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Efficient valuation of exotic derivatives in L´ evy models

Ernst Eberlein and Antonis Papapantoleon

Department of Mathematical Stochastics and Center for Data Analysis and Modeling (FDM) University of Freiburg

Conference on Stochastic Processes: Theory and Applications

• n occasion of the 65th birthday of Wolfgang Runggaldier

Bressanone (Italy), July 16–20, 2007

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Volatility smile and surface

10 20 30 40 50 60 70 80 90 1 2 3 4 5 6 7 8 9 10 10 10.5 11 11.5 12 12.5 13 13.5 14 maturity delta (%) or strike implied vol (%)

2 4 6 8 10 2.5 4.0 6.0 8.0 10.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0 30.0 Maturity (in years) Strike rate (in %)

Volatility surfaces of foreign exchange and interest rate options

• Volatilities vary in strike (smile)
• Volatilities vary in time to maturity (term structure)
• Volatility clustering

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Exponential semimartingale model

Let BT = (Ω, F, F, P) be a stochastic basis, where F = FT and F = (Ft)0≤t≤T. We model the price process of a financial asset as an exponential semimartingale St = eHt , 0 ≤ t ≤ T. (1) H = (Ht)0≤t≤T is a semimartingale with canonical representation H = H0 + B + Hc + h(x) ∗ (µH − ν) + (x − h(x)) ∗ µH. (2) For the processes B, C = Hc, and the measure ν we use the notation T(H|P) = (B, C, ν) which is called the triplet of predictable characteristics of the semimartingale H.

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Alternative model description

E(X) = (E(X)t)0≤t≤T stochastic exponential St = E(e H)t, 0 ≤ t ≤ T dSt = St−d e Ht where e Ht = Ht + 1 2Hct + Z t Z

R

(ex − 1 − x)µH(ds, dx) Note E(e H)t = exp(e Ht − 1 2e Hct) Y

0<s≤t

(1 + ∆e Hs) exp(−∆e Hs) Asset price positive only if ∆e H > −1.

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Martingale modeling

Let Mloc(P) be the class of local martingales.

Assumption (ES)

The process 1{x>1}ex ∗ ν has bounded variation. Then S = eH ∈ Mloc(P) ⇔ B + C 2 + (ex − 1 − h(x)) ∗ ν = 0. (3) Throughout, we assume that P is a (local) martingale measure for S. By the Fundamental Theorem of Asset Pricing, the value of an option on S equals the discounted expected payoff under a martingale measure. We assume zero interest rates.

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Supremum and infimum processes

Let X = (Xt)0≤t≤T be a stochastic process. We denote by X t = sup

0≤u≤t

Xu and X t = inf

0≤u≤t Xu

the supremum and infimum process of X respectively. Since the exponential function is monotone and increasing ST = sup

0≤t≤T

St = sup

0≤t≤T

“ eHt ” = esup0≤t≤T Ht = eHT . (4) Similarly ST = eHT . (5)

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Valuation formulae – payoff functional

We want to price an option with payoff f(XT), where XT = p(Ht, 0 ≤ t ≤ T) is an FT-measurable functional. The functionals we consider are “European style”, and consist of two parts:

1

The payoff function is an arbitrary function f : R → R+; for example f(x) = (ex − K)+ or f(x) = 1{ex >B}, for K, B ∈ R+.

2

The underlying process can be the asset price or the supremum/infimum or an average of the asset price process (e.g. X = H or X = H).

• Exotic options

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Valuation formulae – assumptions

Assumptions: (R1) Assume that R

R e−Rxf(x)dx < ∞ for all R ∈ I1 ⊂ R.

(R2) Assume that MXT (z) = E[ezXT ] < ∞, for all z ∈ I2 ⊂ R. (R3) Assume that I1 ∩ I2 = ∅. Valuation formulae based on Fourier transforms; similar to Raible (2000), but no need for Lebesgue density. Consider the Fourier transform of the payoff function like Borovkov and Novikov (2002); also Hubalek et al. (2006) and ˇ Cern´ y (2007), for hedging. Carr and Madan (1999) and Raible (2000) transform the option price.

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Valuation formulae

Theorem 1

Assume that (R1)–(R3) are in force. Then, the price Vf(X) of an option

• n S = (St)0≤t≤T with payoff f(XT) is given by

Vf(X) = 1 2π Z

R

ϕXT (−u − iR)Ff(u + iR)du, (6) where ϕXT denotes the extended characteristic function of XT and Ff denotes the Fourier transform of f.

