IMPLIED VOLATILITIES FOR MEAN REVERTING SABR MODELS PATRICK S. HAGAN ∗ , ANDREW S. LESNIEWSKI† , AND DIANA E. WOODWARD ‡ Abstract. Here we analyze the mean reverting SABR model: ˜ ˜ ) ˜ 1 ˜ = ( ) ( ˜ = ( ) [ ( ) − ] ˜ ˜ ˜ 2 + ( ) ˜ 1 ˜ 2 = ( ) We use asympotic methods to derive an effective forward equation for the marginal density ˜ (0) = ( ) = E ˜ ( ) − ˜ (0) =
This effective forward equation is not exact, but is accurate through (2 ), the same accuracy as the SABR implied volatility formulas. Since the effective forward equation has one spatial dimension ( ) instead of two ( , ), it can be solved efficiently numerically to obtain the density ( ). European option prices can then be obtained by integrating to find the expected value of the option’s payoff. We then use an effective media analysis to show that for any expiry , we can obtain an effective forward equation with 2 time-independent coefficients that yields the same density ( ) to within ( ). This yields effective SABR parameters , , , and shows that to within 2 , the implied volatilities of European options under the -SABR model are given by the implied volatility formulas for the standard SABR model using these effective SABR parameters.
1. Introduction. 1.1. The standard SABR model. The standard SABR model was developed to manage smile and skew risk of vanilla options [1]-[6]. It is (1.1a) (1.1b) (1.1c)
˜ ˜ ) ˜ 1 ˜ = ( ˜ ˜ 2 ˜ = ˜ 1 ˜ 2 =
where the “backbone” (˜ ) is usually taken to be a power law, (1.1d)
³ ´ (˜ ) = ˜ +
When present, the positive “offset” is included to allow ˜ to become negative, as is needful in the current rate environment. In [14], [15], an effective forward equation was derived for the SABR model. Let n ³ ´ ³ ´¯ o ˜ ) − ¯¯ ˜ (0) = (0) ˜ (1.2a) (0 ; ) = E ˜ ( ) − ( =
be the probability density at ˜ ( ) = , ˜ ( ) = , given that ˜ (0) = , ˜ (0) = . Also define the marginal (reduced) probability density, Z ∞ (0 ; ) (1.2b) ( ) = 0
∗ Gorilla
Science;
[email protected] of Mathematics; Baruch College; One Baruch Way; New York, NY 10010 ‡ Gorilla Science,
[email protected] † Dept
1
˜ ). In [14] it is shown that ( ) so that ( ) is the density at ˜ ( ) = regardless of the value of ( satisfies the effective forward equation (1.3a) (1.3b)
ª ©£ ¤ 2 = 12 2 2 1 + 2 + 2 2 2 2 2 ( ) ( ) → ( − ) as → 0
for 0
through (2 ), where ( ) and are defined by (1.3c)
( ) ≡
1
Z
0 ( 0 )
≡ 0 ( )
The effective forward equation reduces the problem from two dimensions ( and ) to one dimension ( only). This enables option prices to be found efficiently by solving the effective forward equation numerically [14] and then integrating: (1.4)
( ) =
Z
∞
( − ) ( )
Although the effective forward equation, and thus the prices are not exact, they are accurate to within (2 ), the same accuracy as the SABR implied vol formulas [1]. In addition, these prices are exactly arbitrage free [14]. An alternative to this numerical approach is using singular perturbation techniques to analyze the effective forward equation 1.3a, 1.3b. This leads to explicit, closed-form formulas for the implied normal volatilities1 ¡ ( ¢ ) of European options [13]. These formulas agree with the original SABR formulas to within 2 , and are essentially identical under moderate conditions, but appear to be more accurate under extreme conditions. In this paper we establish a similar set of results, but for the mean reverting SABR model. We initially analyze a very general mean reverting SABR model, and then specialize the results to key special cases. 1.2. The (mean reverting) SABR model. The SABR model is effective at managing volatility smiles, volatility as a function of the strike at a single expiry date . For each , the SABR parameters , , are calibrated so that the model’s implied volatility curve ( ) matches the market’s implied volatilities for that . The calibrated model is then used to consolidate and hedge the risks of all options with the same (or similar) exercise dates . The SABR model is less useful for managing volatility surfaces, volatility as a function of the strike at multiple exercise dates , which are needed for handling many exotics. Managing volatility surfaces requires a richer model, such as the dynamic SABR model [16], (1.5a) (1.5b) (1.5c)
˜ ˜ ) ˜ 1 ˜ = ( ) ( ˜ ˜ ˜ = ( )2 ˜ 1 ˜ 2 = ( )
1 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. 2
By allowing , , and to be time dependent, we can calibrate the model to volatility surfaces, matching the observed implied vols ( ) across different strikes and expiries . This allows us to consolidate our risks across different exercise dates, and to deconstruct exotic risks into European option risks. ˜ and thus the moments of ˜ , can become large too It has also been argued that the local volatility , easily in the SABR model. Although the expected value of the local volatility ˜ ( ) is bounded, the variance of ˜ ( ) exhibits slow exponential growth for the SABR model [17]. This renders the original SABR model less suitable for forward volatility options. It may also lead to unrealistic tail properties, making it less suitable for long dated European options. To rectify this, the -SABR model has been proposed [18]-[20], ˜ ˜ ) ˜ 1 ˜ = ( ) ( ˜ ˜ ˜ 2 ˜ = ( ) [ ( ) − ] + ( ) ˜ 1 ˜ 2 = ( )
(1.6a) (1.6b) (1.6c)
where mean reversion counters the growth of ˜ ( ). Not only is the expected value of the local volatility bounded for the -SABR model, the variance of ˜ ( ) is bounded if ( ) 12 2 2 ( )
(1.7a) and exhibits (slow) exponential growth if
( ) 12 2 2 ( )
(1.7b)
