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Symmetrization associated with hyperbolic reflection principle
Pacific Journal of Mathematics for Industry volume 10, Article number: 1 (2018)
Abstract
In this paper, in view of application to pricing of Barrier options under a stochastic volatility model, we study a reflection principle for the hyperbolic Brownian motion, and introduce a hyperbolic version of ImamuraIshigakiOkumura’s symmetrization. Some results of numerical experiments, which imply the efficiency of the numerical scheme based on the symmetrization, are given.
Introduction
Reflection principle and the static hedge of barrier options
The reflection principle of standard Brownian motion relates the probability distribution of a first hitting time to a boundary to the 1dimensional marginal distribution of the process. The formula has a direct application in continuoustime finance, that is, the static hedging of barrier options^{Footnote 1}. The idea is explained roughly as follows. Suppose that we sold a knockout call option^{Footnote 2} (which is a typical barrier option). Its payoff is described as
where

T is the expiry date of the option,

K is the exercise price,

S is the price process of a risky asset, with S_{0}>K,
and

τ:= inf{s>0:S_{ s }<K^{′}}, the first hitting time of S to K^{′}, the knockout boundary, with K^{′}<K.
The static hedge of the knockout option consists of two plainvanilla (=without knockout condition) options, long position of call option with payoff (S_{ T }−K)_{+}, and short position of “put option” whose value

at τ equals the call, and

is zero at T on τ>T.
This simple portfolio hedges the knockout option since it is

zero at T on τ≤T since at τ it is liquidated, and

(S_{ T }−K) at T on τ>T.
This relation can be expressed as
for 0≤t≤T, where \( \{ \mathcal {F}_{t} \} \) is the filtration generated by S. The existence of such an option is ensured by the reflection principle. If S is geometric Brownian motion, it can be the option with payoff (K−S_{ T })_{+} since
by the reflection principle. In general, the property is referred to as (arithmetic) putcall symmetry at K [4], which is weaker than the reflection principle that ensures putcall symmetry for any K.
The interpretation is first proposed in [3], and there are vast literatures since then (see e.g. [1] and references therein). Among these, we just mention a multidimensional extension proposed in [11], where the reflection principle with respect to reflection groups is applied to the pricing of multiasset barrier options, barrier being the boundary of a Weyl chamber. To the best of our knowledge, it is the first attempt to deal with the cases where the barrier(=knockout/in boundary) is not a one point set.
Symmetrization and its application to numerical calculation of the price of barrier options
A new point of view in the literature, the symmetrization, was first introduced in [10], and further generalized in [2]. The symmetrization is a procedure to convert a given diffusion into the one with a weaker version of reflection principle, aiming at obtaining a precise numerical value of the price of barrier options in a reasonable computational time, rather than statichedge in the market described in the previous section.
Let us briefly explain their idea. We work on 1dimensional case for simplicity. Let S be a diffusion process satisfying the following stochastic differential equation:
where, σ and μ are piecewise continuous functions with linear growth. In general we cannot expect the formula (1) to hold. The symmetrization \( \tilde {S} \) of S alternatively satisfies (1). It is defined as a (weak) solution to
where
and
The following is proven in [10].
Theorem 1
(ImamuraIshigakiOkumura [10]) The lawunique solution \( \tilde {S} \) of (4) satisfies the putcall symmetry at K, and \( (\tilde {S}_{t})_{0 \leq t \leq \tau } \) has the same law as (S_{ t })_{0≤t≤τ}.
It then implies
The formula (6), however, is not anymore interpreted as static hedge relation, but it has another application. The Eq. (6) now reads that

