Double Sided Parisian Option Pricing - Semantic Scholar
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Double Sided Parisian Option Pricing J.H.M. Anderluh, J.A.M. van der Weide TUDelft Delft Institute of Applied Mathematics P.O. Box 5031 2600 GA Delft (The Netherlands) e-mail: [email protected] August 2006
Abstract In this paper we derive Fourier transforms for double sided Parisian option contracts. The double sided Parisian option contract is triggered by the stock price process spending some time above an upper level or below some lower level. The double sided Parisian knock-in call contract is the general type of Parisian contract from which all the one-sided contract types follow. We also discuss the Fourier inversion in the paper and conclude with a series of numerical examples, explaining the Parisian optionality and the way prices are affected by the local behavior of Brownian motion in detail.
Key words
Parisian Options – Excursions – Fourier Inversion
JEL Classification: G12, G13 MSC : 60G40, 62L15, 60J65
2
J.H.M. Anderluh, J.A.M. van der Weide
Introduction The Parisian option is a kind of a barrier option with the difference that the contract is not specified in terms of touching a barrier, but in terms of staying or below the barrier for a certain period of time. The interest in these options is motivated by the study of structured products and investment problems. Convertible bonds and problems in real options contain Parisian optionality; the Parisian option contract itself is not exchange traded. Details about the practical differences between standard barrier options and Parisian options are discussed in [8], the first paper on Parisian options. The way Parisian options turn up in real option problems is treated in [12]. The authors in [8] derived Laplace transforms for the one-sided version, which is extended in [12] to a Parisian type of contract that is triggered by staying a period of time above the barrier or hitting a lever exceeding this barrier. Here we treat pricing of the double sided Parisian option and, like the papers previously mentioned, we use Fourier (or Laplace) transforms to achieve this. The calculation of Fourier transform instead of Laplace transforms is motivated by the fact that a lot of numerical Laplace inversion algorithms are using the complex continuation of Laplace transforms to Fourier transforms for the actual inversion, see e.g. [10]. As we want to conclude our paper by a section on numerical examples, we have to invert the Fourier transforms we will calculate. In [13] the authors treat a PDE method approach to solve the Parisian option pricing problem, but convergence turns out to be rather slow. A possible explanation for this is in the local behavior of Brownian motion and we try to illustrate this in our last numerical example. The reason to treat double-sided Parisian options, apart from that there may be practical applications to this type of optionality, is that this contract type is rather general. After analysing the double-sided Parisian knock-in call contract, we are able to give prices for all the one-sided versions as well; we do not need to derive separate formulas for down-and-out calls and up-and-in puts and so on: everything follows from the Fourier transform of the double sided Parisian knock-in call. The concluding numerical examples will show the reader how the various Parisian option types that can be constructed from the double-sided knock-in call behave. The paper is organized as follows. In the first section we introduce the doublesided Parisian option and the relevant notation. In order to price the contract, we rewrite the pricing problem into the problem en of calculating a probability. In the second section we derive Laplace transforms for the double-sided Parisian stopping time and the value of standard Brownian motion at that stopping time. The third section treats the actual Fourier transform calculation, where some technical details are deferred to the appendix. The next section treats the case where the life of the option has started and we have the come up with a value of a Parisian option that has already spent some time above or below one of its triggering levels. The fifth section discusses the Parisian put contract type, where the seventh section summarizes all the contract types that can be derived from the double-sided knock-in call. In section eight we discuss the Fourier inversion and propose an alternative algorithm. In the last section we treat three numerical examples showing various features of the Parisian option and its Fourier transform.
Double Sided Parisian Option Pricing
3
1 The Parisian contract Let (Ω, F, P) be a probability space with filtration {Ft } and (Wt )t≥0 be a standard Brownian motion on this filtration. By (St )t≥0 we denote the risk-neutral stock price process, given by the classical geometric Brownian motion, 1 2 St = S0 e(r− 2 σ )t+σWt ,
where r and σ are the risk-free interest rate and the volatility respectively. In this setup P is the risk-neutral measure or, equivalently, the pricing measure and not the physical measure. Assuming that there exists a bank-account that pays the riskfree interest rate r in a continuously compounded way, the price of an option with (random) payoff Φ is given by its discounted expectation under the pricing measure. The random time γtL (S) measures the last time before T that a process S has been equal to L and is given by, γTL (S) := sup{0 ≤ t ≤ T |St = L},
(1)
Note that γTL is not a stopping time. In case the process is a standard Brownian motion, we suppress the W between brackets. Now we define the double sidedL1 −,L2 + Parisian stopping time TD (S) for the levels L1 < L2 by, 1 ,D2 ´ ³ L1 −,L2 + L1 − L2 + , TD (S) := min T , T ,D D D 1 2 1 2
(2)
L± where the single-sided Parisian stopping time TD is given by, ¾ ½ L± TD = inf t > 0|1 > (t − γtL (S)) > D) . {St k} where k = 1 ln K . {ST > K} = {W σ S0 In the same manner for i = 1, 2 the levels Li transform into li resulting in γTLi (S) = ˜ ). A change of measure allows us to compute the probability Pr (T ) by, γTli (W 1
Pr (T ) = e− 2 m
2
T
¤ £ E emWT 1{WT >k} 1{τ ≤T } ,
(7)
l1 −,l2 + where we used τ as a shorthand notation for TD (W ). The same kind of notation 1 ,D2 l1 − l2 + + − (W ). In (W ) and TD we introduce for τ and τ abbreviating respectively TD 1 2 the next section we derive formulas for the Laplace transforms of the double-sided Parisian stopping times for standard Brownian motion.