Like the dynamic SABR model, the -SABR model is capable of consolidating and adjudicating risks across different expiry dates as well as strikes . In addition, the -SABR model should have better forward volatility and tail properties, but at the price of increased complexity. 2. Results. 2.1. Effective forward equation. In Appendix A singular perturbation methods are used to analyze the -SABR model in the limit ¿ 1. This analysis shows that to within (2 ), the problem can be reduced from two spatial dimensions ( and ) to one ( only) Recall that ( ) is the marginal (reduced) density, n ³ ´¯ o ¯ (2.1) ( ) = E ˜ ( ) − ¯ ˜ (0) = ˜ (0) =
In Appendix A, it is shown that the reduced density solves the effective forward equation 2
= 12 2 2 ( ) 2 ( ) = ( − ) ¡ ¢ through 2 . Here, (2.2a) (2.2b)
(2.3a)
( )
¢ ¡£ ¤ 1 + 2 ( ) + 2 ˜ ( ) 2 2 ( ) as → 0
1 =
Z
for 0
0 ( 0 )
and ( ) is the expected value of the local volatility, (2.3b)
Z ¯ o ¯ ˜ − 0 ( 0 ) 0 ˜ + ( ) ≡ ( )¯ (0) = = n
3
−
(1 )(1 ) 0
1
( 0 ) 0
1
The coefficients can be written explicitly using (2.4a)
( ) ≡
Z
2 ( 0 ) 2 ( 0 ) 0
0
They are 1 ( ) ( ) ( ) 3 ( ) 22 ( ) 44 ( ) ˜( ) = + 2 ( ) − 6 ( ) 2 + 2 ( ) ( ) ( ) ( ) 2 ( ) 5 ( ) ( ) = − [( )Γ0 + ˜ ( )] ( ) + 2 Γ0 = − 0 ( ) ( ) ( ) =
(2.4b) (2.4c) (2.4d) where the integrals are (2.5a)
1 ( ) =
Z
−
2 (1 )(1 ) (1 ) (1 )
0
(2.5b)
2 ( ) =
Z
−
2 (1 ) 2 (1 )
0
(2.5c)
3 ( ) =
Z
1
( 0 ) 0
2
(1 )(1 ) (1 ) (1 )
0
(2.5d)
4 ( ) =
Z
(2.5e)
5 ( ) =
0
Z
−2
2 (1 ) 2 (1 )
1
1
(2 )2 (2 )−
1
(2 )2 (2 )−
2 (1 ) (1 ) (1 ) (1 )
( 0 ) 0
1
1 − ( 0 ) 0
0
Z
Z
1
( 0 ) 0
Z
2
2 1
1
( 0 ) 0
( 0 ) 0
2 1
2 1
(2 )(2 ) (2 ) (2 ) 2 1
1
1
The effective forward equation 2.2a-2.5e is a salient result of this paper. We re-visit this effective forward equation in section §3, where we briefly consider special cases in which the effective forward equation has simpler coefficients. There we also briefly discuss calibration and practical usage of the model. In Appendix C we note that the effective forward equation can be solved numerically using a finite difference scheme. See [14], [16]. 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 procedure provides a very efficient numerical method for finding option prices, since the effective equation has only one spatial dimension ( ) rather than two ( ). Arguments identical to the ones in [14], [16], and [15] show that the option prices obtained this way are arbitrage free. 2.2. Effective media theory. The chances of directly solving the effective forward equation 2.2a-2.2b are poor due to the time dependence of its coefficients. We can sidestep this problem by using an effective media approach developed in [7]. There it is shown that for any specific exercise date , there is an effective 4
ˆ ) agrees with ( ) through forward equation with time-independent coefficients whose solution ( 2 ( ). That is, © ª ˆ ) 1 + (2 ) (2.7) ( ) = ( for all
ˆ where ( ) is the solution of2 (2.8a)
ˆ = 1 2 ∆2 2 ¯ 2
(2.8b)
ˆ = ( − )
´ ³£ ¤ ˆ 1 + 2¯ + 2 ¯ 2 2 ( ) as → 0
where (2.8c)
=
In terms of (2.9a)
( ) ≡
Z
0
1
Z
0 ( 0 )
2
for 0
0
2
0
0
( ) ( )
= ( ) ≡
Z
2 ( ) 2 ( )
0
the “effective constant coefficients” are Z 1 ∆2 ( ) = (2.9b) 2 ( ) 2 ( ) ≡ 0 Z ¯ ( ) = 2 (2.9c) 2 ( ) 2 ( ) ( ) ( ) 2 0 Z 3 (2.9d) 2 ( ) 2 ( ) 2 ( ) ˜ ( ) ¯ ( ) = 3 0 Z Z 18 2 2 ( ) ( ) ( ) 2 (1 )2 (1 ) (1 ) (1 )1 − 3¯2 ( ) + 3 0 0 Z Z 1 ¯ ( ) = 1 (2.9e) 2 ( ) 2 ( )( ) + 2 ( ) 2 ( ) ( ) [˜ ( ) − ¯] 0 0
Note that if ( ), ( ), ( ), ˜ ( ), ( ) were constant, then the effective coefficients would equal these ¯ = . Thus, the effective coefficients represent an unusual type of constants: ∆ = , ¯ = , ¯ = ˜, and average of the coefficients over 0 , so it is unsurprising that the effective constant coefficients are different for different T . We now specialize these general results to the -SABR model by substituting ( ), ˜ ( ), and ( ) from 2.4a - 2.4d. This yields the effective (constant) coefficients (2.10a) (2.10b) (2.10c)
¯ ( ) = 2 3 ( ) 2 Z © ª 3 2 ( ) 2 ( ) 2 ( ) + 12 ( ) + 4 ( ) 4 ( ) 1 − 3¯2 ( ) ¯ ( ) = 3 0 Z ¤ £ ¯ ( ) = − 1 Γ0¯ ( ) + ¯ ( ) + 1 2 ( ) 5 ( ) Γ0 = − 0 ( ) 2 0
∆2 ( ) =
where = ( ) and the integrals 1 through 5 are given by 2.5a-2.5e. In §3 we briefly consider special cases in which the effective forward equation has simpler coefficients. 1 2 ¯ 2 ¯ obtain this result, one needs to set = 1, replace ¯ 2 with ¯, and replace 2 ¯ 2 with 2 ¯ 2 in [7]. Note 2 that these replacements don’t alter our accuracy.
2 To
5
2.3. Effective SABR parameters. The last ¡step ¢ is to find a standard SABR model whose reduced ˆ ) through 2 at the expiry date . Then the reduced density density ( ) matches ( ( ¡ 2¢ ) of the original -SABR model will be identical to the reduced density ( ) to within . Thus, all European option prices and implied volatilities ¡for¢ the -SABR model at will be identical to those of the standard SABR model, at least through 2 . Consider a standard SABR model, ˜ ˜ ) ˜ 1 ˜ = ( ˜ ˜ 2 ˜ = ˜ 1 ˜ 2 =
(2.11a) (2.11b) (2.11c) with ˜ (0) = , and let us choose (2.12a) (2.12b)
√ √ = ¯ ¯ = ∆ ¯ Z 1 ∗ 1 ¯ = − 2 ¯ ( ) + 2 ( ) 5 ( ) 0 1 2
= ∆ 2
¯∗
Γ0 = − 0 ( )
where ∆, ¯, ¯ are defined in 2.10a, 2.10c. The standard SABR model is a special case of the -SABR model with ( ) ≡ 0, ( ) ≡ 1, ( ) ≡ , ( ) ≡ , and ˜ (0) = So for the standard SABR model, ( ) ≡ (see 2.3b) and our analysis yields (see 2.10a-2.10c) (2.13a) (2.13b)
¯ = ∆ ≡ 1 ¯ = − Γ0 2
¯ = 2 2
Eqs. 2.8a-2.8b now show that the reduced density ( ) for the standard SABR model satisfies3 µµ ¶ ¶ 2 ¯ 2 2 2 2 1 2 2 ˆ ˆ (2.14) = + ( ) 1 + 2 2 2 ¡ ¢ ¯ from 2.13a-2.13b yields to within 2 . Substituting our choices for , , , and
(2.15)
¯ 1 2 2 2 ˆ = 2 ∆
³¡ ´ ¢ ˆ 1 + 2¯ + 2 ¯ 2 2 ( )
¡ ¢ ¡ ¢ ¡ ∗ ¢ ¡ ∗ 1 ¢ ¯ = 2 ¯ + 4 . This is identical to the ¯ + ¯ − ¯Γ0 ∆2 = 2 through 2 , since 2 2 effective forward equation 2.8a for the -SABR model. Consequently, all European option prices and implied volatilities for the -SABR model for expiry are to the European option prices and implied ¡ identical ¢ volatilities for the standard SABR model through 2 , provided we use the SABR parameters , , given by 2.12a-2.12a, where ∆ ( ), ¯ ( ), and ¯ ( ) are given by 2.10a-2.10c in terms of (2.16a) (2.16b)
