From Characteristic Functions and Fourier Transforms to PDFs/CDFs and Option Prices

7/30/2019 From Characteristic Functions and Fourier Transforms to PDFs/CDFs and Option Prices

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From Characteristic Functions and FourierTransforms to PDFs/CDFs and Option Prices

Liuren Wu

Zicklin School of Business, Baruch College

Option Pricing

Liuren Wu (Baruch) Fourier Transforms Option Pricing 1 / 22
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3/22

Formalizing the idea

Assume X is a Markov process: Any information about X up to time t issummarized by Xt. Let f(XT|Xt) denote the conditional distribution of XTconditional on time-t information (filtration Ft) under the risk-neutralmeasure Q.

Let (XT) denote the payoff of a contingent claim (derivative), which weassume is a function of XT. Then, its time-t value is

pt = EQt

e

T

trsds(XT)

=

X

e

T

trsds(XT)f(XT|Xt)dXT

Instead of solving PDEs, binomial trees, or simulating the process, we focuson doing the integration based on the risk-neutral densities.

In most of the examples, I assume deterministic interest rates:

pt = er

X

(XT)f(XT|Xt)dXT

where r now is the time-t continuously compounded spot rate of maturityT t.

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4/22

The road map

For most models (processes) discussed in this class on security prices St, wecan derive the characteristic function or Fourier transform of the log securityreturn (ln ST/St) (semi-)analytically.

Given the characteristic function (CF), we just need one numericalintegration to obtain the probability density function (PDF) or cumulativedensity function (CDF).

Given the Fourier transforms (FT), we just need one numerical integrationto obtain the value of vanilla options.

The integration is one-dimensional in both cases no matter how manydimensions/factors the security price St is composed of.

We can apply fast Fourier inversion (FFT or some other methods) to makethe numerical integration (for both PDF and option prices) very fast.

Solving PDEs becomes difficult as the dimension increases.

Simulation works (slowly) for high-dimensional cases and is reserved forpricing exotics.

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5/22

Characteristic function: definition

In probability theory, the characteristic function (CF) of any random variableX completely defines its probability distribution. On the real line it is givenby the following formula:

X(u) E

eiuX

=

eiuxfX(x)dx =

eiuxdFX(x), u R

where u is a real number, i is the imaginary unit, and E denotes the

expected value, fX(x) denotes the probability density function (PDF), andFX(x) denotes the cumulative density function (CDF).

CF is well defined on the whole real line (u).

For option pricing, we extend the definition to the complex plane,u D C, where D denotes the subset of the complex plane on which the

expectation is well defined. X(u) under this extended definition is calledthe generalized Fourier transform.

The generalized Fourier transform includes as special cases the Laplacetransform (when im(u) > 0) and cumulant generating function (whenim(u) < 0) as special cases (when they are well defined).

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6/22

Example: The Black-Scholes model

Under BSM, the log security return follows,

st ln St/S0 =

1

22

t + Wt

The return is normally distributed with mean

12

2

t and variance 2t.

The PDF is fs(x) =1

22texp

(x( 122)t)222t

.

The CF is:

s(u) = E eiust = eiumean12 u

2variance = e(iutiu12

2t 12 u22t)

= e(iut 122(iu+u2)t)

Under Q, = r q.

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7/22

The inversion: From CF to PDF and CDF

There is a bijection between CDF and CFs: Two distinct probabilitydistributions never share the same CF.

Given a CF , it is possible to reconstruct the corresponding CDF:

FX(y) FX(x) = lim

1

2

+

eiux eiuy

iuX(u)du

In general this is an improper integral ...

Another form of the inversion

FX (x) =1

2+

1

2

0

eiuxX (u) eiuxX (u)iu

du.

The inversion formula for PDF:

fX(x) =1

2

+

eiuxX(u)du =

1

0

eiuxX(u)du.

All the integrals here should be understood as a principal value if there is noseparate convergence at the limits.

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8/22

Proofs

Preliminary results:

eiux = cos ux + i sin ux,

1

eiu

iu du = 1

sin uu du = sgn(),

2

0sin u

udu = sgn(),

For fixed x, sgn(y x)dF(y) =

x dF(y) +

x dF(y) = 1 2F(x),

(u)&(u) are complex conjugates.

CDF inversion:

I =

0

eiuxX (u) eiuxX (u)

iu

du

=

0

eiuxeiuz eiuxeiuz

iudF(z)du

=

0

2 sin u(x z)

udF(z)du

=

0

2 sin u(x z)

ududF(z) =

sgn(x z)dF(z)

= (2F(x) 1)

Hence, F(x) = 12 +1

2I.

