IMPLIED VOLATILITY FORMULAS FOR HESTON MODELS PATRICK S. HAGAN ∗ , ANDREW S. LESNIEWSKI† , AND DIANA E. WOODWARD ‡ Abstract. We combine singular perturbation techniques with an effective media argument to analyze the general Heston model, ˜ 12 (˜ ) ˜ 1 ˜ = ( ) Σ ˜ 2 ˜ = ( ) Λ ( ) − Σ ˜ + ( ) Σ ˜ 12 Σ ˜ 1 ˜ 2 = ( )
We first show that the marginal probability density ( ) satisfies an effective 1-d forward equation through 2 . We analyze this 1-d forward equation using an effective media approach. For any given expiry date , this analysis yields effective SABR parameters , , and shows that the marginal density of the SABR model at matches the marginal density of the Heston model at , again to within 2 . Thus, the implied volatility smile ( ) for the Heston model is given by the original SABR implied vol formula with these parameters to within 2 .
1. Introduction. There are two major groups of stochastic volatility models:1 Heston-type models [14]-[17] and SABR-type models [2]-[9]. It is often argued that volatility moments are better behaved for Heston-type models [13], while SABR-type models are more convenient because they possess explicit asymptotic formulas for the implied volatilities of European options [2], [11]. Here we show that the implied volatilities of European options for Heston-type models are given by the same asymptotic formulas that were originally derived for SABR models. Specifically, for any given expiry date , we derive effective SABR parameters , , such that the Heston model’s implied volatility smile at is the same as the volatility ¡smile ¢ of the SABR model with those parameters. This result is not exact, but it is accurate to within 2 , the same accuracy as the explicit SABR formulas themselves.[2] In this paper we work with the generalized Heston model (1.1a) (1.1b) (1.1c)
˜ 1 ˜ 12 (˜ ) ˜ = ( ) Σ h i ˜ = ( ) Λ ( ) − Σ ˜ + ( ) Σ ˜ 12 ˜ 2 Σ ˜ 2 = ( ) ˜ 1
in the limit ¿ 1. We believe all Heston models used in practice are special cases of this model. Like all ˜ ( ) are stochastic, and the variance is Heston models, both the forward ˜ ( ) and instantaneous variance Σ mean reverting. Both the normal Heston model ((˜ ) ≡ 1) and log normal Heston model ((˜ ) ≡ ˜ ) are ˜ special cases of this model and, unlike the original Heston model, we allow the forward ˜ and variance Σ to be correlated. We also allow the volatility factor ( ), mean reversion rate ( ), mean reversion level Λ ( ), volatility of variance ( ), and correlation ( ) to be time dependent. We are not assuming that these parameters are “slowly varying” in . Since vol surfaces usually have very steep smiles for short dated options ( ∼ 1 week), and much milder smiles for longer dated options ( ∼ 1 year), we allow full (1) time dependence believing that this flexibility is needed to fit real volatility surfaces [10]. In this analysis we are also allowing the mean reversion rate to be (1). 2. Results. ∗ Gorilla
Science;
[email protected] of Mathematics; Baruch College; One Baruch Way; New York, NY 10010 ‡ Gorilla Science,
[email protected] 1 A third, less studied group of models are the stochastic-local vol models [17], [18], which seek to combine the vol surface fitting ability of local vol models with the forward volatility features of stochastic vol models. † Dept
1
2.1. The effective 1-d forward equation. Our approach is similar to that used in [10]. We define n ³ ´ ³ ´¯ o ˜ ) − Σ ¯¯ ˜ (0) = Σ(0) ˜ (2.1a) ( Σ) ≡ (0 ; Σ) ≡ E ˜ ( ) − Σ( =
˜ ) = Σ, given that ˜ (0) = , Σ(0) ˜ to be the probability density that ˜ ( ) = , Σ( = . We also let Z ∞ (2.1b) ( ) ≡ (0 ; ) ≡ ( Σ)Σ 0
be the marginal (reduced) probability density. In appendix A singular perturbation techniques are used to analyze the generalized Heston model. There it is shown that the marginal density satisfies an effective 1-d forward equation (2.2a) (2.2b)
2
= 12 2 2 ( ) ˆ ( ) = ( − )
( )
¡ ¢ through 2 , where (2.2c)
¢ ¡£ ¤ 1 + 2 ( ) + 2 ˜ ( ) 2 2 ( ) as → 0
=
1
Z
0 ( 0 )
and where ˆ ( ) is the expected value of the variance, (2.3)
Z ¯ n o 0 0 ˜ ( )¯¯ Σ(0) ˜ ˆ ( ) ≡ Σ = = − 0 ( ) +
−
(1 )Λ (1 )
0
1
(0 )0
Here the coefficients are (2.4a)
( ) =
(2.4b)
˜( ) =
1 ( ) 2 ( )ˆ ( )
3 ( ) 2 2 ( )ˆ ( )
− 3( )
4 ( ) 2 ( ) + 2 2 ( ) ( )ˆ ( ) Γ0 = − 0 ( )
( ) = − [( )Γ0 + ˜ ( )] ( )
(2.4c) and the integrals are (2.5a)
( ) =
Z
2 (1 )ˆ (1 )1
(1 ) =
0
(2.5b)
1 ( ) =
Z
(2.5c)
2 ( ) =
2 (2 )−
1
− (1 ) (1 ) (1 ) ˆ (1 )
0
Z
Z
1
( 0 ) 0
1
(1 ) (1 ) (1 ) ˆ (1 ) (1 ) 1 0
2
2 1
( 0 ) 0
2
1
(2.5d)
3 ( ) =
Z
− 2 (1 )ˆ (1 ) (1 )
(1 ) (1 ) (1 ) ˆ (1 )
−
0
(2.5e)
4 ( ) =
Z
1
0
( 0 ) 0
1
1
( 0 ) 0
Z
(2 ) (2 ) (2 ) 2 1 1
Before continuing, note that the effective forward equation can be solved numerically using a finite difference scheme. See [3], [10], where it is shown that the appropriate boundary conditions are absorbing ones. Once ( ) has been obtained, European option prices can be found by integrating Z ∞ (2.6a) ( − ) ( ) ( ) = ( ) =
(2.6b)
Z
−∞
( − ) ( )