2 Calculating the Laplace Transform of the double sided hitting times (t)
The Brownian meander (mu )0≤u≤1 of time t is the absolute value of a rescaled Brownian motion path of [γt , t] to the time interval [0, 1], where we suppressed the (t) 0 in the notation of γt0 . Here we are only interested in m1 , the final value of the meander which we denote by nt given by, nt = √
1 |Wt |. t − γt
As pointed out in [8] nt is independent of the pair (γt , sgn(Wt )) and for every t > 0 d
we have nt = N , where N has the following density, P[N ∈ dx] = xe−
x2 2
1{x≥0} ,
(8)
and for later on it is useful to define the function Ψc for c ≥ 0 by,
√ £ ¤ c2 z2 Ψc (z) := E ezN 1{N ≥c} = e− 2 +zc + z 2πe 2 N (z − c),
(9)
where N is the CDF of the standard normal distribution. We will abbreviate Ψ0 by Ψ and 1 − Ψc by Ψ˜c . Now consider the meander at time t away from level l and denote its final value by nlt given by, 1{T K; TL2 > d] + P[ST > K; TL2 ≤ d; τ ≤ T ]. We remark that in the first probability on the right-hand side we should add the constraint that T > d, otherwise the Parisian knock-in has not taken place. The reason for leaving this out here, is that we know the value of the Parisian knock-in contract to be zero in case of T < d and we would not invert the Fourier Transform in this situation. We start calculating the Fourier transform φ1 of the first probability on the right-hand side and after that we will compute φ2 , the Fourier transform of the second probability on the right-hand side. First, we restate φ1 into terms of the standard Brownian motion. We want to use the strong Markov property later on, so we have to rewrite the probability and split the Fourier transform into two parts, Z ∞ i h e(iv−α)T E emWt 1{WT >k} (1 − 1{Tl2 ≤d} ) dT = φ1,1 (v) − φ1,2 (v). φ1 (v) = 0
The first part φ1,1 can be computed as follows, Z ∞ Z ∞ x2 1 (iv−α)T √ emx e− 2T dxdt = e φ1,1 (v; k) = 2πT k 0 ( (m−˜vα )k Z ∞ e , k ≥ 0, 1 emx−|x|˜vα dx = v˜α (˜v2α −m) e(m+˜vα )k = v˜α k (˜ v 2 −m2 ) − v ˜α (m+˜ vα ) , k < 0. α
Where we used the same type of arguments as in (18). Now for φ1,2 we get after conditioning on FTl2 and multiple applications of Fubini, iZ ∞ h ¤ £ φ1,2 (v) = eml2 E e(iv−α)Tl2 1{Tl2 ≤d} e(iv−α)T E emWT 1{WT >k−l2 } dT. (25) 0
The integral on the right-hand side of this equation equals φ1,1 (v; k−l). In Appendix 1 we compute the expectation on the right-hand side given in (29). Adding the results gives, ³ ´ φ1 (v) = φ1,1 (v; k) − el2 (m−˜vα ) N (c+ ) + el2 (m+˜vα ) N (c− ) φ1,1 (v; k − l2 ), where
l2 ± v˜α d √ d We still have to compute φ2 , which we can re-write by conditioning on FTl2 . We use the complete, rather elaborate, notation for the Parisian stopping time to explain the strong Markov property in more detail, Z ∞ i h φ2 (v) = e(iv−α)T E emWT 1{WT >k} 1{Tl2 ≤d} 1{τ ≤T } dt 0 Z ∞ i h ¤ £ ml2 e(iv−α)T E 1{Tl2 ≤d} E emWτ 1{Wτ >k−l2 } 1{˜τ ≤t} t=T −T dT, =e c± =
0
l2
Double Sided Parisian Option Pricing
where
l1 −,l2 + τ = TD 1 ,D2
9
and
(l −l )−,0+
τ˜ = TD11,D22
.
Again by substitution and multiple Fubini we get an equation for φ2 like (25), h iZ ∞ ¤ £ (iv−α)Tl2 ml2 e(iv−α)T E emWT 1{WT >k−l2 } 1{˜τ ≤T } dT 1{Tl2 ≤d} φ2 (v) = e E e 0
We recognize immediately the first part of φ1,2 given in (25) in this equation. The integral on the right-hand side is in fact nothing else than the original problem we are solving for different barriers. So finally we have, ³ ´ φ(v) = φ1,1 (v; k)+ el2 (m−˜vα ) N (c+ ) + el2 (m+˜vα ) N (c− ) × (φ(v; k − l2 , l1 − l2 , 0) − φ1,1 (v; k − l2 )) .
For now, we are able to price the double-sided Parisian knock-in call for all combination of initial stock price value, strike and barriers. The next section relates the Fourier transforms computed so far to the double-sided Parisian knock-in put.
5 The Parisian Put For the put option, we need in analogy with (5) to calculate the following probability, h i L1 −,L2 + ≤T . Pr ST ≤ K; TD 1 ,D2
In [8] the authors use an alternative type op put-call parity. Here we suggest the approach taken in [5], i.e. writing the probability as the difference of two probabilities, i h i h L1 −,L2 + L1 −,L2 + ≤ T , S ≥ K; T ≤ T − P P r TD T r D1 ,D2 1 ,D2
where the right-hand side probability is exactly the probability given in (5) and the left-hand side probability can be obtained by taking the limit of (5) for K ↓ 0. So we have to take k → −∞ in equations (20) and (24) resulting in, √ √ 2eml1 Ψ˜ (−m D1 ) 2eml2 Ψ (m D2 ) E (˜ v ) + E− (˜ vα ) for k → −∞. φ(v) = + α v˜α2 − m2 v˜α2 − m2
Now we have Fourier transforms for double-sided Parisian knock-in options, both put and call, we continue discussing the other types of Parisian contract types we can construct from the double-sided Parisian contract.
