Pricing and risk factor evolution for equity and credit under Lévy-driven models

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Pricing and risk factor evolution for equity and credit under L´ evy-driven models

Marco de Innocentis

Credit Suisse and University of Leicester, United Kingdom

Cass Business School Financial Engineering Workshop London, February 4, 2015

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Objectives

1. Highlight increasing practical relevance of jump models in asset pricing and risk management.

2. Outline a method for fast pricing of vanilla options, exotic options, and CDS under L´evy-driven models.

3. Show how L´evy models are needed for risk factor simulation in counterparty risk engines.

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Disclaimer

The views expressed herein are those of the authors only,

no other representation should be attributed.

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Pricing and RFE for equity and credit under L´ evy models

Based on the following

1. M. Boyarchenko, M. de Innocentis and S. Levendorski˘i. Prices of barrier and first-touch digital options in L´evy-driven models, near barrier. International Journal of Theoretical and Applied Finance, 14(7):1045–1090 (2011).

2. M. de Innocentis and S. Levendorski˘i. Pricing discrete barrier options and credit default swaps under L´evy processes.

Quantitative Finance, 14(8):1337-1365 (2014).

3. M. de Innocentis, S. Levendorski˘i and F. Venturini. A new backtesting and risk factor evolution framework for CDS and CD

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Outline

1 Introduction

2 Pricing discretely monitored barrier options and CDS

3 A risk factor evolution model for credit

4 Bibliography

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Outline

1 Introduction

4 Bibliography

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Price comparison for DO barrier option

20000 2500 3000 3500

200 400 600 800 1000

S

V

NIG model Black-Scholes model asymptotic price

The price of a DO put option in the NIG and Black-Scholes (BS) models (NIG parameters from Jeannin and Pistorius, 2007, BS has the same instantaneous variance). The strike is K = 3500, the barrier is H = 2100, the time to maturity is T = 0.25 years, the riskless rate is 3%, and the underlying stock pays no dividends. For NIG, the WHF method of Boyarchenko and Levendorski˘i is used.

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Price comparison for barrier options: empirical evidence

Market prices vs. BS prices of no-touch binary options, from Clark (2011).

BS r ice [p u sp ide]

6 - %

3 - %

% 0

% 3

% 6

% 0

8 40% 0% 40% 80%

BS p ric downsid ]e[ e

Marketprice-BSprice

i B d/Of ef r Tr da de

Approxim ta t e r ne d

1

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The gap at the knock-out barrier

0.01 0.02 0.03 0.04 0.05

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

ln (S/H)

V/H

exact price asymptotic

(a) Price

0.01 0.02 0.03 0.04 0.05

-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

ln (S/H)

Relative error

(b) Relative Error

A large discontinuity at the boundary can occur, e.g. for pure KoBoL of order ν < 1 and positive drift. Example from Asmussen, Madan and Pistorius (2008), calibrated to vanilla options on General Motors: ν = 0.5, λ₋= −5.8031, λ₊= 1.0084, c = 0.2171, µ ≈ 0.2006, riskless rate r = 0.03, T = 1.5. The option parameters are the same as in the previous graph.

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The gap at the knock-out barrier

The effect occurs for pure jump processes when the drift is directed away from the barrier and the process is of finite variation.

If the process is of finite activity, then it is evident that the probability that the process remains above the barrier is separated from 0 at any spot level above the barrier.

If the process is of infinite activity, this is due to the fact that the probability distribution of the infimum process (at an exponentially distributed random time) has an atom at 0.

A qualitatively similar pattern can sometimes be observed in practice, e.g., when a trader who has sold a large position in one-touch options tries to

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An example from counterparty risk

Regulators require risk factor evolution models used in counterparty credit risk to pass historical backtesting. The predicted evolution of simulated risk factors is compared with the historical realisation using statistical tests, e.g. exception counting, Anderson-Darling, Cram´er-von Mises, etc.

We show how introducing jumps in a reduced form risk factor evolution model for CDS improves the backtesting results (see example below).

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Implications for pricing and risk factor simulation

Observed patterns in the prices of knock-out barrier options near the barrier agree with empirical evidence.

In particular, if the underlying is driven by a jump process, then Black-Scholes can underprice knock-out options near the barrier, and overprice them away from the barrier.

The latter result also applies in the case of structural models for credit defaults, with a pure diffusion model underpricing a long CDS (protection buyer) position and overpricing it near default.

The introduction of jumps in a reduced form simulation model for credit produces significant improvement in backtesting performance.

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Outline

1 Introduction

4 Bibliography

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Pricing discretely monitored barrier options and CDS

Joint work with Sergei Levendorski˘i (2014).

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Barrier options

A standard analytical tool to solve boundary problems for backward parabolic equations is to take the Laplace transform w.r.t. time, then use the Wiener-Hopf factorization (WHF) technique to calculate the result. At the last step, the inverse Laplace transform is applied.

In applications to finance, this scheme was realized by Boyarchenko and Levendorski˘i (2002) in analytical form.

In discrete time, for perpetual American options (Bermudans), the WHF was applied by Boyarchenko and Levendorski˘i in (2002).

Fusai et al. (2006, 2010) realized the complete scheme in discrete time, using the z-transform (analogue of the Laplace transform in discrete time) and its inverse.