Proof

Introduce the dampened payoff function g(x) = e−Rxf(x), R ∈ I1. Then Vf(X) = E[f(XT)] = E[eRXT g(XT)] = Z

R

eRxg(x)PXT (dx). (7)

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Proof (cont.)

Under assumption (R1), g has a Fourier transform Fg; inverting it, we get a representation as g(x) = 1 2π Z

R

e−ixuFg(u)du. (8) Returning to the valuation problem (7) we get Vf(X) = Z

R

eRx 1 2π Z

R

e−ixuFg(u)du ! PXT (dx) = 1 2π Z

R

Z

R

ei(−u−iR)xPXT (dx) ! Fg(u)du = 1 2π Z

R

ϕXT (−u − iR)Ff(u + iR)du. (9)

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Valuation formulae II – options

Valuation formulae for options that depend on two functionals of the driving process. Examples: barrier, slide-in or corridor and two-asset correlation option (ST − K)+1{ST >B}; (ST − K)+

N

X

i=1

1{L<STi <H}; (S1

T − K)+1{S2

T >B}.

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Valuation formulae II

Theorem 2

The price Vf,g(X, Y) of an option on S = (St)0≤t≤T with payoff function f(XT)g(YT) is given by Vf,g(X, Y) = 1 4π2 Z

R

Z

R

ϕXT ,YT (−u − iR1, −v − iR2) × Fg(v + iR2)Ff(u + iR1)dvdu, (10) where ϕXT ,YT denotes the extended characteristic function of the random vector (XT, YT).

Proof.

Assumptions and proof are similar to Theorem 1.

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Examples of payoff functions

Example (Call and put option)

Call payoff f(x) = (ex − K)+, K ∈ R+, Ff(u + iR) = K 1+iu−R (iu − R)(1 + iu − R), R ∈ I1 = (1, ∞). (11) Similarly, if f(x) = (K − ex)+, K ∈ R+, Ff(u + iR) = K 1+iu−R (iu − R)(1 + iu − R), R ∈ I1 = (−∞, 0). (12)

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Example (Digital option)

Call payoff 1{ex >B}, B ∈ R+. Ff(u + iR) = −Biu−R 1 iu − R , R ∈ I1 = (0, ∞). (13) Similarly, for the payoff f(x) = 1{ex <B}, B ∈ R+, Ff(u + iR) = Biu−R 1 iu − R , R ∈ I1 = (−∞, 0). (14)

Example (Double digital option)

The payoff of a double digital call option is 1{B<ex <B}, B, B ∈ R+. Ff(u + iR) = 1 iu − R “ Biu−R − Biu−R” , R ∈ I1 = R\{0}. (15)

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Example (Asset-or-nothing digital)

Call payoff f(x) = ex1{ex >B} Ff(u + iR) = − B1+iu−R 1 + iu − R , R ∈ I1 = (1, ∞) Put payoff f(x) = ex1{ex <B} Ff(u + iR) = B1+iu−R 1 + iu − R , R ∈ I1 = (−∞, 1)

Example (Self-quanto option)

Call payoff f(x) = ex(ex − K)+ Ff(u + iR) = K 2+iu−R (1 + iu − R)(2 + iu − R), R ∈ I1 = (2, ∞)

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L´ evy processes

Let L = (Lt)0≤t≤T be a L´ evy process with triplet of local characteristics (b, c, λ), i.e. Bt(ω) = bt, Ct(ω) = ct, ν(ω; dt, dx) = dtλ(dx), λ L´ evy measure.

Assumption (EM)

There exists a constant M > 1 such that Z

{|x|>1}

euxλ(dx) < ∞, ∀u ∈ [−M, M]. Using (EM) and Theorems 25.3 and 25.17 in Sato (1999), we get that E ˆ euLt ˜ < ∞, E ˆ euLt ˜ < ∞ and E ˆ euLt ˜ < ∞ for all u ∈ [−M, M].

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On the characteristic function of the supremum I

Lemma 3

Let L = (Lt)0≤t≤T be a L´ evy process that satisfies assumption (EM). Then, the moment generating function of Lt is defined for all u ∈ (−∞, M] and t ∈ [0, T].

Lemma 4

Let L = (Lt)0≤t≤T be a L´ evy process that satisfies assumption (EM). Then, the characteristic function ϕLt of Lt is holomorphic in the half plane {z ∈ C : −M < ℑz < ∞} and can be represented as a Fourier integral in the complex domain ϕLt (z) = E ˆ eizLt ˜ = Z

R

eizxPLt (dx).