6 Other Types of Parisian Contracts We have started our calculation of the Fourier transform by decomposing the problem into the parts φ+ and φ− , where φ+ and φ− treat respectively the cases in which the knock-in takes place above level L2 and below level L1 . We construct different types of contracts, like the one-sided Parisian option, by different selections of φ + and φ− in the following way: – Case 1. φ = φ+ +φ− . The double-sided Parisian contract that is paying off when S stays longer than consecutive time D1 below level L1 or D2 above level L2 . – Case 2a. φ = φ+ The Parisian contract that pays off when S stays above level L2 for a consecutive period of length D2 , without having been below L1 for a period D1 before.
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J.H.M. Anderluh, J.A.M. van der Weide
– Case 2b. φ = φ− The Parisian contract that pays off when S stays below level L1 for a consecutive period of length D1 , without having been above L2 for a period D2 before. – Case 3a. φ = φ+ + φ− and l1 → −∞ or D1 → ∞. The one-sided Parisian up-and-in call. Taking these limits in the equations for φ give the formulas in [8], where we remark that we compute transforms of the probabilities needed to calculate the Parisian option value, where the authors of [8] compute transforms of the non discounted payoff. – Case 3b. φ = φ+ + φ− and l2 → ∞ or D2 → ∞. Analogously to the previous case, this is the one-sided down-and-in call.
We remark that for given levels L1 , L2 and periods D1 , D2 the double-sided Parisian contract in case 1 is the most expensive contract. The contracts specified in the case 2a and 2b are the cheapest type of Parisian contracts. Less expensive are the onesided Parisian options as described in cases 3a and 3b, where there is only one level for the stock price process that can cause a knock-in of the contract. Even cheaper is the kind of contracts specified in cases 2a and 2b. This type of Parisians do not only have just one stock price level that can cause a knock-in, but it also contains another level for the stock price process that knocks out the contract if the stock price process spends a certain time above or below this level. Note that it is possible to obtain (numerical) values for the one-sided Parisian contract without actually taking the limits as proposed in case 3. For a given time to expiry T the value of the one-sided Parisian down-and-in call can be obtained by inverting the Fourier transform of the double sided Parisian contract for some L2 and D2 > T . Similarly we get the value of the one-sided up-and-in by inverting the transform for some L1 and D1 > T . We will illustrate these remarks in the section on numerical examples. Figure 1 shows the relations between the double-sided and one-sided Parisian contracts, where we abbreviate the double-sided Parisian in and out call by DPIC and DPOC respectively. The one-sided contracts are either up (PU..) or down (PD..) and either in (P.I.) or out (P.O.) contracts. The same type of scheme could be drawn for the Parisian put contracts. Now we have computed and discussed various types of Parisian contracts, we discuss the Fourier inversion in the next section.
PUIC
IO−Par
PUOC L1 ↓ 0, D1 → ∞
L1 ↓ 0, D1 → ∞
φ
IFT
DPIC
IO−Par
L2 ↑ ∞, D2 → ∞
L2 ↑ ∞, D2 → ∞
PDIC
DPOC
IO−Par
Fig. 1 Relations between different types of Parisian contracts
PDOC
Double Sided Parisian Option Pricing
11
7 Fourier Inversion Algorithm 7.1 General Fourier Inversion. Apart from deriving Fourier transforms for the relevant probabilities for the doublesided Parisian option contracts, we are also interested in numerical values for these options. We have to obtain these values by inversion of the Fourier transform for the probabilities we need to construct the contract, where we recall formula (4). In (15) the definition of φ, the Fourier transform of the probability Pr (T ) is stated and values for Pr (T ) can be obtained by the following standard Fourier inversion formula, Z Z 2eaT ∞ eaT ∞ ivT e φ(v)dv = cos(vT ) u∗2 the argument for I is negative, resulting in, Z ∞ √ u2 uem D2 u− 2 Ik−√D2 u−l2 du u∗ 2
! √ e(˜vα +m)(k− D2 u−l2 ) 2˜ vα ue du = − v˜α2 − m2 m + v˜α u∗ 2 i e(˜vα +m)(k−l2 ) h i h √ √ 2˜ vα = 2 E em D2 N 1{N ≥u∗2 } − E e−˜vα D2 N 1{N ≥u∗2 } . 2 v˜α − m m + v˜α Z
∞
√ 2 m D2 u− u2
Ã
Adding up these results gives, Ã √ √ vα D2 ) 2eml2 Ψu∗2 (m D2 ) e(m−˜vα )k+l2 v˜α Ψ˜u∗2 (˜ + φ+ (v) = E+ (˜ vα ) v˜α (˜ vα − m) v˜α2 − m2 ! √ e(˜vα +m)k−l2 v˜α Ψu∗2 (−˜ v α D2 ) k > l2 . − v˜α (˜ vα + m)