An expectation with stopping time is converted to the one without it.
A numerical calculation of an expectation with stopping time often is a tough challenge due to its pathdependent nature. On the other hand, an expectation with respect to one dimensional marginal of a diffusion process is in most cases numerically tractable. Thus the Eq. (6) gives a new insight to the numerical analysis of barrier options/stopping times.
EulerMaruyama approximation of the price of barrier options
The most common technique to numerically approximate an expectation with respect to a diffusion process would be socalled “EulerMaruyama” scheme. Here we briefly recall the scheme.
An EulerMaruyama discretization of (3) with respect to a time partition 0=t_{0}<t_{1}<⋯<t_{ n }=T is given by
where \( \Delta W^{n}_{t_{k}} \sim N (0, t_{k+1} t_{k}) \), mutually independent for k=0,1,⋯,n−1. The stopping time τ is also approximated by
The expectation in the lefthandside of (6) is approximated by (MonteCarlo simulation of)
while the righthandside counterpart is
where \( \tilde {S}^{n} \) is obtained by the same procedure as (7).
The discretization error, by which we mean the difference between the true value of the expectation and its EulerMaruyama approximation like (8) or (9), is known to be of O(n^{−1/2})in general when t_{ k }−t_{k−1}=T/n for all k. It is reported in [5] that the one with stopping time like (8) cannot be improved, while the one with onedimensional marginal like (9) is, provided some continuity of the coefficients, known to be of O(n^{−1}).
The symmetrized drift coefficient (5) may not be continuous in general even if the original one is very smooth, and as far as we know, no existing result ensures the order is of O(n^{−1}) though recently there have been several papers ([12, 13], and [14]) to deal with discontinuous coefficients in line with the problem posed here. In [10], however, they conjecture that it is the case by performing numerical experiments.
References for more detailed and precise results of the order can be found in [10].
SABR model and hyperbolic Brownian motion
In the present paper, we study a hyperbolic version of the symmetrization, with a view to the application of the pricing of barrier options under SABR model, which is known to be transformed to hyperbolic Brownian motion with drift.
The SABR (stochastic alhabetarho) model was introduced in [6]. It is given by
where (W^{1},W^{2}) is a two dimensional Brownian motion, ρ∈(−1,1) and ν is a constant. We note that

A driftless local volatility model is obtained by setting ν=0, and

\( Z_{t} := \psi (S_{t/\nu ^{2}}, V_{t/\nu ^{2}}) + \sqrt {1} V_{t/\nu ^{2}}\) with \( \psi (x,y) = (x\rho y)/\sqrt {1\rho ^{2}} \) is a hyperbolic Brownian motion with drift, a solution to (13) in “Hyperbolic symmetrization” section (for details see [7]).
The following is a “motto” widely accepted among researchers and practitioners in finance (see e.g. [8]): as tractability of one dimensional diffusion processes is attributed to the reduction to the standard Brownian motion with drift by the Lamperti transform, so the analysis of SABR model will be converted to that of hyperbolic Brownian motion with drift, where we can still work on symmetries from linear fractional transformations. We shall observe a realization of this idea in the present paper.
The contents of the present paper
We start with introducing a hyperbolic version of the reflection principle that parallels the one with the standard Brownian motion in “Hyperbolic reflection principle” section. We introduce in “Hyperbolic symmetrization” section a weak version of the reflection principle, which also parallels with the classical putcall symmetry. Associated symmetrization is then introduced. “Numerical experiments” section is devoted to numerical studies. As in the case of the ImamuraIshigakiOkumura’s scheme using classical symmetrization, the error is not proven to be O(n^{−1}) mathematically but the numerical results support the conjecture of the hyperbolic case as well.
Hyperbolic reflection principle
Invariant property of hyperbolic Brownian motion
A Hyperbolic Brownian motion is the unique solution to
where W^{1} and W^{2} are independent Brownian motions. It is defined on the upperhalf plane \(\mathbb {H} = \{(x,y) \in \mathbb {R}^{2} : y>0\)} and we may and sometimes will embed it to \( \mathbb {C} \) by Z_{ t }=X_{ t }+iY_{ t }, where \(i =\sqrt {1}\).
Proposition 1
Let \(f:\mathbb {H}\to \mathbb {H}\) be such that \(f(z):=\frac {az+b}{cz+d}\), where \(\left (\begin {array}{ll} a & b \\ c & d \\ \end {array}\right)\in \text {SL}(2,\mathbb {R})\). Then (f(Z_{ t }))_{t≥0} and (Z_{ t })_{t≥0} are equivalent in law provided that f(Z_{0})=Z_{0}.
Proof
Since Z_{ t }=X_{ t }+iY_{ t }, using Ito’s formula for Z_{ t },
where \(dW^{\mathbb {C}}_{t}=dW^{1}_{t}+idW^{2}_{t}\), which we define to be a complex Brownian motion. On the other hand, since Z_{ t } is a conformal martingale and f is a holomorphic function, we can use Ito’s formula for conformal martingales to get
where \(d\widetilde {W}^{\mathbb {C}}_{t}=\frac {{cZ}_{t}+d^{2}}{({cZ}_{t}+d)^{2}}dW^{\mathbb {C}}_{t}\), which is another complex Brownian motion. Hence Z_{ t } and f(Z_{ t }) are equivalent in law if they start from the same point, as they are defined by the same SDE. □
Hyperbolic reflections
Let be the totality of such isometries π on the upperhalf plane \(\mathbb {H}\) that π^{2}=Id and that the invariant set \(\text {Inv}_{\pi }:=\left \{z \in \mathbb {H}: \pi (z)=z \right \}\) is a geodesic on \(\mathbb {H}\).
Proposition 2
We have that
where \(\Phi _{A}(z)=\frac {az+b}{cz+d}\) for \(A = \left (\begin {array}{ll} a &b \\ c&d \end {array}\right) \in \text {SL}(2,\mathbb {R})\)and \(\Phi _{0}(z):=\overline {z}\).
Proof
It is wellknown that an isometry on \( \mathbb {H} \) is either Φ_{ A } or Φ_{ A }∘Φ_{0} for some \( A \in \text {SL}(2,\mathbb {R})\). By the fundamental theorem of algebra, we know that the equation Φ_{ A }(z)=z has at most two solutions of complex for \(A \in \text {SL}(2,\mathbb {R})\). So .
For \(\Phi _{A}\circ \Phi _{0} \in \text {Isom}(\mathbb {H})\) and for z=x+iy,
By a simple calculation,

