Option price and market instability

Option price and market instability

Accepted Manuscript Option price and market instability Belal E. Baaquie, Miao Yu PII: DOI: Reference: S0378-4371(16)30886-X http://dx.doi.org/10.101...

796KB Sizes 4 Downloads 33 Views

Recommend Documents

No documents
Accepted Manuscript Option price and market instability Belal E. Baaquie, Miao Yu PII: DOI: Reference:

S0378-4371(16)30886-X http://dx.doi.org/10.1016/j.physa.2016.11.080 PHYSA 17736

To appear in:

Physica A

Received date: 15 August 2016 Please cite this article as: B.E. Baaquie, M. Yu, Option price and market instability, Physica A (2016), http://dx.doi.org/10.1016/j.physa.2016.11.080 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Option Price and Market Instability Belal E. Baaquiea and Miao Yu a

∗b

INCEIF,The Global University of Islamic Finance, Lorong Universiti A, 59100 Kuala Lumpur Malaysia b

Department of Physics, National University of Singapore,2 Science Drive 3, 117542, Singapore December 24, 2016

Abstract An option pricing formula, for which the price of an option depends on both the value of the underlying security as well as the velocity of the security, has been proposed in [2]. The FX (foreign exchange) options price was empirically studied in [5], and it was found that the model in general provides an excellent fit for all strike prices with a fixed model parameters – unlike the Black-Scholes option price[11] that requires the empirically determined implied volatility surface to fit the option data. The option price proposed in [2] did not fit the data during the crisis of 2007-2008. We make a hypothesis that the failure of the option price to fit data is an indication of the market’s large deviation from its near equilibrium behavior due to the market’s instability. Furthermore, our indicator of market’s instability is ∗

Corresponding author.

E-mail addresses: [email protected] (Belal E. Baaquie), [email protected] (Miao Yu).

1

shown to be more accurate than the option’s observed volatility. The market prices of the FX option for various currencies are studied in the light of our hypothesis.

1

Introduction

Baaquie introduced the action functional for pricing the Black-Scholes (BS) model in [3] and the Black-Scholes price was given a path integral derivation starting from the action functional. In 2014, Baaquie and Yang postulated another action functional for the option pricing; the Baaquie and Yang (BY) option price depends on the stock price and its velocity – and is essentially Gaussian since the action functional is quadratic in the logarithm of the security (equity) or FX rate [2]. The BY models yields an ATM volatility that fits FX option data quite well – for the market time from 2008 to 2011. The model was further developed and applied to FX options to derive, from the model’s parameters, the full implied volatility surface – for ATM, in-the-money and out-of-the-money [5]. For the period of 2013, the fit of the model with data is quite good [5]. FX options for the exchange rate of various currencies against the US Dollar are studied for the period from 2011 to 2015; the following behavior is observed. • For all currencies, there are ‘normal’ periods where the BY model fits the data fairly well. • There are sudden intermittent periods that punctuate the normal period. And for these periods, the model fails dramatically due to market instability. • The volatility of the underlying FX rate is not a suitable indicator of market instability. Our hypothesis is that the failure of the BY model to fit market data is due to the effects of instability and nonlinearities that are not captured by the BY model – since it is essentially Gaussian. This simplicity of the BY model is used to our advantage by postulating that market instability introduces nonlinear effects

2

causing the model to fail. This very failure of the model in turn is used as a barometer and as a gauge for concluding that the FX market has entered an unstable and nonlinear phase that could also potentially be a crisis phase. The behavior of FX options is an accurate gauge of the state of the international financial system. The FX markets are international and operate 24 hours a day – and are expected to quickly respond to the changing tides of the major economic powers [8]. Furthermore, there is a high volume of daily FX transaction: trading in foreign exchange markets averaged $5.3 trillion per day in April 20161 . High liquidity and the key role of currencies in the major economies, in our opinion, makes FX options a sensitive gauge of the international financial system. The industry benchmark for pricing European call option is the Black-Scholes model. Let the price of security be S = ex . For concreteness, consider a vanilla call option with a payoff function [ex − K]+ and maturing at some future time T . The option price in general has the following time parameters

t : present calendar time ; T : maturity time ; τ = T − t : remaining time The BS option price is given by [9, 12, 14]

CBS (S(t), K, r, σ, (T − t)) = SN (d+ ) − e−r(T −t) KN (d− ) where

d± =

2

1 ln(S/K) + r(T − t) ± σ02 (T − t)/2 p ; N (x) = √ 2π σ0 (T − t)

Z

x

1 2

e− 2 z dz

−∞

Quantum finance formulation

Let x˙ = dx/dt be the velocity of the logarithm of S. In general, the option price C can depend on the price and velocity of the security. A payoff function H(x, v; K) 1

https://en.wikipedia.org/wiki/Foreign exchange market.

3

can depend on both the final stock value and velocity; for remaining time τ = T − t C(x(t), x(t), ˙ τ, K) =

Z

dxdx˙ 0 P (x, x; ˙ x0 , x˙ 0 ; τ )H(x0 , x˙ 0 ; K) ; x˙ =

dx dt

where expression P (x, x; ˙ x0 , x˙ 0 ; τ ) is the conditional probability that the future value is x0 , x˙ 0 at time T , given the value of x, x˙ at present time t. Furthermore, let

0

P (x, x; ˙ x ; τ) =

Z

dx˙ 0 P (x, x; ˙ x0 , x˙ 0 ; τ )

where P (x, x; ˙ x0 ; τ ) is the marginal conditional probability. In the quantum finance formulation of option prices, the conditional probability P (x, x; ˙ x0 , x˙ 0 ; τ ) is given by what is called the transition amplitude

K(x, x; ˙ x0 , x˙ 0 ; τ ) The conditional probability P(x, x; ˙ x0 , x˙ 0 ; τ ) is given by appropriately normalizing the transition amplitude and yields K(x, x; ˙ x0 , x˙ 0 ; τ ) P (x, x; ˙ x0 , x˙ 0 ; τ ) = R 0 0 dx dx˙ K(x, x; ˙ x0 , x˙ 0 ; τ ) Z P (x, x; ˙ x0 ; τ ) = dx˙ 0 P (x, x; ˙ x0 , x˙ 0 ; τ )