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Fluctuation theory for L´ evy processes

Theorem 5 (Wiener–Hopf factorization)

Let L be a L´ evy process. The Laplace transform of L at an independent and exponentially distributed time θ can be identified from the Wiener–Hopf factorization of L via E ˆ e−βLθ˜ = κ(q, 0) κ(q, β) (16) where κ(α, β), α ≥ 0, β ≥ 0, is given by κ(α, β) = k exp „Z ∞ Z ∞ (e−t − e−αt−βx)1 t PLt (dx) dt « . (17) Moreover, κ can be analytically extended to α, β ∈ C with ℜα ≥ 0 and ℜβ ≥ −M.

Proof.

Theorem 6.16 in Kyprianou (2006).

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Linking fixed and exponential times

Lemma 6

Let L = (Lt)0≤t≤T be a L´ evy process that satisfies assumption (EM) and consider β ∈ C with ℜβ ∈ [−M, ∞). The Laplace transforms of Lt, t ∈ [0, T] and Lθ, θ ∼ Exp(q), are related via E ˆ e−βLθ˜ = q Z ∞ e−qtE ˆ e−βLt ˜ dt. (18) Moreover, the Laplace transform of Lθ is finite for β ∈ C with ℜβ ∈ [−M, ∞).

Proof.

An application of Fubini’s theorem yields E ˆ e−βLθ˜ = E h Z ∞ qe−qte−βLt dt i = q Z ∞ e−qtE ˆ e−βLt ˜ dt.

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On the characteristic function of the supremum II

Theorem 7

Let L = (Lt)0≤t≤T be a L´ evy process. The Laplace transform of Lt at a fixed time t, t ∈ [0, T], is given by E ˆ e−βLt ˜ = 1 2π Z

R

et(Y+iv) Y + iv κ(Y + iv, 0) κ(Y + iv, β)dv, (19) for Y > 0. Moreover, the Laplace transform can be extended to the complex plane for β ∈ C with ℜβ ∈ [−M, ∞).

Proof.

Combining eqs. (16) and (18) we get q Z ∞ e−qtE[e−βLt ] dt = κ(q, 0) κ(q, β). (20) Applying Doetsch (1950), we invert the Laplace transform and the claim follows.

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Non-path-dependent options

European option on an asset with price process St = eHt Examples: call, put, digitals, asset-or-nothing, double digitals, self-quanto options − → Xt ≡ HT, i.e. we need ϕHT Generalized hyperbolic model (GH model): ϕH1(u) = eiuµ“ α2 − β2 α2 − (β + iu)2 ”λ/2 Kλ ` δ p α2 − (β + iu)2 ´ Kλ ` δ p α2 − β2´ I2 = (−α − β, α − β) ϕHT (u) = (ϕH1(u))T similar: NIG, CGMY, Meixner

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Non-path-dependent options II

Stochastic volatility L´ evy models: Carr, Geman, Madan, Yor (2003) Stochastic clock Yt = Z t ysds (ys > 0) e.g. CIR process dyt = K(η − yt)dt + λy 1/2

t

dWt Define for a pure jump L´ evy process X = (Xt)t≥0 Ht = XYt (0 ≤ t ≤ T) Then ϕHt (u) = ϕYt (−iϕXt (u)) (ϕYt (−iuϕXt (−i)))iu

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Lookback options

Fixed strike lookback option: (ST − K)+. Combining Theorem 1 and Theorem 7, we get CT(S; K) = 1 2π Z

R

ϕLT (−u − iR) K 1+iu−R (iu − R)(1 + iu − R)du (21) where ϕLT (−u − iR) = 1 2π Z

R

eT(Y+iv) Y + iv κ(Y + iv, 0) κ(Y + iv, iu − R)dv. (22)

• The floating strike lookback option, ( ST − ST)+, is treated by a duality

formula.

