Algorithm

From a numerical perspective we must ultimately consider the processes Xh on a finite state space, which we take to be Sh

M := {x ∈ Zdh : |x| ≤ M} (M > 0, h∈(0, h?)). We let ˆQh denote the sub-Markov generator got fromQh by restriction toS_Mh , and ˆXh be the corresponding Markov chain got by killing Xh at the time T_Mh := inf{t ≥ 0 : |X_th| > M}, sending it to the coffin state ∂ thereafter [Syski, 1992].

Then the basis for the numerical evaluations is the observation that for a (finite state space) Markov chain Y with generator matrix Q, the probability

Py(Yt=z) (respectively the expectationEy[f◦Y], when defined) is given by (etQ)yz (respectively (etQf)(y)). With this in mind we propose the:

Sketch algorithm

(i) Choose {h, M} ⊂(0,∞).

(ii) Calculate, for the truncated sub-Markov generator ˆQh, the matrix exponential exp{tQˆh} or action exp{tQˆh}f thereof (where f is a suitable vector).

(iii) Adjust truncation parameterM, if needed, and discretization parameterh, until sufficient precision has been established.

Two questions now deserve attention: (1) what is the truncation error and (2) what is the expected cost of this algorithm. We address both in turn.

First, with a view to the localization/truncation error, we shall find use of the following:

Proposition 2.34. Let g : [0,∞)→ [0,∞) be nondecreasing, continuous and sub- multiplicative, with lim+∞g= +∞. Let t >0 and denote by

X_t?= sup s∈[0,t]

|Xs|, Xth?= sup s∈[0,t]

|X_sh|,

the running suprema of|X|and of |Xh|, h∈(0, h?), respectively. Suppose further- moreE[g◦ |Xt|]<∞. ThenE[g◦Xt?]<∞ and, moreover, there is someh0∈(0, h?] such that

sup h∈(0,h0)

E[g◦X_th?]<∞.

Remark 2.35. The functiong◦|·|:Rd→[0,∞) is measurable, submultiplicative and

locally bounded, so for a condition on the L´evy measure equivalent toE[g◦|Xt|]<∞

see Proposition 2.31.

We prove Proposition 2.34 below, but first let us show its relation to the truncation error. For a functionf :Zd_h →R, we extend its domain to Zd_h∪ {∂}, by

stipulating thatf(∂) = 0. The following (very crude) estimates may then be made:

Corollary 2.36. Fix t >0. Assume the setting of Proposition 2.34. There is some h0 ∈(0, h?] and then C := suph∈(0,h0)E[g◦X

h?

t ]< ∞, such that the following two claims hold:

(i) For allh∈(0, h0): X

x∈_Zd h

|P(X_th =x)−P( ˆX_th =x)|=P(T_Mh < t)≤C/g(M).

(ii) Letf :Zdh →Rand suppose|f| ≤f˜◦|·|, withf˜: [0,∞)→[0,∞)nondecreasing and such that f /g˜ is (respectively ultimately) nonincreasing. Then for all (respectively sufficiently large) M >0 and h∈(0, h0):

|E[f ◦X_th]−E[f ◦Xˆ_th]| ≤C ˜ f g ! (M). Remark 2.37.

1. With regard to (i), note thatMmay be taken fixed (i.e. independent ofh) and chosen so as to satisfy a prescribed level of precision. In that case such a choice may be verified explicitly at least retrospectively: the sub-Markov generator

ˆ

Qh gives rise to the sub-Markov transition matrix ˆP_th := etQˆh; its deficit (in the row corresponding to state 0) is precisely the probabilityP(T_Mh < t). 2. But, with respect to (ii), M may also be made to depend on h, and then

made to increase to +∞ as h ↓ 0, in which case it is natural to balance the rate of decay of|E[f◦X_th]−E[f ◦Xˆ_th]|against that of|E[f◦Xt]−E[f◦X_th]|

(cf. Proposition 2.32). In particular, since E[g◦ |Xt|] < ∞ ⇔ E[g◦Xt?] ⇔

R

Rd\[−1,1]dg◦ | · |dλ <∞[Sato, 1999, p. 159, Theorem 25.3 & p. 166, Theorem

25.18], this problem is essentially analogous to the one in Proposition 2.32. In particular, Remark 2.33 extends in a straightforward way to account for the truncation error, withM in place ofK(h)−3h/2.