(1)

The description of the stochastic evolution of a security is defined by the Hamiltonian H operator for the security [3]. The transition amplitude, in general, is given by the matrix element of the Hamiltonian operator [4]

K(x, x; ˙ x0 , x˙ 0 ; τ ) = hx, x|e ˙ −τ H |x0 , x˙ 0 i

(2)

The transition amplitude K(x, x; ˙ x0 , x˙ 0 ; T, t) has another representation defined

4

by the (Euclidean) Feynman path integral over all possible paths x(t) of the security from its initial value x0 , x˙ 0 at time τ = 0 to its final value of x, x˙ at time τ [4]. More precisely, K=

Z

DxeS

(3)

Up to a normalization, the path integral measure is given by Z

Dx =

τ Z Y



dx(t)

t=0 −∞

The boundary conditions for all the allowed paths in the Feynman path integral given in Eq. 3 is the following [1]

x(0) = x0 , x(0) ˙ = x˙ 0 ; x(τ ) = x, x(τ ˙ ) = x˙

3

(4)

Transition amplitude K

The model Hamiltonian H for the option price is given by [4]

H=−

∂ 1 1 ∂2 − x˙ + bx˙ 2 + cx2 2 2a ∂ x˙ ∂x 2

(5)

The Hamiltonian given in Eq. 5 yields the following ‘acceleration’ Lagrangian, derived in [4] and given by  1 2 L = − a¨ x + 2b(x˙ + j)2 + cx2 ; S = 2

Z

τ

dtL ; x ¨=

0

dx d2 x ; x˙ = 2 dt dt

(6)

Since the Lagrangian given in Eq. 6 is quadratic, the path integral can be solved exactly using the classical solutions. The stochastic variable x is separated into two parts: the classical solution xc and stochastic part ξ.

x = xc + ξ

5

(7)

with the classical solution xc given by δS[xc ] =0 δx(t) The classical solution has boundary conditions as Eq. 4 and we hence obtain

xc (0) = x0 , x˙ c (0) = x˙ 0 ; xc (τ ) = x, x˙ c (τ ) = x˙

(8)

Hence, Eqs. 7 and 8 yield the boundary condition as below

˙ ˙ ) = 0; ξ(0) = 0; ξ(τ ) = 0 ξ(0) = 0; ξ(τ

(9)

The acceleration action S separates into two parts [2] and is given by

S = S[xc ] + S[ξ]

(10)

Note S[ξ] is independent of x, x, ˙ x0 , x˙ 0 and depends only on τ . The transition amplitude is given by

K=

Z

DxeS =

Z

DξeSξ +Sc = N eSc

(11)

The functional integration is defined by Z

Dx =

τ Z Y



Z

dx(t) ;

t=0 −∞

Dξ =

and the normalization is given by

N (τ ) =

Z

6

DξeSξ

τ Z Y



t=0 −∞

dξ(t)

We obtain the final result that

K(x, x; ˙ x0 , x˙ 0 ; τ ) = N (τ ) exp{Sc (x, x; ˙ x0 , x˙ 0 ; τ )}

4

(12)

BY Model option price

The BY (Baaquie-Yang) model for option pricing – proposed in [5] – is based on the Lagrangian and action given in Eq. 6, and is unlike the Black-Scholes case. The BY price for the European call option, at time t, is given by

CBY (x(t), x(t), ˙ τ, K) =

Z

0

dx0 P (x, x; ˙ x0 ; τ )[ex − K]+

The Lagrangian and action S[xc ] given by Eq. (6) yields [4] 1 S[xc ] = − 2

Z

τ

0

dt(a¨ x2c + 2b(x˙ c + j)2 + cx2c )

(13)

The classical solution S[xc ] with the boundary condition Eq. 4 is solved in the Appendix 12.1. The conditional possibility is given by Eq. 59

0

P (x, x; ˙ x ; τ) =

Z

0

0

0

dv P (x, x; ˙ x , x˙ ; τ ) =

r

2 1 1 exp{− 2 −x0 + ζx + ξ x˙ + j } 2 2πν 2ν

The solution for the classical action Sc yields the following. r and ω are defined in the Appendix Eq. 43 and given below s

r ≡ Re 

 s  √ √ 2 2 b + i ac − b  b + i ac − b  ; ω ≡ Im  a a

7

and 2Ωrω[ω sinh(2rτ ) − r sin(2ωτ )] a (r2 + ω 2 )  ζ = 4Ωrω[ r2 − ω 2 sinh(rτ ) sin(ωτ ) + 2rω cosh(rτ ) cos(ωτ )] ν2 =

ξ = −4Ωrω[ω sinh(rτ ) cos(ωτ ) + r cosh(rτ ) sin(ωτ )]

(14)

where

Ω=

1 (r2

+

ω 2 )2



r2 (r2



3ω 2 ) cos(2ωτ )

− ω 2 (ω 2 − 3r2 ) cosh(2rτ )

(15)

A typical shapes of ν 2 , ξ, ζ as a function of τ , which a = 5; b = 8; c = 100, are shown in Figures 1, 2, 3 and 4. ν2

0.14

0.12 0.02 0.1

0.015 0.08

0.06 0.01

0.04 0.005 0.02

0

0 0

0.5

1

1.5

2

2.5

3

0

0.5

1

1.5

2

2.5

3

τ(year)

τ(year)

Figure 1: ν 2 (τ )

Figure 2:

0.05

1.2

p ν 2 /τ

0 1 −0.05 0.8 −0.1

0.6

−0.15

−0.2

0.4

−0.25 0.2 −0.3 0 −0.35

−0.4

−0.2 0

0.5

1

1.5

2

2.5

3

0

0.5

1

1.5

2

τ(year)

2.5

3

τ(year)