¡ ¢ This yields option prices accurate to within 2 , the same accuracy as the effective forward equation. This numerical approach is discussed briefly in Appendix C. Since the effective equation has only one spatial dimension ( ) rather than two ( Σ), it provides a very efficient method for finding option prices. In addition, European option prices obtained this way are arbitrage free. See Appendix C. 2.2. Effective media theory. Even with the reduction from two spatial dimensions to one, the chances of analytically solving the effective forward equation 2.2a-2.2c are poor due to the time dependence of its coefficients. A nearly identical problem arises from the dynamic SABR model [10], and in [1] an effective media approach has been developed to sidestep the issue of time-dependent coefficients. In [1] it is shown that for any expiry date , we can choose constant coefficients so that the solution of the effective forward equation with constant coefficients and the solution ¡ ¢of the effective forward equation with time-dependent coefficients match for all at to within 2 . Thus, European option prices ¡ ¢calculated from the effective forward equation with constant coefficients are also accurate to within 2 for this particular exercise date. Specifically, suppose that ( ) is the solution of the effective forward equation 2.2a-2.2c. In [1] it is shown that © ¡ ¢ª ¯ ( ) 1 + 2 (2.7) ( ) ≡ for all ¯ ( ) is the solution of2 where (2.8a) (2.8b)
¯ = 1 2 ∆2 2 ¯ 2 ¯ → ( − )
¤ £¡ ¢ ¯ 1 + 2¯ + 2 ¯ 2 2 ( )
for 0
as → 0
where is given by 2.2c as always, and the constant coefficients are given by Z 1 2 2 ( )ˆ ( ) (2.9a) ∆ ( ) = 0 (2.9b)
¯ ( ) = 2 2
Z
( ) ( ) 2 ( ) ˆ ( )
0
obtain this result, one needs to set ≡ 1 and replace 2 ( ) with 2 ( ) ˆ ( ) in [1]. Then ¯ = exp 2 with 1 + ¯ˆ + 2 ¯ˆ2 while still retaining our 2 accuracy can replace 1 + ¯ˆ ¯ + 2 ¯ˆ2 ¯ 2 To
3
1 2
¯ , and we 2
(2.9c)
(2.9d)
Z
3
¯ ( ) =
3
2 ( ) ˜( ) 2 ( ) ˆ ( )
0
18 + 3
Z
¯ ( ) = 1
Z
( ) ( ) ˆ ( ) 2
0
0
Z
0
(1 ) (1 ) 2 (1 ) ˆ (1 ) 1 − 3¯2
[ ( ) + ( ) ˜ ( ) − ( ) ¯] 2 ( ) ˆ ( )
with (2.9e)
( ) ≡
Z
(1 ) ˆ (1 ) 1 2
≡ ( ) =
0
Z
2 ( ) ˆ ( )
0
Note that if ( ), ˜ ( ), and ( ) were constant, then the constant coefficients would equal these constants: ¯ = , ¯ = ˜, and ¯ = . Thus, the constant coefficients represent some sort of average of the coefficients over 0 , so it is unsurprising that the effective constant coefficients are different for different . We can specialize this result to the Heston model by using eqs. 2.4a-2.4c to replace ( ), ˜( ), and ( ) by their values from the Heston model. After some work, this shows that for expiry date , the constant coefficients are Z
(2.10a)
¯ ( ) = 1 2
(2.10b)
3 ¯ ( ) = 3 4
1 ( ) 2 ( ) =
0
Z
2 ( )ˆ ( ) 2 ( ; )
0
3
Z
2 ( ) 4 ( ) − 3¯2 ( ) 0 ¤ £ ¯ ( ) = − 1 Γ0¯ ( ) + ¯ ( ) ( ) Γ0 = − 0 ( ) 2 +
(2.10c) where (2.10d)
Z
( ) =
3
2 (1 )ˆ (1 )1
( ) =
0
(2.10e)
1 ( ) =
Z
(2.10g)
2 ( ) =
(1 ) (1 ) (1 ) ˆ (1 )
4 ( ) =
Z
Z
2 (1 ) −
−
0
(2.10f)
2 ( ) 2 ( )
2 (1 ) 1 ( ) = 0
Z
Z
1
( 0 ) 0
1
( 0 ) 0
1
1
(1 ) (1 ) (1 ) ˆ (1 ) (1 ) 1
0
− (1 ) (1 ) (1 ) ˆ (1 )
0
4
1
( 0 ) 0
Z
1
(2 ) (2 ) (2 ) 2 1
2.3. Effective SABR parameters and implied vols. The marginal density for the standard, constant coefficient ¡SABR ¢ model satisfies an effective 1-d forward equation which is essentially identical to 2.8a, 2.8b through 2 . See[5], [10], [3], [12], among others. Supppose that we can choose ¡ ¢SABR parameters so that the coefficients of the SABR model’s effective forward equation are within 2 of the coefficients of the Heston model’s effective forward equation 2.8a, 2.8b. Then the European option prices under¡ the ¢ Heston model must equal the European option prices under the this SABR model, at least to within 2 . Consequently, the implied volatilities of European options under the model must be given by the ¡ 2Heston ¢ explicit asymptotic formulas for the SABR model, again to within . Note that the explicit asymptotic ¡ ¢ formulas for the SABR model are only accruate to within 2 . So consider the standard SABR model ˜ ˜ ) ˜ 1 ˜ = ( ˜ ˜ 2 ˜ = ˜ ˜ 1 2 =
(2.11a) (2.11b) (2.11c)
where ˜ (0) = . In [5] and [10] it is shown that the marginal probability density ( ) for this model is the solution of £¡ ¢ ¤ 1 2 2 (2.12a) ¯ + 2 2 2 ¯ 2 2 ( ) 1 + 2 ¯ = 2
(2.12b)
→ ( − )
¡ ¢ through 2 , where
as → 0+ 1 2
¯ = − 4
(2.13)
with Γ0 = − 0 ( ). As always, (2.14)
1
=
Z
Γ0
0 ( 0 )
Suppose we choose (2.15)
1 2
= ∆− 4
¯( )∆2
√ = ¯ ¯
√ = ∆ ¯
Then ¡ ¢the coefficients of the SABR effective forward equation 2.12a match the coefficients of 2.8a through 2 . Consequently, European option prices and implied vols under the Heston model 1.1a-1.1c are ¡ 2 ¢identical to the European options prices and implied vols under the SABR model, at least through . Many researchers [2]-[11] have used asymptotic methods to analyze European options under the SABR model. These analyses show that the implied normal volatitities are given by3 (2.16a)
ª © − · · 1 + 2 + · · · 0 () 0) (
( ) = ˜ Z
3 In some markets, prices are quoted in terms of implied Black-Scholes (log normal) vols . Since can be obtained from the normal vol via 2 log 1 = 1 + 24 + ··· −
we consider only normal vols, wolog. 5
¡ ¢ through 2 , where (2.16b)
= ˜
(2.16c)
=
(2.16d)
˜=
Z
0 ( 0 )
() = log
2 2−32 2 + ¯24 {2( ) 00 24 0 1 2 4 ( )
+ +
q 1 + 2 + 2
1 +
( ) − 0 ( ) 0 ( )}