In order to be able to calculate double-sided Parisian call prices for all strikes, we finally have to calculate φ− for k < l1 . The strategy is again to split up the u-integral in (21) into two the parts, i.e. u ≤ u∗1 and u > u∗1 where u∗1 is given by, l1 − k . u∗1 = √ D1
18
J.H.M. Anderluh, J.A.M. van der Weide
We can write for (21), ·Z ∞ ´ ¸ ³ p e(iv−α)ρ h ρ, l1 − D1 N dρ = E 0 ÃZ ∗ ! Z ∞ u1 ml1 √ √ 2 2 e −m D1 u− u2 −m D1 u− u2 √ √ Ik−l1 + D1 u du + Ik−l1 + D1 u du . ue ue v˜α u∗ 0 1 Now for u ≤ u∗1 the argument of I is negative, so we get, Z u∗1 √ u2 ue−m D1 u− 2 Ik−l1 +√D1 u du 0 ! Ã √ Z u∗1 √ 2 2˜ vα e(m+˜vα )(k−l1 + D1 u) −m D1 u− u2 = − ue v˜α2 − m2 m + v˜α 0 h i e(m+˜vα )(k−l1 ) h √ i √ 2˜ vα −m D1 N v ˜α D1 N ∗} − ∗} . = 2 e e E E 1 1 {N ≤u {N ≤u 1 1 v˜α − m2 m + v˜α
For u > u∗1 the argument of I is positive, giving, √ Z ∞ Z ∞ (m−˜ vα )(k−l1 + D1 u) √ √ 2 u2 e −m D u− −m D1 u− u2 1 2 ue Ik−l1 +√D1 u du = du ue v˜α − m u∗ u∗ 1 1 i e(m−˜vα )(k−l1 ) h −˜vα √D1 N E e = 1{N >u∗1 } . v˜α − m Adding up these results and using the special function Ψ gives for ψ− , Ã √ √ e(m−˜vα )k+l1 v˜α Ψu∗1 (−˜ vα D1 ) 2eml1 Ψ˜u∗1 (−m D1 ) + ψ− (v) = E− (˜ vα ) v˜α (˜ vα − m) v˜α2 − m2 ! √ e(m+˜vα )k−l1 v˜α Ψ˜u∗1 (˜ v α D1 ) − k < l1 . v˜α (m + v˜α ) 9.2 The Laplace Transform of Tl occurring before d. We define ψb (λ; d), the Laplace transform of the distribution of the hitting time of level l by a standard Brownian motion W restricted to the set where this hitting time occurs before d by, £ ¤ ψl (λ; d) := E e−λTl 1{Tl ≤d} λ ≥ 0, z2
Now we construct a stopping time S = Tl ∧d and use the martingale Mt = e− 2 t+zWt for our computation. As S is a bounded stopping time, we can use optional sampling to arrive at, h z2 i £ ¤ z2 1 = E[MS ] = ezl E e− 2 Tl 1{Tl ≤d} + e− 2 d E ezWd 1{Tl >d}
The second expectation on the right-hand side can explicitly be calculated as the density P[Wd ∈ dx; Tl > d] is well-known (see e.g. [9]), ¶ µ Z b £ ¤ (x−2l)2 x2 1 ezx e− 2d − e− 2d dx E ezWd 1{Tl >d} = √ 2πd −∞ ¶¸ · µ ¶ µ z2 d l − zd −l − zd 2lz 2 √ √ =e N −e N d d
Double Sided Parisian Option Pricing
19
Here we assumed l ≥ 0. The calculations of l ≤ 0 proceed in the same way and we can write a general result for all l and λ ≥ 0 by, Ã Ã ! ! √ √ √ √ −|l| + 2λd −|l| − 2λd |l| 2λ −|l| 2λ √ √ N N +e (29) ψl (λ; d) = e d d 9.3 Discretization Error. If we define the function f by, f (t) = e−at Pr (t)1{t≥0} , then the Fourier transform φ we compute throughout the paper is in fact the Fourier transform of this function f . Now define the periodic function fp by, ∞ X
fp (t) :=
f
j=−∞
µ
t+
2πj h
¶
.
(30)
This sum is uniformly bounded in t by, fp (t) ≤
∞ X
e−a
≤
j=0
Now fp is periodic with period fp (t) =
2πj h
2π h ,
h 2π
Z
∞
e−ax dx =
− 2π h
h 2πa e h . 2πa
so we obtain its Fourier series by,
∞ X
h cn einht 2π n=−∞
where
cn =
Z
π/h
fp (t)e−inht dt
(31)
−π/h
Calculation of the coefficients cn gives, cn =
Z
π/h
fp (t)e
−inht
dt =
−π/h
=
∞ Z X
j=−∞
π/h
f −π/h
µ
t+
2πj h
π/h
∞ X
f
−π/h j=−∞
µ
¶
e−inht dt =
Z
Z
2πj t+ h ∞
¶
e−inht dt
f (t)e−inht dt = φ(−nh)
(32)
−∞
The interchange of sum and integral is allowed by dominated convergence. Using the coefficients in (32) and the equality between (31) and (30) we obtain the Poisson summation formula, ∞ X
j=−∞
f
µ
2πj t+ h
¶
=
∞ h X φ(−nh)einht . 2π n=−∞
(33)
We can derive the same result for the symmetric function g defined by g(t) = f (|t|). For a fixed t > 0 we can set h = δt and obtain, ¶¶ µ µ ∞ δ X 2π j = φg (−nh)einht g t 1+ δ 2πt n=−∞ j=−∞ ½ µ ¶¾ ∞ δ 2δ X nδ = 0 we have, ½ µ ¶¾ ∞ 2δ X δ nδ cos(nδ) < φ f (t) = g(t) = 0
In order to get an estimate for the error, we need to control the last sum term. Suppose we choose δ < 2π, then we have for this error term ², ¯ ¯¶ µ ¶¶ X µ µ X ¯ 2π ¯¯ 2π ¯ g t 1+ exp −at ¯1 + ²= j ≤ j δ δ ¯ |j|>0 |j|>0 µ µ ¶¶ 2π ∞ X 2π 2e−a( δ −1)t exp −at ≤2 (35) j−1 ≤ 2π δ 1 − e−a δ t j=1 9.4 Truncation Error. If we approximate the integral by a sum, we have Z ∞ ∞ X cos(vj T )
Double Sided Parisian Option Pricing J.H.M. Anderluh, J.A.M. van der Weide TUDelft Delft Institute of Applied Mathematics P.O. Box 5031 2600 GA Delft (The Netherlands) e-mail: [email protected] August 2006
Abstract In this paper we derive Fourier transforms for double sided Parisian option contracts. The double sided Parisian option contract is triggered by the stock price process spending some time above an upper level or below some lower level. The double sided Parisian knock-in call contract is the general type of Parisian contract from which all the one-sided contract types follow. We also discuss the Fourier inversion in the paper and conclude with a series of numerical examples, explaining the Parisian optionality and the way prices are affected by the local behavior of Brownian motion in detail.