Figure 3: ξ(τ )

Figure 4: ζ(τ )

8

4.1

Martingale condition

Consider the case of FX options. Let the domestic currency be $1 and the value of a foreign currency is given by $ex ; let rf and rd be the risk free foreign and the domestic interest rates, respectively. As discussed in [2], the Forex option martingale condition is given by

−rf τ x

e

e =e

−rd τ

Z

+∞

dx0 P (x, x; ˙ x0 ; τ )ex

0

(16)

−∞

The Forex exchange martingale process is shown by Fig. 5

Figure 5: The Forex martingale process. From Eqs.16 and 59, the Forex option martingale condition is given by

−rf τ x

e

e =e

−rd τ

Z

+∞

0

0

x0

dx P (x, x ; x; ˙ τ )e

−∞

  ν2 = exp −rd τ + + ζx + ξv + j (17) 2

and yields the drift given by

j = (rd − rf )τ −

ν2 2

(18)

Hence, the marginal conditional probability distribution, from Eq. 59, is given by

0

P (x, v; x ; τ ) =

r

1 1 exp{− 2 2πν 2 2ν

 2 ν2 −x0 + ζx + ξv + (rd − rf )τ − } 2

(19)

The martingale condition further requires that

ζ = 1; ξ = 0 : violated

9

(20)

When the model is fitted to market data, as given in Figures 21, 22, it is seen that the martingale condition is violated. In particular, ζ, for large τ , deviates from 1 whereas ξ converges to zero. Figure 9 is even more interesting, with ξ going through a minima as one increases τ , and finally converging to 0 for large τ . Only for τ near 0, as in Figures 9 and 11, is ξ ' 0 and ζ ' 1 – as required by the martingale condition given in Eq. 20. The violation of the martingale condition shows that the market in general is not free from arbitrage; only for very short duration options, with τ ' 0 is the martingale condition approximately valid. One possible explanation of the imperfection of the market is that the market has long memory that mitigates against the martingale condition. In fact, in our model the deviation of the parameters ζ, ξ from 1 and 0, respectively, provides a quantitative measure of the degree to which the market is imperfect.

4.2

BY Option: market time

The FX call option BY price CBY (x, x; ˙ K, τ ) is given by CBY (x, x; ˙ τ, K) = e

−rd τ

Z

+∞

−∞

h 0 i dx0 P (x, x0 ; x; ˙ τ ) ex − K

+

= e−rf τ eζx+ξx˙ N (d+ ) − e−rd τ KN (d− ) where d± =

ζx + ξ x˙ − ln(K) + (rd − rf )τ ± ν

ν2 2

.

We introduce two new parameter λ and η such that t t → z = λ( )η λ

(21)

The function z(t) [3] is called market time to differentiate it from calendar time t. Market time is the subjective estimate of time in the minds of the traders whereas calendar time is physical time. The parameters λ, η are a measure of market time, which is greater than calendar time for t < λ and less for t > λ. The difference

10

between t and z, when η < 1, is shown in Figure 6. y

y=t

y=

0

t

Figure 6: The t and z values when η < 1 The transition amplitude is now given by

K(x, x; ˙ x0 , x˙ 0 ; τ ) = hx, x|e ˙ −z0 (τ )H |x0 , x˙ 0 i;

τ z0 = λ( )η λ

(22)

where recall τ = T − t is the time remaining for the maturity of the option. The action is given by

S=

Z

z0

dzL(z)

(23)

0

The parameters λ and η allows us to rescale and dilate calendar time τ and in doing so allows us to fit effective volatilities that have a maximum at a future time that greater than 1.5 years. The recalibration is done in the following manner τ ν 2 (τ ) ⇒ ν 2 (λ( )η ) λ τ η ξ(τ ) ⇒ ξ(λ( ) ) λ τ ζ(τ ) ⇒ ζ(λ( )η ) λ

(24) (25) (26)

The changes of the fitting parameters by different λ and η are shown by the Figures in Group1 and Group2.

11

Table 1: the parameters in fitting Group

a

b

c

λ

η

Group1

1

2.48

19.54

1

1

Group2

1

2.48

19.54

0.3

0.4

Group 1:

Group2:

12

0.09

ν2

0.08 0.05

0.07

0.06

0.04

0.05 0.03

0.04

0.03

0.02

0.02 0.01

0.01

0

0

0

500

1000

1500

2000

2500

3000

0

0.5

1

1.5

2

2.5

3

τ(year)

Figure 7: ν 2 (z)

Figure 8: ν 2 (z)

0.05

0

0

−0.05

−0.05 −0.1

−0.1 −0.15

−0.15 −0.2

−0.2 −0.25

−0.25

−0.3

−0.3

−0.35

−0.35

0

500

1000

1500

2000

2500

3000

0

0.5

1

1.5

2

2.5

3

τ(year)

Figure 9: ξ(z)

Figure 10: ξ(z)

1.2

1

1 0.95

0.8 0.9

0.6 0.85

0.4

0.8

0.2

0.75

0

−0.2

0.7

0

500

1000

1500

2000

2500

3000

0

0.5

1

1.5

2

2.5

3

τ(year)

Figure 11: ζ(z)

Figure 12: ζ(z)

It is obvious that when λ and η change from [1, 1] to [0.3, 0.4], the shape of the parameters [µ, ξ, ζ] also varies significantly. For example, Figures. 15 and 16 can be fit very well with λ = 1, but it is graphs of the type shown in Figure 17 that can only be fitted by the model with λ > 1.

