On the goodness of –t of Kirk™s formula for spread option ...

On the goodness of �t of Kirk�s formula forspread option prices

Elisa Alòs�

Dpt. d�Economia i EmpresaUniversitat Pompeu Fabrac/Ramon Trias Fargas, 25-2708005 Barcelona, Spain

Jorge A. Leóny

Control AutomáticoCINVESTAV-IPN

Apartado Postal 14-74007000 México, D.F., Mexico

Abstract

In this paper we investigate the goodness of �t of the Kirk�s approxi-mation formula for spread option prices in the correlated lognormal frame-work. Towards this end, we use the Malliavin calculus techniques to �ndan expression for the short-time implied volatility skew of options withrandom strikes. In particular, we obtain that this skew is very pronouncedin the case of spread options with extremely high correlations, which can-not be reproduced by a constant volatility approximation as in the Kirk�sformula. This fact agrees with the empirical evidence. Numerical exam-ples are given.Keywords: Spread options, Kirk�s formula, Malliavin calculus, Skorohodintegral.

JEL code: G12, G13Mathematical Subject Classi�cation: 91B28, 91B70, 60H07.

1 Introduction

A spread option is a derivative written as the di¤erence of two underlying assets.Namely, the payo¤ of a call spread option with strike K with time to maturityT is

(S1T � S2T �K)+;where S1 and S2 denote two asset prices. It is well-known that if S1 and S2

are two geometric Brownian motions (that may be correlated) and K = 0;the corresponding option price is given by the Margrabe fomula (see Margrabe(1978)), which can be deduced from the fact that S1=S2 is a log-normal process.Thus, in this case, the spread option value can be expressed as the classicalBlack-Scholes call price with initial asset price S10 ; where we take the strike

�Supported by the grants ECO2011-288755 and MEC FEDER MTM 2009-08869yPartially supported by a CONACyT grant

1

equal to the expected value of S2T and volatility equal toq�2 � 2��0 + (�0)2.

Here � and �0 are the volatility parameters of S1 and S2, respectively, and �denotes the correlation.In the case K 6= 0 the fraction S1=

�S2 +K

�is not log-normal and then the

arguments used in the deduction of Margrabe formula cannot be applied any-more. One proposed solution is to assume the distribution of the spread is ableto be approximated by the normal distribution. This leads us to the Bachelier�smethod (see Shimko (1994)). Even though the approximation of the spreaddistribution by the normal one is poor, this method can be precise in someranges of parameters (see, for example, Carmona and Durrleman (2003)). Amore successful method is suggested by Kirk (1995), who applied the Margrabeformula, aproaching S1=

�S2 +K

�by a log-normal random process. Nowadays

Kirk�s formula is the most popular option pricing approximation expression forspread options due to its accuracy and its simplicity. Recently, some otherauthors have proposed new formulas and methods to estimate spread optionprices. Among them, we can mention Alexander and Venkatramanan (2011),Bjerksund and Stensland (2011), Borovkova et al. (2007), Carmona and Dur-rleman (2003) , and Deng et al. (2008). Numerical evidence has shown thatthese approaches may improve Kirk�s estimates, specially for high correlatedcases (see, for instance, Bjerksund and Stensland (2011)).One interesting tool in the development of an accurate approximation for-

mula is the knowledge of the properties of the corresponding implied volatil-ity. For example, in the case of vainilla options with stochastic volatility, weknow that implied volatilities exhibits smiles and skews (see Renault and Touzi(1996)). Then, we have to take into account that accurate approximationsshould reproduce them and, moreover, we can understand why some formulasfail (for example, we expect that a constant volatility expression will fail for inor out-of-the-money options).Up to our knowledge, there are not similar analytical studies in the case of

options with random strikes (as for example spread options). The main goal ofthis paper is to consider analytically the relation between the implied volatilityand the stock price S1 for options with random strike. By means of the Malli-avin calculus thecniques, we develop an extension of the Margrabe formula thatallows us to �nd an expression for the short-time skew slope for spread options.This analysis of the implied volatility skew evidences that the dependence be-tween the implied volatility and the stock price S1 increases strongly when thecorrelation parameter � is close to 1. Hence we can expect that Kirk�s approxi-mation reduces its accuracy for highly correlated assets due to the fact that inthis approximation the volatility is constant as a function of the stock price S1.This agrees with the numerical evidence (see, for example, Baeva (2011)).The organization of this paper is as follows. In Section 2 we introduce the

framework of this paper. Section 3 is devoted to present a extended Margrabeformula for the case K 6= 0. This formula is used in Section 4 to �gure up anexpression for the derivative of the implied volatility with respect to the log-stock price. The short time behaviour of this derivative is analyzed in Section

2

5. Finally in Section 6 we obtain our results on the goodness of the �t of Kirk�sformula as an application.

2 Statement of the model and notation

In this paper we consider the following model for the log-price of a stockunder a risk-neutral probability measure Q:

dXt =

�r � �

2t

2

�dt+ �t(�dWt +

p1� �2dBt); t 2 [0; T ]: (1)

Here, r is the instantaneous interest rate,W and B are independent standardBrownian motions and � 2 (�1; 1). For the sake of simplicity, we assume thatthe volatility process � is a square-integrable deterministic function which isright-continuous. In the following we denote by FW and FB the �ltrationsgenerated by W and B; respectively. We de�ne F := FW _ FB :In this paper we consider European call options with payo¤ h(XT ) :=�

eXT �KT

�+; where we allow the strike KT to be random. More precisely,

we assume K is a square-integrable, positive, continuous, bounded and FWt -measurable process.Notice that this choice includes some popular classes of options as basket

ones. It is well-known that the price of an European call with strike KT is givenby the formula