( ) = ( ) =
Z
−
0
( 0 ) 0
+
Z
−
(1 )(1 )
0
2 ( 0 ) 2 ( 0 ) 0
and the integrals 1 ( ) through 5 ( ) are given by 2.5a-2.5e. this result is also derived in [14] and [16]. 6
1
( 0 ) 0
= ( )
0
3 Alternatively,
1
2.4. Implied volatilities for European options. Many researchers [1]-[6] have derived explicit asymptotic formulas for the implied normal volatilities of European ¡ ¢ options under the standard SABR model. Since these competing formulas are all accurate through 2 , they are in agreement through this order, but differ at higher order, especially in the extreme tails. For example, in [13] it is shown that the implied normal volatility under the SABR model is
(2.17a)
⎧ ⎨ 1 + 2 Θ () + · · · − · ( ) = · Z · 1 () ⎩ 0 2 1 − Θ () + · · · 0 ( )
¡ ¢ through 2 , where the stochastic volatility terms are (2.17b)
=
Z
0
( 0 )
() = log
+ +
if Θ ≥ 0 if Θ 0
q 1 + 2 + 2
1 +
and the higher order term is (2.17c)
Θ () =
2 24
½ ¾ ½ ¾ ( + ) () − + − () 2 −1 + 3 + 1 − 2 + () () 6 ()
with (2.17d)
() =
q 1 + 2 + 2
= 14 ( ) 00 ( ) −
1 8
2
[ 0 ( )]
The most common ( ) in practice is an offset CEV backbone, ( ) = ( + ) . For this case, (1 − ) ( − ) − = 1− 1− 0 ( + ) − ( + ) 0 ( )
(2.18a)
Z
and (2.18b)
=
( + )1− − ( + )1− 1−
= − 18
(2 − )
( + )2−2
The implied volatility formula in 2.17a - 2.18b appears to be more accurate than the original implied vol formulas for the SABR model, especially in extreme cases when 2 2 isn’t particularly small [1], [13]. In fact, the original formula is essentially identical to 2.17a - 2.18b with Θ() replaced by Θ(0): (2.19)
Θ(0) =
2 − 32 2 2¯ − 3¯2 2 1 + 13 2 = ∆ + 3 ∆2 24 24
Figures 1-3 show an admittedly extreme case. These figures compare the implied volatilities, the time values of European options, and the probability densities derived from these option prices by using the implied volatility formula 2.17a-2.17d (new ), using the original formula with Θ(0) in place of Θ() (orig), and using a finite difference scheme to numerically solve the effective forward equation 2.8a - 2.8c (exact). 7
Fig. 2.1. The implied volatilties for European options with = 100, = 3 and SABR parameters ¯ = 25, ¯ = 055, ¯ = 020. The backbone is ( ) = ( + ) , with = 05 and = 35. Shown is the “exact” ( ).obtained by numerically solving the effective forward equation 2.8a-2.8c, the standard SABR implied vol formula (“orig SABR”) and the new formula 2.17a-2.18b.
Fig. 2.2. The values of out-of-the-money call and put options under the SABR model.for = 100 and = 3. The SABR parameters are the same as Figure 1. Shown are the exact time values obtained by numerically solving the effectove foward equation, along with the values obtained from the original and new SABR formulas.for the implied vol. 8
Fig. 2.3. The probability density implied by the European option prices for options with = 100, = 3 y, and the same SABR parameters as Figure 1. The density computed by numerically solving the effective forward equation (exact) remains positive for all , but probabilities derived from both explicit implied vol formulas lead to negative densities for very low strikes .
3. Conclusions. The effective forward equation (2.2a-2.5e) and the effective SABR parameters (2.12a2.12b) are our principal results. These coefficients and parameters are necessarily complicated, since we started with a very general -SABR model. Often in practice we only need special cases of the -SABR model, cases in which the coefficients simplify. Since represents the equilibrium volatility level, if then the forward volatility curve is expected to drift upward, and if the volatilty curve should drift downward. Often mean reversion in the -SABR model is used to control long term volatilities without a view on whether future volatilities should be higher or lower than current levels. In this case it is natural to use a constant ( ) with ≡ ˜ (0) = , so that (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 ( ) ≡ 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 , , 9
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 -SABR model’s effective SABR parameters to their market values, (3.3)
1 2
= ∆ ( ) 2
¯ ∗ ( )
p = ¯ ( ) ¯ ( )
= ∆ ( )
p ¯ ( )
yields three equations for the three remaining unknowns , , . By bootstrapping = 1, 2, . . . , we can calibrate the piecewise constant -SABR model to the entire volatility surface. Finally, the innate character of the -SABR model can be seen by considering the case in which all the parameters are constant. So assume that ( ), ( ), ( ), ( ), and ( ) are constant, and that ( ) = , so that ( ) ≡ for all . Without loss of generality, we can re-scale so that ≡ 1. This case is considered in Appendix B.3, where it is found that the effective forward equation simplifies to 2
= 12 2 2 = ( − )
(3.4a) (3.4b)
¢ ¡£ ¤ 1 + 2 + 2 ˜ 2 2 ( ) as → 0
where the coefficients are (3.5a) (3.5b)
(3.5c)
1 − − ¡ ¢2 ¢ 2 1 − − ¡ 2 2 1 − (1 + ) − 1 − − 2 +6 2 ˜( ) = 1 − 2 2 2 2 2 2 2 (1 − ) − − −2 +4 2 2 2 ½ ¾ 2 1 − −2 1 − − ( ) = − Γ0 + 2 − ˜ ( ) 2 2 ( ) =
Even though mean reversion does not change the expected value of the instantaneous volatility, ( ), it systematically reduces ( ) and ˜ ( ), and thus the smile, exhibited by the -SABR model. In fact, these coefficients reduce to (3.6a) (3.6b) (3.6c)
1 + t.s.t.’s, ½ ¾ 2 2 1 2 + t.s.t.’s ˜( ) = 2 2 2 1 − + 6 ¾ ½ 2 1 1 − 2 2 ( ) = − − Γ0 + − 6 2 2 + t.s.t.’s 2 ( ) =
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 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 p 3 (1 + 2 ) 2 {1 + · · · } , = (3.7) = p {1 + · · · } for À 1 3 (1 + 2 ) Appendix A. Derivation of the effective forward equation. 10