An efficient numerical procedure for the continuous monitoring case was developed by Boyarchenko and Levendorski˘i (2009), and refined in subsequent papers.

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Barrier options

For some barrier options, the breaching of the barrier level is only observed at a finite set of times (almost always, once a day).

Known as discretely monitored, or discrete barrier options.

In the equity market, most barrier options are monitored

discretely. In addition, discrete monitoring is increasingly popular in FX for pairs against EUR, monitored upon daily ECB fixing at 14:15 CET, cf. Castagna (2009), Becker and Wystup (2009).

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L´ evy processes: general definitions

A L´evy process is a RCLL stochastic process with stationary and independent increments over non-overlapping time intervals.

Every L´evy process X = (X_t) has a characteristic exponent ψ. In 1D, this is defined by

E e^{i ξX}^t = e^−tψ(ξ),

where ξ ∈ R. The L´evy-Khintchine formula gives the following expression for ψ

ψ(ξ) = σ²

2 ξ²− i µξ + Z

R\{0}

(1 − e^{ix ξ}+ ix ξ1^|x|<1(x ))F (dx ), where F (dx ), the L´evy measure, satisfies ^R_R\{0}min{|x |², 1}F (dx ) < ∞.

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L´ evy processes: general definitions

E.g. for a KoBoL process¹, F (dx ) takes the form

F (dx ) = c₊x^−ν−1e^λ⁻^x1{x>0}dx + c−|x|^−ν−1e^λ⁺^|x|1{x<0}dx , where ν ∈ [0, 2). We will take c₊ = c− = c. The λ± are known as steepness parameters.

Other model classes: Normal Tempered Stable, with ν ∈ (0, 2) (the ν = 1 case being Normal Inverse Gaussian), Generalized Hyperbolic, β-family, etc.

The L´evy density has asymptotic form O(|x |^−ν−1) for x → 0, where ν is the order of the process. The process is of finite (infinite) variation if ν < 1 (ν ≥ 1).

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Model processes

We assume that

the option underlying follows an exponential L´evy process St = e^X^t, under a risk-neutral measure Q chosen for pricing.

The characteristic exponent ψ(ξ) admits the analytic

continuation into the complex plane with the cuts i (−∞, λ−] and i [λ₊, +∞), where λ−< −1 < 0 < λ₊, and satisfies certain growth conditions in this region. These are satisfied for most models used in finance.

In order for ψ to be the characteristic exponent under Q, it must satisfy the EMM condition: r + ψ(−i ) = q, where q is the dividend rate.

We take both r and q to be constant.

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IFT method for European options

The (inverse) Fourier transform (IFT) method for European options was introduced to finance by Heston (1993) and Eydeland (1994) for diffusion models.

It was later applied to wide classes of stochastic volatility models; in particular, the Lewis-Lipton pricing formula for puts and calls in the Heston model was derived by Lewis (2000) and Lipton (2002).

In the context of L´evy models, it was used by S. Boyarchenko and Levendorski˘i (1998) and Carr and Madan (1999) and, in the context of affine term structure models, by Duffie, Pan and Singleton (2000) and Chacko and Das (2000).

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IFT method for European options: L´ evy processes

The value function V (τ, X_t) of a European option (without barrier) with payoff at maturity G (X_T) and time to maturity τ = T − t is given by

V (τ, x ) = E[e^{−r τ}G (X_T)|X_t = x ].

If the Fourier transform bG (ξ) =R

Re^{−i ξx}G (x )dx of G satisfies the decay condition | bG (ξ)| = O(ξ^−1−δ), for some δ > 0, or its derivative does, then Fubini’s theorem applies, and we can write

V (τ, x ) = e^{−r τ} 2π E

Z

Im ξ=ω

e^iX^T^ξG (ξ)d ξb

X_t = x

= e^{−r τ} 2π

Z

Im ξ=ω

eix ξ−τ ψ(ξ)

G (ξ)d ξ.b

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IFT method for European options: L´ evy processes

Several modifications of the IFT method were introduced by S.

Boyarchenko and Levendorski˘i (2011).

Of these, parabolic IFT, which relies on deforming the contour of integration under a conformal map, has been shown to be the most efficient, especially in the case of short maturity options, cf. S.

Boyarchenko and Levendorski˘i (2011), de Innocentis (2011).

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IFT method for European options: OTM vs ITM

We consider only the case of put options².

Write x⁰ = log(S_t/K ) + µτ , where τ = T − t denotes time to maturity. We refer to a put option as

out-of-the-money (OTM) if x⁰ > 0, in-the-money (ITM) if x⁰ < 0, at-the-money (ATM) if x⁰ = 0.

2Calls can be treated as put options by using the change of variables η = i − ξ.

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IFT method for European options: parabolic IFT

Let x⁰ ≥ 0 (OTM/ATM case). For α ∈ [1, 2], consider the conformal map χ⁺:= χ⁺_α defined on the half-plane Im ξ < λ₊ by

χ⁺(ξ) = i λ₊− i(λ₊+ i ξ)^α

λ^α−1₊ := i λ₊− i λ^1−α₊ exp[α ln(λ₊+ i ξ)].

If α ∈ (1, 2), the image is the obtuse angle

{i λ₊+ z | z 6= 0, arg z 6∈ [π/2 − π(1 − α/2), π/2 + π(1 − α/2)]}.