Therefore, the implied normal volatilities ¡ ¢ for European options under the Heston model are also given by eqs. 2.16a-2.16d, at least through 2 . variations of the implied vol formula exist in the literature. Although they agree to within ¡ Several ¢ 2 , different implied vol formulas have different tail properties. In particular, in [11] a more complicated implied volatility formula has been derived, which may be more accurate under extreme conditions. 3. Conclusions. In this paper we used asympotic methods to derive an effective 1-d forward equation 2.2a-2.5e for the Heston model. This equation permits ¡very ¢ efficient numerical pricing of European options. Even though these prices are only accurate through 2 , they are guaranteed to be arbitage free. We also derived effective SABR parameters for the general Heston model. From this result we conclude that using Heston models in lieu of SABR models (or SABR models in lieu of Heston models)¡ doesn’t ¢ substantially increase the gamut of smiles available, as all the Heston model smiles are within 2 of a SABR model smile. The coefficients in the effective 1-d forward equation, and the effective SABR parameters are necessarily complicated, since we started with a very general Heston model. Often in practice we only need special cases of the Heston model, cases in which the coefficients simplify. For example, since Λ ( ) represents the equilibrium variance level, if Λ ( ) Σ (0) ≡ then the forward volatility curve is expected to drift upward, ˜ ( ) should trend and if Λ ( ) the volatility curve should drift downward. Without a view on whether Σ ˜ (0) ≡ for all . It then higher or lower, it is natural to use a constant value for Λ ( ) with Λ ( ) ≡ Σ follows that ˆ ( ) ≡
(3.1)
for all ,
The effective forward equation and effective SABR parameters for this case are derived in Appendix B.1. In section B.2 we assume that ˆ ( ) ≡ as above, and we also assume that is constant and that ( ), ( ), and ( ) are all piecewise constant: (3.2a)
( ) =
( ) =
( ) =
for −1
where (3.2b)
0 = 0 1 · · ·
The formulas for this case are derived in Appendix B.2. This is an important case in practice: it allows us to use to control the long term volatility, and the parameters , , can be used to calibrate the model to volatility smiles at a discrete set of exercise dates 1 , 2 , , corresponding to the liquid expiry dates. Specifically, suppose that and the backbone function ( ) have been set, and suppose that , , have been determined for = 1 2 − 1. Also suppose that we calibrate the market’s smile ( ) to the standard SABR model to determine , , . Since , , are known for = 1 2 − 1, matching the Heston’s model’s effective SABR parameters to their market values, p p 2 1 2 (3.3) = ∆ − 4 ¯( )∆ = ¯ ¯ ( ) = ∆ ¯ ( ) 6
yields three equations for the three remaining unknowns , , . By bootstrapping = 1, 2, . . . , we can calibrate the piecewise constant Heston model to the entire volatility surface. Finally, the innate character of the Heston model can be seen by considering the case in which all the parameters are constant. So assume that ( ), ( ), ( ), ( ), and Λ( ) are constant, and that Λ( ) = , so that ˆ ( ) ≡ for all . Without loss of generality, we can re-scale so that ˆ ≡ 1. This case is considered in Appendix B.3, where it is found that the effective forward equation simplifies to ¢ ¡£ ¤ 2 = 12 2 2 ( ) ( ) 1 + 2 ( ) + 2 ˜ ( ) 2 2 ( ) (3.4a) (3.4b) = ( − ) as → 0 ¡ 2¢ through , where Z 1 0 (3.4c) = ( 0 ) and where the coefficients are (3.5a)
(3.5b)
(3.5c)
( ) =
1 − − 2
˜( ) =
¡ ¢2 3 2 2 − 1 + − 1 − − 2 1 − − − 2 2 2 4 2 2 2 2 2 2 1 − (1 + ) − + 2 2 2
( ) = − [( )Γ0 + ˜ ( )] 2
Γ0 ≡ −
0 (0) = 0 ( ) (0)
Even though mean reversion does not change the expected value of the instantaneous variance, ˆ ( ), it systematically reduces ( ) and ˜ ( ), and thus the smile, exhibited by the Heston model. In fact, these coefficients reduce to 1 (3.6a) ( ) = + t.s.t.’s 2 (3.6b)
˜( ) =
2 1 − 62 3 2 2 1 + + t.s.t.’s 2 42 2 2 2 3 3
when À 1, where “t.s.t.’s” stands for “transcendentally small terms.” Since ( ) decays like 1 and ˜( ) decays like 12 2 , we conclude that the Heston model’s mean reversion causes an innate (but slow) relaxation of the skew and smile. This is confirmed by the effective SABR parameters for this case. These are also derived in Appendix B.3, where it is shown that √ √ 2 1 2 (3.7) = − 4 ¯( ) = ¯ ¯ = ¯ where (3.8a) (3.8b)
− ¯ ( ) = − 1 + 2 2 ¡ ¡ ¢2 ¢2 2 − − 1 − − 2 1 + 2 − 2 − − 2 2 2 +3 2 ¯ ( ) = 3 2 3 4 83 4
7
For long expiries, (3.9)
2 = √ {1 − · · · } , 3
√ 3 = {1 − · · · } 2
for À 1
Since → 0 for long expiries, the Heston model’s innate smile flattens for long dated options. Appendix A. Derivation of the effective forward equation. A.1. Derivation for ( ) ≡ 1. We first consider the generalized Heston model for the special case of ( ) ≡ 1. With ( ) ≡ 1, the model is (A.1a) (A.1b) (A.1c)
˜ 12 (˜ ) ˜ 1 ˜ = Σ h i ˜ = ( ) Λ ( ) − Σ ˜ + ( ) Σ ˜ 12 ˜ 2 Σ ˜ 2 = ( ) ˜ 1
We analyze this model in the limit ¿ 1. In this analysis, we do not assume that the parameters ( ), Λ( ), ( ), ( ) are “slowly varying” in . Instead we allow full (1) time dependence so that we can fit the model to real volatility surfaces. After obtaining the effective forward equation for this model, we use the transformation Z (A.2) = 2 ( 0 ) 0 0
to obtain the effective forward equation for the Heston model with non-constant ( ). A.1.1. Expected variance. Before analyzing equations A.1a-A.1c, we define the expected value of the instantaneous variance, ¯ n o ˜ ( )¯¯ Σ() ˜ (A.3) ( ; ) ≡ Σ =
Taking the expected value of A.1b yields (A.4a)
= ( ) [Λ ( ) − ]
for
so (A.4b)
( ; ) = −
( 0 ) 0
+
Z
−
(1 )Λ (1 )
1
( 0 ) 0
1
Consequently, (A.4c)
0 0 = − () [Λ () − ] − ( )
for
˜ ˜ ( ) to be : We also define = (; ) as the initial value of Σ() needed for the expected value of Σ (A.5a)
= ( ; ) if and only if = (; )
From A.4b then, (A.5b)
(; ) =
( 0 ) 0
−
Z
1
(1 )Λ (1 )
8
( 0 ) 0
1
More generally, (A.6)
if = ( ; ) then ( ; ) = ( ; ) for all n o ˜ ( ) goes from ( ) to ( ), then ( ( ; )) = ( ( ; )) is also on the I.e., if the path of Σ path for all between and . Note that = () [Λ () − ] 0 0 = −( ) [Λ ( ) − ] ( )
(A.7a) (A.7b) for , as is needed later.