If a=d, for any b and c, (10) is satisfied.
Since a^{2}−bc=1, we have \(c=\frac {a^{2}1}{b}\) if b≠0, that is;
For b=0, c is an arbitrary real number and a=±1 from a^{2}=1;

If a≠d, the Eq. (10) is
We get a=−d and b=c=0.
Finally, we should find that the invariant set is geodesic. A geodesics of upper half plane is a line perpendicular to the real line, or a halfcircle orthogonal to the real line.

If
$$A= \left(\begin{array}{ll} a & b \\ \frac{a^{2}1}{b} & a \\ \end{array}\right), $$
and if a≠±1,
The last equation means that it is a half circle, with center \(\left (\frac {ab}{a^{2}1},0\right)\) and radius of \(\left \frac {b}{a^{2}1}\right \).
If a=±1,
The equation means that the invariant set is lines perpendicular to the real line.

If
$$A= \left(\begin{array}{cc} \pm 1 & 0 \\ c & \pm 1 \\ \end{array}\right), $$
and if c≠0, without loss of generality, we may set a=1;
The invariant set is a circle with center \(\left (\frac {1}{c},0\right)\) and the radius \(\frac {1}{c}\).
If c=0, the invariant set is the lines perpendicular to the real line. □
Hyperbolic reflection principle
Let . Then, \( \mathbb {H} = D_{+} \cup \text {Inv}_{\pi } \cup D_{} \), where D_{±} are the connected components of \( \mathbb {H} \setminus \text {Inv}_{\pi } \).
Proposition 3
(Hyperbolic Reflection Principle) Let Z_{0}∈D_{+} and τ=inf{t≥0:Z_{ t }∉D_{+}}=inf{t≥0:Z_{ t }∈Inv_{ π }}. If we put \(\widetilde {Z_{t}}=Z_{t} 1_{\left \{t<\tau \right \}}+\pi \left (Z_{t}\right)1_{\left \{t\geq \tau \right \}}\), then we have \( \left (Z_{t}\right)=(\widetilde {Z_{t}}) \) in law.
Proof
It suffices to show that if π is a reflection of \(\mathbb {H}\), then (π(Z_{ t }))_{t≥0}=(Z_{ t })_{t≥0} in law if Z_{0}∈Inv_{ π } since Z is a strong Markov process and Z_{ τ }∈Inv_{ π }. As we have seen that π=Φ_{ A }∘Φ_{0} for some specific \( A \in \text {SL} (2, \mathbb {R}) \), and by Proposition 1, we only need to check that \( (\overline {Z}_{t}) \) is identically distributed as (Z_{ t }) as a stochastic process, but this is obvious since (X_{ t }) is identically distributed as (−X_{ t }). □
Hyperbolic symmetrization
Hyperbolic putcall symmetry
Let . Then, by Proposition 2, we know that
for
A Hyperbolic Brownian motion with drift is a unique solution in \(\mathbb {H}\) (if it exists) to
where W^{1} and W^{2} are independent Brownian motions and μ_{1} and μ_{2} are measurable functions. If we use complex coordinate, the SDE (12) is rewritten as
where \( W^{\mathbb {C}} := W^{1} + i W^{2} \) and μ(Z)=μ_{1}(Re(Z),Im(Z))+iμ_{2}(Re(Z),Im(Z)).
Theorem 2
Let and we write
to unify the two classes in the expression (11). Suppose that μ satisfies
and (13) has a unique weak solution. Then (π(Z_{ t })) and (Z_{ t })have the same law as a stochastic process, provided that Z_{0}∈Inv_{ π }
Proof
Using Itô formula for π(Z_{ t }), we have
where we have used assumption (14) in the last line and
which is another complex Brownian motion. Now Theorem follows by the lawuniqueness of the SDE (13). □
Symmetrization
Here we present a hyperbolic version of the symmetrization introduced in [2] and [10].
Theorem 3
We keep the setting of “Hyperbolic reflection principle” section and Theorem 2 except for the drift function μ. We let
Then,