13

5

Mapping BY Model to data

Every option has a remaining time τ = T − t. Options can be studied for which remaining time τ remains constant, namely a collection of options issued at different calendar time and maturing at a constant time in the future, and shown in Figure 13(a). Another class of options has a maturity time at some fixed calendar T , and shown in Figure 13(b). In this paper, only options for which τ is held fixed are analyzed – and shown in Figure 13(a). For each instant t, the value of the security is represented by the model variable x(z(τ )), with τ being fixed. Since we are holding τ fixed, for each t a new instrument is being issued with maturity determined by remaining time τ . The price of a security contains information both about when the option is issued and when it matures. Hence, we extend the notation x(z(τ )) to x(t, z(τ )) to make the mapping of the model variables to market data more clear. Let t be calendar time and T = τ + t be the sliding maturity time, as shown in Figure 13(a). We adopt the notation that τ x(z) = x(t, z(τ )) : z(τ ) = λ( )η λ

: τ = constant

For the case of options with remaining time τ that is fixed, as shown in Figure 13(a), the notation x(t, z(τ )) means the value of the stock price at time t for an option maturing at future time t + τ . For options that mature at some fixed time T in the future, as shown in Figure 13(b), we have the notation

x(z) = x(t, z(T − t)) : z(τ ) = λ(

T −t η ) ; τ = T − t : constant T λ

14

t(calendar time)

t(calendar time) T

x(t’,τ)

t’

x(t,τ)

t

0

t’ t

τ

x(t’,τ) x(t,τ)

0

future time

x(t’,τ(t’))

(a)

x(t,τ(t))

(b)

T

future time

Figure 13: (a) The model variable x(t, τ ) with different calendar times t, t0 but with the same remaining time τ . (b) Model variable for fixed maturity time T , with remaining time τ (t) 6= τ (t0 ). Let xD (t) be the value of the data at time t. The calibration of the model is based on the following mapping

xD (t) = x(t, z(τ )) : constant τ

(27)

xD (t) = x(t, z(T − t)) : constant T

(28)

The mapping from t to z, and the connection of data xD (t) to its representation in the model x(z(t)), is shown in Figure 14. Eqs. 27 and 28 may look a bit strange since xD (t) does not depend on τ or T ; what we need to keep in mind is that the data xD (t) is being used to generate option prices, and options carry maturity time. The role played by τ in mapping the model to data will become more clear when we compute the velocity of the model. The velocity in the model is defined by dx dt dx = dz dz dt From the definition of market time we have dz dt

=

d  T − t η τ λ( ) = −η( )η−1 dt λ λ

15

To relate the model velocity to the velocity given by data, we compare the value of data at two nearby instants given by t and t − δ – as shown in Figure 14. For fixed remaining time τ , as shown in Figure 14(a), we have the following dx(t, z(τ )) dz

−1 dx(t, z) dt dx  τ = −η( )η−1 dz dt λ dt  τ η−1 −1 x(t, z(τ )) − x(t − δ, z(τ )) = −η( ) λ δ =

From the mapping given in Eq. 27 we obtain dx dx(t, z(τ )) 1  τ 1−η xD (t) − xD (t − δ) = =− · dz dz η λ δ

(29)

In the analysis of this paper δ is chosen equal to 1 day=1/360 years.

t(calendar time)

t t-δ

0

t(calendar time)

xD(t)=x(t,τ)

t

xD(t-δ))=x(t-δ,τ)

t-δ

x(t-δ,τ)

......

T1

xD(t)=x(t,τ) x(T,Z(τ))

xD(t-δ))=x(t-δ,τ)

ε

0

future time

x(T,Z(T-t+δ))

T1

......

future time

(b)

(a)

Figure 14: (a) Model velocity for fixed remaining time τ . (b) Model velocity for fixed maturity time T is found by comparing x(t, z(τ ) to x(t − δ, z(τ + δ). The mapping for the option with fixed maturity time T, τ = T − t is given in Eq. 28 and yields dx x(t, z(T − t)) − x(t − δ, z(T − t + δ)) = dz z(T − t) − z(T − t + δ) x(t, z(T − t)) − x(t − δ, z(T − t + δ)) =  xD (t) − xD (t − δ) 1  τ 1−η xD (t) − xD (t − δ) = =− ·  η λ δ

16

(30)

since τ τ +δ η τ  = z(T − t) − z(T − t + δ) = λ( )η − λ( ) = −η( )η−1 δ + O(δ 2 ) λ λ λ where  is shown in Figure 14(b). From Eq. 30, note that a change of δ in calendar time is equal to a change of  in market time z(T − t). Note that the result given in Eq. 30 for an option with a fixed maturity T is the same as the result for constant remaining time τ given in Eq. 29. The mapping of the model velocity to data is given in Eq. 29. Note that dx/dz is the velocity for an option maturing after remaining time τ , as shown in Figure 14. The options for different remaining times have the empirical velocity dxD /dt scaled by the remaining time τ for obtaining the effective model velocity dx/dz. The scaling factor is the same for options with constant remaining time and with a fixed maturity time, and is the following 1  τ 1−η dxD dx =− dz η λ dt This result shows the consistency of the definition of market time. Since η < 1 we see that the model’s velocity is enhanced for large remaining time τ , with the effect of market time become more significant for τ >> λ. Note for η → 1, z(τ ) → τ , and hence market time becomes equal to remaining calendar time, and the scaling factor becomes 1 as expected.

17

6

Calibration of the BY Model

The fitting of data is with the price of call option is given by Eq. 21

2

Cdata (τ ) = CBY (rf , rd , z(τ ), x, x) ˙

(31)

The market price Cdata is obtained from the Black-Sholes formula

Cdata (τ ) = CBS (S, K, rf , rd , σAT M , T − t0 )

(32)

Where σAT M is provided by the market. The testing and calibration of the BYmodel is given by using Eqs. 31 and 32. The at-the-money (ATM) options are often used to calculate the implied volatility because they are the most traded contracts; implied volatility differs with strike prices and time to expiration as well as depending on calendar time. We use the data from Bloomberg to obtain implied volatility σAT M in the form of a dimensionless number τ G2data (τ ), with at-the-money implied volatility given by

σAT M =

q τ G2data (τ )

The daily FX volatility data is downloaded from Bloomberg and the following fixed remaining times are chosen as

{τn |n = 1, 2, ...8} = [0.0833, 0.1667, 0.25, 0.5, 1, 1.5, 2, 3] years. R-square and root mean square (RMSE) error are chosen to measure the goodness of fit. For each calendar date t, there is a fit of volatility Cdata , so the R-square 2

The calibration of the model in [2] is for market ATM volatility Gdata and uses the formula ν 2 (τ η ) = G2data (τ )τ

The shortcoming of this calibration is that the parameters λ is not used. This leads to the maximum of the effective volatility in the paper being fixed to be at around 1.5 years in the future, and hence is unable to fit many cases.