A.1.2. Conservation law. Let
n ³ ´ ³ ´¯ o ˜ ) − ¯¯ ˜ () = Σ() ˜ ( ; ) = E ˜ ( ) − Σ( =
(A.8)
˜ ( ) = , given that ˜ () = and Σ ˜ () = . The forward be the probability density at ˜ ( ) = , Σ Kolmogorov equation for the Heston model A.1a-A.1c is [19] ¤ £ (A.9a) = − [(Λ − ) ] + 12 2 2 ( ) + 2 [ ( )] + 12 2 2 [ ] for , with (A.9b)
= ( − )( − )
Let us define the volatility moments ()
(A.10)
( ; ) ≡
Z
∞
as →
( ; )
0
Integrating the forward Kolmogorov equation over all yields the conservation equation h i (0) = 12 2 2 ( )(1) (A.11a) for
(0)
(A.11b)
(1)
(1)
= ( − )
as →
where is ( ; ) In the following analysis, singular perturbation techniques will be used to analyze the backward equations for the moments () ( ; ). This analysis will show that for any 0, £ ¤ 2 (A.12) (1) (0 ; ) = 1 + 2 + 2 ˜ 2 (0) (0 ; ) through (2 ), and will determine the coefficients ( ), ( ), ˜ ( ), and ( ). Substituting this into the conservation law A.11a, A.11b at = 0 then yields the effective forward equation (A.13a) (A.13b)
2
= 12 2 = ( − )
¤ £¡ ¢ 1 + 2 + 2 ˜ 2 2 ( ) as → 0
for 0
where ( ) is today’s current marginal density, (A.14)
( ) = (0) (0 ; ) =
Z
0
∞
(0 ; )
¡ 2¢ (1) (0) Since the relation between and is accurate through , the effective forward equation is also ¡ 2¢ accurate through . 9
A.1.3. Changes of variables. Let us now choose an arbitrary, but specific, . For this , the backward Kolmogorov equation for () ( ; ) is [20] (A.15a)
()
−
()
()
2 2 () 1 2 1 2 2 = (Λ − ) () + 2 ( ) + ( ) + 2
for , subject to () ( ; ) → ( − )
(A.15b)
as →
We start by using a series of tranformations to make the backwards equations for (1) and (0) as similar as possible. Define (A.16)
≡ ( ; )
= (; )
and switch variables from to . Then (A.17a) (A.17b)
0 0 → + ( ; ) = − () [Λ() − (; )] − ( ) 0 0 → ( ; ) = − ( )
so the backwards equation becomes (A.18a) (A.18b)
()
−
n o () () () = 12 2 (; ) 2 ( ) + 2˜ ( ) + ˜ 2
() ( ; ) → ( − )
as →
since = (; ) → as → . Here we have used ˜ () ≡ ˜ (; ) ≡ ()−
(A.19)
for
( 0 ) 0
Next we switch variables from to , where (A.20a)
=
1
Z
0 ( 0 )
For clarity, we also introduce (A.20b)
() ≡ ( )
Γ() ≡
0 () = − 0 ( ) ()
Then (A.21a) (A.21b)
1 −1 2 −→ 2 2 −→ () 2 () 1 () ( − ) = (0)
½
2 − Γ() 2
¾
so the backward equation becomes (A.22a)
()
−
= (; )
n
1 () 2
()
()
− 12 Γ()() + 12 2 ˜ 2 − ˜ 10
o
for , with () ( ; ) →
(A.22b)
() (0)
as →
˜ () ( ; ) by We define () ( ; ) =
(A.23)
˜ () ( ; ) (0)
˜ () ( ; ) satisfies Then (A.24a)
˜ () = (; ) −
½
1 ˜ () 2
˜ () 1 2 ˜ 2 ˜ () ˜ () − ˜ − 12 Γ () + 2 ( − 1) 2 2
˜ ˜ () 1 2 2 ˜ () ˜ () + 2 −˜ ˜ + 2
for , with ˜ () ( ; ) → ()
(A.24b)
as →
Finally, we define the function Z Z (A.25a) (; ) ≡ (1 ; )1 = ( ) −
(1 ) Λ (1 ) ( 1 ) 1
where (A.25b)
(; ) =
Z
1
( 0 ) 0
1
and change variables from to , where (A.26a)
≡ (; )
We also define ≡ ˜(; ) as the inverse of (; ): = ˜(; ) ⇔ = (; )
(A.26b) Then (A.27a) (A.27b) (A.27c)
−→ (˜; ) = −(˜; ) −→ + (˜; ) = + (˜; ) 2 2 2 2 2 −→ + 2 (˜; ) + (˜; ) 2 2 2
In terms of the backwards equation becomes (A.28a)
³ ´ () () () () 1 ˜ () = 1 ˜ () ˜ ˜ ˜ ˜ − Γ () − ˜ + ˜ 2 2 + 12 2 2 2 ˜ ˜ () 1 2 ˜ 2 ˜ () ˜ 2 ˜ () + 2 + 2 ( − 1) 2 2 ˜ ˜ () 1 2 2 ˜ () ˜ () +2 + 2 ˜ + 2 ˜ 2
−
11
¾
for 0, with ˜ () ( ; ) → ()
(A.28b)
as → 0
˜ () = ˜ () + ˜ () + 2 ˜ () + · · · , we would find that If we were to expand 0 1 2 (A.29a)
˜ () = 0
(A.29b)
˜ () 0
1 ˜ () 2 0
→ ()
for 0 as → 0
so to leading order ˜ () = √ 1 −2 2 0 2
(A.30)
˜ () , 2 ˜ () , and 2 ˜ () are all (3 ) ˜ () is independent of to leading order, so the terms 2 We see that 2 or smaller. As we are only working through ( ), we neglect these terms, obtaining ³ ´ () () () () ˜ () = 1 ˜ () ˜ ˜ ˜ ˜ (A.31a) − 1 Γ () − ˜ + ˜ 2 2 + 1 2 2
2
2
2
− (A.31b)
˜ ˜ () 1 2 ˜ ˜ () ˜ 2 ˜ () + 2 + 2 ( − 1) 2
˜ () ( ; ) → ()
as → 0
Here, , ˜ , and are short for ≡ (; ) ≡ (˜(; )) ˜ ≡ ˜ (; ) ≡ ˜ (˜(; ) ; ) ≡ (; ) ≡ (˜(; ) ; )
(A.32a) (A.32b) (A.32c)
A.1.4. Equating = 0 and = 1. The equations for = 1 and = 0 are ³ ´ (0) (0) (0) 1 ˜ (0) 1 ˜ (0) ˜ ˜ ˜ ˜ (0) (A.33a) = − Γ () − ˜ + ˜ 2 2 + 1 2 2
(A.33b)
2
2
³ ´ 1 ˜ (1) 1 ˜ (1) ˜ (1) + ˜ (1) ˜ (1) ˜ (1) ˜ 2 2 + 12 2 = 2 − 2 Γ () − ˜ −
˜ ˜ (1) ˜ 2 ˜ (1) + 2
for 0, with (A.33c)
˜ (0) ( ; ) → ()
˜ (2) ( ; ) → ()
as → 0
We now set (A.34)