Proof. (i) follows from the estimate P

x∈_Zd

h|P(X

h

t = x)−P( ˆXth = x)| = P(TMh < t) =P(X_th?> M)≤ E[g◦Xth?]

g(M) , which is an application of Markov’s inequality. When

it comes to (ii), we have for all (respectively sufficiently large) M >0:

|E[f◦X_th]−E[f◦Xˆ_th]| ≤E h |f| ◦X_th 1(T_Mh < t) i ≤E h ˜ f ◦ |X_th|1(T_Mh < t) i ≤ Ehf˜◦X_th?1(T_Mh < t)i=E " ˜ f g ! ◦X_th? ! g◦X_th?1(X_th?> M) # ≤ f˜ g ! (M)E[g◦X_th?],

whence the desired conclusion follows.

Proof of Proposition 2.34. We refer to [Sato, 1999, p. 166, Theorem 25.18] for the proof that E[g◦X_t?] < ∞. Next, by right continuity of the sample paths of X, we may choose b > 0, such that P(X_t∗ ≤ b/2) > 0 and we may also insist on b/2 being a continuity point of the distribution function of X_t? (there being only denumerably many points of discontinuity thereof). Now, Xh → X as h ↓ 0 with respect to the Skorokhod topology on the space of c`adl`ag paths. Moreover, by [Jacod and Shiryaev, 2003, p. 339, Proposition 2.4], the mapping Φ := (α 7→

sup_s∈[0,t]|α(s)|) :D([0,∞),Rd) →R is continuous at every point α in the space of

c`adl`ag pathsD([0,∞),Rd), which is continuous att. In particular, Φ is continuous,

a.s. with respect to the law of the processX on the Skorokhod space [Sato, 1999, p. 59, Theorem 11.1]. By the Portmanteau Theorem and since weak convergence is preserved under continuous mappings, it follows that there is someh0 ∈(0, h?] such that infh∈(0,h0)P(X

h?

t ≤b/2)>0.

Moreover, from the proof of [Sato, 1999, p. 166, Theorem 25.18], by letting ˜

g : [0,∞) → [0,∞) be nondecreasing, continuous, vanishing at zero and agreeing withg on restriction to [1,∞), we may then show for each h∈(0, h?) that:

E[˜g◦(X_th?−b);X_th?> b]≤E[˜g◦ |X_th|]/P(X_th?≤b/2).

Now, since E[g◦ |Xt|]< ∞, by Proposition 2.31 (cf. Remark 2.35), there is some h0 ∈(0, h?] such that suph∈(0,h0)E[g◦ |X

h

t|]<∞, and thus suph∈(0,h0)E[˜g◦ |X

h t|]<

∞.

Combining the above, it follows that for someh0 ∈ (0, h?], suph∈(0,h0)E[˜g◦

(X_th?−b);Xh?

t > b]<∞and thus suph∈(0,h0)E[g◦(X

h?

t −b);Xth? > b]<∞. Finally, an application of submultiplicativity ofg allows us to conclude.

Having thus dealt with the truncation error, let us briefly discuss the cost of our algorithm.

The latter is clearly governed by the calculation of the matrix exponential, or, respectively, of its action on some vector. Indeed, if we consider as fixed the generator matrix ˆQh, and, in particular, its dimension n∼(M/h)d, then this may typically require O(n3) [Moler and Loan, 2003; Higham, 2005], respectively O(n2) [Al-Mohy and Higham, 2011], floating point operations. Note, however, that this is a notional complexity analysis of the algorithm. A more detailed argument would ultimately have to specify precisely the particular method used to determine the (respectively action of a) matrix exponential, and, moreover, take into account how

ˆ

Further analysis in this respect goes beyond the desired scope of this thesis. We finish off by giving some numerical experiments in the univariate case. To compute the action of ˆQh_{on a vector we use the MATLAB functionexpmv.m[Al-} Mohy and Higham, 2011], unless ˆQh is sparse, in which case we use the MATLAB functionexpv.mfrom [Sidje, 1998].

We begin with transition densities. To shorten notation, fix the time t= 1 and allow p := p1(0,·) and ph := _h1Pˆ1h(0,·) ( ˆPh being the analogue of Ph for the

process ˆXh). Note that to evaluate the latter, it is sufficient to compute (eQˆht)0·=

(e( ˆQh)0t1{0})0, where ( ˆQh)0 denotes transposition.

Example 2.38. Consider first Brownian motion with drift, σ2 = 1, µ = 1, λ = 0 (scheme 1, V = 0). We compare the density p with the approximation ph (h ∈ {1/2n : n ∈ {0,1,2,3}}) on the interval [0,2] (see Figure 2.2 on p. 65), choosing M = 5. The vector of deficit probabilities (P(T_M1/2n < t))3_n=0corresponding to using this truncation was (5.9·10−4,1.5·10−4,5.8·10−5,4.4·10−5). In this case the matrix

ˆ Qh is sparse. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.2 0.25 0.3 0.35 0.4 p p1 p1/2 p1/4 p1/8

Figure 2.2: Convergence of ph _{to p (as h ↓ 0) on the interval [0,2] for Brownian} motion with drift (σ2 = µ = 1, λ = 0, scheme 1, V = 0). See Example 2.38 for details.