Vt = e�r(T�t)E

�(eXT �KT )+jFt

�: (2)

In the sequel, we will make use of the following notation:

� MTt := E

�KT j FWt

�: Observe that, by the martingale representation the-

orem,

MTt = E (KT ) +

Z t

0

m(T; s)dWs;

for some FW�measurable and adapted process m (T; �) :

� vt :=�

YtT�t

� 12

; with Yt :=R Tta2sds; where a

2sds := �2sds � 2

dhMT ;Xis

MTs

+

dhMT ;MT is

(MTs )

2 . Note that

a2s = �2s � 2��sm(T; s)

MTs

+m2(T; s)

(MTs )

2

= �2s�1� �2

�+

��s �

m(T; s)

MTs

�2is a positive quantity. Although the right-hand-side of the last equalitydepends on T , we denote it by a2s in order to simplify the notation. It iseasy to see that for all � 2 (�1; 1), a2s � C (�)�2s:

3

� BS(t; x;K; �) denotes the price of an European call option under the clas-sical Black-Scholes model with constant volatility �, current log stock pricex, time to maturity T � t; strike price K and interest rate r: Rememberthat in this case:

BS(t; x;K; �) = exN(d+)�Ke�r(T�t)N(d�);

where N denotes the cumulative probability function of the standard nor-mal law and

d� :=x� ~x�t�pT � t

� �2

pT � t;

with ~x�t := lnK � r(T � t):

� LBS��2�stands for the Black-Scholes di¤erential operator, in the log

variable, with volatility � :

LBS��2�= @t +

1

2�2@2xx + (r �

1

2�2)@x � r�

It is well known that LBS��2�BS(�; �; �) = 0:

Now we describe some basic notation that is used in this article. For this, weassume that the reader is familiar with the elementary results of the Malliavincalculus, as given for instance in Nualart (2006).Let us consider a standard Brownian motion Z = fZt; t 2 [0; T ]g de�ned on a

complete probability space (;F ; P ): The set D1;2Z is the domain of the derivativeoperator DZ in the Malliavin calculus sense. D1;2Z is a dense subset of L2()and DZ is a closed and unbounded operator from L2() into L2([0; T ]�):Wealso consider the iterated derivatives DZ;n; for n > 1; whose domains is denotedby Dn;2W :

The adjoint of the derivative operator DZ , denoted by �Z ; is an extension ofthe Itô integral in the sense that the set L2a([0; T ]�) of square integrable andadapted processes (with respect to the �ltration generated by Z) is includedin Dom�Z and the operator �Z restricted to L2a([0; T ] � ) coincides with theItô integral. We make use of the notation �Z(u) =

R T0utdZt: We recall that

Ln;2 := L2([0; T ];Dn;2W ) is included in the domain of �Z for all n � 1:

3 A decomposition result

Before proving an extension of the Hull and White formula, we state the fol-lowing result, which is nedeed in the remaining of the paper.

4

Lemma 1 Let KT be bounded, 0 � t � s < T and Gt := Ft _ FWT . Assume� 2 (�1; 1) :Then, for any n � 0, there exists C = C(n; �) such that��E �(@n+2x � @n+1x )BS(s;Xs;M

Ts ; vs)jGt

�� C

Z T

t

�2�d�

!� 12 (n+1)

:

Proof: In order to show this result, we proceed as in the proof of Lemma4.1 in Alòs, León and Vives (2007) and we use the fact that K is a boundedand FW�measurable and adapted process to obtain that��E �(@n+2x � @n+1x )BS(s;Xs;M

Ts ; vs)jGt

�� C

(1� �2)

Z s

t

�2�d� +

Z T

s

a2�d�

!� 12 (n+1)

:

We know that for all � 2 (�1; 1), a2s � C (�)�2s for some positive constant C (�).Then

R Tsa2�d� � C (�)

R Ts�2�d�; from where the result follows.

Now we are able to prove the main result of this section, the extended Hulland White formula. We will need the following hypothesis:(H1) The process a2 2 L1;2:

Theorem 2 Consider the model (1) and assume that hyptothesis (H1) holds.Then it follows that

Vt = E�BS(t;Xt;M

Tt ; vt)

��Ft�+1

2E

(�

Z T

t

e�r(s�t)�@3xxx � @2xx

�BS(s;Xs;M

Ts ; vs)�s�

Ws ds

+

Z T

t

e�r(s�t)@K�@2xx � @x

�BS(s;Xs;M

Ts ; vs)�

Ws m(T; s)ds

��Ft); (3)

where �Ws :=hDWs

R Tsa2(r)dr

i:

Proof: This proof is similar to the one of the main theorem in Alòs, Leónand Vives (2007), so we only sketch it. Notice that BS(T;XT ;MT

T ; vT ) = VT :Then, from (2), we have

e�rtVt = E(e�rTBS(T;XT ;KT ; vT )jFt):

Now, using the Itô�s formula to the process

t! e�rtBS(t;Xt;MTt ; vt)

5

and proceeding as in Alòs, León and Vives (2007) (see also Alòs and Nualart(1998), Alòs (2006) or Nualart (2006)), we can write

e�rTBS(T;XT ;MTT ; vT )

= e�rtBS(t;Xt;MTt ; vt)

+

Z T

t

e�rsLBS(v2s)BS(s;Xs;MTs ; vs)ds

+

Z T

t

e�rs@xBS(s;Xs;MTs ; vs)�s(�dWs +

p1� �2dBs)

+

Z T

t

e�rs@KBS(s;Xs;MTs ; vs)dM

Ts

+

Z T

t

e�rs@2xKBS(s;Xs;MTs ; vs)d

MT ; X

�s

+1

2

Z T

t

e�rs@�BS(s;Xs;MTs ; vs)

v2s � a2svs(T � s)

ds

+

Z T

t

e�rs@2x�BS(s;Xs;MTs ; vs)