˜ (1) ( ; ) = ( ; )2+2 2 +2
and seek to find () ≡ (; ) () ≡ (; ) and () ≡ (; ) so that the equation for is ˜ (0) , at least through (2 ). This will allow us to conclude that ˜ (0) ( ; ) ≡ identical to the equation for 12
˜ (2) and ˜ (0) at = 0 ( ; ) through (2 ). Evaluating at = 0 will then yield the relation between for this particular . Since is arbitrary, this then yields the relationship needed to obtain the effective forward equation for all . We have ¡ © ¢ ª 2+2 2 +2 ˜ (1) (A.35a) + 2 + 2 2 + 2 = (A.35b) (A.35c) (A.35d) (A.35e)
ª © 2+2 2 +2 ˜ (1) + 2 + (2 ) =
¡ ¢ ¢ ª © ¡ 2+2 2 +2 ˜ (1) + 4 + 2 + 2 42 + 2 + (3 ) = ª © 2+2 2 +2 ˜ (1) + 2 + 2 ( + ) + (2 ) =
ª © ˜ (1) = 2+2 2 +2 + 2 + 2 ( + ) + (2 )
We now substitute eqs. A.35a-A.35e into the backward equation A.33b, and discard terms which are smaller than (2 ). In doing this, we note that to leading order 2 1 =√ − 2 2
(A.36a)
Therefore, is independent of to leading order, so all derivatives are () or smaller. Also (A.36b) (A.36c)
+ = (), µ ¶ 1 2 − 1 = (). − 2
We obtain (A.37a)
( + ) + 12 2 ˜ 2 2 = 12 − 12 Γ − ˜ ½ ¾ ˜ 2 − + − 2 ½ ¾ ˜ ˜ 2 2 2 − + 2 − 2˜ ( + ) + − 2 ½ ¾ ˜ ˜ ˜2 2 + − + + 2 − 2 − Γ0 − 2˜ ( + ) + − 2 2 µ ¶ ³ ˜ ´ + 2 − +
through (2 ), with the initial condition (A.37b)
→ ()
as → 0
¡ ¢ , since it is no larger than¡ ¢ 3 . We have also set Γ0 ≡ Γ (0) and Here we have dropped at −22 ˜ replaced 2 Γ () with 2 Γ0 without losing our through 2 accuracy. 13
¡ ¢ Resolution through (). We resolve the + term through (2 ). This will enable us to rewrite it as a linear combination of 2 2 and 2 terms, which can be eliminated by our choice of and . We first choose (A.38a)
+ =
˜ 2
Thus, (A.38b)
() =
1 1 (; ) 2
where (A.38c)
1 (; ) =
Z
(1 )˜ (1 ) 1 =
0
Z
˜(; )
(1 )˜ (1 ; ) (1 ; )1
With this choice, equation A.37a reduces to − 12 = − 12 Γ0 − ˜ for 0 → () as → 0 ¡ ¢ through (). We have neglected the ˜ and + terms, since these are actually (2 ), and have replaced Γ () with the constant Γ0 ≡ Γ(0). Expanding (A.39a) (A.39b)
= 0 + 1 + · · ·
(A.40) yields
0 =0 0 − 12
(A.41a)
for 0
0
→ ()
(A.41b)
as → 0
The solution is 2 1 0 = √ − 2 2
(A.42) so (A.43a)
0
= − 0
0
=
µ
2 1 − 2
¶
0
0
µ 3 ¶ = − 3 + 3 2 0
and (A.43b)
µ
+
¶
1 0 = − 0
µ
+
¶
0 =−
3 2
µ
¶ 2 − 1 0
At () we have (A.44a) (A.44b)
1 0 0 = − 12 Γ0 0 − ˜ = − 12 Γ0 0 − 12 ˜ 1 − 12
1 → 0
as → 0
Solving yields (A.45)
0 1 = − 12 Γ0 0 − 12 2 ()
14
where (A.46)
2 ( ) =
Z
(1 )˜ (1 ) (˜(1 ; ) ; )1
0
= =
Z
˜(; ) Z ˜(; )
(1 )˜ (1 ; ) (1 ; ) (1 ; )1 (1 ) (1 ) (1 ; )
Z
1
−
2 1
( 0 ) 0
2 1
Therefore (A.47)
³ ³ ´ ´ + = 2 1 + 1 + · · · µ ¶ 32 () 1 32 () 2 2 0 2 0 − − Γ = 2 0 + ··· 22 22
Identifying with (0) . We substitute equation A.47 into equation A.37a, and use 2 − = ˜
(A.48) to simplify the results. This yields
( + ) + 12 2 ˜ 2 2 = 12 − 12 Γ () − ˜ ½ ¾ 2 ˜ ˜ 2 2 2 − + 2 + 3 − 2˜ ( + ) + − 2 ½ ˜ ˜ + − + + 22 − Γ0 − 2 2 ¾ ˜ 2 ˜2 ( + ) + +3 − 2˜ − 2 2 ¡ ¢ through (2 ). We choose and to set the coefficients of 2 2 and 2 to zero. So including A.38a, we have chosen
(A.49)
(A.50a) (A.50b) (A.50c)
2 () = ˜ ¡ 2 ¢ ˜2 = −3 (2 ) + 2˜ () + + 2˜ 2 ˜ ˜ = − () + 22 − Γ0 − 2 2
With these choices, the equation for is (A.51a) (A.51b)
( + ) + 12 2 ˜ 2 2 = 12 − 12 Γ () − ˜ → () as → 0
¡ ¢ ˜ (0) . Therefore ≡ ˜ (0) , at least through 2 , so through (2 ). This is identical to equation A.33a for (A.52)
¡ £ ¤ ¢ (0) ˜ (0) = 2 1 + 2 + 2 + 22 2 ˜ ˜ (1) = 2+2 2 +2 15
¡ ¢ through 2 . Thus, from A.23 we have
¡ ¢ 2 (1) ( ; ) = 1 + 2 + 2 ˜ 2 (0) ( ; )
(A.53)
¡ ¢ through 2 , with
for all 0
˜ = + 22
(A.54)
where (A.55a) (A.55b) (A.55c)
() = ˜ 2 ¡ 2 ¢ ¢ ¡ ˜2 ˜ = [ () + ] + 2 2 2 − 3 (2 ) + 2˜ 2 = − (˜ ) − Γ0 ()
A.1.5. Coefficients. We now solve A.55a - A.55c to obtain (; ), ˜(; ), and (; ). As noted earlier, the solution of A.55a is (A.56a)
(; ) =
1 1 (; ) 2
where (A.56b)
1 (; ) =
Z
−
˜(; )
(1 ) (1 ) (1 ; )
1
( 0 ) 0
1
Here we replaced ˜ (; ) by ()−
(A.57)
( 0 ) 0
so that the coefficients are expressed in terms of the original variables. To solve A.55b for ˜, we need to re-express () + . From A.56a, A.56b we have ¡ ¢ ¡ ¢ 1 (˜; ) (˜)˜ ˜; () + = (A.58) ˜; − 2 2 2 ¡ ¢ 1 ˜ ˜ (; ) − ()˜ ˜; (˜; ) 2 Z 1 + (1 ) (1 ) 1 2 ˜(; ) where
˜(; ) is the derivative with respect to at constant . But differentiating = (˜(; ) ; )