Key words
Parisian Options – Excursions – Fourier Inversion
JEL Classification: G12, G13 MSC : 60G40, 62L15, 60J65
2
J.H.M. Anderluh, J.A.M. van der Weide
Introduction The Parisian option is a kind of a barrier option with the difference that the contract is not specified in terms of touching a barrier, but in terms of staying or below the barrier for a certain period of time. The interest in these options is motivated by the study of structured products and investment problems. Convertible bonds and problems in real options contain Parisian optionality; the Parisian option contract itself is not exchange traded. Details about the practical differences between standard barrier options and Parisian options are discussed in [8], the first paper on Parisian options. The way Parisian options turn up in real option problems is treated in [12]. The authors in [8] derived Laplace transforms for the one-sided version, which is extended in [12] to a Parisian type of contract that is triggered by staying a period of time above the barrier or hitting a lever exceeding this barrier. Here we treat pricing of the double sided Parisian option and, like the papers previously mentioned, we use Fourier (or Laplace) transforms to achieve this. The calculation of Fourier transform instead of Laplace transforms is motivated by the fact that a lot of numerical Laplace inversion algorithms are using the complex continuation of Laplace transforms to Fourier transforms for the actual inversion, see e.g. [10]. As we want to conclude our paper by a section on numerical examples, we have to invert the Fourier transforms we will calculate. In [13] the authors treat a PDE method approach to solve the Parisian option pricing problem, but convergence turns out to be rather slow. A possible explanation for this is in the local behavior of Brownian motion and we try to illustrate this in our last numerical example. The reason to treat double-sided Parisian options, apart from that there may be practical applications to this type of optionality, is that this contract type is rather general. After analysing the double-sided Parisian knock-in call contract, we are able to give prices for all the one-sided versions as well; we do not need to derive separate formulas for down-and-out calls and up-and-in puts and so on: everything follows from the Fourier transform of the double sided Parisian knock-in call. The concluding numerical examples will show the reader how the various Parisian option types that can be constructed from the double-sided knock-in call behave. The paper is organized as follows. In the first section we introduce the doublesided Parisian option and the relevant notation. In order to price the contract, we rewrite the pricing problem into the problem en of calculating a probability. In the second section we derive Laplace transforms for the double-sided Parisian stopping time and the value of standard Brownian motion at that stopping time. The third section treats the actual Fourier transform calculation, where some technical details are deferred to the appendix. The next section treats the case where the life of the option has started and we have the come up with a value of a Parisian option that has already spent some time above or below one of its triggering levels. The fifth section discusses the Parisian put contract type, where the seventh section summarizes all the contract types that can be derived from the double-sided knock-in call. In section eight we discuss the Fourier inversion and propose an alternative algorithm. In the last section we treat three numerical examples showing various features of the Parisian option and its Fourier transform.
Double Sided Parisian Option Pricing
3
1 The Parisian contract Let (Ω, F, P) be a probability space with filtration {Ft } and (Wt )t≥0 be a standard Brownian motion on this filtration. By (St )t≥0 we denote the risk-neutral stock price process, given by the classical geometric Brownian motion, 1 2 St = S0 e(r− 2 σ )t+σWt ,
where r and σ are the risk-free interest rate and the volatility respectively. In this setup P is the risk-neutral measure or, equivalently, the pricing measure and not the physical measure. Assuming that there exists a bank-account that pays the riskfree interest rate r in a continuously compounded way, the price of an option with (random) payoff Φ is given by its discounted expectation under the pricing measure. The random time γtL (S) measures the last time before T that a process S has been equal to L and is given by, γTL (S) := sup{0 ≤ t ≤ T |St = L},
(1)
Note that γTL is not a stopping time. In case the process is a standard Brownian motion, we suppress the W between brackets. Now we define the double sidedL1 −,L2 + Parisian stopping time TD (S) for the levels L1 < L2 by, 1 ,D2 ´ ³ L1 −,L2 + L1 − L2 + , TD (S) := min T , T ,D D D 1 2 1 2
(2)
L± where the single-sided Parisian stopping time TD is given by, ¾ ½ L± TD = inf t > 0|1 > (t − γtL (S)) > D) . {St k} where k = 1 ln K . {ST > K} = {W σ S0 In the same manner for i = 1, 2 the levels Li transform into li resulting in γTLi (S) = ˜ ). A change of measure allows us to compute the probability Pr (T ) by, γTli (W 1
Pr (T ) = e− 2 m
2
T
¤ £ E emWT 1{WT >k} 1{τ ≤T } ,
(7)
l1 −,l2 + where we used τ as a shorthand notation for TD (W ). The same kind of notation 1 ,D2 l1 − l2 + + − (W ). In (W ) and TD we introduce for τ and τ abbreviating respectively TD 1 2 the next section we derive formulas for the Laplace transforms of the double-sided Parisian stopping times for standard Brownian motion.