Example 2.39. Consider now α-stable L´evy processes, σ2 = 0, µ = 0, λ(dx) = dx/|x|1+α _{(scheme 2, V = 1). We compare the density p with p}h _{on the interval}

[0,1] (see Figure 2.3 on p. 67). Computations are made for the vector of alphas given by (αk)4_k=1 := (1/2,1,4/3,5/3) with corresponding truncation parameters (Mk)4k=1 = (500,100,30,20) resulting in the deficit probabilities (uniformly over

the h considered) of (P(T_Mh

k < t))

4

k=1 = (1.7·10

−1_,2.0·10−2_{,(from 1.7 to 1.8)·}

10−2,(from 0.94 to 1.01)·10−2). The heavy tails of the L´evy density necessitate a relatively high value ofM. Nevertheless, excluding the case α = 5/3, a reduction of M by a factor of 5 resulted in an absolute change of the approximating densi- ties, which was at most of the order of magnitude of the discretization error itself. Conversely, forα= 1/2, when the deficit probability is highest and appreciable, in- creasingM by a factor of 2, resulted in an absolute change of the calculated densities of the order 10−6 (uniformly overh∈ {1,1/2,1/4}). Finally, note thatα = 1 gives rise to the Cauchy distribution, whereas otherwise we use the MATLAB function stblpdf.mto get a benchmark density against which a comparison can be made. Example 2.40. A particular VG model [Carr et al., 2002; Madan et al., 1998] has σ2 = 0, µ= 0, λ(dx) = e−|_|xx_||1_R\{0}(x)dx (scheme 2, V = 1). Again we comparep

withph (h∈ {1/2n:n∈ {0,1,2,3}}) on the interval [0,1] (see Figure 2.4 on p. 68), choosingM = 5. The vector of deficit probabilities (P(T_M1/2n < t))3_n=0corresponding to using this truncation was (5.2·10−3,6.4·10−3,7.2·10−3,7.6·10−3). The density pis given explicitly by (x7→e−|x|/2).

Finally, to illustrate convergence of expectations, we consider a particular option pricing problem.

Example 2.41. Suppose that, under the pricing measure, the stock price process S = (St)t≥0 is given by St=S0ert+Xt,t≥0, where S0 is the initial price, r is the

interest rate, andX is a tempered stable process with L´evy measure given by:

λ(dx) =c e −λ+x x1+α 1(0,∞)(x) + e−λ−|x| |x|1+α1(−∞,0)(x) ! dx.

To satisfy the martingale condition, we must haveE[eXt_{]≡1, which in turn uniquely}

determines the drift µ(we have, of course,σ2 = 0). The price of the European put option with maturityT and strike K at time zero is then given by:

P(T, K) =e−rTE[(K−ST)+].

We choose the same value for the parameters as [Poirot and Tankov, 2006], namely S0 = 100, r = 4%, α = 1/2, c = 1/2, λ+ = 3.5, λ− = 2 and T = 0.25, so that we

0 0.2 0.4 0.6 0.8 1 0.023 0.0235 0.024 0.0245 0.025 0.0255 α=1/2 p p1 p1/2 p1/4 p1/8 0 0.2 0.4 0.6 0.8 1 0.086 0.088 0.09 0.092 0.094 0.096 0.098 0.1 0.102 α=1 _p p1 p1/2 p1/4 p1/8 0 0.2 0.4 0.6 0.8 1 0.11 0.112 0.114 0.116 0.118 0.12 0.122 0.124 0.126 0.128 α=4/3 p p1 p1/2 p1/4 p1/8 p1/16 0 0.2 0.4 0.6 0.8 1 0.108 0.11 0.112 0.114 0.116 0.118 0.12 0.122 α=5/3 p p1 p1/2 p1/4 p1/8 p1/16 p1/32

Figure 2.3: Convergence of ph to p (as h ↓ 0) on the interval [0,1] for α- stable L´evy processes (σ2 = 0, µ = 0, λ(dx) = dx/|x|1+α_{, scheme 2, V = 1),} α ∈ {1/2,1,4/3,5/3}. See Example 2.39 for details. Note that convergence be- comes progressively worse asα↑, which is precisely consistent with Figure 2.1 and the theoretical order of convergence, this being O(h(2−α)∧1) (up to a slowly vary- ing factor log(1/h), whenα = 1; and noting that Orey’s condition is satisfied with =α). For example, whenα= 5/3 each successive approximation should be closer to the limit by a factor of 1₂1/3 = 0.8, as it is..