(A.59) with respect to at constant yields (A.60) Using (A.61)
¡ ¢ ¡ ¢ ˜(; ) 0 = ˜; + ˜; ¡ ¢ ¡ ¢ ˜; = ˜;
(; ) =
Z
16
(1 ; )1
this becomes ¡ ¢ ˜ (; ) ˜; = (˜; )
(A.62)
Consequently, the first and third terms on the right hand side of A.58 cancel out. Hence Z 1 (˜; ) 1 (A.63) () + = − + (1 ) (1 ) 1 2 2 2 ˜(; ) so (A.64)
Z
˜ 2˜ [ () + ] = −2˜ +
(1 ) (1 ) 1
˜(; ) Z
¢ ¡ ˜ = −2 2 2 +
(1 ) (1 ) 1
˜(; )
where we are abbreviating (; ) as () for brevity. The equation for ˜ now becomes (A.65) Solving yields (A.66a)
¡ 2 ¢ ˜2 ˜ ˜ = − 3 (2 ) + 2
Z
˜(; )
(1 ) (1 ) 1
3 1 1 3 (; ) − 2 (; )2 (; ) + 2 4 (; ) 22
˜(; ) =
where (A.66b)
(A.66c)
3 (; ) =
4 (; ) =
Z
Z
−
˜(;)
2 (1 )(1 )
1
−
˜(; )
(1 ) (1 ) (1 )
( 0 ) 0
Z
−
1
1
( 0 ) 0
Z
2 1
( 0 ) 0
2 1
(2 ) (2 ) 2 1
1
Lastly, (A.67)
(; ) (; ) = −(; )Γ0 − ˜
A.1.6. The effective forward equation. Let us now switch from back to the initial condition . In terms of we have £ ¤ 2 (A.68a) (1) ( ; ) = 1 + 2 + 2 ˜ 2 (0) ( ; ) where the coefficients are given by (A.68b) (A.68c) (A.68d)
1 1 ( ; ) 2 1 3 1 ˜( ; ) = 2 3 ( ; ) − 2 2 ( ; ) + 2 4 ( ; ) 2 ( ; ) = −Γ0 − ˜ ( ; ) =
17
Here, (A.69a)
(A.69b)
≡ ( ; ) = ( ; ) =
Z
Z
(1 ; )1 ≡
(1 ; )1
To obtain the effective forward equation, we need the relationship between (1) and (0) only for = 0. Let (A.70a)
ˆ ( ) ≡ ( ; 0 ) = −
0
( 0 ) 0
+
Z
−
(1 )Λ(1 ) 0
1
( 0 ) 0
1
and (A.70b)
( ) ≡ ( ; 0 ) =
Z
ˆ (1 ) 1 = 0
Z
(1 ; 0 ) 1
0
Then (A.71a) where now (A.71b)
£ ¤ 2 (1) (0 ; ) = ˆ ( ) ( ) 1 + 2( ) + 2 ˜ ( ) 2 (0) (0 ; ) ( ) ≡ ( ; 0 )
˜( ) ≡ ˜( ; 0 )
( ) ≡ ( ; 0 )
Substituting this into the conservation law A.11a yields our effective forward equation. Since we are denoting (0) (0 ; ) as ( ), the effective forward equation is (A.72a) (A.72b)
2
= 12 2 ˆ ( ) = ( − )
( )
¡£
¢ ¤ 1 + 2 ( ) + 2 ˜ ( ) 2 2 ( ) as → 0
where (A.73a)
( ) =
(A.73b)
˜( ) =
1 ( ) 2 ( )ˆ ( ) 3 ( ) 2 2 ( )ˆ ( )
− 3( )
4 ( ) 2 ( ) + 2 ( ) 2 ( )ˆ ( )
( ) = − [( )Γ0 + ˜ ( )] ( )
(A.73c) Here, (A.74a)
(A.74b)
1 ( ) =
Z
2 ( ) =
−
(1 ) (1 ) ˆ (1 ) 0
Z
1
( 0 ) 0
1
(1 ) (1 ) ˆ (1 ) (1 ) 1
0
18
for 0
(A.74c)
3 ( ) =
Z
− 2 (1 )ˆ (1 )
0
(A.74d)
4 ( ) =
Z
( 0 ) 0
1
− (1 ) (1 ) ˆ (1 )
0
1
(1 ) 1
( 0 ) 0
Z
(2 ) (2 ) 2 1 1
where (A.74e)
Z
( ) =
ˆ (1 ) 1 ,
(1 ) =
0
Z
1
−
2 1
( 0 ) 0
2
A.2. Effective forward equation for non-constant ( ). We now extend the effective forward equation to the general Heston model, ˜ 12 (˜ ) ˜ 1 ˜ = ( ) Σ h i ˜ = ( ) Λ ( ) − Σ ˜ + ( ) Σ ˜ 12 ˜ 2 Σ
(A.75a) (A.75b)
˜ 2 = ( ) ˜ 1
(A.75c)
The forward Kolmogorov (Fökker-Planck) equation for this model is [19] ¤ £ (A.76a) = − [(Λ − ) ] + 12 2 2 2 ( ) + 2 [ ( )] + 12 2 2 [ ] for 0, with (A.76b)
= ( − )( − )
as → 0
We change variables from to (A.77)
( ) =
Z
2 (1 )1
0
The forward equation becomes (A.78)
= −
2 ¤ £ 2 2 1 2 1 2 [(Λ − ) ] + ( ) + + [ ] [ ( )] 2 2 2 2
This is identical to the original forward equation A.9a, except that has been replaced by , the mean reversion rate has been replaced by 2 , and has been replaced by . Therefore, we re-write the effective forward equation in terms , 2 , and then switch variables from back to . In this way we find that the for the general Heston model, the marginal density ( ) satisfies the effective 1-d forward equation ¢ ¡£ ¤ = 12 2 2 ( ) ˆ ( )( ) 1 + 2 ( ) + 2 ˜ ( ) 2 2 ( ) (A.79a) (A.79b) = ( − ) as → 0 ¡ ¢ through 2 , where (A.79c)
=
1
Z
19
0 ( 0 )
and (A.80a)
ˆ ( ) = −
(A.80b)
( ) =
(A.80c)
˜( ) =
0
( 0 ) 0
+
Z
(1 )Λ(1 )
−
0
1 ( ) 2 ( )ˆ ( )
1
( 0 ) 0
1
3 ( ) 2 2 ( )ˆ ( )
− 3( )
4 ( ) 2 ( ) + 2 ( ) 2 ( )ˆ ( )
( ) = − [( )Γ0 + ˜ ( )] ( )
(A.80d)
exactly as before. Now, however, the integrals are given by (A.81a)
(A.81b)
( ) =
Z
1 ( ) =
2 (1 )ˆ (1 )1
(1 ) =
0
Z
2 ( ) =
2 (2 )−
1
− (1 ) (1 ) (1 ) ˆ (1 )
0
(A.81c)
Z
Z
1
( 0 ) 0
2 1
( 0 ) 0
2
1
(1 ) (1 ) (1 ) ˆ (1 ) (1 ) 1
0
(A.81d)
3 ( ) =
Z
− 2 (1 )ˆ (1 ) (1 )
0
(A.81e)
4 ( ) =
Z
1
− (1 ) (1 ) (1 ) ˆ (1 )
0
( 0 ) 0
1
1
( 0 ) 0
Z
(2 ) (2 ) (2 ) 2 1
1
For completeness, we note that (A.82)
Γ0 ≡ −
0 (0) = 0 ( ) (0)
Appendix B. Special cases. Appendix A analyzes a very general Heston model which leads to complicated effective forward equations, A.79a-A.82. Here we specialize these results to three key special cases. B.1. Constant expected variance: ˆ ( ) ≡ 1.
