curve that reproduces the smile and skew effects using a parsimonious number of parameters.

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Vanilla options in the QNV model

RE-PARAMETRIZATION : dY

_t

= σ

qY

_t

+ (1 − q)x

₀

+ 1

2 s (Y

_t

− x

₀

)

²

x

0

dW (t).

PRICING:

Closed-form solutions for vanilla derivatives available (see Andersen, 2011);

Solutions depend on the roots of the polynomial in (3);

Even the implementations of closed form solutions requires some care;

CALIBRATION: must allow for the possibility to switch from the real roots case to the complex roots case without

constraints;

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Numerical results: pricing of Vanilla options

Given stationary quantization grids, pricing is immediate: the price (with r = 0) of an European Vanilla Put option on Y with

maturity T and strike K , given an N-quantization grid

y = (y

₁

, . . . , y

_N

) at t = T and associated optimal quantizer b Y

_T

, is

E[(K − Y

T

)

⁺

] =

N

X

i =1

(K − y

i

)

⁺

P

Y b

T

= y

i

.

The dimension of every grid is 20 and we have 10 time steps;

Parameters as in Andersen, 2011;

Error measure: sum of the squared differences between implied volatilities (“Res-norm”) on 7 strikes, from 85% to 115% of the spot initial value (y

₀

= 100) and 6 maturities, from 2 months to 2 years.

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Results obtained using Matlab on a CPU 2.4 GHz and 8 Gb memory laptop.

Real roots Complex roots

Res. Norm. 3.5025e − 04 2.7504e − 04

Comp. Time (closed form) 0.0295 sec 9.8383 sec Comp. Time (quantization) 0.8804 sec 0.8338 sec

Figure: Quantization grids, an example.

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Numerical results: calibration

We work on European Vanilla Call-Put option on the Dax Index, as of 19 June 2014;

Calibration is done via a standard non-linear least-squares optimizer that minimizes

X

n

σ

^imp_{n, market}

− σ

^imp_n,model

2

.

Using closed form formulas, it turns out that the implied volatility smile produced by the market is fitted better when the two roots are complex.

Closed form formulas do not perform well for short maturities.

Short maturities from 2 to 5 months, long maturities from 1 to 3 years, 7 strikes.

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Short maturities Long maturities Closed formulas Res. Norm. - 5.6292e − 04

Comp. Time - 339.1503 sec

Quantization Res. Norm. 3.7074e − 04 4.2690e − 04 Comp. Time 141.6354 sec 168.3524 sec

Figure: Calibration via quantization: squared errors of the implied

volatilities. On the left long maturities, on the right short maturities.

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Numerical results: pricing of barrier options

Same model data as before, we fix T =

¹₃

and K = 100.

We compare the price of up-and-out put options obtained via the quantization method, with 100-dimensional quantizers, with Monte Carlo, with 5 ∗ 10

⁵

simulations. The time step equal to

₃₆₀¹

.

The benchmark price is computed via Monte Carlo, with 10

³

discretization points in the time grid and 10

⁶

simulations.

Benchmark price Quantization price Monte Carlo price

L = 101.25 122.83447 141.06537 144.33757

L = 102.5 205.96689 216.54927 220.69586

L = 103.75 264.11461 268.67753 273.11058

L = 105 299.71130 299.93827 304.81308

L = 106.25 318.74848 316.32067 321.23110

Computational time - 13.98614 sec 50.06883 sec

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Future developments

Increasing dimensionality: fast quantization in stochastic volatility models, like Heston, SABR, multi-Heston, Wishart;

Increasing derivatives’s complexity: pricing of (more) exotic derivatives;

Application to more general discretization schemes?

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Thank you for your attention