Now, in the present case, X is a process of finite variation, i.e. κ(0) < ∞, hence convergence of densities is of order O(h), since Orey’s condition holds with = 1/2 (scheme 2,V = 1). Moreover,1_R\[−1,1]·λintegrates (x7→e2|x|), whereas the

function (x 7→(K−ert+x)+) is bounded. Pursuant to (2) of Remark 2.37 we thus chooseM =M(h) := 1₂log(1/h)

∨1, which by Corollary 2.36 and Proposition 2.32 (withK(h) =M(h)) (cf. also ((3)b) of Remark 2.33) ensures that:

|Pˆh(T, K)−P(T, K)|=O(hlog(1/h)),

where ˆPh(T, K) := e−rTE[(K−S0erT+ ˆX

h

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.25 0.3 0.35 0.4 0.45 0.5 p p1 p1/2 p1/4 p1/8

Figure 2.4: Convergence ofph top(ash↓0) on the interval [0,1] for the VG model (σ2 = 0,µ= 0,λ(dx) = e−|_|x|x|1_R\{0}(x)dx, scheme 2,V = 1). Note that in this case

Orey’s condition fails, but (at least as evidenced numerically) linear convergence does not. See Example 2.40 for details.

convergence on the decreasing sequencehn:= 1/2n,n≥1.

In particular, we wish to emphasize that the computations were all (reason- ably) fast. For example, to compute the vector ( ˆPhn_{(T, K))}9

n=1 with K = 80, the

times (in seconds; entry-by-entry) (0.0106, 0.0038, 0.0044, 0.0078, 0.0457, 0.0367, 0.0925, 0.4504, 2.4219) were required on an Intel 2.53 GHz processor (times ob- tained using MATLAB’stic-tocfacility). This is much better than, e.g., the Monte Carlo method of [Poirot and Tankov, 2006] and comparable with the finite difference method of [Cont and Voltchkova, 2005] (VG2 model in [Cont and Voltchkova, 2005, p. 1617, Section 7]).

In conclusion, the above numerical experiments serve to indicate that our method behaves robustly when the Blumenthal-Getoor index of the L´evy measure is not too close to 2 (in particular, if the pure-jump part has finite variation). It does less well if this is not the case, since then the discretisation parameterh must be chosen small, which is expensive in terms of numerics (viz. the size of ˆQh_).

K→ 80 85 90 95 100 105 110 115 120 P(T, K)→ 1.7444 2.3926 3.2835 4.5366 6.3711 9.1430 12.7631 16.8429 21.1855 n Pˆhn_{(T, K)−P(T, K)} 1 0.6411 0.5422 0.2006 -0.5033 -1.7885 -0.8227 0.0970 0.5570 0.7542 2 -0.1089 0.2816 0.4295 0.2151 -0.5806 0.0975 0.5341 0.5109 0.2250 3 -0.2271 -0.1596 -0.1928 0.0920 -0.2046 0.1405 0.0348 -0.4356 -0.3937 4 -0.0904 -0.0753 -0.0517 -0.0442 0.0652 0.1487 0.0057 -0.1511 -0.1838 5 -0.0411 -0.0338 -0.0193 -0.0053 0.0679 0.0569 -0.0073 -0.0616 -0.0833 6 -0.0184 -0.0163 -0.0081 0.0022 0.0347 0.0314 -0.0033 -0.0244 -0.0384 7 -0.0079 -0.0069 -0.0040 0.0019 0.0152 0.0109 -0.0034 -0.0108 -0.0164 8 -0.0034 -0.0029 -0.0016 0.0011 0.0072 0.0053 -0.0012 -0.0048 -0.0070 9 -0.0014 -0.0012 -0.0007 0.0006 0.0033 0.0026 -0.0004 -0.0020 -0.0030

Table 2.4: Convergence of the put option price for a CGMY model (scheme 2, V = 1). See Example 2.41 for details.

Chapter 3

Some fluctuation results in the

theory of L´evy processes

The class of L´evy processes for which overshoots are almost surely con- stant quantities is precisely characterized. A fluctuation theory and, in particular, a theory of scale functions is developed for upwards skip-free L´evy chains, i.e. for right-continuous random walks embedded into con- tinuous time as compound Poisson processes. This is done by analogy to the spectrally negative class of L´evy processes.

Throughout this chapter we work on a filtered probability space (Ω,F,F=

(Ft)t≥0,P), which satisfies the standard assumptions (see Definition 1.16). We let

X = (Xt)t≥0 be a L´evy process on this space (X is assumed to be F-adapted and

to have independent increments relative to F) with characteristic triplet (σ2, λ, µ)c˜

relative to some cut-off function ˜c. Recall the notation regarding the supremum and infimum processesXandX, as well as the first passage timesTx, ˆTx andTx−,x∈R for the processX (see Definitions 1.20 and 1.22).