2 Calculating the Laplace Transform of the double sided hitting times (t)
The Brownian meander (mu )0≤u≤1 of time t is the absolute value of a rescaled Brownian motion path of [γt , t] to the time interval [0, 1], where we suppressed the (t) 0 in the notation of γt0 . Here we are only interested in m1 , the final value of the meander which we denote by nt given by, nt = √
1 |Wt |. t − γt
As pointed out in [8] nt is independent of the pair (γt , sgn(Wt )) and for every t > 0 d
we have nt = N , where N has the following density, P[N ∈ dx] = xe−
x2 2
1{x≥0} ,
(8)
and for later on it is useful to define the function Ψc for c ≥ 0 by,
√ £ ¤ c2 z2 Ψc (z) := E ezN 1{N ≥c} = e− 2 +zc + z 2πe 2 N (z − c),
(9)
where N is the CDF of the standard normal distribution. We will abbreviate Ψ0 by Ψ and 1 − Ψc by Ψ˜c . Now consider the meander at time t away from level l and denote its final value by nlt given by, 1{T K; TL2 > d] + P[ST > K; TL2 ≤ d; τ ≤ T ]. We remark that in the first probability on the right-hand side we should add the constraint that T > d, otherwise the Parisian knock-in has not taken place. The reason for leaving this out here, is that we know the value of the Parisian knock-in contract to be zero in case of T < d and we would not invert the Fourier Transform in this situation. We start calculating the Fourier transform φ1 of the first probability on the right-hand side and after that we will compute φ2 , the Fourier transform of the second probability on the right-hand side. First, we restate φ1 into terms of the standard Brownian motion. We want to use the strong Markov property later on, so we have to rewrite the probability and split the Fourier transform into two parts, Z ∞ i h e(iv−α)T E emWt 1{WT >k} (1 − 1{Tl2 ≤d} ) dT = φ1,1 (v) − φ1,2 (v). φ1 (v) = 0
The first part φ1,1 can be computed as follows, Z ∞ Z ∞ x2 1 (iv−α)T √ emx e− 2T dxdt = e φ1,1 (v; k) = 2πT k 0 ( (m−˜vα )k Z ∞ e , k ≥ 0, 1 emx−|x|˜vα dx = v˜α (˜v2α −m) e(m+˜vα )k = v˜α k (˜ v 2 −m2 ) − v ˜α (m+˜ vα ) , k < 0. α
Where we used the same type of arguments as in (18). Now for φ1,2 we get after conditioning on FTl2 and multiple applications of Fubini, iZ ∞ h ¤ £ φ1,2 (v) = eml2 E e(iv−α)Tl2 1{Tl2 ≤d} e(iv−α)T E emWT 1{WT >k−l2 } dT. (25) 0
The integral on the right-hand side of this equation equals φ1,1 (v; k−l). In Appendix 1 we compute the expectation on the right-hand side given in (29). Adding the results gives, ³ ´ φ1 (v) = φ1,1 (v; k) − el2 (m−˜vα ) N (c+ ) + el2 (m+˜vα ) N (c− ) φ1,1 (v; k − l2 ), where
l2 ± v˜α d √ d We still have to compute φ2 , which we can re-write by conditioning on FTl2 . We use the complete, rather elaborate, notation for the Parisian stopping time to explain the strong Markov property in more detail, Z ∞ i h φ2 (v) = e(iv−α)T E emWT 1{WT >k} 1{Tl2 ≤d} 1{τ ≤T } dt 0 Z ∞ i h ¤ £ ml2 e(iv−α)T E 1{Tl2 ≤d} E emWτ 1{Wτ >k−l2 } 1{˜τ ≤t} t=T −T dT, =e c± =
0
l2
Double Sided Parisian Option Pricing
where
l1 −,l2 + τ = TD 1 ,D2
9
and
(l −l )−,0+
τ˜ = TD11,D22
.
Again by substitution and multiple Fubini we get an equation for φ2 like (25), h iZ ∞ ¤ £ (iv−α)Tl2 ml2 e(iv−α)T E emWT 1{WT >k−l2 } 1{˜τ ≤T } dT 1{Tl2 ≤d} φ2 (v) = e E e 0
We recognize immediately the first part of φ1,2 given in (25) in this equation. The integral on the right-hand side is in fact nothing else than the original problem we are solving for different barriers. So finally we have, ³ ´ φ(v) = φ1,1 (v; k)+ el2 (m−˜vα ) N (c+ ) + el2 (m+˜vα ) N (c− ) × (φ(v; k − l2 , l1 − l2 , 0) − φ1,1 (v; k − l2 )) .
For now, we are able to price the double-sided Parisian knock-in call for all combination of initial stock price value, strike and barriers. The next section relates the Fourier transforms computed so far to the double-sided Parisian knock-in put.
5 The Parisian Put For the put option, we need in analogy with (5) to calculate the following probability, h i L1 −,L2 + ≤T . Pr ST ≤ K; TD 1 ,D2
In [8] the authors use an alternative type op put-call parity. Here we suggest the approach taken in [5], i.e. writing the probability as the difference of two probabilities, i h i h L1 −,L2 + L1 −,L2 + ≤ T , S ≥ K; T ≤ T − P P r TD T r D1 ,D2 1 ,D2
where the right-hand side probability is exactly the probability given in (5) and the left-hand side probability can be obtained by taking the limit of (5) for K ↓ 0. So we have to take k → −∞ in equations (20) and (24) resulting in, √ √ 2eml1 Ψ˜ (−m D1 ) 2eml2 Ψ (m D2 ) E (˜ v ) + E− (˜ vα ) for k → −∞. φ(v) = + α v˜α2 − m2 v˜α2 − m2
Now we have Fourier transforms for double-sided Parisian knock-in options, both put and call, we continue discussing the other types of Parisian contract types we can construct from the double-sided Parisian contract.