�s��Ws

2vs(T � s)ds

+

Z T

t

e�rs@2K�BS(s;Xs;MTs ; vs)

�Ws m(T; s)

2vs(T � s)ds

+1

2

Z T

t

e�rs�@2xx � @x

�BS(s;Xs;M

Ts ; vs)

��2s � v2s

�ds

+1

2

Z T

t

e�rs@2KKBS(s;Xs;MTs ; vs)d

MT ;MT

�s:

6

Hence, the fact that LBS(v2s)BS(s;Xs;MTs ; vs) = 0; multiplying by e

rtand tak-ing conditional expectations we can establish

E�e�r(T�t)BS(T;XT ;M

TT ; vT )

��Ft�= E

(BS(t;Xt;M

Tt ; vt) +

Z T

t

e�r(s�t)@2xKBS(s;Xs;MTs ; vs)d

MT ; X

�s

+1

2

Z T

t

e�r(s�t)@�BS(s;Xs;MTs ; vs)


ds

+

Z T

t

e�r(s�t)@2x�BS(s;Xs;MTs ; vs)

�s��Ws

2vs(T � s)ds

+

Z T

t

e�r(s�t)@2K�BS(s;Xs;MTs ; vs)

�Ws m(T; s)

2vs(T � s)ds

+1

2

Z T

t

e�r(s�t)�@2xx � @x

�BS(s;Xs;M

Ts ; vs)

��2s � v2s

�ds

+1

2

Z T

t

e�r(s�t)@2KKBS(s;Xs;MTs ; vs)d

MT ;MT

�s

��Ft):

Consequently, the classical relationships between the greeks:

@2xxBS � @xBS = @�BS1

�(T � t)

@2xKBS = �@�BS1

K�(T � t)

@2KKBS = @�BS1

K2�(T � t)give


TT ; vT )

��Ft�= E

(BS(t;Xt;M

Tt ; vt)�

Z T

t


1

MTs vs(T � s)

dMT ; X

�s

+1

2

Z T

t



ds

+

Z T

t


�s��Ws

2vs(T � s)ds

+

Z T

t


�Ws m(T; s)

2vs(T � s)ds

+1

2

Z T

t


��2s � v2s

� 1

vs(T � t)ds

+1

2

Z T

t


1

M2s vs(T � s)

dMT ;MT

�s

��Ft):

7

That is,


TT ; vT )

��Ft�= E

(BS(t;Xt;M

Tt ; vt) +

Z T

t

e�r(s�t)@�BS(s;Xs;M

Ts ; vs)

vs(T � t)

�"�dMT ; X

�s

MTs

+1

2

�v2s � a2s

�ds+

1

2

��2s � v2s

�ds+

1

2

dMT ;MT

�s

(MTs )

2

#

+

Z T

t


�s��Ws

2vs(T � s)ds

+

Z T

t


�Ws m(T; s)

2vs(T � s)ds

��Ft):

Since, a2sds := �2sds� 2

dhMT ;Xis

MTs

+dhMT ;MT i

s

(MTs )

2 we obtain


TT ; vT )

��Ft�= E

(BS(t;Xt;M

Tt ; vt) +

Z T

t


�s��Ws

2vs(T � s)ds

+

Z T

t


�Ws m(T; s)

2vs(T � s)ds

��Ft);

as we wanted to prove.

Example 3 Assume the model (1) with constant volatility �: We consider acall spread option with strike equal to KT = S

0T +K, where K is a non-negative

deterministic constant and S0 is another stock price of the form S0t = exp (X0t) ;

where

dX 0t =

r � (�

0)2

2

!dt+ �0dWt; t 2 [0; T ];

for some positive constant �0. Then we can easily check thatm (T; �) = exp(r(T��))S0��

0; MT� = exp(r(T � �))S0� +K; and then

a2� := �2 � 2��

0 exp(r(T � �))S0�exp(r(T � �))S0� +K

+(�0)

2(exp(r(T � �))S0�)

2

(exp(r(T � �))S0� +K)2 :

Notice that, if K = 0

a2� := �2 � 2��0 + (�0)2

and DWs a

2(�) = 0: Then, the equality (3) reduces to

Vt = BS

�t;Xt; exp (r (T � t))S0t;

q�2 � 2��0 + (�0)2

�;

from where we recover the well-known Margrabe formula (see Margrabe (1978)).

8

Remark 4 Notice that, in the context of the previous example, when K is neg-ative, the call option on the spread ST � S0T is equivalent to the correspondingput option on the spread S0T � ST with positive strike �K. Then, without lossof generality, we can assume that the spread option is written with a positive K.

4 Derivative of the implied volatility

Let It(Xt) denote the implied volatility process, which satis�es by de�nitionVt = BS(t;Xt;Mt; It(Xt)): In this section we prove a formula for its at-the-money derivative that we use in Section 5 to study the short-time behavior ofthe implied volatility.