A.1. Derivation for ( ) ≡ 1. We first derive the effective forward equation for the special case of ( ) ≡ 1. The mean-reverting SABR model is then ˜ ˜ ) ˜ 1 ˜ = ( ˜ ˜ ˜ 2 ˜ = ( ) [ ( ) − ] + ( ) ˜ 1 ˜ 2 = ( )
(A.1a) (A.1b) (A.1c)
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 general mean reverting model 1.5a-1.5c. A.1.1. Expected volatilities. Before analyzing equations A.1a-A.1c, we define the expected value of the volatility, ¯ n o ¯ ˜ (A.3) ( ; ) ≡ ˜ ( )¯ () =
Taking the expected value of A.1b yields
= ( ) [ ( ) − ]
(A.4a)
for
so (A.4b)
( ; ) = −
( 0 ) 0
+
Z
−
(1 ) (1 )
1
( 0 ) 0
1
Consequently, 0 0 for = − () [ () − ] − ( ) Let us also define = (; ) as the initial value of ˜ () needed for the expected value of ˜ ( ) to be
(A.4c)
: (A.5a)
= ( ; ) if and only if = (; )
From A.4b then, (A.5b)
(; ) =
( 0 ) 0
−
Z
(1 ) (1 )
1
( 0 ) 0
1
More generally, (A.6)
if = ( ; ) then ( ; ) = ( ; ) for all n o I.e., if the path of ˜ ( ) goes from ( ) to ( ), then ( ( ; )) = ( ( ; )) is also on the path for all between and . Note that (A.7a) (A.7b)
= () [ () − ] 0 0 = −( ) [ ( ) − ] ( )
for , as is needed later. 11
A.1.2. Conservation law. Recall that n ³ ´ ³ ´¯ o ˜ ) − ¯¯ ˜ () = () ˜ = (A.8) ( ; ) = E ˜ ( ) − (
is the probability density at ˜ ( ) = , ˜ ( ) = , given that ˜ () = and ˜ () = . The forward Kolmogorov equation for the -SABR model is [21] ¤ £ ¤ ¤ £ £ (A.9a) = − [( − ) ] + 12 2 2 2 ( ) + 2 2 ( ) + 12 2 2 2 for , with (A.9b)
= ( − )( − )
as →
Let us define the volatility moments () ( ; ) ≡
(A.10)
Z
∞
( ; )
0
Integrating the forward Kolmogorov equation over all yields the conservation equation h i (0) = 12 2 2 ( )(2) (A.11a) for
(0) = ( − )
(A.11b)
as →
where (2) is (2) ( ; ) 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) (2) (0 ; ) = 2 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 2
= 12 2 2 = ( − )
(A.13a) (A.13b)
¤ £¡ ¢ 1 + 2 + 2 ˜ 2 2 ( ) as → 0
for 0
where ( ) is today’s current marginal density, ( ) = (0) (0 ; ) =
(A.14)
Z
∞
(0 ; )
0
¡ 2¢ (2) (0) Since the relation between ¡ 2 ¢ and is accurate through , the effective forward equation is also accurate through . A.1.3. Changes of variables. Let us now choose an arbitrary, but specific, . For this , the backward Kolmogorov equation for () ( ; ) is [22] (A.15a)
()
−
()
()
() () = ( − ) + 12 2 2 2 ( ) + 2 2 ( ) + 12 2 2 2
for , subject to (A.15b)
() ( ; ) → ( − ) 12
as →
We start by using a series of tranformations to make the backwards equations for (2) and (0) as similar as possible. Define ¯ n o ¯ ˜ (A.16a) = ˜ ( )¯ () = ≡ ( ; )
and switch variables from to . In terms of we have (A.16b)
= (; )
and (A.17a) (A.17b)
0 0 → + ( ; ) = − () [() − (; )] − ( ) 0 0 → ( ; ) = − ( )
Define ˜() ≡ ˜(; ) ≡ ()−
(A.18)
( 0 ) 0
Then the backwards equation becomes (A.19a) (A.19b)
()
−
n o () () = 12 2 2 (; ) 2 ( ) + 2˜ ( ) + ˜2 ()
() ( ; ) → ( − )
for
as →
since = (; ) → as → . 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) for , with (A.22b)
()
−
= 2 (; )
n
1 () 2
1 2 2 () − 12 Γ()() − ˜ () ˜ + 2
() ( ; ) →
() (0) 13
as →
o
˜ () ( ; ) by We define ˜ () ( ; ) (0)
() ( ; ) =
(A.23) ˜ () ( ; ) satisfies Then (A.24a)
()
˜ −
= 2 (; )
½
1 ˜ () 2
˜ ˜ () 1 2 ˜2 ˜ () + 2 ( − 1) 2 ¾ 2 ˜ ˜ () 1 2 2 ˜ () 2 ˜ () −˜ + + ˜ 2
˜ () − − 12 Γ ()
for , with ˜ () ( ; ) → ()
(A.24b)
as →
Finally, we define the function (A.25a)
(; ) =
Z
2 (1 ; )1
and change variables from to , where (A.25b)
≡ (; )
We also define ≡ ˜(; ) as the inverse of (; ) in the sense that (A.25c)
= ˜(; ) ⇔ = (; )
Then (A.26a) (A.26b) (A.26c)
−→ (˜; ) = − 2 (˜; ) −→ + (˜; ) 2 2 2 ˜; ) −→ + 2 ( 2 2 2 +2 (˜; ) 2 + (˜; )
where (A.27a) (A.27b)
(˜; ) = 2 (˜; ) = 2
Z Z
(1 ; )
˜
2
˜
1
1
( 0 ) 0
( 0 ) 0
1
1
We emphasize that (˜; ) and (˜; ) are the partial derivatives of (; ) with respect to with held fixed, evaluated at = ˜(; ). In particular, (A.28)
(˜; ) 6=
ª © ˜ ( (; ) ; ) 14
In terms of the backward equation becomes ³ ´ ³ ´ 1 ˜ () 1 ˜ () ˜ () ˜ () + 1 2 ˜2 2 ˜ () ˜ () ˜ () + (A.29a) = 2 − 2 Γ () − ˜ + 2 ˜ ˜ () ˜2 ˜ () 1 2 ˜2 ˜ () + 2 + 2 ( − 1) 2 ³ ´ 2 ˜ ˜ () ˜ () + ˜ () + 1 2 ˜2 2 +2 2 −
for 0, with
˜ () ( ; ) → ()
(A.29b)
as → 0
˜ () = ˜ () + ˜ () + 2 ˜ () + · · · , we would find that If we were to expand 0 1 2 (A.30a)
˜ () = 0
(A.30b)
˜ () 0
1 ˜ () 2 0
→ ()
for 0 as → 0
so to leading order ˜ () = √ 1 −2 2 0 2
(A.31)
2 ˜ () 3 ˜ () , 2 ˜ () ˜ () is independent of to leading order, so the terms 2 We see that , and are all ( ) 2 or smaller. As we are only working through ( ), we neglect these terms, obtaining ³ ´ ³ ´ () () () 2 ˜ () () 1 ˜ () 1 1 2 2 ˜ () ˜ ˜ ˜ ˜ = − Γ () − ˜ + ˜ + (A.32a) + 2 2 2
−
˜ ˜ () ˜2 ˜ () 1 2 ˜2 ˜ () + 2 + 2 ( − 1) 2
for 0, with ˜ () ( ; ) → ()
(A.32b)
as → 0
Here, () and ˜ () are shorthand for () ≡ (; ) ≡ (˜(; )) ˜ () ≡ ˜(; ) ≡ ˜(˜(; ) ; )
(A.33a) (A.33b)
A.1.4. Equating = 0 and = 2. The equations for = 2 and = 0 are ³ ´ ³ ´ (0) (0) (0) 2 ˜ (0) (0) 1 ˜ (0) 1 1 2 2 ˜ (0) ˜ ˜ ˜ ˜ (A.34a) = − Γ () − ˜ + ˜ + + 2
(A.34b)
2
2
³ ´ ³ ´ (2) (2) (2) 2 ˜ (2) (2) 1 ˜ (2) 1 1 2 2 ˜ (2) ˜ ˜ ˜ ˜ = − Γ () − ˜ + ˜ + + 2 2 2 −2
˜2 ˜ (2) ˜2 ˜ (2) ˜ ˜ (2) 2 + 22 + 2
for 0, with (A.34c)
˜ (0) ( ; ) → ()
˜ (2) ( ; ) → () 15
as → 0
We now set (A.35)
˜ (2) ( ; ) = ( ; )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 2 (2) ˜ and ˜ (0) at = 0 ( ; ) through ( ). 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 ˜ (2) (A.36a) + 2 + 2 2 + 2 = ª © 2+2 2 +2 ˜ (2) (A.36b) + 2 + (2 ) = ¡ ¢ ¢ ª © ¡ 2+2 2 +2 ˜ (2) (A.36c) + 4 + 2 + 2 42 + 2 + (3 ) = ª © 2+2 2 +2 ˜ (2) (A.36d) + 2 + 2 ( + ) + (2 ) = ª © 2+2 2 +2 ˜ (2) (A.36e) + 2 + 2 ( + ) + (2 ) = We now substitute eqs. A.36a-A.36e into the backward equation, A.34b, and discard terms which are smaller than (2 ). In doing this, we note that to leading order 2 1 =√ − 2 2
(A.37a)
Therefore, is independent of to leading order, so all derivatives are () or smaller. Also + = ().