PDF inversion:

f(x) = F

(x) =1

2

0

e

iux (u) + e

iux (u)

du =1

0eiux (u) du

Reference: Kendalls Advanced Theory of Statistics, Volume I, chapter 4

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Proofs: The option transform

Ic (u)

e

iukdc (k) = e

iukc (k)

c (k) iue

iukdk

Checking the boundary conditions, we have c () = 0 (when strike is infinity) and c () = (Ft 0)/F0 = 1 when the

strike is zero. Hence, eiuc () = 0 and we will carry the other non-convergent limit eiu.

Ic (u) = e

iu

c (k) iue

iukdk

= eiu

iu

e

st ek

1stkdF (s)

e

iukdk

= e

iu

iu

e

st

e

k1stke

iuk

dk

dF (s)

= eiu

iu

st

e

iuk+st e(iu+1)k

dk

dF (s)

= eiu

iu

est eiuk

iu

e(iu+1)k

iu + 1

st

dF (s) .

We need to check the boundary again. limk e(iu+1)k = 0 given the real component e. The other boundary isnon-convergent esteiu , which we pull out and take the expectation to have

iu

esteiu

iudF (s) = e

iu,

which cancels out the other nonconvergent term.

I

c (u) = iu

e(iu+1)stiu

e(iu+1)st

iu + 1

dF (s) =

e(iu+1)st

iu + 1 dF (s) =

(u i)

iu + 1 .

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12/22

Proofs: The option transform inversion

The proof for the option transform inversion is similar to that for the CDF. In particular, our scaled option value c(k) behavesjust like a CDF: c () = 0 (when strike is infinity), and c () = 1 (when strike is zero). Hence, the inversion formula is

I

0

eiux (u) eiux (u)

iudu

= (as before) =

sgn (x z) dF (z) = (1 2c (x)) .

Thus,

c (x) =1

2+

1

2I.

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13/22

II. The PDF analog

Treat c(k) analogous to a PDF.

The option transform:

IIc (z)

eizkc(k)dk =s (z i)

(iz) (iz + 1)

with z = u i, D R+ for the option transform to be welldefined.

The range of depends on payoff structure and model.The exact value choice of is a numerical issue.Carr and Madan (1999, Journal of Computational Finance) refer to as thedampening coefficient.Given the transform on return s(u), we know the transform on call.

The inversion is analogous to that for a PDF:

c(k) =1

2

i+i

eizkIIc (z)dz =ek

0

eiukIIc (u i)du.

References: Carr&Wu, Time-Changed Levy Processes and Option Pricing, JFE, 2004, 17(1), 113141.

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Proofs

II

(z)

e

izkc (k) dk

=

e

st ek

1stkdF (s)

eizk

dk

=

e

st ek

1stkeizk

dk

dF (s)

=

st

e

izk+st e(iz+1)k

dk

dF (s)

=

est eizkiz

e(iz+1)k

iz + 1

st

dF (s)We need to consider the boundary conditions at k = . limk e

(iz+1)k = 0 as long as the real component of iz is

greater than 1. limk eizk = 0 as long as the real component of iz is greater than 0. Hence, taken together, we need

the real component of iz to be greater than zero. If we write z = u i, with both u and real, we have iz = iu + . Hence,we need > 0 for the above boundary condition to converge. Given that ui > 0, we have

II

(z) =

e(iz+1)st

iz

e(iz+1)st

iz + 1

dF (s)

=

e(iz+1)st

iz (iz + 1)dF (s) =

(z i)

iz (iz + 1).

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15/22

Fast Fourier Transform (FFT)

FFT is an efficient algorithm for computing discrete Fourier coefficients.

The discrete Fourier transform is a mapping of f = (f0, , fN1) on thevector of Fourier coefficients d = (d0, , dN1), such that

dj =1

N

N1m=0

fmejm 2

Ni, j = 0, 1, ,N 1.

FFT allows the efficient calculation of d if N is an even number, sayN = 2n, n N. The algorithm reduces the number of multiplications in therequired N summations from an order of 22n to that of n2n1, a veryconsiderable reduction.

By a suitable discretization, we can approximate the inversion of a PDF(also option price) in the above form to take advantage of thecomputational efficiency of FFT.

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16/22

Return PDF inversion

Compare the PDF inversion with the FFT form:

fX(x) =1

0e

iux

X(u)du. dj =1

N

N1m=0

fmejm 2

Ni

Discretize the integral using the trapezoid rule:fX(x)

1

N1m=0 me

iumxX(um)u, m = 12 when m = 0 and 1 otherwise.

(trapezoid rule:b

a h(x)dx =h(a)+h(b)

2 +N1k=1 h(a + kx)x.)

Set = u, um = m.

Set xj = b+ j with = 2/(N) being the return grid and b being aparameter that controls the return range.

