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In Chapters 3 and 4 we adapt existing numerical pricing procedures by Fusai et al. (2016) and Feng and Linetsky (2008). Chapter 3 looks at the effect of the Gibbs phenomenon on the error performance of the pricing method for discrete barrier options. Chapter 4 extends the calculation of the Spitzer identities to continuous monitoring using option pricing as a motivating example. In addition, for the methods based on the pricing procedures by Fusai et al. (2016), we perform a detailed error analysis which gives additional insight into their original method. For a comprehensive background to the techniques we refer the interested reader to the aforementioned papers. However in order to provide a self contained guide to the work done in later chapters we include here brief summaries of the original implementation of the three pricing methods that we modify and improve in this thesis.

2.3.4.1 Spitzer based method for single-barrier options

We describe the pricing procedure devised by Fusai et al. (2016) for single-barrier down- and-out options as an example, but the use of the Spitzer identities is equally applicable to other types of barrier options and also to lookback options; the pricing formulae described by Green et al. (2010) include methods for single-barrier up-and-out and knock-in options.

1. Compute the characteristic function of the underlying asset process Ψ(ξ + iαd, ∆t),

where αd is the damping parameter introduced in Section 2.2.3, Eq. (2.91).

2. Use the Plemelj-Sokhotsky relations with the sinc-based Hilbert transform to fac- torise

Φ(ξ, q) := 1 − Ψ(ξ + iαd, ∆t) = Φ⊕(ξ, q)Φ (ξ, q) (2.105)

with q selected for N −2 dates according to the criteria specified by Abate and Whitt (1992b) for the inverse z-transform.

3. Decompose with respect to l

P (ξ, q) := Ψ(ξ + iαd) Φ (ξ, q) = Pl+(ξ, q) + Pl−(ξ, q), (2.106) and calculate e_b p(ξ, q) := Ψ(ξ + iαd) Pl+(ξ, q) Φ⊕(ξ, q) . (2.107)

4. Calculate the option price as

v(0, 0) := F_ξ→x−1 h b φ∗(ξ)Z_{q→N −2}−1 ep(ξ, q)_b i (0), (2.108)

2.3. NUMERICAL METHODS

function given in Eq. (2.92).

Notice that in the numerical implementation described above, the Spitzer identity is calculated for N − 2 dates, with the inclusion of the characteristic function Ψ(ξ + iαd, ∆t) in Eqs. (2.106) and (2.107) applying the first and final dates respectively. In Fusai et al.

(2016), results were presented showing exponential error convergence for general L´evy

processes and in Section 3.1.1 of Chapter 3 we explore the reason for this and develop bounds for the error convergence.

2.3.4.2 Spitzer based method for double-barrier options

The pricing procedure by Fusai et al. (2016) for double-barrier options is very similar to the method for the single-barrier options described in Section 2.3.4.1, in that it uses Wiener-Hopf factorisation and decomposition to compute the appropriate Spitzer identity. However, the major difference in this case is that the equations cannot be solved directly and so require the use of a fixed-point algorithm. The steps in the pricing procedure are the same as those for single-barrier down-and-out options described in Section 2.3.4.1 with the exception of Step 3 which is now replaced by the fixed-point algorithm

3 (a) Set J−(ξ, q) = Jl−(ξ, q) = 0. (b) Decompose with respect to l

Pc(ξ, q) :=

Ψ(ξ + iαd) − Φ⊕(ξ, q)Ju+(ξ, q) Φ (ξ, q)

= Pl+(ξ, q) + Pl−(ξ, q), (2.109)

and set Jl−(ξ, q) := Pl−(ξ, q). (c) Decompose with respect to u

Qc(ξ, q) := Ψ(ξ + iαd) − Φ (ξ, q)Jl−(ξ, q) Φ⊕(ξ, q) = Qu+(ξ, q) + Qu−(ξ, q), (2.110) and set Ju+(ξ, q) := Qu+(ξ, q). (d) Calculate e b p(ξ, q) := Ψ(ξ + iαd) Ψ(ξ + iαd) − Φ (ξ, q)Jl−(ξ, q) − Φ⊕(ξ, q)J−(ξ, q) Φ(ξ, q) . (2.111)

(e) If the difference between the new and the old value of ep(ξ, q) is less than a prede-b fined tolerance or the number of iterations is greater than a certain threshold then continue, otherwise return to step (b). Numerical tests have shown that an iteration threshold of 5 is sufficient, as higher values do not yield improvements.

Similarly to the procedure for single-barrier options described in Section 2.3.4.1, the Spitzer identity is calculated for N − 2 dates and the first and last date are applied using the

characteristic function in Eqs. (2.109)–(2.111). In contrast to the method for pricing single- barrier options, Fusai et al. (2016) present result showing polynomial error convergence only. The reason for this is explored in more detail in Section 3.1.2 and we present an updated method with exponential error convergence in Section 3.2.

2.3.4.3 Feng and Linetsky method

As shown in Section 2.1, we can use Eqs. (2.26)–(2.28) to obtain the Fourier transform of the part of a function above or below a barrier or between two barriers. This property of the Hilbert transform was used by Feng and Linetsky (2008) to price discrete barrier options exploiting the relationship between the price at two successive monitoring dates:

v(x, tn−1) =

Z u

l

v(x0, tn)k(x − x0, ∆t)dx0. (2.112)

Here v(x, tN) = φ(x)e−αdx, i.e. the payoff of the option, and k(·, ∆t) denotes the transition density of the underlying process with step size ∆t and Ψ(ξ, ∆t) is its characteristic func- tion. Using the convolution theorem together with the Hilbert transform the relationship between the price at two successive dates can be expressed as

b v(ξ, tn−1) = 1 2 n Ψ(ξ + iαd, ∆t)bv(ξ, tn) + e ilξ_iHh_e−ilξ Ψ(ξ + iαd, ∆t)v(ξ, tb n) io (2.113) for a single-barrier down-and-out option and

b

v(ξ, tn−1) = 1 2

n

eilξiHhe−ilξΨ(ξ + iαd, ∆t)bv(ξ, tn) i

− eiuξiHhe−iuξΨ(ξ + iαd, ∆t)bv(ξ, tn) io

(2.114) for a double-barrier option.

Feng and Linetsky (2008) presented results for this method showing exponential error convergence for underlying processes with exponential characteristic functions and poly- nomial error convergence for the VG process. This was explained in detail in the original paper and in Section 3.1.3 we present a new method with improved error performance for the VG process.