20
B.1.1. Effective forward equation. In the Heston model, Λ ( ) represents the equilibrium value for ˜ If Σ ˜ (0)Λ we expect volatilities to drift higher, and if Σ ˜ (0) Λ we expect volatilities the local variance Σ. to drift lower. Commonly one does not have a view on whether future volatilities should drift higher or ˜ (0), lower. In this case it is natural to take Λ ( ) to be constant with the value Σ ˜ (0) Λ ( ) ≡ Σ
(B.1)
˜ (0) By using the invariance Then the expected value of the variance is constant: ˆ ( ) ≡ ≡ Σ √ ˜ ( ) → Σ ˜ ( ) , (B.2) Σ Λ ( ) → Λ ( ) , ( ) → ( ) ( ) → ( ) for any constant , we can now set ˆ ( ) ≡ 1
(B.3)
for all
without loss of generality. For this special case the effective forward equation 2.2a-2.5e simplifies to ¢ ¡£ ¤ 2 (B.4a) = 12 2 2 ( ) ( ) 1 + 2 ( ) + 2 ˜ ( ) 2 2 ( ) (B.4b) = ( − ) as → 0 ¡ 2¢ through , where Z 1 0 (B.4c) = ( 0 ) Here the coefficients are (B.5a)
( ) =
1 ( ) 2 ( )
3 ( ) 2 ( ) 4 ( ) − 3( ) 2 + 2 2 ( ) ( ) 2 ( )
˜( ) =
( ) = − [( )Γ0 + ˜ ( )] ( )
(B.5b)
Γ0 ≡ −
0 (0) = 0 ( ) (0)
and the integrals are (B.6a)
(B.6b)
(B.6c)
( ) =
Z
1 ( ) =
2 ( ) =
2
(1 )1
(1 ) =
0
Z
(1 ) (1 ) (1 )
−
0
Z
1
Z
2 (2 ) −
1
( 0 ) 0
2 1
( 0 ) 0
2
1
(1 ) (1 ) (1 ) (1 )1
0
(B.6d)
3 ( ) =
Z
−
2 (1 )
0
(B.6e)
4 ( ) =
Z
0
1
( 0 ) 0
(1 )1
−
(1 ) (1 ) (1 )
1
( 0 ) 0
Z
1
21
(2 ) (2 ) (2 ) 2 1
B.1.2. Effective SABR parameters. For constant effective variance (ˆ ( ) ≡ 1), the effective SABR parameters at are 1 2
= ∆− 4
(B.7)
¯( )∆2
√ = ¯ ¯
√ = ∆ ¯
as before. However, the “effective constant coefficients” in 2.10a-2.10g now simplify to (B.8a) (B.8b)
Z 1 ¯ ( ) = 2 ( ) ≡ 2 ( ) 0 2 ( ) Z Z 3 3 2 ( )2 ( ; ) + 3 2 ( ) 4 ( ) − 3¯2 ( ) ¯ ( ) = 3 4 0 0
∆2 ( ) =
where the integrals are given by B.6a-B.6e above. B.2. Piecewise constant volatility surfaces. ˜ (0) for all , and thus set B.2.1. Effective forward equation. We again assume that Λ ( ) ≡ Σ ˆ ( ) ≡ 1 without loss of generality. We also assume that is constant and that ( ), ( ), and ( ) are all piecewise constant: (B.9a)
( ) =
( ) =
=
for −1
where 0 = 0 1 · · · ≡
(B.9b)
This is the simplest case which allows us to calibrate the model to volatility smiles at a discrete set of exercise dates 1 , 2 , , . For this case, the effective forward equation and effective coefficients are given by B.4a-B.4c and B.5a, B.5b, exactly as in the preceding case. However, the formulas for ( ) and the integrals 1 ( ), 2 ( ), . . . , 5 ( ) can be written differently. To express these integrals succinctly, let us assume that is in the interval, −1
(B.10a) and define (B.10b) (B.10c)
= − −1 = − −1
for = 1 2 − 1
Then (B.11a)
( ) =
X
2
=1
and (B.11b)
1 ( ) = −
X =1
1 − − 22
(B.11c)
2 ( ) =
X
3
=1
+
X
−1
2
=2
(B.11d)
3 ( ) = −
− 1 + − 2
X
1 − − −−1 X 1 − − =1
2 2
=1
+−
X
(1 − − ) 22
2 −−1
=2
(B.11e)
4 ( ) = −
X
2 2 2
=1
+−
X
=2
2
−1 1 − − X 2 2 1 − −2 2 =1
1 − (1 + ) − 2 −1 X
=1
1 − −
B.2.2. Effective SABR parameters. For constant effective variance (ˆ ( ) ≡ 1), the effective SABR parameters at are (B.12)
1 2
= ∆− 4
¯( )∆2
p = ¯ ¯ ( )
p = ∆ ¯ ( )
as before, but now the “effective constant coefficients” are
¯ ( ) = 2 ( ) 2 ( ) 3 3 ¯ ( ) = 3 5 ( ) + 3 6 ( ) − 3¯2 ( ) 4
∆2 ( ) ≡
(B.13a) (B.13b)
where ( ) and 2 ( ) are given by B.6a and B.6c, and (B.14)
5 ( ) =
Z
2 ( ) 2 ( ; )
6 ( ) ≡
0
Z
0
To express and the integrals succinctly, define (B.15)
≡ − −1
for = 1 2
where we recall that 0 = 0 and = . Then (B.16a)
≡ ( ) ≡
Z
2 ( ) =
0
X =1
23
2
2 ( ) 4 ( )
(B.16b)
2 ( ) ≡ =
Z
0 X
2 ( )1 ( )
3
=1
+
X
− 1 + − 2
2 −−1
=2
(B.16c)
5 ( ) ≡ =
Z
0 X
2 ( )2 ( ; )
2 4
=1
1 + 2 − (2 − − ) 23
2
2 (1 − − ) X 2 −−1 1 − − 2 =1 =+1 ⎛ ⎞2 −1 − X 1 − −2 X 1 − ⎠ 2 2 −−1 + 2 ⎝ 2 =1 =+1
+
(B.16d)
6 ( ) ≡ =
Z
0 X
−1 X
−1 1 − − X 1 − − =1
2 ( ) 4 ( )
2 2 4
=1
+
X X
−1
X
=3
1 − (1 + ) − −−1 X 1 − − 2 =1 −1
2
=2
+
−2 + + (2 + ) − 3
3
=2
+
2 2
1 − − −−1 X 2 2 2 1 − (1 + )− 2 =1 −1
2
−1
X 1 − − −−1 X 1 − − =2 =1
B.3. Constant model parameters. ˜ (0) for all , and thus set B.3.1. Effective forward equation. We again assume that Λ ( ) ≡ Σ ˆ ( ) ≡ 1 without loss of generality. We also assume that all other model parameters ( ), ( ), ( ), ( ) are constant. By finding the volatility surfaces that can be generated without varying the parameters, this case illustrates the innate behavior of the Heston model. It also allows us to fit the volatility smile at a single exercise , while using the constant to roughly mimic the rest of the volatility surface. The effective forward equation for constant parameters is once again given by eqs. B.4a-B.4c, but now the coefficients simplify to (B.17a)
( ) =
1 − − 2 24
¡ ¢2 2 1 − − 3 2 2 − 1 + − 1 − − ˜( ) = 2 − 42 2 2 2 2 2 2 2 1 − (1 + ) − + 2 2 2
(B.17b)
( ) = − [( )Γ0 + ˜ ( )] 2
(B.17c)
Γ0 ≡ −
0 (0) = 0 ( ) (0)
B.3.2. Effective SABR parameters. With all the model parameters constant and ˆ ( ) ≡ 1, the effective SABR parameters are √ √ 2 1 2 = ¯ ¯ = ¯ (B.18) = − 4 ¯( ) where now (B.19a) (B.19b)
− ¯ ( ) = − 1 + 2 2 ¡ ¡ ¢2 ¢2 2 − − 1 − − 2 1 + 2 − 2 − − 2 2 2 +3 2 ¯ ( ) = 3 2 3 4 83 4
For short expiries this reduces to (B.20a)
{1 − · · · } ∼ p 1 − 2
∼ 12 {1 − · · · }
for ¿ 1
and for long expiries, (B.20b)
2 = √ {1 − · · · } , 3
=
√ 3 {1 − · · · } 2
for À 1
Since → 0 for long expiries, the smile naturally flattens for long dated options under the Heston model. Appendix C. Boundary conditions and numerical option pricing. The effective forward equation only has one spatial dimension, so options can be priced efficiently by solving the effective forward equation numerically to obtain the density ( ) over an appropriate domain min max , and then integrating to find the expected value of the payoff. See [3], [10]. Most backbones ( ) have (min ) = 0 at some point min . This acts as a natural diffusive barrier, and then one assumes ˜ ( ) ≥ min for all . For example, with a shifted CEV backbone ( ) = ( + ) , the barrier is at min = −. Usually there are no diffusive barriers for large , so max needs to be picked large enough that its value doesn’t materially affect option prices. Numerically solving the effective forward equation for min max requires boundary conditions at = min and = max . Boundary conditions are discussed in [3], where it is found that absorbing boundary conditions, (C.1)