18

and RMSE error are functions of calendar time t. R2 is defined as

R2 (t) = 1 −

8 X

[Cdata (t, τn ) − Cfit (t, τn )]2

n=1 8 X

(33)

[Cdata (t, τn ) − C¯data (t, τn )]

2

n=1

where τn is the remaining time and C¯data (t, τn ) is the mean of Cdata (t) at calendar time t. Higher R2 means better fit, and the exact fit has an R2 value equal to 1. RMS error is defined by v u  N  u1 X Cdata (t, τn ) − Cfit (t, τn ) 2 t . RMSE(t) = N Cdata (t, τn )

(34)

n=1

7

Fitting Results

In general, the data results in three typical shapes that can be fitted by the model are shown in Figures 15 to 17 and the irregular shape that have no fit is shown Figure 18. The pattern C is in fact the same as pattern A; with the difference that for pattern C the maximum has been pushed out far into the future. Pattern C can be fitted by choosing a suitable λ.

19

21

12

volatility σ / percent per year

volatility σ / percent per year

σ data σ fit 11.5

11

10.5

10 0

0.5

1

1.5

2

2.5

σ data σ fit

20 19 18 17 16 15 0

3

0.5

1

1.5

2

2.5

Expiration time / year

Expiration time / year

Figure 15: Pattern A, 2009-09-23

Figure 16: Pattern B, 2009-02-02

volatility σ / percent per year

9.5 σ data σ fit

9 8.5 8 7.5 7 6.5 0

0.5

1

1.5

2

2.5

3

Expiration time / year

Figure 17: The fourth pattern C, 2013-12-18 The graph below shows a set of data that is irregular and can not be fitted. 10.48 σ data σ fit

volatility σ / percent per year

10.46 10.44 10.42 10.4 10.38 10.36 10.34 10.32 10.3 0

0.5

1

1.5

2

2.5

3

Expiration time / year

Figure 18: Irregular data, 2008-08-28 The model has five free parameters, as shown in Table 1, and options with 8 different remaining time are being fitted. The fits are usually very good, except for

20

3

exceptional periods. To test our hypothesis, we mark those periods for which the fit fails, namely for which R2 < 0.99. We analyze the exchange rate of the major international currencies against the US Dollar, namely the Euro, the Pound (GBP), the Japanese Yen, the Swiss Franc (CHF), the Australian Dollar (AUD), the Canadian Dollar and the New Zealand Dollar. Of these we analyze the option of the exchange rate of five major currencies against the US Dollar, which are the Euro, GBP, CHF, Yen, AUD. The Canadian Dollar is highly correlated with the US Dollar and the New Zealand Dollar to the AUD, and hence their analysis does not give any new insights. The scheme of our analysis is the following. • We find the periods for which the model fails. • We seek to explain the reason of their failure as being due to market turbulence. • If the failure of model for a time period occurs only for a particular a country, we look for reasons such as policy changes or international developments that specifically impact on that country. • If for a time period the model fails for all the major currencies, we seek an explanation that originates in the international financial system. The graphs below shows one example of the EURUSD fitting fpr 2, January, 2008.

Table 2: The parameters in the fitting R2

a

b

c

λ

η

0.9992

23.96

0.801

2.48

0.843

0.325

21

0.11

0.035 data fit

0.1

0.03 0.09 0.025 0.08

0.02

0.07

0.06

0.015

0.05 0.01 0.04 0.005 0.03

0.02 0

0.5

1

1.5

2

2.5

0

3 (year)

0

0.5

1

Remaining time

1.5

2

2.5

3 (year)

2.5

3(year)

Remaining time

Figure 20: ν 2

Figure 19: Option price fitting −0.5

1.001

−0.6 1 −0.7 0.999

−0.8

−0.9 0.998 −1 0.997 −1.1

−1.2

0.996

−1.3 0.995 −1.4

−1.5 0

0.5

1

1.5

2

2.5

0.994

3 (year)

0

0.5

1

1.5

2

Remaining time

Remaining time

Figure 21: ξ

Figure 22: ζ

22

The graphs below shows the parameters fitting for EURUSD from 2009/01/01 to 2009/04/21. −4

x 10 1

0.9999 5 0.9999

0.9998

4

0.9998 3 0.9997

0.9997

2 0

10

20

30

40

50

60

70

0

10

20

30

40

50

60

70

time lag(day)

time lag(day)

Figure 23: Option price fitting R2

Figure 24: Option fitting rmse

0.7

0.5

0.45 0.6 0.4 0.5

0.35

0.3 0.4 0.25 0.3 0.2

0.15

0.2

0.1 0.1 0.05

0

0 0

10

20

30

40

50

60

70

0

10

20

30

40

50

time lag(day)

60

70

time lag(day)

Figure 25: r

Figure 26: ω

3

0.46

0.44 2.5 0.42

2 0.4

0.38 1.5

0.36 1 0.34

0.5

0.32 0

10

20

30

40

50

60

70

0

10

20

30

40

50

60

70

(time lag(day)

time lag(day)

Figure 27: λ

Figure 28: η

23

8

Global crisis

Our focus is from 2008 to the present (2016). Before embarking on a study of specific currencies, we expect that events that have had a major impact on the international financial system should be enumerated and we should be able to find signals of these globally. This would provide a check on our analysis as well as show the consistency of using FX option price as a gauge of the international financial system. Two international events with a potentially significant impact on almost all countries of the world, and of three regional events, are discussed below. The period of international significance is the 2007-2008 financial crisis.3 Our study also shows many other instabilities, some of which are country specific.