For α = 2, the image is the complex plane with the cut i [λ+, +∞).

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IFT method for European options: parabolic IFT

With parabolic IFT, the map χ⁺_α is used to deform the contour of integration in the IFT integral for V_put(τ, x⁰)

V_put(τ, x⁰) = e^{−r τ} 2π

Z

L⁺_ω;χ

e^ix⁰^{ξ−τ ψ}⁰^(ξ)G (ξ)d ξ,b where L⁺_ω;χ is the image of Im ξ = ω ∈ (0, λ₊) under χ⁺_α, and ψ⁰(ξ) = i µξ + ψ(ξ).

Along the new contour, the integrand decays much more rapidly³.

3A similar idea was later used by S. Boyarchenko and Levendorski˘i (2013), M. Boyarchenko and Levendorski˘i (2014) and Levendorski˘i (2014) for the pricing of continuously monitored barrier options and CDS using the WHF.

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Backward induction for barrier options

Used by Eydeland (1994) in the diffusion framework.

Variations of this method for L´evy models were used by Feng and Linetsky (HT method, 2008), Lord, Fang, Bervoets and Oosterlee (convolution or CONV method, 2008), Fang and Oosterlee (cosine or COS method, 2009).

The time-0 price of a discretely monitored DO barrier option with terminal payoff G (XT), maturity T , monitoring interval ¯∆ = T /N, barrier H is given by

V (T , x ) = e^−rTE^x[1(h,∞)(X∆¯)1(h,∞)(X_{2 ¯}_∆) . . .1(h,∞)(X_T)G (X_T)], where x = ln S , h = ln H, E^x[·] = E[·|X = x ].

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Backward induction for barrier options

By the law of iterated expectations, this can be calculated as follows i. Calculate

V_N−1(x ) = e^{−r ¯}^∆1[h,∞)(x ) · E^x[1(h,∞)(X∆¯)G (X∆¯)]

as an ordinary European option.

ii. For s = N − 2, N − 3, . . . , 0, calculate

V_s(x ) = e^{−r ¯}^∆1[h,∞)(x ) · E^x[V_s+1(X∆¯)].

For model processes of order ν > 0, can show that V_s ∈ C^∞([h, ∞)).

Problem: we do not know the Fourier transforms of the functions V_s.

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Backward induction for barrier options

As in Eydeland (1994), we approximate V_s(x ) on a grid ~x = (x_j)^M_{j =1}, with x_j = h + (j − 1)∆, by using interpolation. Write

V_s(x ) ≈ V_s(x ) =

M−1

X

j =1 m

X

`=0

f_{j ,`}^m,s

x − x_j xj +1− xj

`

1[xj,xj +1),

We obtain

Vb_s(ξ) =

m

X

`=0

∆^−`

M

X

j =1

W_{j ,`}^m,se^−ix^j^ξ(−i ξ)^−`−1,

where the W_{j ,`}^m,s can be expressed in terms of the interpolation coefficients f^m,s.

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Backward induction for barrier options

We obtain for V_s(x_k)

V_s(x_k) ≈ e^{−r ¯}^∆ 2π

Z

Im ξ=ω

e^{i (x}^k^{+µ ¯}^{∆)ξ− ¯}^∆ψ⁰^(ξ)Vb_s+1(ξ)d ξ

=

M

X

j =1 m

X

`=0

∆^−`W_{j ,`}^m,s+1(f^m,s+1)e^{−r ¯}^∆ 2π × Z

Im ξ=ω

e^{i (x}^k^−x^j^{+µ ¯}^{∆)ξ− ¯}^∆ψ⁰^(ξ)(−i ξ)^−`−1d ξ.

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Backward induction for barrier options

By introducing the notation I_`(n) = e^{−r ¯}^∆

2π Z

Im ξ=ω

e^{i (n∆+µ ¯}^{∆)ξ− ¯}^∆ψ⁰^(ξ)(−i ξ)^−`−1d ξ, where ` = 0, 1, . . . , m, and ω ∈ (0, λ₊), we can write

V_s(x_k) =

M

X

j =1 m

X

`=0

∆^−`W_{j ,`}^m,s+1(f^m,s+1)I_`(k − j ).

The oscillatory integrals I_`(n) can be calculated very efficiently using parabolic IFT, and the double summation in the expression for V_s(x_k) can be computed using the fast convolution algorithm, used in M.

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Algorithm

In the case of constant monitoring interval ¯∆ and the same grid ~x used at each step of backward induction:

1. Calculate I0(n), n = 0, 1, . . . , M − 1, and I`(n),

n = 1 − M, 2 − M, . . . , M − 1, ` = 1, . . . , m, using parabolic IFT.

2. Calculate V_N−1 over the grid ~x using parabolic IFT.

3. In a cycle w.r.t s = N − 2, N − 3, . . . , 0, calculate an

approximation to V_s(x ) over the grid ~x using fast convolution.

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Error bounds

Three main sources of errors:

a. truncation in the state space at x = x_M b. interpolation

c. discretization and truncation of the IFT integral in the dual space.

As we shall see, in exponential L´evy models, the error of (a) decays exponentially and can be efficiently controlled independently of other errors. With some minor exceptions (VG, KoBoL of order ν close to zero), the error of (c) also decays exponentially or faster, and can be controlled easily.