(i) the law unique solution of the SDE, if it exists,
$${dZ}_{t} = \text{Im} (Z_{t}) dW^{\mathbb{C}} + \tilde{\mu} (Z_{t}) \,dt\qquad $$satisfies (π(Z_{ t }))=(Z_{ t }) in law, provided that Z_{0}∈Inv_{ π }.

(ii) Let Z_{0}∈D_{+} and τ=inf{t≥0:Z_{ t }∉D_{+}}=inf{t≥0:Z_{ t }∈Inv_{ π }}. If we put \(\widetilde {Z_{t}}=Z_{t} 1_{\left \{t<\tau \right \}}+\pi \left (Z_{t}\right)1_{\left \{t\geq \tau \right \}}\), then we have \( \left (Z_{t}\right)=(\widetilde {Z_{t}}) \) in law.

(iii) [Conversion Formula] Suppose that F is a bounded measurable function on \( \mathbb {H} \) with support in D_{+}. Then,
$$\begin{aligned} &E [F(Z_{t}) 1_{\{\tau > t\}}]\\ &= E [F(Z_{t}) ]  E [F(\pi(Z_{t})) ]. \end{aligned} $$
Proof
(i) and (ii) are direct consequences of Theorem 2 and Proposition 3. (iii) can be proven in the same manner as in [10]. □
Example 1
Let Z be the unique solution to (13), \(\pi (z)=\frac {1}{\bar {z}}\) and Inv_{ π }={z=1}. We let \(D_{+} := \{z\in \mathbb {H}: z> 1 \}\) and
where c is a constant, then the symmetrization \(\tilde {\mu }\) in Theorem 3 is
Numerical experiments
In the hyperbolic symmetrization proposed in the present paper the symmetrized drift may not be continuous in general, as in the case of the symmetrization in [10]. This means that no rigorous mathematical result guarantees the efficiency— (high) order of convergence— in EulerMaruyama approximation. In [10], it is claimed, however, that numerical experiments show the efficiency. In this section we present some simulation results of the Example 1 with c=1, t=1, and F(z)=(z−1)_{+}∧N with N=10^{4}, which suggest that in the hyperbolic case the conjecture is still likely to be true.
We work on EulerMaruyama discretization scheme with MonteCarlo simulation, described below.