(A.37b) We obtain (A.38a)
¡ ¢ ( + ) + 12 2 ˜2 2 + = 12 − 12 Γ − ˜ ½ ¾ ˜ 2 2 2 − + − ( + )} − { + 2 − 2˜ ½ ¾ ˜ ˜2 + − + + 22 − 4 + 2 − Γ0 − 2˜ ( + ) 2 µ ¶ ³ µ ¶ ´ ˜ ˜2 +2 − + − 2 ˜ − 2
through (2 ), with the initial condition (A.38b)
→ ()
as → 0
¡ ¢ Here we have dropped at −22 ˜ , since it is no larger than¡ ¢ 3 . We have also set Γ0 ≡ Γ (0) and 2 replaced 2 Γ () with 2 Γ0 without losing our¡ through ¢ accuracy. Resolution through (). We now resolve the + term through (2 ). We can then express this term and the 2 term in terms of 2 2 , and 2 . This will enable us to choose , , and to eliminate the , 2 2 , and 2 terms. We first choose (A.39a)
+ = 16
˜
Thus, (A.39b)
() =
1 1 (; )
where Z
1 (; ) =
(A.39c)
(1 )˜ (1 ) 1 =
0
Z
˜(;)
(1 )˜ (1 ; ) 2 (1 ; )1
With this choice, equation A.38a reduces to − 12 = − 12 Γ0 − ˜ for 0 → () as → 0 ¡ ¢ through (). We have neglected the ˜ and + terms since these are actually (2 ). As before, we have replaced Γ () with the constant Γ0 ≡ Γ(0). Expanding (A.40a) (A.40b)
= 0 + 1 + · · ·
(A.41) yields
0 0 − 12 =0
(A.42a)
for 0
0
→ ()
(A.42b)
as → 0
The solution is 2 1 0 = √ − 2 2
(A.43) so (A.44a) and (A.44b)
0 µ
= − 0 +
¶
0
0
=
µ
1 = − 0
2 1 − 2 µ
¶
0
0
¶
+
0
µ 3 ¶ = − 3 + 3 2 0
3 =− 2
µ
¶ 2 − 1 0
At () we have (A.45a) (A.45b)
1 0 0 1 − 12 = − 12 Γ0 0 − ˜ = − 12 Γ0 0 − 12 ˜
1 → 0
as → 0
Solving yields 0 1 = − 12 Γ0 0 − 3 ()
(A.46) where (A.47)
3 ( ) = = =
Z
1 2
(1 )˜ (1 ) (˜(1 ; ) ; )1
0
1 2
Z
Z
˜(;)
˜(;)
(1 )˜ (1 ; ) 2 (1 ; ) (1 ; )1
(1 ) (1 ) 2 (1 ; )
Z
1
17
(2 ; )−
2 1
( 0 ) 0
2 1
See A.18, A.27a. Therefore ³ ´ ³ ´ (A.48) + = 2 1 + 1 + · · · ¶ µ 33 () 2 2 33 () 1 2 − Γ = − 2 0 + ··· 2 2 Identifying with (0) . We substitute equation A.48 and 2 =
(A.49)
1 2 2 1 − 2 + · · · 2 2
into equation A.38a, and use (A.50)
=
˜ −
to simplify the results. This yields ¡ ¢ ( + ) + 12 2 ˜2 2 + = 12 − 12 Γ () − ˜ ½ µ ¾ 2 2 ¶ 3 ˜2 − + 2 + 6 − 2˜ − ˜ + − 2˜ ½ ˜ 3 ˜2 ˜ + − + + 22 − 4 + 2 − Γ0 + 6 µ ¾ 2¶ ˜ − 2˜ − ˜ + − 2˜ 2 ¡ ¢ through (2 ). We choose and to set the coefficients of 2 2 and 2 to zero. So including A.39a, we have chosen (A.51)
(A.52a) (A.52b) (A.52c)
() = ˆ µ ¶ ¡ 2 ¢ ˜2 = −6 3 + 2˜ () − 3˜ + + 2˜
˜ ˜ 3 ˜2 + 2 − Γ0 + 6 µ ¶ ˜2 + − 2˜ () − 3˜ − 2˜
= + 22 − 4
With these choices, the equation for is (A.53a) (A.53b)
¡ ¢ ( + ) + 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.34a for (A.54)
¡ £ ¤ ¢ (0) ˜ (2) = 2+2 2 +2 ˜ (0) = 2 1 + 2 + 2 + 22 2 ˜
¡ ¢ through 2 . Thus, from A.23 we have (A.55)
2
(2) ( ; ) = 2
¡ ¢ 1 + 2 + 2 ˜ 2 (0) ( ; ) 18
for all 0
¡ ¢ through 2 , with
˜ = + 22
(A.56)
where (A.57a)
() = ˜ ¡ 2 ¢ ¡ ¢ ˜2 ˜ = [() + ] + 2 2 2 − 6 (3 ) + 2˜ ˜2 ) = − () Γ0 + 2 − (˜
(A.57b) (A.57c)
A.1.5. Coefficients. We now solve A.57a - A.57c to obtain (; ), (; ), and (; ). As noted earlier, the solution of A.57a is (A.58a)
1 1 (; )
(; ) =
where (A.58b)
1 (; ) =
Z
−
˜(;)
(1 ) (1 ) 2 (1 ; )
Note that we have replaced ˜() ≡ ˜(; ) by ()−
(A.59)
( 0 ) 0
1
( 0 ) 0
1
so that the coefficients are expressed in terms of the original variables. To solve A.57b for ˜, we need to re-express () + . From A.58a, A.58b we have (A.60)
where
¢ 1 (; ) 1 ()˜ (; ) ¡˜ + ; − 1 (; ) 2 ¢ 1 (; ) ()˜ (; ) ¡˜ = ; − 2 ()˜ (; ) 2 ˜ − ( (; ) ; ) ˜(; ) Z 2 − ( 0 ) 0 + (1 ) (1 ) (1 ; ) (1 ; ) 1 1 ˜(;)
() + () =
˜ (; )
is the derivative with respect to at constant . But differentiating = (˜(; ) ; )
(A.61) with respect to at constant yields (A.62) Since (A.63)
¡ ¢ ¡ ¢ ˜(; ) 0 = ˜; + ˜; (; ) =
Z
2 (1 ; )1
19
¡ ¢ ¡ ¢ we see that ˜; = − 2 ˜; , so
¢ ¢ ¡ ¡ ˜(; ) = ˜; 2 ˜;
(A.64)
Consquently, the first and third terms on the right hand side of A.60 cancel out. Noting that (A.65)
(1 ; ) =
( 0 ) 0
1
we obtain (A.66)
˜ 4˜ 2˜ [() + () ] = −2 +
Z
¢ ¡ 4˜ = − 2 2 +
˜(;) Z
(1 ) (1 ) (1 )1
˜(;)
(1 ) (1 ) (1 )1
where we are abbreviating (; ) as () for brevity. The equation for ˜ now becomes (A.67)
¡ ¢ ¡ 2 ¢ ˜2 4˜ + 2 2 − 6 (3 ) + ˜ =
Z
˜(;)
(1 ) (1 ) (1 )1
Solving yields (A.68a)
(; )3 (; ) 4 2 2 (; ) + 2 (; ) − 6 + 2 4 (; ) 2 2
˜(; ) =
where (A.68b)
(A.68c)
