局部波动率模型推导

Substitute back into Equation (15) and re-arrange terms to obtain the …rst
identity
Z1
C
K @C
K ST f (S; T )dS = P (t; T )
: P (t; T ) @K
(16)
2.3.2 Second Identity
(St; t). We need the following preliminaries:
Discount factor P (t; T ) = exp
RT
t
rsds
:
Fokker-Planck equation. Denote by f (St; t) the probability density function of the underlying price St at time t. Then f satis…es the equation
(14)
Before evaluating these two integrals we need the following two identities.
3
2.3 Two Useful Identities
2.3.1 First Identity
From the call price Equation (8), we obtain
2.1 Required Derivatives
We need the following derivatives of the call C(St; t).
2
First derivative with respect to strike
@C
Z1 @
@K
=
P (t; T )
KZ
@K (ST
1
K)f (S; T )dS
K
Z1
T Sf (S; T )dS
K
We have assumed lim (S K)Sf (S; T ) = 0. Substitute the …rst identity (16)
S!1
to obtain the …rst integral I1
I1
=
TC P (t; T )
+
TK P (t; T )
@P @T
=
rT P (t; T ) so we can write (11)
@C
Z1
@
@T =
rT C + P (t; T ) (ST
K
K) @T [f (S; T )] dS:
(12)
2.2 Main Equation
In
Equation (12) substitute the Fokker-Planck
price at t = T it is understood that S = ST ; f (S; T ) = f (ST ; T ) and dS = dST : We sometimes omit the subscript for notational convenience.
2 Derivation of the General Dupire Equation (1)
(ST
1
K)1(ST >K)
= P (t; T ) (ST K)f (S; T )dS:
K
where 1(ST >K) is the Heaviside function and where E [ ] = E [ jFt]. In the all the integrals in this Note, since the expectations are taken for the underlying
We also present the derivation of local volatility from Black-Scholes implied
volatility, outlined in [1]. We will derive the following three equations that involve local volatility = (St; t) or local variance vL = 2:
@C
2=
(K; T )2
=
@T
1 2
K
2
@2C @K2
:
(5)
p The local volatility is then vL = 2(K; T ): In this Note the derivation of
these equations are all explained in detail.
S=K
= [0 0]
2S2f (S; T )
1 Local Volatility Model for the Underlying
The underlying St follows the process
dSt = tStdt + (St; t)StdWt
(6)
= (rt qt) Stdt + (St; t)StdWt:
We sometimes drop the subscript and write dS = Sdt + SdW where =
C
Z1
P (t; T )
=
(ST ZK1
K)f (S; T )dS Z1
(15)
=
ST f (S; T )dS K f (S; T )dS:
K
K
From
the
expression
for
@C @K
in
Equation
(9)
we
obtain
Z1
1 @C
f (S; T )dS =
:
K
P (t; T ) @K
S!1
First derivative with respect to maturity–use the chain rule
@C
@C
Z1
@T
=
P (t; T )
@T
Z
K 1
(ST
K)f (S; T )dS + @
(11)
P (t; T )
(ST
K
K) @T [f (S; T )] dS:
Note
that
Dupire local volatility is derived. In Equation (13) have two integrals to evaluate
Z1
@
I1 I2
= =
T
(ST
Z
K 1
K)@S @2
(STKK)@Fra bibliotek2[Sf (S; T )] dS; 2S2f (S; T ) dS:
2.4.1 First integral
Use integration by parts with u = ST
K; u0
=
1; v0
=
@ @S
[Sf (S;
T
)]
;
v
=
Sf (S; T )
I1 = [ T (ST = [0 0]
K)ST f (S; T )]SS==1 K Z1
T Sf (S; T )dS:
@2 @K
+
i:
K
@2 @K2
Solving for the local variance in Equation (1), we obtain
@C
2 = (K; T )2 = @T
(rT qT ) C
1 2
K
2
@2C @K2
K
@C @K
:
(4)
1
If we set the risk-free rate rT and the dividend yield qT each equal to zero, Equations (1) and (2) can each be solved to yield the same equation involving local volatility, namely
@f
@
1 @2
= @t
[ @S
Sf (S; t)] + 2 @S2
2S2f (S; t) :
(7)
Time-t price of European call with strike K, denoted C = C(St; K)
C = P (t; T )E (ST K)+
(8)
=
P (t; T )E Z
@C
= @T
K (rT
@C qT ) @K
qT C
+
1 K2E 2
2 T
jST
=
K
@2C @K2 :
(2)
3. Local volatility as a function of Black-Scholes implied volatility, = (K; T ) (appears in [1]) expressed here as the local variance vL
We
use
the
expression for
@2C @K2
in Equation (10) to obtain
the
second
identity
1 @2C
f (K; T ) = P (t; T ) @K2 :
(17)
2.4 Evaluating the Integrals
We can now evaluate the integrals I1 and I2 de…ned in Equation (14).