Hence, the scheme behaves as if the interpolation were the only

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Error bounds: truncation error

Lemma 1

For any ω₋ ∈ (λ₋, 0) and ω₊ ∈ (0, λ₊), the truncation error in the state space, for the time-0 option price at X₀ = x , admits an upper bound via

(K − e^h)e^{−(r +ψ(i ω}⁺^))T· e^−ω⁺^(x^M^{−ln K )+ω}⁻^(x^M^−x) sup

τ ∈[0,T ]

e^{−τ (ψ(i ω}⁻^{)−ψ(i ω}⁺⁾⁾. Note that this estimate is independent of the number N of

monitoring dates.

If at least one of the steepness parameters λ−, λ₊ is not small in absolute value, then a moderate xM − x should suffice to satisfy a very small error tolerance.

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Error bounds: interpolation error

Lemma 2

If X is a model process of order ν > 0, then the total error in the time-0 option price due to piece-wise polynomial interpolation of order m admits the following approximate bound

Err_int.tot. ≤ e^−rTCm+1∆^m+1(N − 1)(K − H)||p_∆^(m+1)_¯ ||L1, where C₂ = 1/8, C₃ = 1/6, C₄ = 1/24.

where p∆¯ is the transition probability density of X .

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Error bounds: interpolation error

The following result gives a bound for the norm ||p_∆⁽ⁿ⁾||_L₁ Proposition 3

For model processes of order ν > 0, we have the following approximate bound for ||p^(m+1)_¯

∆ ||_L₁

||p_∆⁽ⁿ⁾_¯ ||_L₁ ≤ 2Γ(n/ν) (d₊⁰)^n/νπνD(n)

∆¯^−n/ν,

where, for n ∈ Z+,

D(n) = sup

φ∈(^0,min{^π2,_2ν^π})

(cos(φν))^n/νcos(φ − π/2).

For KoBoL of order ν > 0+, d₊⁰ = −2cΓ(−ν) cos(πν/2).

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Error bounds: interpolation error

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

x 10^-6 0

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

∆³

Maximum absolute error

Set A Set B

Maximum absolute error due to piece-wise quadratic interpolation in the interval 1.01 · H ≤ S ≤ K versus ∆³, where ∆ is the mesh in log-S space, for a

down-and-out put option with strike K = 100, barrier H = 80, T = 0.25, for

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Other methods

With the Hilbert transform (HT) method of Feng and Linetsky (2008), all calculations in the backward induction loop take place in the dual space.

The authors give a prescription for the numerical integration mesh, and accuracy depends on the number M of points in the dual space grid.

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Other methods

The COS method of Fang and Oosterlee (2009) relies on a cosine expansion of a truncated version of the terminal payoff. Accuracy depends both on the number N of terms in the cosine expansion, and on the interval (a, b) used for truncation in x -space at each step.

In turn, a and b are determined according to a = c1+ x − L

q

c2+√ c4

b = c1+ x + L q

c2+√ c4

where c₁, c₂, c₄ are the cumulants of X .

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Other methods

Fusai, Abrahams and Sgarra (2006) used the WHF to obtain a closed form expression for discretely monitored barrier options under the Black-Scholes model.

Green, Fusai and Abrahams (2010) used the WHF in the context of random walks to obtain pricing formulas for discretely monitored barrier options and first-touch digitals, generalizing the previous results to a wide class of L´evy processes.

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Barrier Options: Numerical Results (Set A, 1 Year)

S Price _abs _rel Time ∆ M ζ L N

Benchmark 80.89757 0.08130903 185.23

Quadratic 80.89757 0.08129447 1.46 · 10⁻⁵ 1.80 · 10⁻⁴ 0.051 0.005579

HT 80.89757 0.08221303 9.04 · 10⁻⁴ 1.11 · 10⁻² 0.468 5000 0.47278

HT 80.89757 0.08092782 3.81 · 10⁻⁴ 4.69 · 10⁻³ 6.801 50000 0.13465 HT 80.89757 0.0814066 9.76 · 10⁻⁵ 1.20 · 10⁻³ 130.7 1000000 0.02628

COS 80.89757 0.10289052 2.16 · 10⁻² 2.65 · 10⁻¹ 0.108 8 1000

COS 80.89757 0.0813639 5.49 · 10⁻⁵ 6.70 · 10⁻⁴ 0.107 15 1000

Benchmark 100 0.60133743 185.23 0.000223

Quadratic 100 0.60118634 1.51 · 10⁻⁴ 2.50 · 10⁻⁴ 0.051 0.005579

HT 100 0.60123454 1.03 · 10⁻⁴ 1.70 · 10⁻⁴ 0.109 1000 1.13742

COS 100 0.62570898 2.44 · 10⁻² 4.05 · 10⁻² 0.107 8 1000

COS 100 0.60136022 2.28 · 10⁻⁵ 4.00 · 10⁻⁵ 0.106 15 1000

Comparison of prices for a DO put option with barrier e^h= 80 strike K = 100, and maturity T = 1, for x = h + 50∆0and x = ln K , where ∆0= (ln K − h)/1000 ≈ 2.23 · 10⁻⁴. Time:

CPU time (seconds). For HT, we used the Feng-Linetsky prescription for ζ.