Figure 29: TED and the financial crisis in 2008. 4

1. The TED spread is an acronym for the difference between the interest rates on interbank loans and on short-term U.S. government debt. The TED spread is calculated in the market by measuring the difference between the threemonth LIBOR and the three-month US Treasury-bills interest rate. A large 3

https://en.wikipedia.org/wiki/Financial crisis of 2007%E2%80%9308.

24

TED spread indicates an increase in the risk of short term commercial loans and a lack of liquidity. The sudden spike in TED reaching almost 5%, as shown in Figure 29 (where USGG3M is the US 3-month treasury bill), is a reflection of the 2008 financial meltdown. We examine the impact of the 2007-2008 financial separately for the five major currencies. Three events of regional and local significance are the following. 1. United States debt-ceiling crisis of 2013 lasted January-October, 2013.

5

Our

study shows that the crisis had a strong impact on Europe and Japan but little impact on Australia and Switzerland;a possible explanation is that both Australia and Switzerland are not that closely tied to the US financial policy as are Europe and Japan. 2. According to our model based on the pricing of FX options for 2014-2015, the Euro, British and Swiss Franc were impacted by the drastic fall in oil-prices and the deterioration of the relation of the US and Europe with Russian. 3. The black swan event for the Swiss Franc was caused internally by removing the pegging of the Swiss Franc to the Euro on 15 January, 2015.

9

Result for five major FX options

From the graphs we can see that all of Europe except the Swiss Franc were heavily influenced by the 2008 financial crisis. Each graph for the FX options is alongside another graph showing the volatility of the FX rate. We compare these graphs to see if the FX volatility is also an accurate gauge of market instability. We find that the volatility of the FX rate is not a very precise criterion of financial instability. For example, as shown in Fig. 30 (b), although FX rate had a high volatility during the 2008 financial crisis, there are other periods of low 5

https://en.wikipedia.org/wiki/United States debt-ceiling crisis of 2013.

25

volatility even though there was marked market instability for the Euro, as in the post-2014 period.

9.1

Euro

The model in [2] showed that the Euro FX option price data could also be used to gauge the market. The model could correctly reflect the occurrence of the global financial crisis on the EURUSD exchange rate. This result is re-produced in Figure

2

Exchange Volatility

1.005

Option price R

2

30 and indicated by I.

1

0.995

0.99

500 450 400 350 300 250 200

(I)

0.985

(II)

150 100

0.98 50

0.975 2008

0

2009

2010

2011

2012

2013

2014

2015 Date(year)

(a)

2008

2009

2010

2011

2012

2013

2014

2015

(b)

Figure 30: (a) R2 of EURUSD and (b) Fx volatility of EURUSD The price of EURUSD option was disordered for a short period after the 2008 financial crisis. The 2008 financial crisis, however, triggered sovereign debt crisis in Europe in 2013, and this is marked as II in Figure 30. The crisis denoted by II could also have had contributions from the US financial crisis of 2013. Hence, financial instability was correctly gauged by the failure of the model.

26

Date(year)

Australia Dollar Exchange Volatility

Option price R

2

2

9.2 1

0.9 0.8 0.7 0.6

800 720 640 560 480

0.5

400

0.4

320

0.3

240

0.2

160

0.1

80 0

0

2008

2009

2010

2011

2012

2013

2014

2015 Date(year)

(a)

2008

2009

2010

2011

2012

2013

2014

2015

(b)

Figure 31: (a) R2 of AUDUSD and (b) Fx volatility of AUDUSD The graph shows that for the period of the 2008 crisis, the model correctly reflects the financial meltdown.

9.3

Swiss Franc

The behavior of the model for the case of the Swiss Franc is one of the most interesting. To start with, the 2007-2008 financial crisis left the Swiss Franc untouched since it did not take part in the leveraging and high risk instruments that primarily the US and UK banks, and to a lesser extent the European banks were engaged in. The fittings above confirms our expectation that the Swiss Franc was not affected much in 2008; in constrast, the Euro, including the British Pound, were highly impacted. The Swiss dollar has a very big FX volatility from 2010 to 2012 as shown in Fig 32 (b). This is because of a policy of the Swiss government. After the crisis broke out in 2008, there was a flight to safety in Europe, with large flows of money to Swiss Bank. This raised the value of the Swiss Franc and led to the Swiss policy, announced in September 2011, that set an upper limit to the valuation of the CHF to EURO to be capped at 1.2. This led to a short burst of instability, marked as I in Figure 32(a). The fluctuations marked II in Figure 32(b) describes this high

27

Date(year)

Exchange Volatility

Option price R

2

2

volatility period. 1

0.95

500 450 400 350 300

0.9

250

(II)

200

0.85

(I)

150 100

0.8

50 0

0.75 2008

2009

2010

2011

2012

2013

2014

2015 Date(year)

2008

2009

2010

2011

(a)

2012

2013

2014

2015

Date(year)

(b)

Figure 32: (a) R2 of CHFUSD and (b) Fx volatility of CHFUSD On 15th January, 2015 the Swiss suddenly canceled the upper limit of CHF against the Euro, which is a rare and unpredictable event and can be called a ”Black Swan” event. This rare event is correctly captured by the failure of the model – indicated by a spike in Figure 32(a) for 2015.

British Pound

1

Exchange Volatility

Option price R

2

2

9.4

0.995 0.99

0.985 0.98

500 450 400 350 300

0.975

250

0.97

200

0.965

150

0.96

100

0.955

50

0.95 2008

0

2009

2010

2011

2012

2013

2014

2015 Date(year)

(a)

2008

2009

2010

2011

2012

2013

2014

2015

(b)

Figure 33: (a) R2 of GBPUSD and (b) Fx volatility of GBPUSD The British Pound, as can be seen from Figure 33, was highly unstable in 2008 leading to the crash of 2008. This is because British Pound is closely tied to the

28

Date(year)

behavior of the US Dollar. The British Pound was fairly stable after 2008, becoming unstable in 2013-2014. It is relatively stable now, compared to other major currencies. The volatility of the FX rate does not provide any sign of instability, in contrast to the instability shown by the failure of the model for the period of 2013 and end of 2014.