6 Other Types of Parisian Contracts We have started our calculation of the Fourier transform by decomposing the problem into the parts φ+ and φ− , where φ+ and φ− treat respectively the cases in which the knock-in takes place above level L2 and below level L1 . We construct different types of contracts, like the one-sided Parisian option, by different selections of φ + and φ− in the following way: – Case 1. φ = φ+ +φ− . The double-sided Parisian contract that is paying off when S stays longer than consecutive time D1 below level L1 or D2 above level L2 . – Case 2a. φ = φ+ The Parisian contract that pays off when S stays above level L2 for a consecutive period of length D2 , without having been below L1 for a period D1 before.
10
J.H.M. Anderluh, J.A.M. van der Weide
– Case 2b. φ = φ− The Parisian contract that pays off when S stays below level L1 for a consecutive period of length D1 , without having been above L2 for a period D2 before. – Case 3a. φ = φ+ + φ− and l1 → −∞ or D1 → ∞. The one-sided Parisian up-and-in call. Taking these limits in the equations for φ give the formulas in [8], where we remark that we compute transforms of the probabilities needed to calculate the Parisian option value, where the authors of [8] compute transforms of the non discounted payoff. – Case 3b. φ = φ+ + φ− and l2 → ∞ or D2 → ∞. Analogously to the previous case, this is the one-sided down-and-in call.
We remark that for given levels L1 , L2 and periods D1 , D2 the double-sided Parisian contract in case 1 is the most expensive contract. The contracts specified in the case 2a and 2b are the cheapest type of Parisian contracts. Less expensive are the onesided Parisian options as described in cases 3a and 3b, where there is only one level for the stock price process that can cause a knock-in of the contract. Even cheaper is the kind of contracts specified in cases 2a and 2b. This type of Parisians do not only have just one stock price level that can cause a knock-in, but it also contains another level for the stock price process that knocks out the contract if the stock price process spends a certain time above or below this level. Note that it is possible to obtain (numerical) values for the one-sided Parisian contract without actually taking the limits as proposed in case 3. For a given time to expiry T the value of the one-sided Parisian down-and-in call can be obtained by inverting the Fourier transform of the double sided Parisian contract for some L2 and D2 > T . Similarly we get the value of the one-sided up-and-in by inverting the transform for some L1 and D1 > T . We will illustrate these remarks in the section on numerical examples. Figure 1 shows the relations between the double-sided and one-sided Parisian contracts, where we abbreviate the double-sided Parisian in and out call by DPIC and DPOC respectively. The one-sided contracts are either up (PU..) or down (PD..) and either in (P.I.) or out (P.O.) contracts. The same type of scheme could be drawn for the Parisian put contracts. Now we have computed and discussed various types of Parisian contracts, we discuss the Fourier inversion in the next section.
PUIC
IO−Par
PUOC L1 ↓ 0, D1 → ∞
L1 ↓ 0, D1 → ∞
φ
IFT
DPIC
IO−Par
L2 ↑ ∞, D2 → ∞
L2 ↑ ∞, D2 → ∞
PDIC
DPOC
IO−Par
Fig. 1 Relations between different types of Parisian contracts
PDOC
Double Sided Parisian Option Pricing
11
7 Fourier Inversion Algorithm 7.1 General Fourier Inversion. Apart from deriving Fourier transforms for the relevant probabilities for the doublesided Parisian option contracts, we are also interested in numerical values for these options. We have to obtain these values by inversion of the Fourier transform for the probabilities we need to construct the contract, where we recall formula (4). In (15) the definition of φ, the Fourier transform of the probability Pr (T ) is stated and values for Pr (T ) can be obtained by the following standard Fourier inversion formula, Z Z 2eaT ∞ eaT ∞ ivT e φ(v)dv = cos(vT ) u∗2 the argument for I is negative, resulting in, Z ∞ √ u2 uem D2 u− 2 Ik−√D2 u−l2 du u∗ 2
! √ e(˜vα +m)(k− D2 u−l2 ) 2˜ vα ue du = − v˜α2 − m2 m + v˜α u∗ 2 i e(˜vα +m)(k−l2 ) h i h √ √ 2˜ vα = 2 E em D2 N 1{N ≥u∗2 } − E e−˜vα D2 N 1{N ≥u∗2 } . 2 v˜α − m m + v˜α Z
