CEV

Constant Elasticity of Variance (CEV) OptionPricing Model:Integration and Detailed Derivation

Ying-Lin Hsu

Department of Applied MathematicsNational Chung Hsing University

Co-authors: T. I. Lin and C. F. Lee

Oct. 21, 2008

Ying-Lin Hsu (NCHU) Oct. 21, 2008 1 / 46

Outline

Introduction

Transition Probability Density Function

Noncentral Chi-Square Distribution

The Noncentral Chi-Square Approach to Option Pricing ModelDetailed Derivations of C1 and C2 Approaches to 2Some Computational ConsiderationsSpecial Cases

Concluding Remarks


Introduction

CEV modelThe CEV option pricing model is defined as

dS = Sdt+ S/2dZ, < 2,

where dZ is a Wiener process and is a positive constant.


Introduction

The elasticity is 2 since the return variance (S, t) = 2S2 withrespect to price S has the following relationship

d(S, t)/dS

(S, t)/S= 2,

which implies that d(S, t)/(S, t) = ( 2)dS/S.


Introduction

If = 2, then the elasticity is zero and the stock prices are lognormallydistributed as in the Black and Scholes model (1973).If = 1, then the elasticity is -1. The model proposed by Cox and Ross(1976).


Introduction

We will focus on the case of < 2 since many empirical evidences(see Campbell (1987), Glosten et al. (1993), Brandt and Kang (2004))have shown that the relationship between the stock price and its returnvolatility is negative. The transition density for > 2 is given byEmanuel and Macbeth (1982)



Consider the constant elasticity of variance diffusion,

dS = (S, t) + (S, t)dZ,

where (S, t) = rS aS, and (S, t) = S/2, 0 < 2. Then

dS = (r a)Sdt + S/2dZ.



Let Y = Y (S, t) = S2. By Itos Lemma with

Y

S= (2 )S1 , Y

t= 0,

2Y

S2= (2 )(1 )S ,

we have

dY =[(r a)(2 )Y + 1

22( 1)( 2)

]dt+ 2(2 )2Y dZ.



The Kolmogorov forward equation for Y becomes

P

t=

1

2

2

Y 2[2(2 )Y P ]

Y

{[(ra)(2)Y+1

22(1)(2)]P}.

Then f(ST | St, T > t) = f(YT | yt, T > t) | J | where J = (2 )S1 .



By Fellers Lemma with a = 122(2 )2, b = (r a)(2 ),h = 12

2( 2)(1 ), x = 1/T , x0 = 1/t and t = = (T t), we have

f(ST | St, T > t) = (2)k1/(2)(xz1)1/(2(2))exzI1/(2)(2(xz)1/2)

where

k =2(r a)

2(2 )[e(ra)(2) 1] , x = kS2t e

(ra)(2) , z = kS2T .



Cox (1975) obtained the following option pricing formula:

C = Ster

n=0

exxnG(n+ 1 + 1/(2 ), kK2)

(n+ 1)

Kern=0

exxn+1/(2)G(n+ 1, kK2

)(n+ 1 + 1/(2 )) ,

where G(m, ) = ((m))1 e

uum1du is the standardcomplementary gamma distribution function.



If Z1, . . . , Z are standard normal random variables, and 1, . . . , areconstants, then

Y =

i=1

(Zi + i)2

is the noncentral Chi-square distribution with degrees of freedomand noncentrality parameter =

j=1

2j , and is denoted as

2

().When j = 0 for all j, then Y is distributed as the central Chi-squaredistribution with degrees of freedom, and is denoted as 2 .



The cumulative distribution function of 2 () is

F (x; , ) = P (2

() x)

= e/2j=0

(/2)j

j!2/2(/2 + j)

x0y/2+j1e

y

2 dy, x > 0.



An alternative expression for F (x; , ) is

F (x; , ) =

j=0

((/2)je/2

j!

)P (2+2j x).

The complementary distribution function of 2 () is

Q(x; , ) = 1 F (x; , ).



The probability density function of 2 () can be expressed as amixture of central Chi-square probability density functions.

p2 ()

(x) = e/2j=0

(12)j

j!p2+2j (x)

=e(x+)/2

2/2

j=0

x/2+j1j

(/2 + j)22jj!.



An alternative expression for the probability density function of 2 () is

p2 ()

(x) =1

2

(x

)(2)/4exp

{12(+ x)

}I(2)/2(

x), x > 0,

where Ik is the modified Bessel function of the first kind of order k andis defined as

Ik(z) =

(1

2z

)k j=0

(z2/4

)jj!(k + j + 1)

.


The Noncentral Chi-Square Approach to Option Pricing Model

Following Schroder (1989) with the transition probability densityfunction, the option pricing formula under the CEV model is

C = E(max(0, ST K)

), = T t

= er K

f(ST | St, T > t)(ST K)dST

= er K

ST f(ST | St, T > t)dST erK K

f(ST | St, T > t)dST= C1 C2.