To center return around zero, set b = N/2.The PDF becomes

fX(xj) 1

N

N1m=0

fmejm 2N i, fm = m

N

eium bX(um). (1)

with j = 0, 1, ,N 1. The summation has the FFT form and can henceLiuren Wu (Baruch) Fourier Transforms Option Pricing 16 / 22

C ll l i i
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17/22

Call value inversion

Compare the call inversion (method II) with the FFT form:

c(k) =ek

0e

iuk

IIc (u i)du. dj =

1

N

N1m=0

fmejm 2

Ni

Discretize the integral using the trapezoid rule:

c(k) ek

Nm=0 me

ium kIIc (um i)u

Set = u, um = m.Set kj = b+ j with = 2/(N) being the return grid and b being aparameter that controls the return range.

To center return around zero, set b = N/2.

The call value becomes

c(kj) 1

N

N1m=0

fmejm 2

Ni, fm = m

N

ekj+iumbIIc (um i).

with j = 0, 1, ,N 1. The summation has the FFT form and can hencebe computed efficiently.


FFT i l i
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FFT implementation

To implement the FFT, we need to fix the following parameters

N = 2n

: The number of summation grids. Setting it to be the power of 2can speed up the FFT calculation.

= u: The discrete summation grid width. The smaller the grid, thebetter the approximation.

However, given N, also determines the strike grid = 2/(N). The finer thesummation grid , the coarser the strike spacing returned from the FFTcalculation. There is a trade off: If we want to have more option value calculatedat a finer grid of strikes, we would need to use a coarser summation grid andhence less accuracy.The lower and upper bound truncation b = N/2 is also determined by the

summation grid choice.FFT generates option values at N strikes simultaneously. However, if the strikegrid is larger, many of the returned strikes are out of the interesting region.


III F i l FFT
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III. Fractional FFT

Fractional FFT (FRFT) separates the integration grid choice from the strikegrids. With appropriate control, it can generate more accurate option values

given the same amount of calculation.

The method can efficiently compute,

dj =

N1m=0

fmejmi, j = 0, 1, ...,N 1,

for any value of the parameter .

The standard FFT can be seen as a special case for = 2/N. Therefore,we can use the FRFT method to compute,

c(kj) 1

N

N1m=0

fmejmi, fm = mN

ekj+iumbIIc (um).

without the trade-off between the summation grid and the strike spacing .

We require = 2/N under standard FFT.


F ti l FFT i l t ti
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20/22

Fractional FFT implementation

Let d = D(f, ) denote the FRFT operation, with D(f) = D(f, 2/N) beingthe standard FFT as a special case.

An N-point FRFT can be implemented by invoking three 2N-point FFTprocedures.

Define the following 2N-point vectors:

y = fn

ein2N1

n=0, (0)

N1n=0

, (2)z =

ein

2N1

n=0,

ei(Nn)2N1

n=0

. (3)

The FRFT is given by,

Dk(h, ) =

eik2N1

k=0 D1k (Dj(y) Dj(z)) , (4)

where D1k () denotes the inverse FFT operation and denoteselement-by-element vector multiplication.


F ti l FFT i l t ti
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21/22

Fractional FFT implementation

Due to the multiple application of the FFT operations, an N-point FRFTprocedure demands a similar number of elementary operations as a 4N-point

FFT procedure.

Given the free choices on and , FRFT can be applied more efficiently.Using a smaller N with FRFT can achieve the same option pricing accuracyas using a much larger N with FFT.

The accuracy improvement is larger when we have a better understanding ofthe model and model parameters so that we can set the boundaries moretightly.

Caveat: The more freedom also asks for more discretion and caution inapplying this method to generate robust results in all situations. This

concern becomes especially important for model estimation, during whichthe trial model parameters can vary greatly.

Reference: Chourdakis, 2005, Option pricing using fractional FFT, JCF, 8(2).


IV Fourier cosine series expansions
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IV. Fourier-cosine series expansionsFang & Oosterlee, A novel pricing method for European options based on Fourier-cosine series expansions, 2008.

Given a characteristic function (u), the density function can be numerically obtained via the Fourier-cosine seriesexpansion,

f(x) = 12

R

eiux(u)du

N1j=0

j cos

(x a)uj

Vj

where uj =j

ba, Vj =

2ba

Re

(uj)eiuja

, and [a, b] denotes a truncation of the return range. Choosing the range

to be 10 standard deviation away from the mean seems to work well: b, a = 10.

Applying the expansion to the option valuation, we have

C(K, t) Kert N1

j=0

jRe

s

uj

eiuj(k+a)Uj

where Uj =2

ba

j(0, b) j(0, b)

with

j(c, d) =1

1 + u2j

cos((d a)uj)ed cos((c a)uj)e

c+ uj sin((d a)uj)e

d uj sin((c a)uj)e

c

,

j(c, d) =

sin((d a)uj) sin((c a)uj)

/uj j = 0

(d c) j = 0

Works well. Some constraints on how [a, b] are chosen.

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From Characteristic Functions and Fourier Transforms to PDFs/CDFs and Option Prices