∞
√ 2 m D2 u− u2
Ã
Adding up these results gives, Ã √ √ vα D2 ) 2eml2 Ψu∗2 (m D2 ) e(m−˜vα )k+l2 v˜α Ψ˜u∗2 (˜ + φ+ (v) = E+ (˜ vα ) v˜α (˜ vα − m) v˜α2 − m2 ! √ e(˜vα +m)k−l2 v˜α Ψu∗2 (−˜ v α D2 ) k > l2 . − v˜α (˜ vα + m)
In order to be able to calculate double-sided Parisian call prices for all strikes, we finally have to calculate φ− for k < l1 . The strategy is again to split up the u-integral in (21) into two the parts, i.e. u ≤ u∗1 and u > u∗1 where u∗1 is given by, l1 − k . u∗1 = √ D1
18
J.H.M. Anderluh, J.A.M. van der Weide
We can write for (21), ·Z ∞ ´ ¸ ³ p e(iv−α)ρ h ρ, l1 − D1 N dρ = E 0 ÃZ ∗ ! Z ∞ u1 ml1 √ √ 2 2 e −m D1 u− u2 −m D1 u− u2 √ √ Ik−l1 + D1 u du + Ik−l1 + D1 u du . ue ue v˜α u∗ 0 1 Now for u ≤ u∗1 the argument of I is negative, so we get, Z u∗1 √ u2 ue−m D1 u− 2 Ik−l1 +√D1 u du 0 ! Ã √ Z u∗1 √ 2 2˜ vα e(m+˜vα )(k−l1 + D1 u) −m D1 u− u2 = − ue v˜α2 − m2 m + v˜α 0 h i e(m+˜vα )(k−l1 ) h √ i √ 2˜ vα −m D1 N v ˜α D1 N ∗} − ∗} . = 2 e e E E 1 1 {N ≤u {N ≤u 1 1 v˜α − m2 m + v˜α
For u > u∗1 the argument of I is positive, giving, √ Z ∞ Z ∞ (m−˜ vα )(k−l1 + D1 u) √ √ 2 u2 e −m D u− −m D1 u− u2 1 2 ue Ik−l1 +√D1 u du = du ue v˜α − m u∗ u∗ 1 1 i e(m−˜vα )(k−l1 ) h −˜vα √D1 N E e = 1{N >u∗1 } . v˜α − m Adding up these results and using the special function Ψ gives for ψ− , Ã √ √ e(m−˜vα )k+l1 v˜α Ψu∗1 (−˜ vα D1 ) 2eml1 Ψ˜u∗1 (−m D1 ) + ψ− (v) = E− (˜ vα ) v˜α (˜ vα − m) v˜α2 − m2 ! √ e(m+˜vα )k−l1 v˜α Ψ˜u∗1 (˜ v α D1 ) − k < l1 . v˜α (m + v˜α ) 9.2 The Laplace Transform of Tl occurring before d. We define ψb (λ; d), the Laplace transform of the distribution of the hitting time of level l by a standard Brownian motion W restricted to the set where this hitting time occurs before d by, £ ¤ ψl (λ; d) := E e−λTl 1{Tl ≤d} λ ≥ 0, z2
Now we construct a stopping time S = Tl ∧d and use the martingale Mt = e− 2 t+zWt for our computation. As S is a bounded stopping time, we can use optional sampling to arrive at, h z2 i £ ¤ z2 1 = E[MS ] = ezl E e− 2 Tl 1{Tl ≤d} + e− 2 d E ezWd 1{Tl >d}
The second expectation on the right-hand side can explicitly be calculated as the density P[Wd ∈ dx; Tl > d] is well-known (see e.g. [9]), ¶ µ Z b £ ¤ (x−2l)2 x2 1 ezx e− 2d − e− 2d dx E ezWd 1{Tl >d} = √ 2πd −∞ ¶¸ · µ ¶ µ z2 d l − zd −l − zd 2lz 2 √ √ =e N −e N d d
Double Sided Parisian Option Pricing
19
Here we assumed l ≥ 0. The calculations of l ≤ 0 proceed in the same way and we can write a general result for all l and λ ≥ 0 by, Ã Ã ! ! √ √ √ √ −|l| + 2λd −|l| − 2λd |l| 2λ −|l| 2λ √ √ N N +e (29) ψl (λ; d) = e d d 9.3 Discretization Error. If we define the function f by, f (t) = e−at Pr (t)1{t≥0} , then the Fourier transform φ we compute throughout the paper is in fact the Fourier transform of this function f . Now define the periodic function fp by, ∞ X
fp (t) :=
f
j=−∞
µ
t+
2πj h
¶
.
(30)
This sum is uniformly bounded in t by, fp (t) ≤
∞ X
e−a
≤
j=0
Now fp is periodic with period fp (t) =
2πj h
2π h ,
h 2π
Z
∞
e−ax dx =
− 2π h
h 2πa e h . 2πa
so we obtain its Fourier series by,
∞ X
h cn einht 2π n=−∞
where
cn =
Z
π/h
fp (t)e−inht dt
(31)
−π/h
Calculation of the coefficients cn gives, cn =
Z
π/h
fp (t)e
−inht
dt =
−π/h
=
∞ Z X
j=−∞
π/h
f −π/h
µ
t+
2πj h
π/h
∞ X
f
−π/h j=−∞
µ
¶
e−inht dt =
Z
Z
2πj t+ h ∞
¶
e−inht dt
f (t)e−inht dt = φ(−nh)
(32)
−∞
The interchange of sum and integral is allowed by dominated convergence. Using the coefficients in (32) and the equality between (31) and (30) we obtain the Poisson summation formula, ∞ X
j=−∞
f
µ
2πj t+ h
¶
=
∞ h X φ(−nh)einht . 2π n=−∞
(33)
We can derive the same result for the symmetric function g defined by g(t) = f (|t|). For a fixed t > 0 we can set h = δt and obtain, ¶¶ µ µ ∞ δ X 2π j = φg (−nh)einht g t 1+ δ 2πt n=−∞ j=−∞ ½ µ ¶¾ ∞ δ 2δ X nδ = 0 we have, ½ µ ¶¾ ∞ 2δ X δ nδ cos(nδ) < φ f (t) = g(t) = 0
In order to get an estimate for the error, we need to control the last sum term. Suppose we choose δ < 2π, then we have for this error term ², ¯ ¯¶ µ ¶¶ X µ µ X ¯ 2π ¯¯ 2π ¯ g t 1+ exp −at ¯1 + ²= j ≤ j δ δ ¯ |j|>0 |j|>0 µ µ ¶¶ 2π ∞ X 2π 2e−a( δ −1)t exp −at ≤2 (35) j−1 ≤ 2π δ 1 − e−a δ t j=1 9.4 Truncation Error. If we approximate the integral by a sum, we have Z ∞ ∞ X cos(vj T )