(A.68d)
2 (; ) =
Z
3 (; ) =
4 (; ) =
2
˜(;)
Z
2
(1 ) (1 )
−
1
˜(;)
Z
(1 ) (1 ) 2 (1 )
Z
Z
−
(1 ) (1 ) 2 (1 )
(2 )−
1
(2 )−
1
˜(;)
( 0 ) 0
1
2 1
( 0 ) 0
Z
2 1
( 0 ) 0
( 0 ) 0
2 1
(2 ) (2 ) (2 )2 1
1
Lastly, (A.69a)
(; ) + (; ) = −(; )Γ0 − ˜
1 5 (; 0 ) 2
with (A.69b)
5 (; ) =
Z
˜(;)
−2
2 (1 ) 2 (1 ; ) 20
1
2 1
( 0 ) 0
1
A.1.6. The effective forward equation. Let us now switch from back to the initial condition . In terms of we have (A.70a)
2
(2) ( ; ) = 2
where the coefficients are given by (A.70b) (A.70c) (A.70d)
£ ¤ 1 + 2 + 2 ˜ 2 (0) ( ; )
1 1 ( ; ) 2 6 4 ˜( ; ) = 2 2 ( ; ) + 2 − 2 3 ( ; ) + 2 4 ( ; ) 1 ( ; ) = −Γ0 − ˜ + 2 5 ( ; ) Γ0 = − 0 ( ) ( ; ) =
Here, (A.71a)
(A.71b)
= (; ) ≡ ( ; ) = ( ; ) =
Z
2 (1 ; )1 ≡
Z
2 (1 ; )1
To obtain the effective forward equation, we need the relationship between (2) and (0) only for = 0. Let (A.72)
( ) = ( ; 0 ) = −
0
( 0 ) 0
+
Z
−
(1 )(1 )
0
1
( 0 ) 0
1
and (A.73)
( ) =
Z
2
(1 ) 1 = 0
Z
(1 ; 0 ) 1 0
Then (A.74a)
2
(2) (0 ; ) = 2 ( )
( )
where (A.74b)
( ) ≡ ( ; 0 )
£ ¤ 1 + 2( ) + 2 ˜ ( ) 2 (0) (0 ; )
˜( ) ≡ ˜( ; 0 )
( ) ≡ ( ; 0 )
Substituting this into the conservation law A.13a now yields our effective forward equation. Since we are denoting (0) (0 ; ) as ( ), the effective forward equation is (A.75a) (A.75b)
2
= 12 2 ( ) 2 ( ) = ( − )
¡£ ¢ ¤ 1 + 2 ( ) + 2 ˜ ( ) 2 2 ( ) as → 0
with =
1
Z
21
0 ( 0 )
for 0
Here the coefficients are given by 1 ( ) ( ) ( ) 3 ( ) 22 ( ) 44 ( ) ˜( ) = + 2 ( ) − 6( ) 2 + ( ) 2 ( ) ( ) ( ) 2 ( ) 5 ( ) ( ) = − [( )Γ0 + ˜ ( )] ( ) + 2 ( ) ( ) =
(A.76a) (A.76b) (A.76c)
where the integrals are given by Z 2 (1 ) 1 (A.77a) ( ) =
Z
(1 ) =
0
(A.77b)
Z
1 ( ) =
2 (1 )(1 ) (1 )
−
0
(A.77c)
Z
2 ( ) =
−
2 (1 ) 2 (1 )
0
(A.77d)
3 ( ) =
Z
1
1
(2 )−
1
( 0 ) 0
( 0 ) 0
2 1
( 0 ) 0
2
1
(1 ) 1
2 (1 )(1 ) (1 ) (1 ) 1
0
(A.77e)
(A.77f)
Z
4 ( ) =
5 ( ) =
Z
−
2
(1 ) (1 ) (1 ) 0
−2
2 (1 ) 2 (1 )
0
1
1
( 0 ) 0
Z
(2 )(2 ) (2 ) 2 1
1
( 0 ) 0
1
A.2. Effective forward equation for non-constant ( ). We now extend the effective forward equation to the general -SABR model, (A.78a) (A.78b) (A.78c)
˜ ˜ ) ˜ 1 ˜ = ( ) ( ˜ ˜ ˜ = ( )2 ˜ 1 ˜ 2 = ( )
The forward Kolmogorov (Fökker-Planck) equation for this model is [21] ¤ £ ¤ ¤ £ £ (A.79a) = − [( − ) ] + 12 2 2 2 2 ( ) + 2 2 ( ) + 12 2 2 2
for 0, with (A.79b)
= ( − )( − )
as → 0
We change variables from to (A.80)
( ) =
Z
2 (1 )1 0
22
The forward equation then becomes (A.81)
= −
2 £ £ 2 2 ¤ ¤ ¤ £ 2 2 1 2 1 2 2 . [( − ) ] + ( ) + ( ) + 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 -SABR model, the marginal density ( ) satisfies the effective 1-d forward equation ¡£ ¢ ¤ 2 (A.82a) = 12 2 ( ) 2 ( ) 2 ( ) 1 + 2 ( ) + 2 ˜ ( ) 2 2 ( ) (A.82b) = ( − ) as → 0 where (A.82c)
=
Z
1
0 ( 0 )
and ( ) = −
(A.83a)
(A.83b)
( ) =
(A.83c)
0
( 0 ) 0
+
Z
−
(1 )(1 )
0
( ) = − [( )Γ0 + ˜ ( )] ( ) +
0
1 ( ) =
Z
2 ( ) =
Z
−
2 (1 )(1 ) (1 ) (1 )
(A.84d)
3 ( ) =
Z
−
2 (1 ) 2 (1 )
0
1
5 ( ) 2 ( )
(2 )2 (2 )−
1
0
(A.84c)
( 0 ) 0
3 ( ) 4 ( ) 22 ( ) + 2 ( ) − 6( ) 2 +4 ( ) 2 ( ) ( ) ( ) 2 ( )
exactly as before. However, now the integrals are given by Z Z 2 ( 0 ) 2 ( 0 ) 0 (1 ) = (A.84a) ( ) = (A.84b)
1
1 ( ) ( ) ( )
˜( ) =
(A.83d)
1
( 0 ) 0
1
( 0 ) 0
1
(1 ) 1
2 (1 )(1 ) (1 ) (1 ) (1 ) 1 0
23
2 1
( 0 ) 0
2
(A.84e)
4 ( ) =
Z
2
(1 ) (1 ) (1 ) (1 )
−
0
(A.84f)
5 ( ) =
Z
2 (1 ) 2 (1 ) 0
−2
1
( 0 ) 0
1
( 0 ) 0
Z
(2 )(2 ) (2 ) (2 ) 2 1
1
1
For completeness, we note that (A.84g)
Γ0 ≡
0 (0) = − 0 ( ) (0)
Appendix B. Special cases. Appendix A analyzes a very general -SABR model which leads to complicated effective forward equations, A.82a-A.84g. Here we specialize these results to three key special cases. B.1. Constant expected volatility. B.1.1. Effective forward equation. If the -SABR model is being used to control long term volatilities without a view on whether future volatilities should be higher or lower than current levels, then it is natural to take the equilibrium volatilty ( ) to be constant for all , with ( ) ≡ ˜ (0) ≡ for all . It then follows that (B.1)
( ) ≡
for all .