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Floating strike lookback options (1)

Payoff of a put: “ β sup

0≤t≤T

St − ST ”+ for a 0 < β ≤ 1 Assume H′ = (H′

t )0≤t≤T satisfies

Law “ H′

T − inf t≤T H′ t |P′”

= Law “ sup

t≤T

H′

t |P′”

(holds for L´ evy processes), then PT(β sup S; S) = βC′

T

“ sup S′; 1 β ” Value of a floating strike lookback put → value of a fixed strike lookback call

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Floating strike lookback options (2)

Payoff of a call: “ ST − α inf

0≤t≤T St

”+ for an α ≥ 1 Assume H′ = (H′

t )0≤t≤T satisfies the reflection principle

Law “ sup

t≤T

H′

t − H′ T|P′”

= Law(− inf

t≤T H′ t |P′)

(holds for L´ evy processes), then CT(S; α inf S) = αP′

T

“ 1 α; inf S′” Value of a floating strike lookback call → value of a fixed strike lookback put

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Floating strike lookback options (3)

Proof

CT ` S; α inf S ´ = E ˆ` ST − α inf

t≤T St

´+˜ = E » ST „ 1 − α inft≤T St ST «+– = E′h“ 1 − αeinft≤T Ht −HT ”+i = E′h“ 1 − αeH′

T −supt≤T H′ t

”+i The process H′ = (H′

t )0≤t≤T satisfies the reflection principle:

Law “ sup

t≤T

H′

t − H′ T | P′”

= Law “ − inf

t≤T H′ t | P′”

CT ` S; α inf S ´ = αE′h“ 1 α − einft≤T H′

t

”+i = αE′h“ 1 α − inf

t≤T S′ t

”+i = αP′

T

“ 1 α; inf S′”

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One-touch options

One-touch call option: 1{ST >B}. Combining Theorem 1, Theorem 7 and the example for digital options, we get DCT(S; B) = 1 4π2 Z

R

Z

R

eT(Y+iv) Y + iv κ(Y + iv, 0) κ(Y + iv, iu − R) Biu−R R − iu dvdu. (23) Similarly for the one-touch put option: 1{ST ≤B}. DPT(S; B) = 1 4π2 Z

R

Z

R

eT(Y+iv) Y + iv b κ(Y + iv, 0) b κ(Y + iv, iu − R) Biu−R iu − R dvdu. (24)

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Equity default swap (EDS)

• Fixed premium exchanged for payment at “default”
• default: drop of stock price by 30% or 50% of S0 → first passage

time

• fixed leg pays premium K at times T1, . . . , TN, if Ti ≤ τB
• if τB ≤ T: protection payment, paid at time τB
• premium of the EDS chosen such that initial value equals 0; hence

K = E ˆ e−rτB1{τB≤T} ˜ PN

i=1 E

ˆ e−rTi 1{τB>Ti } ˜. (25)

• Calculations similar to touch options, since 1{τB≤T} = 1{ST ≤B}.

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Options on two assets

Two-asset correlation options: Payoff of a correlation call: (S1

T − K)+1{S2

T >B}

Measurement asset S2 in the money − → call on a payment asset S1 Asset price processes Si

t = exp(Li t)

i = 1, 2 where L = (L1, L2) is a time-inhomogeneous L´ evy process TACT(S1, S2; K, B) = 1 4π2 Z

R

Z

R

ϕLT (−u − iR1, −v − iR2) × K 1+iu−R1 (iu − R1)(1 + iu − R1) Biu−R2 R2 − iu dvdu

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References

• Borovkov, K. and A. Novikov (2002). On a new approach to

calculating expectations for option pricing. J. Appl. Probab. 39, 889–895.

• Carr, P

. and D. B. Madan (1999). Option valuation using the fast Fourier transform. J. Comput. Finance 2 (4), 61–73.

• ˇ

Cern´ y, A. (2007). Optimal continuous-time hedging with leptokurtic returns. Math. Finance 17, 175–203.

• Eberlein, E. and A. Papapantoleon (2007). Valuation of

exotic and credit derivatives in L´ evy models. FDM Preprint 97, University of Freiburg

• Eberlein, E., A. Papapantoleon, and A. N. Shiryaev (2006).

On the duality principle in option pricing: semimartingale

• setting. FDM-Preprint Nr. 92, University of Freiburg

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References (cont.)

• Hubalek, F

., J. Kallsen and L. Krawczyk (2006). Variance-optimal hedging for processes with stationary independent increments. Ann. Appl. Probab. 16, 853–885.

• Kyprianou, A. E. (2006). Introductory lectures on fluctuations
• f L´

evy processes with applications. Springer.

• A. Papapantoleon (2007). Applications of semimartingales

and L´ evy processes in finance: duality and valuation. Ph.D. thesis, University of Freiburg.

• Raible, S. (2000). L´

evy processes in finance: theory, numerics, and empirical facts. Ph.D. thesis, University of Freiburg.

• Sato, K.-I. (1999). L´

evy processes and infinitely divisible

• distributions. Cambridge University Press.