( min ) = 0
( max ) = 0
must be used to preserve the Martingale property of ˜ ( ). Since the boundaries are absorbing, the probability density will develop -functions at the boundaries min , max . Crudely speaking, ⎧ at = min ⎨ ( )( − min ) ( ) for min max (C.2) ( ) = ⎩ ( )( − max ) at = max 25
In [3] it is shown that conservation of probability requires that the probability accumulates at min and max according to the flux reaching the boundaries. Therefore, we must solve ¤ £ (C.3a) for min max = 12 2 ( ) =0 at = min (C.3b) =0 at = max (C.3c) (C.3d) = ( − ) as → 0 where (C.3e)
2
2 ( ) ≡ 2 2 ( ) 2 ( )
( )
¤ £ 1 + 2 ( ) ( ) + 2 ˜ ( ) 2 ( ) 2 ( )
to obtain the continuous part of the distribution, and then integrate
¤ £ 2 ( )
(C.4a)
= lim+ →min
(C.4b)
= − lim− →max
1 2
1 2
with initial conditions (0) = 0
(C.4c)
¤ £ 2 ( ) (0) = 0
to obtain the probability tied up in the boundaries. The effective forward equation, the boundary conditions, and the initial condition, form a well-posed problem for the density . Stable, exactly arbitrage free, numerical methods for solving this problem are developed in [?]. With these methods, the numerical solution satisfies (C.5a) (C.5b)
( ) + min ( ) +
Z
max
min max
Z
( ) + ( ) = 1 ( ) + max ( ) =
min
exactly, so the combined probability totals unity and ˜ ( ) is a Martingale to within round-off error. Of course max should be increased if the probability ( ) at the upper boundary is significant. Once the numerical solution has been obtained, European option values can be obtained by numerically integrating Z max ( ) = (C.6a) ( − ) ( ) + (max − ) ( )
(C.6b)
( ) =
Z
min
( − ) ( ) + ( − min ) ( )
Note that solving the effective forward equation from 0 to yields the terminal density ( ) for all , and thus yields the option prices for all strikes for this expiry . In [3] it is also shown that the maximum principle [23] guarantees that the probability density ( ) is strictly positive, as are ( ) and ( ). Since ˜ ( ) is a Martingale, this shows that these option prices are arbitrage free [24], [25]. 26
The -functions at the boundaries are surprising at first sight. To understand how it can arise naturally, and how they can be resolved by finer-scale modeling, consider a modified backbone ½ ( ) for min + (C.7) ( ) = () ( − min ) for min min + for ¿ 1. As long as 0, the boundary = min is inaccessible, so no boundary condition is needed, and there is no flux of probability to the boundary, so no delta function develops [21]. Nevertheless, a boundary layer analysis in [?] shows that in the limit → 0, the probability density ( ) obeys absorbing boundary conditions at min +, and the total probability in min min + remains finite as → 0, and increases according to C.4a-C.4b. Thus the delta function arises as limit as as → 0. REFERENCES [1] Hagan, P. S., Lesniewski, A.S., and Woodward, D.E. (2017). Effective media analysis for stochastic volatility models. Wilmott Magazine, to appear. [2] Hagan, P. S., Kumar, D., Lesniewski, A.S., and Woodward, D.E. (2002). Managing smile risk. Wilmott Magazine, 2002: 84-108. [3] Hagan, P. S., Kumar, D., Lesniewski, A.S., and Woodward, D.E. (2013). Arbitrage Free SABR. Wilmott Magazine, January, 2013: 1-16. [4] Obloj J. (2008). Fine-tune your smile: Correction to Hagan et al. Wilmott Magazine, May 2008: 102-109. [5] Hagan, P. S., Lesniewski, A.S., and Woodward, D.E.. Implied volatilities for mean reverting SABR models, In preparation [6] Antonov, A., Konikov, M., and Spector, M., (2013). SABR spreads its wings. Risk Magazine, 2013: 58-63. [7] Antonov, A. and Spector, M., (2012). Advanced analytics for the SABR model. SSRN, March 2012: 2026350. [8] Andreasen J. and Huge, B.N. (2013) Expanded forward volatility. Risk Magazine, Jan 2013: 101-107. [9] Paulot L. (2015). Asymptotic implied volatility at the second order with application to the SABR model. In: Friz P., Gatheral J., Gulisashvili A., Jacquier A., Teichmann J. (eds) Large Deviations and Asymptotic Methods in Finance. Springer Proceedings in Mathematics & Statistics 110. [10] Hagan, P. S., Lesniewski, A.S., and Woodward, D.E. (2017). Managing vol surfaces. Wilmott Magazine, to appear. [11] Hagan, P. S., Kumar, D., Lesniewski, A.S., and Woodward, D.E. (2016). Universal smiles. Wilmott Magazine, July 2016: 40-55. [12] Balland, P. and Q. Tran, Q. (2013). SABR goes normal. Risk Magazine, 2013: 76-81. [13] Andersen, L.B.G. and Piterbarg, V.V. (2007). Moment explosions in stochastic volatility models. Finance and Stochastics, 2007:11:29—50. [14] Heston, S.L. (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. of Fin. Studies 6 no 2, 1993: 327-343. [15] Carr, P. and Madan, D. (1999) Option valuation using the fast Fourier transform, J. Comp Fin 1999: 61-73. [16] Grzelak, L.A. and Oosterlee, C.W. (2011) On the Heston model with stochastic interest rates, SIAM J Fin Math 2011: 255-286. [17] van der Stoep, A.W., Grzelak, L.A. and Oosterlee, C.W. (2014) The Heston stochastic-local volatility model, Int. J. Theor. & App. Finance 17 no 7 2014: 1450045. [18] Henry-Labordère, P. (2009) Calibration of local stochastic volatility models, Risk, Sept. 2009: 112-117 [19] Kadanoff, L. P. (2000) Statistical Physics: statics, dynamics and renormalization, 2000, World Scientific, Singapore [20] Feller, W. (1971) An Introduction to Probability Theory and Its Applications, Vol. 2, 1997 Wiley, New York [21] Sobczyk, K. (2001) Stochastic Differential Equations, 2001 Berlin:Springer [22] Neu, J. C. (1978) Nonlinear oscillations in discrete and continuous systems, Thesis, California Institute of Technology, 1978. [23] Protter, M.H. and Weinberger, H.F. (1984) Maximum Principles in Differential Equations, 1984 Springer, New York [24] Dupire, B. (1994) Pricing with a smile, Risk 7 1994: 18-20. [25] Derman, E. and Kani, I. (1994) Riding a smile, Risk 7 1994: 139-145.
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