Inspection of the effective forward equation in 2.2a-2.5e shows that multiplying by a constant and dividing ( ) by the same constant leaves the effective forward equation unchanged. We can thus set ( ) ≡ 1 without loss of generality. The effective forward equation 2.2a-2.5e for this case simplifies to ¡£ ¢ ¤ 2 = 12 2 2 ( ) 1 + 2 + 2 ˜ 2 2 ( ) = ( − ) as → 0
(B.2a) (B.2b) In terms of (B.3a)
( ) ≡
Z
2 ( 0 ) 0
0
the coefficients are (B.3b) (B.3c)
1 ( ) 3 ( ) 22 ( ) 44 ( ) ˜( ) = 2 + 2 ( ) − 6 ( ) 2 + 2 ( ) ( ) ( ) ( ) ( ) = − [( )Γ0 + ˜ ( )] ( ) + 5 ( ) Γ0 = − 0 ( ) ( ) =
where the integrals are now (B.4a)
1 ( ) =
Z
(1 ) (1 ) (1 )
−
0
(B.4b)
2 ( ) =
Z
0
2 (1 )
−
1
( 0 ) 0
1
( 0 ) 0
1
(1 ) 1 24
(B.4c)
3 ( ) =
Z
(1 ) (1 ) (1 ) (1 ) 1
0
(B.4d)
Z
4 ( ) =
(1 ) (1 ) (1 )
−
0
(B.4e)
5 ( ) =
Z
−2
2 (1 )
0
1
( 0 ) 0
( 0 ) 0
1
Z
(2 ) (2 ) (2 ) 2 1
1
1
with (B.4f)
(1 ) =
Z
2 (2 )−
1
2 1
( 0 ) 0
2
B.1.2. Effective SABR parameters. For constant effective volatility ( ( ) ≡ 1), the effective SABR parameters at are, p p 1 2 ¯∗ = ∆ ( ) 2 (B.5a) = ¯ ( ) ¯ ( ) = ∆ ¯ ( ) Z 1 ∗ 1 ¯ ( ) = − 2 ¯ ( ) + (B.5b) 2 ( ) 5 ( ) 0 where the “effective constant coefficients” are Z 1 ¯ ( ) = 2 3 ( ) ∆2 ( ) = (B.6a) ≡ 2 ( ) 0 2 Z © ª 3 (B.6b) 2 ( ) 22 ( ) + 12 ( ) + 44 ( ) 1 − 3¯2 ( ) ¯ ( ) = 3 0 with
(B.6c)
( ) ≡
Z
2 ( 0 ) 0
0
= ( ) ≡
Z
2 ( )
0
B.2. Piecewise constant volatility surfaces. B.2.1. Effective forward equation. We again assume that ( ) ≡ ˜ (0) ≡ , and thus ( ) ≡ for all . Consequently we set ≡ 1 without loss of generality. We also assume that is constant and that ( ), ( ), and ( ) are all piecewise constant: (B.7a)
( ) =
( ) =
( ) =
for −1
where (B.7b)
0 = 0 1 · · · ≡
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.2a-B.3c, exactly as in the preceding case. However, the formulas for ( ) and the integrals 1 ( ), 2 ( ), . . . , 5 ( ) are different. To express these integrals succinctly, let us assume that is in the interval, (B.8a)
−1 25
and define (B.8b) (B.8c)
= − −1 = − −1
for = 1 2 − 1
Then (B.9a)
( ) =
X
2
=1
and (B.9b)
−
1 ( ) =
X
=1
(B.9c)
2 ( ) =
−
X
2 2
=1
+−
X
1 − −
¡ ¢2 1 − −
2 −−1
=2
(B.9d)
3 ( ) =
X
3
=1
+
X
1 2
=2
(B.9e)
4 ( ) = −
X
−1
− − −−1 X 1 − − =1 2 2 2
X
=2
(B.9f)
5 ( ) = −2
X =1
−1 1 − − X 2 2 1 − −2 2 =1
− 1 + − 2
=1
+−
22
2 2
1 − (1 + ) − 2 −1 X
=1
1 − −
1 − −2 2
B.2.2. Effective SABR parameters. The effective SABR parameters for this case are given by p p 1 2 ¯∗ (B.10) = ∆ ( ) 2 = ¯ ( ) ¯ ( ) = ∆ ¯ ( )
as before, but now (B.11a) (B.11b) (B.11c)
¯ ( ) = 2 3 ( ) 2 1 ¯ ( ) = 3 {66 ( ) + 37 ( ) + 128 ( )} − 3¯2 ( ) ¯ ∗ ( ) = − 1 ¯ ( ) + 1 9 ( ) 2
∆2 ( ) =
26
To express and the integrals succinctly, define ≡ − −1
(B.12)
for = 1 2
Then ≡ ( ) ≡
(B.13a)
3 ( ) ≡
(B.13b)
=
Z
Z
3
=
0 X
=1
X
2 −−1
−1 1 − − X 1 − − =1
2 ( ) 2 ( )
1 4 2
=1 X
2
− 1 + − 2
=2
Z
0
X
0
X
+
6 ( ) ≡
2 ( ) =
2 ( )1 ( )
=1
(B.13c)
¡ ¢2 + 2 − 2 − −
¡ 1 − −
43 ¢2
−1 X
1 − −2 2 2 =1 =2 ¡ ¢ −1 − 2 − X X 21− −−1 2 2 1 − + 22 =1 =2
+
+
X
4
2
=3
(B.13d)
7 ( ) ≡ =
2
−2−1
2 2
−1 −1 1 − − −−1 X 2 1 − − −−1 X 2 2 1 − −2 2 =2 =1
Z
2 ( ) 12 ( )
0
X
1 2 2 4
=1
X
¡ ¢2 + 2 − 2 − −
23 ¡ ¢ − 2 3 1−
−1 X
1 − − 2 =1 =2 ⎛ ⎞2 −1 −2 − X X 1 − 1 − ⎠ 2 + −2−1 ⎝ 2 =1 +
=2
27
−−1
(B.13e)
8 ( ) ≡ =
Z
0 X
2 ( ) 4 ( )
2 2 4
=1
+
X
−2 + + (2 + ) − 3 −1
3
=2
+
X
1 2
=2
+
X
1 2
=3
(B.13f)
9 ( ) ≡ =
Z
− 1 − (1 + ) − −−1 X 1 − 2 =1
−1 − − −−1 X 2 2 2 1 − (1 + )− 2 =1
−1 −1 X − − −−1 X 1 − − =2 =1
2 ( ) 5 ( )
0
X
2 2
=1
+
X
=2
2
−1 + 2 + −2 42 −1 1 − −2 −2−1 X 2 2 1 − −2 2 2 =1
B.3. Constant model parameters. B.3.1. Effective forward equation. For the last special case, we suppose that all the model parameters ( ), ( ), ( ), ( ), ( ) are constant with ( ) ≡ . As before, ( ) is then constant and we choose ( ) ≡ = 1 without loss of generality. By finding the volatility surfaces that can be generated without varying the parameters, this case illustrates the innate behavior of the 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.2a-B.2b, but now the coefficients simplify to (B.14a) (B.14b)
(B.14c)
1 − − ¡ ¢2 ¢ 2 1 − − ¡ 2 2 1 − (1 + ) − 1 − − + 6 ( ) = 1 − 2 2 2 2 2 2 2 2 2 (1 − ) − − −2 +4 2 2 2 ½ ¾ 1 − − 2 1 − −2 ( ) = − Γ0 − ( ) + 2 2 2 ( ) =
B.3.2. Effective SABR parameters. With all the model parameters constant and = 1, the effective SABR parameters for this case are given by p p 1 2 ¯∗ (B.15) = 2 = ¯ ( ) ¯ ( ) = ¯ ( ) 28
where now −
¯ ( ) = 2 − 1 + 2 2
(B.16a)
¡ ¢ − 2 ¢ 2 ¡ 2 1 + 2 − 2 − ¯ ( ) = 3 2 1 + 3 23 ¡ ¢2 2 2 − − 1 − − 2 2 +12 2 4 4 ½ ¾ 2 2 − 1 + −2 ∗ 1 ¯ ( ) = − 2 ¯ ( ) + 2 2 2 42
(B.16b)
(B.16c)
For short expiries this reduces to n o 2 (B.17a) ∼ 1 + 1−3 + · · · 24
n = 1 −
and for long expiries, (B.17b)
2 {1 + · · · } , =p 3 (1 + 2 )
3−2 8
+ ···
p 3 (1 + 2 ) = {1 + · · · }
o
for ¿ 1
for À 1