Proposition 5 Assume that the model (1) holds with a 2 L1;2 and that, forevery �xed t 2 [0; T ) ;

�R Tt�2�d�

��1<1. Then it follows that

@It@Xt

(x�t ) =E(R Tte�r(s�t)(@xF (s;Xs;M

Ts ; vs)� 1

2F (s;Xs;MTs ; vs))dsjFt)

@�BS(t; x�t ;MTt ; It(x

�t ))

��Xt=x�t

; a.s.

where

F (s;Xs;MTs ; vs) =

1

2

��@3xxx � @2xx

�BS(s;Xs;M

Ts ; vs)�s��

Ws

+@K�@2xx � @x

�BS(s;Xs;M

Ts ; vs)�

Ws m(T; s)

�and x�t = ln(M

Tt )� r(T � t):

Proof: Using Theorem 2 and the expression Vt = BS(t;Xt;MTt ; It(Xt)) we

obtain

@Vt@Xt

= @xBS(t;Xt;MTt ; It(Xt)) + @�BS(t;Xt;M

Tt ; It(Xt))

@It@Xt

(Xt): (4)

and

Vt = E(BS(t;Xt;MTt ; vt)jFt) + E(

Z T

t

e�r(s�t)F (s;Xs;MTs ; vs)dsjFt);

which implies that

@Vt@Xt

= E(@xBS(t;Xt;MTt ; vt)jFt) + E(

Z T

t

e�r(s�t)@xF (s;Xs;MTs ; vs)dsjFt):

(5)We can check that the conditional expectationE(

R Tte�r(s�t)@xF (s;Xs;M

Ts ; vs)dsjFt)is

well de�ned and �nite a.s. due to the fact that�R T

t�2�d�

��1< 1. Thus, (4)

9

and (5) imply

@It@Xt

(x�t ) (6)

=1


�t ))

�E(@xBS(t; x

�t ;M

Tt ; vt)jFt)� @xBS(t; x�t ;MT

t ; It(x�t ))

+E(

Z T

t

e�r(s�t)@xF (s;Xs;MTs ; vs)dsjFt)

#��Xt=x�t

:

Notice that

E(@xBS(t; x�t ;M

Tt ; vt)jFt)

= @xE(BS(t; x;MTt ; vt)jFt)

��x=x�t

= @xBS(t; x;MTt ; I

0t (x))jx=x�t ; (7)

where, by the Hull and White formula, I0t (Xt) is the implied volatility of calloption with constant strike MT

t , for a certain stochastic volatility model where� = 0 and the volatility process is given by at. Thus,

@x(BS(t; x;MTt ; I

0t (x))

��x=x�t

= @xBS(t; x�t ;M

Tt ; I

0t (x

�t )) + @�BS(t; x

�t ;M

Tt ; I

0t (x

�t ))@I0t@x

(x�t ) : (8)

From Renault and Touzi (1996) we know that @I0t@x (x

�t ) = 0: Then, (6), (7) and

(8) imply that

@It@Xt

(x�t ) (9)

=1


�t ))

�@xBS(t; x

�t ;M

Tt ; I

0t (x

�t ))� @xBS(t; x�t ;MT

t ; It(x�t ))

+E(

Z T

t

e�r(s�t)@xF (s;Xs;MTs ; vs)dsjFt)

#��Xt=x�t

:

On the other hand, straightforward calculations lead us to

@xBS(t; x�t ;M

Tt ; �) = e

x�tN(1

2�pT � t)

and

BS(t; x�t ;MTt ; �) = e

x�t (N(1

2�pT � t)�N(�1

2�pT � t)):

Then@xBS(t; x

�t ;M

Tt ; �) =

1

2(ex

�t +BS(t; x�t ;M

Tt ; �));

which yields

10

@xBS(t; x�t ;M

Tt ; I

0t (x

�t ))� @xBS(t; x�t ;MT

t ; It(x�t ))

=1

2(BS(t; x�t ;M

Tt ; I

0t (x

�t ))�BS(t; x�t ;MT

t ; It(x�t )))

=1

2E(BS(t; x�t ;M

Tt ; vt)� VtjFt)

= �12E(

Z T

t

e�r(s�t)F (s;Xs;MTs ; vs)dsjFt)

��Xt=x�t

:

This, together with (9), implies that the result holds.

5 Short-time behaviour

The pourpose of this section is to study the limit of @It@Xt

(x�t ) as T # t: Thefollowing result is part of the tool needed for our results.

Lemma 6 Assume the model (1) is satis�ed. Then It(x�t )pT � t ! 0 a.s. as

T ! t:

Proof: Notice that the fact that K is a square-integrable and continuousrandom process and the dominated convergence theorem lead to get

VtjXt=x�t= E(e�r(T�t)(eXT �KT )+jFt)

��Xt=x�t

= E (e�r(T�t)(eXT�Xte�r(T�t)MTt �KT )+jFt)

��Xt=x�t

� E ((eXT�XtMTt �KT e

r(T�t))+jFt)��Xt=x�t

= E�((eXT�Xt � er(T�t))MT

t + er(T�t)(MT

t �KT ))+jFt)��Xt=x�t

� E (jeXT�Xt � er(T�t)jMTt jFt)

��Xt=x�t

+E (jMTt �KT jer(T�t)jFt)

��Xt=x�t

� MTt E (jeXT�Xt � er(T�t)jjFt)

��Xt=x�t

+E (jMTt �KT jer(T�t)jFt)

��Xt=x�t

! 0 a:s:;

as T ! t. Hence, taking into account that, in the at-the-money case, VtjXt=x�t=

BS(t; x�t ;MTt ; It(x

�t )); we deduce that

BS(t; x�t ;MTt ; It(x

�t )) = 2M

Tt e

�r(T�t)�N

�I(x�t )

pT � t2

�� 12

��! 0 a:s:;

11

and this allows us to complete the proof.Henceforth we consider the following hypotheses:

(H1�) a2 2 L2;2 and, moreover, there exists a positive constant C such that , forall 0 < s < � < r < T; ��DW

s ar��+ ��DW

� DWs ar

�� C:Notice that this hypotheses implies that (H1) holds.

(H2) There exist two positive constants c1; c2 such that for all r 2 [0; T ] c1 ��r � c2: Notice that, as for all � 2 (�1; 1), a2s � C (�)�2s for some positiveconstant C (�) ; this hypothesis implies that a2s is lower bounded.