Parameters of KoBoL: λ−= −8, λ+= 9, ν = 1.2, second moment m2= 0.16, riskless rate r = 0.03 and dividend rate q = 0 (parameter set A, used by Boyarchenko and Levendorski˘i, 2011). absand reldenote absolute and relative errors, respectively. Benchmarks were calculated using cubic interpolation with state space mesh ∆0.

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Barrier Options: Numerical Results (Set B, 1 Year)

Benchmark 80.89757 0.34160881 193.74 0.000223

Quadratic 80.89757 0.34153123 7.76 · 10⁻⁵ 2.30 · 10⁻⁴ 0.223 0.001116 Quadratic 80.89757 0.34155933 4.940 · 10⁻⁵ 1.40 · 10⁻⁴ 0.061 0.002231

HT 80.89757 0.34038686 1.22 · 10⁻³ 3.58 · 10⁻³ 0.094 1000 5.69475

HT 80.89757 0.34136607 2.42 · 10⁻³ 7.10 · 10⁻⁴ 0.25 6000 2.72307

COS 80.89757 0.63866249 2.97 · 10⁻¹ 8.70 · 10⁻¹ 0.031 8 200

COS 80.89757 0.356945 1.53 · 10⁻² 4.49 · 10⁻² 0.031 15 200

COS 80.89757 0.34221689 6.08 · 10⁻⁴ 1.78 · 10⁻³ 0.046 20 300

Benchmark 100 1.90919403 193.74 0.000223

Quadratic 100 1.90929924 1.05 · 10⁻⁴ 6.00 · 10⁻⁵ 0.061 0.002231

HT 100 1.91088427 1.69 · 10⁻³ 8.90 · 10⁻⁴ 0.031 1000 5.69475

COS 100 1.28 · 10⁶³ 1.28 · 10⁶³ 6.70 · 10⁶² 0.049 8 400

COS 100 1.91185873 2.66 · 10⁻³ 1.40 · 10⁻³ 0.032 15 200

COS 100 1.91047424 1.28 · 10⁻³ 6.70 · 10⁻⁴ 0.024 20 100

Comparison of prices for a DO put option with barrier e^h= 80 strike K = 100, and maturity T = 1, for x = h + 50∆0and x = ln K , where ∆0= (ln K − h)/1000 ≈ 2.23 · 10⁻⁴. Time: CPU time (seconds). For HT, we used the Feng-Linetsky prescription for ζ. Parameters of KoBoL:

λ−= −60, λ+= 50, ν = 0.7, c = 4, with riskless rate r = 0.03 and dividend rate q = 0 (parameter set B, used both by Feng and Linetsky, 2008, and by Fang and Oosterlee, 2009).

absand reldenote absolute and relative errors, respectively. Benchmarks were calculated using cubic interpolation with state space mesh ∆0. Settings for quadratic interpolation: state space truncation xM= h + 1, and ζ = 5, Λ = 200 used for all applications of parabolic IFT.

Calculations carried out in MATLAB R2011b (win64), on Intel i5-2410M, 2.30 GHz 4 GB RAM, running Windows 7 Professional (64 bit).

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Relative errors of COS

82 84 86 88 90 92 94 96 98

10^-4 10^-2 10⁰ 10²

S

Relative error

1 month 3 months 6 months 1 year 2 years

(a) Parameter Set A

82 84 86 88 90 92 94 96 98

10^-2 10⁰ 10²

S

Relative error

1 month 3 months 6 months 1 year 2 years

(b) Parameter Set B

Logarithmic plots of relative errors of COS with N = 10⁶for the price of a DO option with H = 80, K = 100, daily monitoring, under KoBoL, over the interval

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Discretely Monitored Barrier Options and CDS:

Conclusions

Introduced a new method to price discretely monitored barrier options, CDS and defaultable bonds, using a backward induction procedure based on piece-wise polynomial interpolation of the option price at each step, fast convolution, and parabolic IFT.

An analysis of the errors incurred by approximations in both state and dual spaces shows that the only important source of error is the interpolation. Theoretical bounds were derived for this and other errors.

While COS can be faster for some choices of parameters, its accuracy depends on the choice of truncation interval as well as the number of terms in the cosine expansion.

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Discretely Monitored Barrier Options and CDS:

Conclusions

The prescriptions for L given by Fang and Oosterlee (2009) and Fang et al. (2009) for barrier options and CDS, respectively, can result in very large errors for daily monitoring, even when using the KoBoL parameters in the original studies.

As shown by S. Boyarchenko and Levendorski˘i (2011), an appropriate choice of truncation interval for COS and similar methods can be highly non-trivial, even in the simpler case of European options.

The accuracy of the HT method depends only on the number M of points in the dual space grid.

However, for typical choices of parameters which arise from

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Outline

1 Introduction

4 Bibliography

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A risk factor evolution model for credit derivatives

Work in progress in collaboration with Sergei Levendorski˘i

and Fabrizio Venturini.

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Counterparty credit risk

Counterparty credit risk (CCR) is the risk that a counterparty to an OTC transaction will fail to meet its obligations.

Key ingredients in CCR modelling are Loss given default (LGD)

Probability of default (PD) Exposure at default (EAD).

Under the Basel III Internal Model Method (IMM) approach, EAD can be computed using the output of the bank’s internal Monte Carlo model of future exposure.