2

Japanese Yen

1

Exchange Volatility

Option price R2

9.5

0.99

0.98

0.97

500 450 400 350 300 250

0.96

200 150

0.95

100 0.94

50 0.93 2008

2009

2010

2011

2012

2013

2014

0

2015 Date(year)

2008

(a)

2009

2010

2011

2012

2013

2014

2015

(b)

Figure 34: (a) R2 of JPYUSD and (b) Fx volatility of JPYUSD According to Figures 30(a) and 33(a), from 2008 to 2009, the EUR and GBP was immediately disordered when the crisis of 2007-2008 occurred. The Japanese Yuan option, as seen in Figures 34, was affected between 2009 and 2010. This fact seems to show that although Japan has the same financial policy as the Western countries, the crisis had a slightly delayed effect on Japan.

10

Conclusion

The model for FX options is based on a Gaussian Lagrangian. The hypothesis that the option price cannot fit the market’s behavior when the market is undergoing large scale fluctuations and changes seems to be borne out by data. The accuracy of the fit of the model to the market value of the FX option price is seen to be quite

29

Date(year)

sensitive to the markets characteristics, with the model’s accuracy falling suddenly with the onset of instability. The volatility of the FX rates do not seem to be an accurate gauge of the market’s instability, with the FX volatility providing no clear signal of the onset of market instability. The BY option model is sensitive enough to represent the differences in the various markets. For example, the BY model for the FX options for CHF behaves quite differently from those for the AUD. The fittings demonstrate that the model could accurately describe the trend of each country.

11

Acknowledgements

We thank Prof Abbas Mirakhor for useful discussions that helped to sharpen the earlier result that option volatility is a gauge for the behavior of the financial markets.We thank Prof. Obiyathulla for a critical reading of the paper. We thank Cao Yang for many useful discussions.

12 12.1

Appendix Classical Solution

From Eq. 6, the lagrangian is given by

L=−

 1 2 a¨ x + 2b(x˙ + j)2 + cx2 ; S = 2

Z

τ

dtL

(35)

0

The transition amplitude in Feynman path integral is given by Eq. 12

B.C :

K = N eSC

(36)

x(0) = x0 , x(0) ˙ = v 0 ; x(τ ) = x, x(τ ˙ )=v

(37)

30

The classical solution satisfies the equation below ∂S[xc (t)] =0 ∂x(t)

(38)

From the Euler-Lagrangian equation, the classical solution xc (t) satisfies equation as below .... a x c (t) − 2b¨ xc (t) + cxc (t) = 0.

(39)

According to the market data, the solution should is the complex branch of Eq. 39; hence

b2 − ac < 0

(40)

Define y as the four conjugate roots of the equation

ay 4 − 2by 2 + c = 0

(41)

The four complex solution are as below

y = ±r ± iω

(42)

where s

r ≡ Re 



b + i ac − a

b2



s

 ; ω ≡ Im 

 √ 2 b + i ac − b  a

Then the relationship from [a, b, c] to [r, ω] is as below

b = −a(r2 − ω 2 ) ; c = a(r2 + ω 2 )2

31

(43)

Using the notation of r and ω, the general solution of xc (t) is given by

xc (t) = ert (a1 sin ωt + a2 cos ωt) + e−rt (a3 sin ωt + a4 cos ωt)

(44)

where a1,2,3,4 are constants solved by the boundary conditions 37. The action S yields

S = S[xc + ] Z   1 τ dt a(¨ xc + ¨)2 + 2b(x˙ c + ˙ + j)2 + c(xc + )2 =− 2 0 = S[xc ] + S[] + R

(45)

where Sc is the classical action Z 1 τ S[xc ] = − dt 2 0 Z 1 τ S[] = − dt 2 0

 

a¨ x2c + 2b(x˙ c + j)2 + cx2c a¨ 2c + 2b˙2 + c2 + 4bj



(46)



The residual term R is

R=−

Z

τ

dt (a¨ xc ¨ + 2bx˙ c ˙ + cxc ) Z τ ... .... = (−a¨ xc ˙ − 2bx˙ c  + x c )|τ0 − dt (a x − 2b¨ xc + cxc ). 0

(47)

0

From Eqs. 39 and 9, R = 0.

(48)

Integrating the classical action Sc in Eq. 46 by part, and applying the equations

32

of motion, the action can be expressed only in terms of the boundary conditions Z 1 τ ... .... Sc = − dt d(−a x c xc + a¨ xc x˙ c + 2bx˙ c xc + 4bjxc + bj 2 ) + xc (a x c − 2b¨ xc + cxc ) 2 0  τ 1  ... 2 =− −a x c xc + a¨ xc x˙ c + 2bx˙ c xc + 4bjxc + bj 2 0 =−

4 1 X xI MIJ xJ − 2bjx1 + 2bjx4 − bj 2 τ. 2

(49)

I,J=1

To find out coefficient MIJ , assuming j = 0 yields

Sc = −

4 1 X xI MIJ xJ 2

(50)

I,J=1

We rewrite the xi , vi , xf , vf as

xi = x1 ; vi = x2 ;

(51)

xf = x3 ; vf = x4

(52)

From Eq. 49, the derivatives of Sc yield MIJ given by

MIJ = −

∂2S ∂xI ∂xJ

(53)

According to the symmetry of the transition amplitude discussed in [4]

M11 = M33 ; M22 = M44 ; M12 = −M34 ; M14 = −M23

33

(54)