(H3) The processm(T; �) 2 L1;2 and moreover, there exists a positive Ft�adaptedprocess Ct such that for all T > s > r > t;

E�jm(T; r)j2

��Ft�+ E ��DWs m(T; r)

��2��Ft� � Ct:Proposition 7 Assume that the model (1) and Hypotheses (H1�)-(H3) hold.Also assume that there is a constant c > 0 such that c < Kt; for all t 2 [0; T ].Then

@�BS(t; x�t ;M

Tt ; It(x

�t ))

@It@Xt

(x�t )

=1

2E

�

�@x �

1

2

��@3xxx � @2xx

�BS(t; x�t ;M

Tt ; vt)

Z T

t

�s�Ws ds

+@K

�@x �

1

2

��@2xx � @x

�BS(t; x�t ;M

Tt ; vt)

Z T

t

�Ws m(T; s)ds

��Ft!

+O(T � t):

as T ! t.

Proof: Proposition 5 gives us that

@�BS(t; x�t ;M

Tt ; It(x

�t ))

@It@Xt

(x�t )

=1

2E

�

Z T

t

e�r(s�t)(@x �1

2)�@3xxx � @2xx

�BS(s;Xs;M

Ts ; vs)�s�

Ws ds

+

Z T

t

e�r(s�t)(@x �1

2)@K

�@2xx � @x

�BS(s;Xs;M

Ts ; vs)

� �Ws m(T; s)ds��Ft��

Xt=x�t

=: T1 + T2:

Now the proof is decomposed into two steps.

12

Step 1. Here we see that

T1 =�

2E

L(t; x�t ;M

Tt ; vt)

Z T

t

�s�Ws dsjFt

!+O (T � t) ; (10)

where L(s;Xs;MTs ; vs) =

�@x � 1

2

� �@3xxx � @2xx

�BS(s;Xs;M

Ts ; vs): In fact, ap-

plying Itô formula to

�e�rsL(s;Xs;MTs ; vs)(

Z T

s

�r�Wr dr)

as in the proof of Theorem 2 and taking conditional expectations with respectto Ft; we obtain that

�

2E(

Z T

t

e�r(s�t)L(s;Xs;MTs ; vs)�s�

Ws dsjFt)

=�

2E

L(t;Xt;M

Tt ; vt)(

Z T

t

�s�Ws ds)jFt

!

+�2

4E(

Z T

t

e�r(s�t)(@3xxx � @2xx)L(s;Xs;MTs ; vs)�s�

Ws

� Z T

s

�r�Wr dr

!dsjFt)

+�

4E(

Z T

t

e�r(s�t)@K(@2xx � @x)L(s;Xs;MT

s ; vs) �Ws m(T; s)

� Z T

s

�Wr �rdr

!dsjFt)

+�2

2E(

Z T

t

e�r(s�t)@xL(s;Xs;MTs ; vs) �s

� Z T

s

(DWs �

Wr )�rdr

!dsjFt)j

+�

2E(

Z T

t

e�r(s�t)@KL(s;Xs;MTs ; vs) m(T; s)

� Z T

s

(DWs �

Wr )�rdr

!dsjFt)

=1

2E

L(t;Xt;M

Tt ; vt)(

Z T

t

��s�Ws ds)jFt

!+S1 + S2 + S3 + S4:

13

Using the proof of Lemma 1 and Hypotheses (H1�) and (H2), we can write

jS1j =

��24 E(Z T

t

e�r(s�t)E�(@3xxx � @2xx)L(s;Xs;MT

s ; vs)��Gt�

�(Z T

s

�Wr �rdr)�Ws �sdsjFt)

�� C

6Xk=4

E

24Z T

t

Z T

s

a2�d�

!� k2

j(Z T

s

�Wr �rdr)�Ws �sjdsjFt

35� C

6Xk=4

E

"Z T

t

(T � s)�k2 j(Z T

s

�Wr �rdr)�Ws �sjdsjFt

#Hence, using Hypotheses (H1�), (H2), and (H3), we can write

jS1j � C6X

k=4

(T � t)�k2+4 = O(T � t):

Similarly, we have

jS2j =

��4E Z T

t

e�r(s�t)E�@K(@

2xx � @x)L(s;Xs;MT

s ; vs)��Gt�

�(Z T

s

�Wr �rdr)�Ws m(T; s)dsjFt

!�� :Therefore, the relation

@2BS(t; x;K; �)

@x@K=1

k

�@BS(t; x;K; �)

@x� @

2BS(t; x;K; �)

@x2

�;

togheter with the hypotheses of the Proposition, implies

jS2j � Ct6X

k=3

E

Z T

t

(T � s)��2+3 jm(T; s)j

��Ft!= O(T � t):

In a similar way,

jS3j =�2

2E(

Z T

t

e�r(s�t)@xL(s;Xs;Ms; vs) �s

� Z T

s

(DWs �

Wr )�rdr

!dsjFt)jXt=x�t

� C4X

k=3

��E Z T

t

(T � s)��2 �s

Z T

s

(DWs �

Wr )�rdr

!ds

��Ft!��

� C4X

k=3

Z T

t

(T � s)��2+2

!= O(T � t):

14

Finally, the same arguments give us that

jS4j = O(T � t):

Step 2. In order to �nish the proof we only need to proceed as in Step 1.Here we see that

T2 =1

2E

P (t; x�t ;M

Tt ; vt)

Z T

t

�Ws m(T; s)dsjFt

!+O (T � t) ; (11)

where P (s;Xs;MTs ; vs) = (@x� 1

2 )@K�@2xx � @x

�BS(s;Xs;M

Ts ; vs): In fact, ap-

plying Itô formula to

e�rsP (s;Xs;MTs ; vs)(

Z T

s

m(T; r)�Wr dr)

as in the proof of Theorem 2 and taking conditional expectations with respectto Ft; we obtain that

1

2E

Z T

t

e�r(s�t)P (s;Xs;MTs ; vs)�

Ws m(T; s)ds

��Ft!