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Counterparty credit risk

Regulators require IMM exposure models to pass backtesting at both risk factor evolution (RFE) and portfolio level.

Backtesting attempts to assess the predictive power of the

model by comparing its past predictions with realized

outcomes.

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RFE backtesting example

For a range of historical dates (e.g. 1 Jan 2008 - 31 Dec 2013) the RFE model is calibrated, e.g. at monthly intervals.

For each calibration, we calculate the model CDF of the realized values at different time horizon (e.g. 1 month, 3 months, 6 months, 1 year).

Using various statistical hypothesis tests (e.g. exception counting, Anderson-Darling, Cram´er-von Mises), we calculate the p-value corresponding to a given time horizon and assign a red/amber/green colour according to whether it lies in [0, 0.01), [0.01, 0.05), or [0.05, 1], respectively.

See Anfuso et al. (2013) and Kenyon and Stamm (2012) for details.

For a wide class of L´evy processes the CDF can be computed using the FFT (Feng and Lin, 2013; Ballotta and Kyriakou, 2014), or by using the parabolic IFT method of S. Boyarchenko and Levendorski˘i (see, e.g., M.

Boyarchenko, 2012).

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A credit RFE model

Objective: simulate the MtM of the bank’s CDS portfolio in the MC exposure engine.

Both reduced form and structural approaches have problems when the underlying process is a diffusion.

In structural models driven by diffusions defaults are always predictable.

Under reduced form models, defaults are typically not associated with sudden increases in spreads. Historical evidence shows that some defaults (e.g. Argentina, 2001) were preceded by a rally in credit spreads, while others were not.

In both cases, the situation can be improved by introducing jumps.

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A credit RFE model

We consider a reduced form evolution model driven by a diffusion process.

As will be seen, this produces a large number of exceptions in backtesting.

To address the impact of jumps, we compared the backtesting results from the original model with those obtained after replacing the underlying process with a superposition of a diffusion and a compound Poisson process P_t=PNt

i =0Y_i, where N_t is a Poisson process, and Y_i ∼ N(0, s²).

We fit the model to the same set of underlying CDS names with and without jumps, and assessed the backtesting results using exception counting at 1% and 99%, the Anderson-Darling (AD) and the Cram´er-von Mises (CvM) tests.

We calibrated the model each month end between January 2008 and June 2013 (66 months), and backtested survival probabilities with tenors between 1 year and 5 years, for 1 month, 3 months, 6 months and 1 year horizons.

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Backtesting results 1/5

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Backtesting results 2/5

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Backtesting results 3/5

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Backtesting results 4/5

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Backtesting results 5/5

Note that, while the inclusion of additional parameters generally decreases the in-sample calibration error, it can also lead to additional out-of-sample errors.

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Backtesting results: Name 11

In the picture above, the full lines represent the 1% and 99% percentiles of the distribution in the jump-diffusion case, the dashed lines above show corresponding percentiles in the no jump case, and the dots are the realized values.

The width of the distribution, measured by the difference between the 99% and the 1% percentile, is much larger in the jump-diffusion than in the pure diffusion case. This greatly reduces the number of backtesting exceptions.

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Analysis of worst performers: Name 13

In the case of Name 13, the problem arises due to extremely high levels of observed survival probabilities at the beginning and end of periods, as a result of

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Analysis of worst performers: Name 13

This causes the AD test to fail. Exception counts and CvM still pass.

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Analysis of worst performers: Name 21

In the case of Name 21, the volatility introduced by jumps produces an excess of

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Analysis of worst performers: Name 21

This causes distributional tests to fail, as the model is exceedingly conservative.

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Conclusions

Gaussian model fails RFE backtesting, especially at short horizons.

By replacing the process for the log-hazard rate with a superposition of a diffusion and a compound Poisson process, substantially thicker tails in the hazard rate and survival probability distribution can be achieved, in agreement with empirical evidence.

The model with jumps passes backtesting, apart from two names failing the AD test (exception counting is passed in all cases).

Work is under way to develop a robust calibration procedure for a

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Outline

1 Introduction

4 Bibliography

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Bibliography I

F. Anfuso, D. Karyampas, and A. Nawroth.

A sound Basel III compliant framework for backtesting Credit Exposure Models.

Working paper, May 2013.

Available at SSRN: http://papers.ssrn.com/abstract=2264620.

S. Asmussen, D. Madan, and M.R. Pistorius.

Pricing Equity Default Swaps under an approximation to the CGMY L´evy model.

Journal of Computational Finance, 11:79–93, 2008.

L. Ballotta and I. Kyriakou.

Monte carlo simulation of the CGMY process and option pricing.

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Bibliography II

C. Becker and U. Wystup.

On the cost of delayed currency fixing announcements.

Annals of Finance, 5:161–174, March 2009.

F. Black and J.C. Cox.

Valuing corporate securities: some effects of bonds indenture provisions.

Journal of Finance, 31:351–367, 1976.

M. Boyarchenko.

Fast simulation of L´evy processes, August 2012.

To appear in Quantitative Finance. Available at SSRN:

http://ssrn.com/abstract=2138661 or http://dx.doi.org/10.2139/ssrn.2138661.