The Noncentral Chi-Square Approach to Option Pricing Model Detailed Derivations of C1 and C2

Making the change of variable w = kS2T , we have

dST = (2 )1k1/(2)w(1)/(2)dw.

Let y = kK2.



C1 = er

y

exw(x/w)1/(42)I 12

(2xw)(w/k)1/(2)dw

= er (x/k)1/(2) y

exw(x/w)1/(42)I 12

(2xw)dw

= erSte(ra)

y

exw(w/x)1/(42)I 12

(2xw)dw

= eaSt

y

exw(w/x)1/(42)I 12

(2xw)dw,



C2 = Ker

y

x1

42w(12+22)/(42)exwI 12

(2xw)dw

= Ker y

exw(x/w)1/(4)I 12

(2xw)dw.



Recall that the probability density function of the noncentral Chi-squaredistribution with noncentrality and degree of freedom is

p2 ()

(x) =1

2(x/)(2)/4I(2)/2(

x)e(+x)/2

= P (x; , ).

Let Q(x; , ) =x p2 ()

(y)dy. Then letting w = 2w and x = 2x.



C1 = Stea

y

e(x+w)/2(wx

)1/(42)I 1

2

(2xw)dw

= Stea

2y

e(x

+w

)/2

(w

x

)1/(42)I 1

2

(2xw

) 12dw

= SteaQ(2y; , x

),

= SteaQ

(2y; 2 +

2

2 , 2x),



C2 = Ker

y

exw( xw

)1/(42)I 1

2

(2xw)dw

= Ker

2ye(x

+w

)/2

(x

w

)1/(42)I 1

2

(2x

w

) 12dw

= Q

(2y; 2 2

2 , 2x)



Thus,

C = SteaQ(2y; 2 +

2

2 , 2x)KerQ(2y; 2 2

2 , 2x).

It is noted that 2 2/(2 ) can be negative for < 2. Thus furtherwork is needed.



Using the monotone convergence theorem and the integration byparts, we have

y

P (2y; 2, 2k)dk

=

y

ezk(z/k)1kz

1n=0

(zk)n

n!(n+ 1 + 1)dk

=

y

ezzn+1

(n+ )

y

ekkn

(n+ 1)dk

=

n=0

g(n + , z)G(n + 1, y)

=n=0

g(n + 1, z)i=1

g(i, y).



G(n, y) =

y

ekkn1

(n)dk

= y

kn1

(n)dek

=yn1ey

(n)+

y

kn2ek

(n 1)dk

=

ni=1

yi1ey

(i)

=

ni=1

g(i, y).



Next, applying the monotone convergence theorem, we have

Q(z; , k)

=

z

1

2

(yk

)(2)/4I 2

2

(ky)e(k+y)/2dy

=

z

1

2

(yk

)(2)/4 (12

ky

)(2)/2 n=0

(ky/4)n

n!(+2n

2

)e(k+y)/2dy=

n=0

ek/2kn(

12

)n (n+ 1)

z

ey/2y+2n

21(

12

)(+2n)/2(+2n

2

)dy=

n=0

ek/2(k/2)n

(n+ 1)

z

(1/2)(+2n)/2

(+2n

2

) ey/2y +2n2 1dy=

n=0

ek/2(k/2)n

(n+ 1)Q(z; + 2n, 0),



where

Q(z; + 2n, 0) =

z

(1/2)(+2n)/2

(+2n

2

) ey/2y +2n2 1dy=

z/2

1

(+2n

2

)eyy +2n2 1dy= G (n+ /2, z/2) .



Furthermore

Q(z; , k) =n=0

g (n+ 1, k/2)G (n+ /2, z/2)

=

n=0

g (n, k/2)G

(n+

22

, z/2

).

Hence

Q(2z; 2, 2k) =

n=0

g (n, k)G (n+ 1, z) .



Again

Q(2z; 2, 2k)

= g (1, k)G (, z) + g (2, k)G ( + 1, z) + g (3, k)G ( + 2, z) + = g (1, k) [G ( 1, z) + g (, z)] + g (2, k) [G ( 1, z) + g ( + 1, z)]

+g (3, k) [G ( 1, z) + g (, z) + g ( + 1, z) + g ( + 2, z)] + = G ( 1, z) + g (, z) + g ( + 1, z) [1 g (1, k)]

+g ( + 2, z) [1 g (1, k) g (2, k)] +

= G ( 1, z) +n=0

g ( + n, z) g ( + 1, z) [g (1, k)]

g ( + 2, z) [g (1, k) + g (2, k)] + = 1 g ( + 1, z) [g (1, k)]

g ( + 2, z) [g (1, k) + g (2, k)] .



We conclude that

Q(2z; 2, 2k) = 1n=1

g (n+ , z)

ni=1

g (i, k) .

And y

P (2z; 2, 2k)dk = 1Q (2z; 2( 1), 2y) .



Thus

C2 = Ker

y

P

(2x; 2 +

2

2 , 2w)dw

= KerQ

(2y; 2 2

2 , 2x)

= Ker(1Q(2x; 2

2 , 2y)).