=1

2E

P (t;Xt;M

Tt ; vt)(

Z T

t

m(T; s)�Ws ds)jFt

!

+�

4E(

Z T

t

e�r(s�t)(@3xxx � @2xx)P (s;Xs;MTs ; vs)�s�

Ws

� Z T

s

m(T; r)�Wr dr

!dsjFt)

+1

4E(

Z T

t

e�r(s�t)@k(@2xx � @x)P (s;Xs;MT

s ; vs) �Ws m(T; s)

� Z T

s

�Wr m(T; r)dr

!dsjFt)j

+�

2E(

Z T

t

e�r(s�t)@xP (s;Xs;MTs ; vs) �s

� Z T

s

DWs

��Wr m(T; r)

�dr

!dsjFt)

+1

2E(

Z T

t

e�r(s�t)@KP (s;Xs;MTs ; vs) m(T; s)

� Z T

s

DWs

��Wr m(T; r)

�dr

!dsjFt):

Now, following the same arguments as in Step 1 the proof is complete.

15

Remark 8 This proof only needs some integrability and regularity conditions.So, depending on the coe¢ cients of the model (1) and the process K, Hypotheses(H1�)-(H3) can be substituted by appropiate integrability conditions.

Now we can state the main result of this paper. Toward this end, we needto state the following assumptions:

(H4) Assume that m(�; �) has continous paths and that, for each t 2 [0; T ] �xed,

supt<�^s^r<T

E

��sar �

�t~ata2�

��Ft�! 0 as T ! t; a.s.

and

supt<�^s^r<T

E

�m(T; s)ar �

m(t; t)

~ata2�

��Ft�! 0 as T ! t; a.s.

where, by convention,

~at := �2t � 2��t

m(t; t)

Kt+m2(t; t)

K2t

(H5) There exists a Ft -measurable random variable D+t at such that, for every

�xed t > 0;

supt<s<r<T

��E ��DWs ar �D+

t at��Ft��! 0; a.s.

as T ! t:

Theorem 9 Consider the model (1) and suppose that Hypotheses (H1�)-(H5)hold and there exists a positive constant c such that c < K: Then

limT!t

@It@Xt

(x�t ) =1

2

�m(t; t)

Kt� ��t

�D+t at~a2t

: (12)

Proof: We can write

@�BS(t; x�t ;M

Tt ; It(x

�t )) =

MTt e

�r(T�t)e�It(x�t )

2(T�t)8

pT � tp

2�;

�@x �

1

2

��@3xxx � @2xx

�BS(t; x�t ;M

Tt ; vt) = �MT

t e�r(T�t) 1p

2�e�

v2t (T�t)8 v�3t (T�t)� 3

2

and

@K

�@x �

1

2

��@2xx � @x

�BS(t; x�t ;M

Tt ; vt)

= MTt e

�r(T�t) 1p2�e�

v2t (T�t)8 v�1t (T � t)� 1

2

�1

MTt v

2t (T � t)

�:

16

Then we can write, due to Lemma 6 and Proposition 7,

@It@Xt

(x�t )

= ��2eIt(x

�t )2(T�t)8 (T � t)�2E(e�

v2t (T�t)8 v�3t

Z T

t

�s�Ws dsjFt)

+1

2eIt(x

�t )2(T�t)8 (T � t)�1E

e�

v2t (T�t)8 v�1t

�1

MTt v

2t (T � t)

�Z T

t

�Ws m(T; s)dsjFt

!+O(T � t) 12

= : S1 + S2 +O(T � t)12 :

By Lemma 6, we know that It(x�t )2(T � t)! 0 a.s. as T ! t: Then,

limT!t

S1 = ��

2limT!t

"(T � t)�2E(e�

v2t (T�t)8 v�3t

Z T

t

�s�Ws dsjFt)

#and

limT!t

S2 =1

2limT!t

�(T � t)�1E(e�

v2t (T�t)8 v�1t

�1

MTt v

2t (T � t)

��Z T

t

�Ws m(T; s)dsjFt)#: (13)

Now, let us see that

limT!t

�S1 +

��t2~a2t

D+t at

�= 0 a.s.: (14)

In fact, we can establish

limT!t

�S1 +

��t2~a2t

D+t at

�= lim

T!tE

�ATBT +

��t2~a2t

D+t at

��Ft�where

AT := � exp

��v

2t (T � t)8

�1

vt

and

BT := �1

v2t (T � t)2Z T

t

Z T

s

ar�sDWs ardrds:

Consequently

limT!t

E

�ATBT +

��t2~a2t

D+t at

��Ft�= lim

T!tE

��AT �

�

~at

�BT

��Ft�+ �

~atlimT!t

E

�BT +

�t2~at

D+t at

��Ft�= lim

T!tU1 +

�

~atlimT!t

U2:

17

Applying Schwartz inequality for conditional expectation, it follows that

U1 �"E

�AT �

�

~at

�2��Ft!# 1

2 �E�B2T��Ft�� 12 :

From the dominated convergence theorem and (H2), it is easy to see that

E

��AT � �

~at

�2��Ft� tends to zero a.s. as T ! t; and a simple calculation

gives us that (H1�) and (H2) imply that E�B2T��Ft� is bounded, from where we

deduce that limT!t U1 = 0:Observe that we also have,

jU2j =

�� 1

(T � t)2E Z T

t

Z T

s

��sarv2t

DWs ar �

�t~atD+t at

�drds

��Ft!��

� C

(T � t)2

��E Z T

t

Z T

s

��sarv2t

� �t~at

�DWs ardrds

��Ft!��

+C

(T � t)2

��E Z T

t

Z T

s

�DWs ar �D+

t at��Ft

!drds

��= : jU2;1j+ jU2;2j :

Using Hypotheses (H1�) and (H2) we obtain that

jU2;1j � C

(T � t)2E Z T

t

Z T

s

��sarv2t � �t~at

�� drds��Ft!