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Bibliography III

M. Boyarchenko, M. de Innocentis, and S.Z. Levendorski˘i.

Prices of barrier and first-touch digital options in L´evy-driven models, near barrier.

International Journal of Theoretical and Applied Finance, 14(7):1045–1090, November 2011.

M. Boyarchenko and S.Z. Levendorski˘i.

Prices of barrier and first-touch digital options in L´evy-driven models.

International Journal of Theoretical and Applied Finance, 12(8):1045–1090, December 2009.

Refined and enhanced fast Fourier transform techniques, with an application to the pricing of barrier options.

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Bibliography IV

Ghost calibration and pricing barrier options and cds in spectrally one-sided L´evy models: the parabolic Laplace inversion method.

Working paper, June 2014.

S. Boyarchenko and S. Levendorski˘i.

Efficient variations of Fourier transform in applications to option pricing.

Journal of Computational Finance, 18(2):57–90, 2015.

S.I. Boyarchenko and S.Z. Levendorski˘i.

On rational pricing of derivative securities for a family of non-Gaussian processes.

Preprint 98/7, Institut f¨ur Mathematik, Universit¨at Potsdam, 1998.

Available at http://opus.kobv.de/ubp/volltexte/2008/2519/.

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Bibliography V

Option pricing for truncated L´evy processes.

International Journal of Theoretical and Applied Finance, 3(3):549–552, July 2000.

Non-Gaussian Merton-Black-Scholes Theory, volume 9 of Adv. Ser.

Stat. Sci. Appl. Probab.

World Scientific Publishing Co., River Edge, NJ, 2002.

Efficient Laplace inversion, Wiener-Hopf factorization and pricing lookbacks.

International Journal of Theoretical and Applied Finance,

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Bibliography VI

J. Cariboni and W. Schoutens.

Pricing credit default swaps under L´evy models.

P. Carr, H. Geman, D.B. Madan, and M. Yor.

The fine structure of asset returns: an empirical investigation.

Journal of Business, 75:305–332, 2002.

P. Carr and D.B. Madan.

Option valuation using the Fast Fourier Transform.

A. Castagna.

The Hedging Costs of Discrete Monitoring of FX Barrier Options.

Working paper, January 2009.

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Bibliography VII

G. Chacko and S. Das.

Pricing interest rate derivatives: a general approach.

Review of Financial Studies, 15(1):195–241, 2002.

I. Clark.

Foreign Exchange Option Pricing - A Practitioner’s Guide.

Wiley Finance. John Wiley & Sons, Ltd., Chichester, 2011.

M. de Innocentis.

Fast calculation of prices and sensitivities of European options under Variance Gamma.

Working paper, August 2011.

Available at SSRN:

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Bibliography VIII

M. de Innocentis and S.Z. Levendorski˘i.

Pricing discrete barrier options and credit default swaps under L´evy processes.

Quantitative Finance, 14(8):1337–1365, 2014.

Available online at

http://dx.doi.org/10.1080/14697688.2013.826814.

D. Duffie, J. Pan, and K. Singleton.

Transform Analysis and Asset Pricing for Affine Jump Diffusions.

Econometrica, 68:1343–1376, 2000.

A. Eydeland.

A fast algorithm for computing integrals in function spaces: financial applications.

Computational Economics, 7:277–285, 1994.

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Bibliography IX

F. Fang, H. J¨onsson, C.W. Oosterlee, and W. Schoutens.

Fast valuation and calibration of credit default swaps under L´evy dynamics.

Working paper, 2009.

Available at: SSRN:

http://papers.ssrn.com/abstract=1628672.

F. Fang and C.W. Oosterlee.

Pricing early-exercise and discrete barrier options by Fourier-cosine series expansions.

Numerische Mathematik, 114(1):27–62, 2009.

L. Feng and X. Lin.

Inverting analytic characteristic functions and financial applications.

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Bibliography X

G. Fusai, I.D. Abrahams, and C. Sgarra.

The Wiener-Hopf technique and discretely monitored path-dependent option pricing.

Finance and Stochastics, 10(1):1–26, January 2006.

H. Geman and D.B. Madan.

Risks in returns: a pure jump perspective.

In Exotic Option Pricing and Advanced L´evy Models. Wiley, 2005.

R. Green, G. Fusai, and I.D. Abrahams.

The Wiener-Hopf technique and discretely monitored path-dependent option pricing.

Mathematical Finance, 20(2):259–288, April 2011.

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Bibliography XI

S. L. Heston.

A closed-form solution for options with stochastic volatility with applications to bond and currency options.

The Review of Financial Studies, 6(2):327–343, 1993.

M. Jeannin and M.R. Pistorius.

A transform approach to calculate prices and greeks of barrier options driven by a class of L´evy processes.

King’s College London and Nomura Models and Methodology Group Preprint, 2007.

Available at: http://arxiv.org/abs/0812.3128.

C. Kenyon and R. Stamm.

Discounting, LIBOR, CVA and Funding.

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Bibliography XII

O. Kudryavtsev and S.Z. Levendorski˘i.

Fast and accurate pricing of barrier options under L´evy processes.

Finance and Stochastics, 13(4):531–562, 2009.

S. Levendorski˘i.

Efficient pricing and reliable calibration in the Heston model.

International Journal of Theoretical and Applied Finance, 15(7), 2012.

125050 (44 pages).