We obtain

C = SteaQ(2y; 2 +

2

2 , 2x)Ker

(1Q(2x; 2

2 , 2y)),

where y = kK2, x = kS2t e(ra)(2) ,k = 2(r a)/(2(2 )(e(ra)(2) 1)) and a is the continuousproportional dividend rate.



When > 2 (see, Emanuel and MacBeth (1982), Chen and Lee(1993)), the call option formula is as follows:

C = SteaQ(2x;

2

2 , 2y)Ker

(1Q(2y; 2 + 2

2 , 2x)).


The Noncentral Chi-Square Approach to Option Pricing Model Approaches to 2

We consider that the noncentral Chi-square distribution will approachlog-normal as tends to 2. Since when either or approaches toinfinity, the standardized variable

2

() ( + )2( + 2)

tends to N(0,1) as either or .



lim2

2

() ( + )2( + 2)

= lim2

2kS2T ( + )2( + 2)

= lim2

2rS2T (1 )2(er(2) 1) 2rS2t er

(2)

2(er(2) 1)

2(er(2) 1)/(2 )(1 )2(er(2) 1) + 4rS2t er(2)

=lnST [lnSt + (r 2/2) ]

,

where r = r a.



Thus,lnST | lnSt N(lnSt + (r a 2/2), 2)

as 2. Similarly, it also holds when 2+. And we also obtainthe same result for > 2.


The Noncentral Chi-Square Approach to Option Pricing Model Some Computational Considerations

First initialize the following four variables (with n = 1)

gA =ezz

(1 + )= g(1 + , z),

gB = ek = g(1, k),

Sg = gB,

R = 1 (gA)(Sg).



Then repeat the following loop beginning with n = 2 and increaseincrement n by one after each iteration. The loop is terminated whenthe contribution to the sum, R, is declining and is very small.

gA = gA

(z

n+ 1)

= g(n + , z),

gB = gB

(k

n 1)

= g(n, k),

Sg = Sg + gB = g(1, k) + g(n, k)

R = R (gA)(Sg) = the nth partial sum.



As each iteration, gA equals g(n + , z), gB equals g(n, k) and Sgequals g(1, k) + + g(n, k). The computation is easily done.



Using the approximation

Q(z; , k) = Pr(2

> z)

= Pr

(

2

+ k>

z

+ k

)

= Pr

( 2

+ k

)h>

(z

+ k

)h

=

1 hP [1 h+ 0.5(2 h)mP ]

(z

+k

)hh

2P (1 +mP )

,

where h = 1 23( + k)( + 3k)( + 2k)2, P = ( + 2k)/( + k)2 andm = (h 1)(1 3h).


The Noncentral Chi-Square Approach to Option Pricing Model Special Cases

Q(z, 1, k) =

z

fY (y)dy

= 1 FY (z)= 1 P (Y z)= 1 P ((z + )2 z)= 1 P (|z + | z)= 1 P ( z z +z)= 1 [N (z ) N ( z)]= N (z

k) + 1N (z

k)

= N (k z) +N (

k z).



Q(z, 3, k) = Q(z, 1, k) +12

j=0

(k/4)j zj+ 12j!

(12 + j + 1

) e(k+z)/2

= Q(z, 1, k) +22pi

1k

j=0

((kz)1/2

)2j+1(2j + 1)!

e(k+z)/2

= Q(z, 1, k) +1

k2pi

j=0

(kz)2j+1

(2j + 1)!e(k+z)/2

= Q(z, 1, k) +1k

[12pi

ekz 1

2piekz

] e(z+k)/2

= Q(z, 1, k) +1k

[12pi

e(kz)2/2 1

2pie(

k+z)2/2

]

= Q(z, 1, k) +1k

[N (

k z)N (

k +

z)].



Q(z; 5, k)

= Q(z; 1, k) + k1/2[N (

k z)N (

k +

z)]

+(k1z k3/2)N (

k z) + (k1z + k3/2)N (

k +

z)

= Q(z; 1, k) + (k1/2 k3/2 + k1z)N (k z)

(k1/2 1 k1z)N (k +

z)

= Q(z; 1, k)

+k3

2

[(k 1 +

kz)N (

k z) (k 1

kz)N (

k +

z)].


Concluding Remarks

The option pricing formula under the CEV model is quite complexbecause it involves the cumulative distribution function of thenoncentral Chi-square distribution Q(z; , k).

Some computational considerations are given in the article whichwill facilitate the computation of the CEV option pricing formula.

We can use the function pchisq(q, df , ncp) in free software R tocalculate the cumulative distribution function of the noncentralChi-square distribution, where q is the quantile, df is the degree offreedom which non-integer values are allowed and ncp is thenoncentrality parameter.


Concluding Remarks

THE END.

Thanks For Your Attentions!


Main PartIntroductionTransition Probability Density FunctionNoncentral Chi-Square DistributionThe Noncentral Chi-Square Approach to Option Pricing ModelDetailed Derivations of C1 and C2 Approaches to 2Some Computational ConsiderationsSpecial Cases

Concluding Remarks

CEV

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