Since → 0 for long expiries, the smile naturally flattens for long dated options under the -SABR 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 [14], [16]. Most backbone functions ( ) have (min ) = 0 at some point min which acts as a diffusive barrier. It is then natural to take ˜ ( ) ≥ min . For example, with a shifted CEV backbone, ( ) = ( + ) , the barrier is at min = −. Usually there are no diffusive barriers for large , so max is usually picked large enough so that it’s 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 [14], 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
In [14] 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) = ( − ) as → 0 29
where (C.3d)
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 [14]. With these methods, the numerical solution satisfies (C.5a) (C.5b)
( ) + min ( ) +
Z
max
min Z max
( ) + ( ) = 1 ( ) + max ( ) =
min
exactly, so the combined probability totals unity and ˜ ( ) is a Martingale to within round-off error. Of course, if the probability ( ) at the upper boundary is significant, max should be increased. 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 [14] 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 [26], [27]. The -functions at the boundaries are surprising at first sight. To understand how it can arise naturally, and how it 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 [24]. Nevertheless, a boundary layer analysis in [14] shows that in the limit → 0, the probability density ( ) obeys absorbing boundary 30
conditions at min + , and the total probability in min min + remains finite as and increases according to C.4a-C.4b. Thus, in the limit → 0, we have a finite probability in an infinitesimal region, matching our idea of a delta function. So apparently, delta functions just represents a thin layer which could be resolved by finer scale modeling. REFERENCES [1] Hagan, P. S., Kumar, D., Lesniewski, A.S., and Woodward, D.E. (2002). Managing smile risk. Wilmott Magazine, 2002: 84-108. [2] Obloj J. (2008). Fine-tune your smile: Correction to Hagan et al. Wilmott Magazine, May 2008: 102-109. [3] Antonov, A., Konikov, M., and Spector, M., (2013). SABR spreads its wings. Risk Magazine, 2013: 58-63. [4] Antonov, A. and Spector, M., (2012). Advanced analytics for the SABR model. SSRN, March 2012: 2026350. [5] Andreasen J. and Huge, B.N. (2013) Expanded forward volatility. Risk Magazine, Jan 2013: 101-107. [6] 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. [7] Hagan, P. S., Lesniewski, A.S., and Woodward, D.E. (2017). Effective media analysis for stochastic volatility models. Wilmott Magazine, to appear. [8] Dean, D.S., Drummond, I.T., Horgan, R.R. and Lefevre, A. (2004) Perturbation theory for the effective diffusion constant in a medium of random scatterer. J. Phys. A. 37, 2004: 10459. [9] Chen, G., Che, H.-A. and Gu, G.-Q. (1994) Perturbation expansion method and effective medium approximation for effective conductivity of nonlinear nomposite media, Com Theor Physics 22 no 3, 1994: 265-272. [10] Walsh, J.B. , Brown, S.R. and Durham,W.B. (1997) Effective medium theory with spatial correlation for flow in a fracture, J. Geophys. Res. 102 no 22 Oct 1997: 587-594. [11] Maurel, A. and Pagn, V. (2008) Effective propagation in a perturbed periodic structure, Phys Rev B 78 no 5 2008: 052301 [12] Ammari, H. and Kang, H. (eds) (2006) Inverse Problems, Multi-scale Analysis and Effective Medium Theory, Contemporary Math. 408 AMS, Providence [13] Hagan, P. S., Kumar, D., Lesniewski, A.S., and Woodward, D.E. (2016). Universal smiles. Wilmott Magazine, July 2016: 40-55. [14] Hagan, P. S., Kumar, D., Lesniewski, A.S., and Woodward, D.E. (2013). Arbitrage Free SABR. Wilmott Magazine, January, 2013: 1-16. [15] Balland, P. and Q. Tran, Q. (2013). SABR goes normal. Risk Magazine, 2013: 76-81. [16] Hagan, P. S., Lesniewski, A.S., and Woodward, D.E. (2017). Managing vol surfaces. Wilmott Magazine, to appear. [17] Andersen, L.B.G. and Piterbarg, V.V. (2007). Moment explosions in stochastic volatility models. Finance and Stochastics, 2007:11:29—50. [18] Henry-Labordère, P. (2009) .Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing (2009) Taylor & Francis Boca Raton [19] Kennedy, J.E. and Pham, D. (2014) On the approximation of the SABR with mean reversion model: A probabilistic approach, App Math Fin 21 no 5, 2014:451-481 [20] Zhang, S., Mazzucato, A.L., and Victor Nistor,V. (2016). Heat kernels, solvable Lie groups, and the mean reverting SABR stochastic volatility model, arXiv:1605.03097 [21] Kadanoff, L. P. (2000) Statistical Physics: statics, dynamics and renormalization, 2000, World Scientific, Singapore [22] Feller, W. (1971) An Introduction to Probability Theory and Its Applications, Vol. 2, 1997 Wiley, New York [23] Protter, M.H. and Weinberger, H.F. (1984) Maximum Principles in Differential Equations, 1984 Springer, New York [24] Sobczyk, K. (2001) Stochastic Differential Equations, 2001 Berlin:Springer [25] Neu, J. C. (1978) Nonlinear oscillations in discrete and continuous systems, Thesis, California Institute of Technology, 1978. [26] Dupire, B. (1994) Pricing with a smile, Risk 7 1994: 18-20. [27] Derman, E. and Kani, I. (1994) Riding a smile, Risk 7 1994: 139-145.
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