S.Z. Levendorski˘i.

Method of paired contours and pricing barrier options and CDS of long maturities.

To appear. Available at SSRN:

http://papers.ssrn.com/abstract=2267107.

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Bibliography XIII

S.Z. Levendorski˘i.

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Working paper, June 2014.

A. Lewis.

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Bibliography XIV

E. Schl¨ogl.

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In Advances in Finance and Stochastics: Essays in honour of Dieter Sondermann, pages 197–218. Springer-Verlag, 2002.

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Appendix: Barrier Options: Numerical Results (Set A, 3 Months)

Benchmark 80.89757 0.58657346 184.8 0.000223

Quadratic 80.89757 0.58657684 3.38 · 10⁻⁶ 1.00 · 10⁻⁵ 0.026 0.005579

HT 80.89757 0.57137568 1.52 · 10⁻² 2.59 · 10⁻² 0.031 1000 1.13742

HT 80.89757 0.59306974 6.50 · 10⁻³ 1.11 · 10⁻² 0.14 5000 0.47278

HT 80.89757 0.58113611 5.44 · 10⁻³ 9.27 · 10⁻³ 0.281 8000 0.36587

HT 80.89757 0.58383383 2.74 · 10⁻³ 4.67 · 10⁻³ 1.794 50000 0.13465

HT 80.89757 0.58850986 1.94 · 10⁻³ 3.30 · 10⁻³ 4.165 100000 0.09226 HT 80.89757 0.58727466 7.01 · 10⁻⁴ 1.20 · 10⁻³ 34.057 1000000 0.02628

COS 80.89757 0.58728375 7.10 · 10⁻⁴ 1.21 · 10⁻³ 0.012 8 300

Benchmark 100 2.59027151 184.8 0.000223

Quadratic 100 2.5912833 1.01 · 10⁻³ 3.90 · 10⁻⁴ 0.026 0.005579

HT 100 2.5895322 7.39 · 10⁻⁴ 2.90 · 10⁻⁴ 0.031 1000 1.13742

COS 100 2.59042878 1.57 · 10⁻⁴ 6.00 · 10⁻⁵ 0.006 8 100

Comparison of prices for a DO put option with barrier e^h= 80 strike K = 100, and maturity T = 0.25, for x = h + 50∆0and x = ln K , where ∆0= (ln K − h)/1000 ≈ 2.23 · 10⁻⁴. Time:

Parameters of KoBoL: λ−= −8, λ+= 9, ν = 1.2, second moment m2= 0.16, riskless rate r = 0.03 and dividend rate q = 0 (parameter set A, used by Boyarchenko and Levendorski˘i, 2011). _absand _reldenote absolute and relative errors, respectively. Benchmarks were calculated using cubic interpolation with state space mesh ∆0.

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Appendix: Barrier Options: Numerical Results (Set B, 3 Months)

Benchmark 80.89757 1.77132603 192.24 0.000223

Quadratic 80.89757 1.77103311 2.93 · 10⁻⁴ 1.70 · 10⁻⁴ 0.041 0.002231

HT 80.89757 1.76453336 6.80 · 10⁻² 3.83 · 10⁻³ 0.047 1000 5.69475

HT 80.89757 1.76962911 1.70 · 10⁻² 9.60 · 10⁻⁴ 0.172 6000 2.72307

COS 80.89757 1.89363042 1.22 · 10⁻¹ 6.91 · 10⁻² 0.029 8 1000

COS 80.89757 1.78247147 1.11 · 10⁻² 6.29 · 10⁻³ 0.03 10 1000

COS 80.89757 1.7724825 1.16 · 10⁻³ 6.50 · 10⁻⁴ 0.03 15 1000

Benchmark 100 3.11359622 192.24 0.000223

Quadratic 100 3.11373903 1.43 · 10⁻⁴ 5.00 · 10⁻⁵ 0.041 0.002231

HT 100 3.11668802 3.09 · 10⁻³ 9.90 · 10⁻⁴ 0.062 2000 4.28076

COS 100 1.19 · 10¹³ 1.19 · 10¹³ 3.83 · 10¹² 0.029 8 1000

COS 100 3.13122123 1.76 · 10⁻² 5.66 · 10⁻³ 0.029 10 1000

COS 100 3.11697328 3.38 · 10⁻³ 1.08 · 10⁻³ 0.029 15 1000

Comparison of prices for a DO put option with barrier e^h= 80 strike K = 100, and maturity T = 0.25, for x = h + 50∆0and x = ln K , where ∆0= (ln K − h)/1000 ≈ 2.23 · 10⁻⁴. Time:

Parameters of KoBoL: λ−= −60, λ+= 50, ν = 0.7, c = 4, with riskless rate r = 0.03 and dividend rate q = 0 (parameter set B, used both by Feng and Linetsky, 2008, for HT, and by Fang and Oosterlee, 2009, for COS). absand reldenote absolute and relative errors, respectively. Benchmarks were calculated using cubic interpolation with state space mesh ∆0. Settings for quadratic interpolation: state space truncation xM= h + 1, and ζ = 5, Λ = 200 used for all applications of parabolic IFT.

Calculations were carried out with MATLAB R2011b (win64), on Intel i5-2410M, 2.30 GHz with 4 GB RAM, running Windows 7 Professional (64 bit).