The result for MIJ is given as below

M11 M12 M13 M14 M22 M23 M24

   2arω r2 + ω 2 −1 + e4rτ ω + 2e2rτ r sin[2τ ω] = 2 ω + e4rτ ω 2 − 2e2rτ (r2 + ω 2 ) + 2e2rτ r2 cos[2τ ω]     −2a 1 + e4rτ r2 ω 2 + b ω 2 + e4rτ ω 2 − 2e2rτ r2 + ω 2 + 2e2rτ r2 b + 2aω 2 cos[2τ ω] =− 2 ) + 2e2rτ r 2 cos[2τ ω]) (ω 2 + e4rτ ω 2 − 2e2rτ (r2 + ω     4aerτ rω r2 + ω 2 −1 + e2rτ ω cos[τ ω] + 1 + e2rτ r sin[τ ω] =− ω 2 + e4rτ ω 2 − 2e2rτ (r2 + ω 2 ) + 2e2rτ r2 cos[2τ ω]   4aerτ −1 + e2rτ rω r2 + ω 2 sin[τ ω] = 2 ω + e4rτ ω 2 − 2e2rτ (r2 + ω 2 ) + 2e2rτ r2 cos[2τ ω]  2arω ω − e4rτ ω + 2e2rτ r sin[2τ ω] =− 2 ω + e4rτ ω 2 − 2e2rτ (r2 + ω 2 ) + 2e2rτ r2 cos[2τ ω]   4aerτ −1 + e2rτ rω r2 + ω 2 sin[τ ω] =− 2 ω + e4rτ ω 2 − 2e2rτ (r2 + ω 2 ) + 2e2rτ r2 cos[2τ ω]    4aerτ rω − −1 + e2rτ ω cos[τ ω] + 1 + e2rτ r sin[τ ω] = ω 2 + e4rτ ω 2 − 2e2rτ (r2 + ω 2 ) + 2e2rτ r2 cos[2τ ω]

Hence we obtain Sc given by 1 1 Sc (xi , vi , xf , vf ) = − M11 (x2i + x2f ) − M22 (vi2 + vf2 ) − M13 xi xf − M24 vi vf (55) 2 2 −M12 xi vi − M34 xf vf − M14 xi vf − M23 xf vi − 2bjxi + 2bjxf − bj 2 τ (56) Expression of xi , vi , xf , vf are transferred to the earlier notation

x0 = xi ,

v 0 = vi ;

x = xf ,

v = vf

(57)

According to the definition for the kernel K

0

0

K(x, v; x0 , v 0 , τ ) = N eSc (x,v;x ,v ,τ )

1 1 Sc (x, v; x0 , v 0 , τ ) = − M11 (x02 + x2 ) − M22 (v 02 + v 2 ) − M13 x0 x − M24 v 0 v 2 2 −M12 x0 v 0 − M34 xv − M14 x0 v − M23 xv 0 − 2bjx0 + 2bjv − bj 2 τ The conditional probability distribution is given by 0

0

eSc (x,v;x ,v ,τ ) P (x, v; x , v , τ ) = R 0 0 S (x,v;x0 ,v0 ,τ ) dx dv e c 0

0

34

(58)

The marginal conditional probability distribution is given by

0

P (x, v; x ; τ ) = =

Z

r

dv 0 P (x, v; x0 , v 0 ; τ ) 2 1 1 exp{− 2 −x0 + ζx + ξv + j } 2 2πν 2ν

(59)

where 2Ωrω[ω sinh(2rτ ) − r sin(2ωτ )] a (r2 + ω 2 )  ζ = 4Ωrω[ r2 − ω 2 sinh(rτ ) sin(ωτ ) + 2rω cosh(rτ ) cos(ωτ )] ν2 =

ξ = −4Ωrω[ω sinh(rτ ) cos(ωτ ) + r cosh(rτ ) sin(ωτ )]

(60)

and

Ω=

1 (r2

+

ω 2 )2



r2 (r2



3ω 2 ) cos(2ωτ )

− ω 2 (ω 2 − 3r2 ) cosh(2rτ )

(61)

References [1] B. E. Baaquie and Cui Liang. American option pricing for interest rate caps and coupon bonds in quantum finance. Physica A, 38:285–316, 2007. [2] B. E. Baaquie and Cao Yang. Option volatility and the acceleration lagrangian. Physica A, 393:337–363, 2014. [3] Belal E Baaquie. Quantum finance: Path integrals and Hamiltonians for options and interest rates. Cambridge University Press, 2004. [4] Belal E. Baaquie. Quantum Field Theory and Superstrings: A Primer. to be published, UK, 1st edition, 2014. [5] Belal E Baaquie, Xin Du, and Jitendra Bhanap. Option pricing: Stock price, stock velocity and the acceleration lagrangian. Physica A: Statistical Mechanics and its Applications, 416:564–581, 2014.

35

[6] Ole E Barndorff-Nielsen. Econometric analysis of realized volatility and its use in estimating stochastic volatility models. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 64(2):253–280, 2002. [7] Jean-Philippe Bouchaud and Marc Potters. Theory of financial risk and derivative pricing: from statistical physics to risk management. Cambridge university press, 2003. [8] Mark B. Garman and Steven W. Kohlhagen. Foreign currency option values. Journal of International Money and Finance, 2(3):231 – 237, 1983. [9] J Orlin Grabbe. The pricing of call and put options on foreign exchange. Journal of International Money and Finance, 2(3):239–253, 1983. [10] Emmanuel Haven and Andrei Khrennikov. Quantum social science. Cambridge University Press, 2013. [11] JOHN HULL and ALAN WHITE. The pricing of options on assets with stochastic volatilities. The Journal of Finance, 42(2):281–300, 1987. [12] Lishang Jiang and Canguo Li. Mathematical modeling and methods of option pricing. World Scientific, 2005. [13] Rosario N Mantegna and H Eugene Stanley. Introduction to econophysics: correlations and complexity in finance. Cambridge university press, 1999. [14] Paul Wilmott, Jeff Dewynne, and Sam Howison. Option pricing: mathematical models and computation. Oxford financial press, 1993.

36

! The highlights of the paper: • • • •

The empirical study of statistical pricing model is investigated.! The time evolution of Option price is made in this model.! The pricing of Forex Option is made by the empirical model.! The empirical results of Forex Option for different countries in the period of 2008-2015 is figured out by the model.!