1.
Let n be the number of discretization; we put t_{ k }=k/n, k=0,1,⋯,n.

2.
Let Z be the original process and \( \widetilde {Z} \) be the symmetrized one. We approximate Z and \( \widetilde {Z} \) by Z^{n}=(X^{n},Y^{n}) and \( \widetilde {Z}^{n} = (\widetilde {X}^{n}, \widetilde {Y}^{n}) \), defined as
$$\begin{aligned} & X^{n}_{t_{k}}X^{n}_{t_{k1}} \\ &= Y^{n}_{t_{k1}}\Delta W^{n}_{t_{k}} + \mu \left(Y^{n}_{t_{k1}}\right) n^{1}, \\ Y^{n}_{t_{k}} &= Y^{n}_{t_{k1}} \text{exp} \left(\Delta W^{n}_{t_{k}}  (2n)^{1}\right), \\ \,k&=1,2,\cdots,n \end{aligned} $$and
$$\begin{aligned} \widetilde{X}^{n}_{t_{k}}\widetilde{X}^{n}_{t_{k1}} =& \widetilde{Y}^{n}_{t_{k1}}\Delta W^{n}_{t_{k}} + \tilde{\mu}_{1}\\ & \left(\widetilde{X}^{n}_{t_{k1}}, \widetilde{Y}^{n}_{t_{k1}}\right) n^{1}, \\ \widetilde{Y}^{n}_{t_{k}}\widetilde{Y}^{n}_{t_{k1}} =& \widetilde{Y}^{n}_{t_{k1}}\Delta W_{t_{k}}^{n} +\tilde{\mu_{2}}\\ &\left(\widetilde{X}^{n}_{t_{k1}}, \widetilde{Y}^{n}_{t_{k1}}\right) n^{1}, \\ k=&1,2,\cdots,n, \end{aligned} $$where \(\tilde {\mu _{1}}\) and \(\tilde {\mu }_{2}\) are such that \(\tilde {\mu } = \tilde {\mu }_{1} + i \tilde {\mu _{2}}\). Here \( \left \{ \Delta W^{n}_{t_{k}}: k=1,2, \cdots, n\right \} \) simulates, by pseudo random numbers, independent copies of centered Gaussian random variables with variance n^{−1}.

3.
The MonteCarlo simulation of PathWise EulerMaruyama approximation of E[F(Z_{1})1_{{τ>1}}] is obtained by
$$\begin{aligned} &\text{PWEM} (n) \\ &:= \frac{1}{M} \sum_{m=1}^{M} F\left(Z^{n,m}_{1}\right) 1_{\{\tau^{n,m} > 1 \} }, \end{aligned} $$where Z^{n,m} stands for the mth simulation of Z^{n}, and
$$\tau^{n,m} = \text{min} \left\{ t_{k} : Z^{n,m}_{t_{k}} \leq 1 \right\} $$ 
4.
The MonteCarlo simulation of \( E[F(\widetilde {Z}_{1}) ]  E [F(\pi (\widetilde {Z}_{1}))] \) is given by
$$\begin{aligned} & \text{Symmetrization} (n) \\ &:= \frac{1}{M} \sum_{m=1}^{M} \left(F\left(\widetilde{Z}^{n,m}_{1}\right)  F \left(\pi \left(\widetilde{Z}^{n,m}_{1}\right)\right)\right), \end{aligned} $$ 
5.
The “true” value Tr(n) is set to be Symmetrization(n) for some large n.

6.
The errors are calculated accordingly as
$$\begin{aligned} &\text{PW EM Error} (n)\\ &:= \text{log} \text{Tr}(n)  \text{PW EM} (n) \end{aligned} $$and
$$\begin{aligned} &\text{Sym Error} (n)\\ &:= \text{log} \text{Tr}(n)  \text{Symmetrization} (n). \end{aligned} $$
The results are visualized as follows. The Figures 1 and 2 show the results when (X_{0},Y_{0})=(0.75,0.7) and (X_{0},Y_{0})=(1.0,1.0), respectively, and the “true" value is calculate for n=1000. The Tables 1 and 2 are the values of the dotted points in the Figures 1 and 2, respectively. The tangent of the regression line corresponds to the order of the convergence, which may suggest that it is of order 1 in the case of symmetrization.
Notes
 1.
A barrier option is a financial derivative with an additional condition that is made active when the underlying price process goes beyond/below a certain level. For details, see e.g. [9].
 2.
An option is called of “knockout” type if the payoff becomes zero if the underlying price process hits a certain value.
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The experiments are reproducible except the pseudo random numbers used in the MonteCarlo simulation.
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Introduced a hyperbolic version of ImamuraIshigakiOkumura’s symmetrization, and by numerical experiments showed efficiency of the scheme. All authors read and approved the final manuscript.
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Correspondence to Yuuki Ida.
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Keywords
 Hyperbolic Brownian motion
 Reflection principle
 Symmetrization
 Barrier option
 EulerMaruyama scheme