� C

(T � t)2E Z T

t

Z T

s

��sar � �t~at v2t�� drds

��Ft!

=C

(T � t)2E Z T

t

Z T

s

��sar � �t~at(T � t)

Z T

t

a2�d�

�� drds��Ft!

� C

(T � t)3Z T

t

Z T

s

Z T

t

E

��sar � �t~at a2��Ft� d�drds;

which tends to zero, a.s. as T ! t, because of Hypothesis (H4). Similarly,

jU2;2j �C

(T � t)2

��Z T

t

Z T

s

E��DWs ar �D+

t at��Ft� drds

�� ;which tends to zero by Hypothesis (H5). Thus we have proved (14) is true.On the other hand, by (13) we can write

limT!t

�S2 �

m(t; t)

2Kt~a2tD+at

�= lim

T!tE

�ATBT �

m(t; t)

2Kt~a2tD+at

��Ft�

18

but now

AT := exp

��v

2t (T � t)8

�1

MTt vt

and

BT :=1

v2t (T � t)2Z T

t

Z T

s

arm(T; s)DWs ardrds:

Finally, proceeding similarly as before, we have (13) yields that S2 converges tom(t;t)2Kt~a2t

D+at, which, together with (14), implies that (12) is satis�ed.

6 Application to the study of spread options

This section is devoted to apply the previous results to study the implied volatil-ity behaviour for spread options. This study allows us to predict when the Kirk�sapproximation formula for spread options may fail. Moreover, we will see howthis analysis gives us a tool to improve Kirk�s formula.

6.1 Short-time behaviour of the implied volatility for spreadoptions

Consider an spread option with KT = S0

T +K as in Example 3. For the sake ofsimplicity we will assume the interest rate r = 0. Then it is easy to see that

a2t := �2 � 2��0 S0t

S0t +K+ (�0)

2 (S0t)2

(S0t +K)2 :

We can easily check that, for � < t

DW� a

2t =

�2��0 K

(S0t +K)2 + 2 (�

0)2�

S0tS0t +K

�K

(S0t +K)2

!�0S0t

= 2 (�0)2�� + �0

�S0t

S0t +K

��S0tK

(S0t +K)2 :

Hence, we deduce that

DW� at = DW

�

qa2t =

DWr a

2t

2pa2t

=1pa2t

�� + �0

�S0t

S0t +K

��(�0)

2 S0tK

(S0t +K)2 :

19

Then, from Theorem 9, we get

limT!t

@It@Xt

(x�t )

=1

2

�m(t; t)

Kt� ��

�D+t at~a2t

=1

2

��0�

S0tS0t +K

��

�21�pa2t

�3 (�0)2 S0tK

(S0t +K)2 : (15)

Remark 10 Notice that the above quantity is allways positive. In the followingexamples we will study its behaviour as a function of K and �:

Example 11 In Figure 1 we plot limT!t@It@Xt

(x�t ) as a function of K for � = 0:9(solid) and � = 1 (dash), and with St = 100; � = 0:5 and �0 = 0:4: We canobserve the limit skew limT!t

@It@Xt

(x�t ) is zero in the case K = 0. This wasexpected from Example 3, where we found that in this case the implied volatilityis constant, and then @It

@Xt= 0: Notice also that, even this skew increases with

K, this increment seems to be clearly bigger in the completely correlated case� = 1:

0 1 2 3 4 50.00

0.01

0.02

K

implied volatiltiy skew

Figure 1: limT!t@It@Xt

(x�t ) as a function of K for � = 0:9 (solid)and � = 1 (dash). Here � = 0:5; �0 = 0:4:


(x�t ) as a function of � for K = 5(solid) and K = 10 (dash) and for the same parameter values as in Fig.1. Wecan observe the limit skew limT!t

@It@Xt

(x�t ) has its maximum in the completelycorrelated case � = 1: Notice that this means that the constant volatility approx-imation given by Kirk�s formula is expected to be less accurate in this case. Thisfact is consistent numerical empirical evidence (see for example Baeva (2011)and Borovkova (2007)).

20

1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0

0.01

0.02

0.03

0.04

0.05

0.06

ro

iimplied volatility skew


(x�t ) as a function of � for K = 5 (solid) and K = 10(dash). Here � = 0:5; �0 = 0:4:


(x�t ) as a function of � and K forthe same parameter values as in Fig.1 and Fig. 2. Notice that this limit skewis substantially bigger near the case � = 1:

0.5correlation

0.51.0

0.0

K 0 1.04

810

62

0.00

0.01

0.02

0.03z

0.04

0.05


(x�t ) as a function of � and K:

6.2 Applications to the study of the accuracy of Kirk�sformula

With the above notations, the Kirk approximation for an spread option can bewritten as:

BS(t;Xt;MTt ;qa2t ):

It is well-known that Kirk�s formula is a very accurate approximation for spreadoptions given its simplicity (see for example Baeva (2011), Bjerksund and Stens-land (2011) or Carmona and Durrleman (2011)). Nevertheless, it is well-knownit may fail for highly correlated assets (see for example Baeva (2011)). Theresults in the above sections give an analytical reason for this phenomenon.

21

In fact, notice thatpa2t (the volatility parameter in the Kirk�s formula) is a

process that does not depend on Xt nor on the time to maturiry T � t: Then,Kirk�s formula may not reproduce the short-time volatility skews that we haveseen appear in the highly correlated case (� close to 1) and we can expect it canfail when � is near one. In the following examples, we compare Kirk�s approxi-mation prices with the ones obtained with a Monte-Carlo simulation procedurewith 100,000 trials.We use these simulation results as the benchmark for thetrue spread option value

Example 14 In the following table we can compare the prices given by Kirk�sformula and by the Monte-Carlo simulations, for di¤erent values for K and�: Here Xt = ln(100); S0t = 100; r = 0; T � t = 0:5; � = 0:5 and �0 = 0:4.The di¤erence is given in % of the Monte-Carlo price. As expected from ouranalytical study, the errors increase strongly when the correlation � is close to1.

K=� 0:60 0:98 0:99 0:999

5Monte-Carlo

KirkError

9; 45649; 4176�0 ; 410%

2; 18902; 21591 ; 230%

1; 83861; 87752 ; 117%

1; 50111; 54202 ; 725%

10Monte-Carlo

KirkError

7; 64047; 5988�0 ; 545%

1; 27141; 33264 ; 814%

1; 02071; 10157 ; 913%

0; 79340; 884811 ; 516%

Notice that our study of the volatility skew gives not only a theoreticalunderstanding of it goodness-of-�t of Kirk�s approximation, but also gives ussome hints to improve it. In fact, from (15) and by using Taylor expansions wecan expect that for small times to maturities and for near-the-money options,the expression

It(Xt) :=qa2t +

1

2

��0�

S0tS0t +K

��

�21�pa2t

�3 (�0)2 S0tK

(S0t +K)2 (Xt � x

�t )

can be a reasonable approximation for the implied volatility It(Xt). In fact, letus consider the modi�ed Kirk approximation given by

BS(t;Xt;MTt ; It(Xt)):

In the following example we will check numerically the goodness-of-�t of thisapproximation.

Example 15 In the following table we can compare the prices given by themodi�ed Kirk�s formula and by the Monte-Carlo simulations, for di¤erent valuesfor K and � and for the same parameters as in Example 14. Notice that wehave reduced signi�catively the error of approximation with respect to the resultsin this last example, specially in the case of highly correlated assets.

22

K=� 0:60 0:98 0:99 0:999

5Monte-CarloModi�ed Kirk

Error

9; 45649; 4255�0 ; 327%

2; 18902; 20670 ; 809%

1; 83861; 83090 ; 804%

1; 50111; 48290 ; 414%

10Monte-CarloModi�ed Kirk

Error

7; 64047; 6060�0 ; 451%

1; 27141; 28881 ; 368%

1; 02071; 03671 ; 660%

0; 79340; 82101 ; 400%

7 Conclusions

We have used the Malliavin calculus techniques to �nd an expression for theshort-time implied volatility skew of options with random strikes. In particular,we have seen that this analytical study gives us a key to understand why someapproximation formulas may fail for some sets of parameters. As an application,we have seen that this skew is very pronounced for spread options in the highcorrelation case, wich explains why a constant volatility approximation as Kirk�sformula cannot be accurate in this case. Finally, our approach gives us somehints to improve this estimate, by introducing a skew in the approximation ofthe corresponding implied volatility. Our preliminar numerical analysis showsthat this can be a natural and e¢ cient way to improve Kirk�s formula. A precisedevelopement of this improvement is left for future research.

References

[1] C. Alexander and A. Venkatramanan: Closed form approximations forspread options. Applied Mathematical Finance 18 (5), 447-472 (2011)

[2] E. Alòs: A generalization of the Hull and White formula with applicationsto option pricing approximation. Finance Stoch. 10, 353-365 (2006).

[3] E. Alòs, J.A. León and J. Vives: On the short-time behavior of the im-plied volatility for jump-di¤usion models with stochastic volatility. FinanceStoch. 11, 571-589 (2007).

[4] E. Alòs and D. Nualart: An extension of Itô�s formula for anticipatingprocesses. J. Theor. Probab. 11, 493-515 (1998)

[5] T. Baeva: On the pricing and sensitivity of spread options on two correlatedassets. Preprint (2011).

[6] P. Bjerksund and G. Stensland: Closed form spread option valuation.Quantitative Finance iFirst, 1-10 (2011).

[7] S. Borovkova, F. J. Permana and H. v.d.Weide: A closed form approachto the valuation and hedging of basket and spread option. The Journal ofDerivatives 14 (4), 8-24 (2007).

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[8] R. Carmona andV. Durrleman: Pricing and hedging spread options. SIAMRev. 45 (4), 627-685 (2003).

[9] S. Deng, M. Li and J. Zhou: Closed-form approximations for spread optionprices and greeks. Preprint (2008)

[10] J.-P. Fouque, G. Papanicolaou, R. Sircar and K. Solna: Maturity cycles inimplied volatility. Finance Stoch. 8 (4), 451-477 (2004).

[11] J. Jacod and P. Protter: Risk-neutral compatibility with option prices.Finance Stoch. 14 (2), 285-315 (2010).

[12] E. Kirk: Correlation in the energy markets. In Managing Energy Price Risk(First Edition). London: Risk Publications and Enron, pp. 71-78.(1995)

[13] W. Margrabe: The value of an option to exchange one asset for another.The Journal of Finance 33 (1), 177-186 (1978).

[14] A. Medvedev and O. Scaillet: Approximation and calibration of short-termimplied volatilities under jump-di¤usion stochastic volatility. Rev. FinanceStud. 20 (2), 427-459 (2007).

[15] D. Nualart: The Malliavin Calculus and Related Topics. Second Edition.Springer-Verlag, Berlin (2006).

[16] E. Renault and N. Touzi: Option hedging and implied volatilities in astochastic volatility model. Math. Finance 6 (3) 279-302 (1996).

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