Managing Smile Risk Patrick S

Home , Forward price, Implied volatility, Local volatility, Volatility smile

wilm003.qxd 7/26/02 7:05 PM Page 84

Managing Smile Risk Patrick S. Hagan*, Deep Kumar†, Andrew S. Lesniewski‡, and Diana E. Woodward§

Abstract the smile. We apply the SABR model to USD interest rate options, and Market smiles and skews are usually managed by using local volatility find good agreement between the theoretical and observed smiles. models a la Dupire. We discover that the dynamics of the market smile pre- Key words. smiles, skew, dynamic hedging, stochastic vols, volga, dicted by local vol models is opposite of observed market behavior: when vanna the price of the underlying decreases, local vol models predict that the smile shifts to higher prices; when the price increases, these models predict that the smile shifts to lower prices. Due to this contradiction between 1 Introduction model and market, delta and vega hedges derived from the model can be unstable and may perform worse than naive Black-Scholes’ hedges. European options are often priced and hedged using Black’s model, or, To eliminate this problem, we derive the SABR model, a stochastic equivalently, the Black-Scholes model. In Black’s model there is a one-to- volatility model in which the forward value satisfies one relation between the price of a European option and the volatility

ˆ ˆβ parameter σB . Consequently, option prices are often quoted by stating dF = aˆF dW1 the implied volatility σ , the unique value of the volatility which yields the ˆ = ˆ B da νadW2 option’s dollar price when used in Black’s model. In theory, the volatility ˆ and the forward F and volatility aˆ are correlated: dW1dW2 = ρdt. We use σB in Black’s model is a constant. In practice, options with different singular perturbation techniques to obtain the prices of European strikes K require different volatilities σB to match their market prices. See options under the SABR model, and from these prices we obtain explicit, figure 1. Handling these market skews and smiles correctly is critical to closed-form algebraic formulas for the implied volatility as functions of fixed income and foreign exchange desks, since these desks usually have today’s forward price f = Fˆ(0 ) and the strike K. These formulas immedi- large exposures across a wide range of strikes. Yet the inherent contra- ately yield the market price, the market risks, including vanna and volga diction of using different volatilities for different options makes it diffi- risks, and show that the SABR model captures the correct dynamics of cult to successfully manage these risks using Black’s model.

*[email protected]; Bear-Stearns Inc, 383 Madison Ave, New York, NY 10179 †BNP Paribas; 787 Seventh Avenue; New York NY 10019 ‡BNP Paribas; 787 Seventh Avenue; New York NY 10019 §Societe Generale; 1221 Avenue of the Americas; New York NY 10020

84 Wilmott magazine wilm003.qxd 7/26/02 7:05 PM Page 85

TECHNICAL ARTICLE 4

The development of local volatility models by Dupire [2], [3] and Derman- M99 Eurodollar option Kani [4], [5] was a major advance in handling smiles and skews. Local 30 volatility models are self-consistent, arbitrage-free, and can be calibrated to precisely match observed market smiles and skews. Currently these mod- 25 els are the most popular way of managing smile and skew risk. However, as we shall discover in section 2, the dynamic behavior of smiles and skews predicted by local vol models is exactly opposite the behavior observed in 20 the marketplace: when the price of the underlying asset decreases, local vol

models predict that the smile shifts to higher prices; when the price increas- Vol (%) 15 es, these models predict that the smile shifts to lower prices. In reality, asset prices and market smiles move in the same direction. This contradiction 10 between the model and the marketplace tends to de-stabilize the delta and vega hedges derived from local volatility models, and often these hedges perform worse than the naive Black-Scholes’ hedges. 5 To resolve this problem, we derive the SABR model, a stochastic 92.0 93.0 94.0 95.0 96.0 97.0 volatility model in which the asset price and volatility are correlated. Strike Singular perturbation techniques are used to obtain the prices of Fig. 1.1 Implied volatility for the June 99 Eurodollar options. Shown are European options under the SABR model, and from these prices we close-of-day values along with the volatilities predicted by the SABR model. obtain a closed-form algebraic formula for the implied volatility as a Data taken from Bloomberg information services on March 23, 1999 function of today’s forward price f and the strike K. This closed-form for-

mula for the implied volatility allows the market price and the market Here the expectation E is over the forward measure, and “ |F0” can be inter- risks, including vanna and volga risks, to be obtained immediately from pretted as “given all information available at t = 0.” Martingale pricing the- Black’s formula. It also provides good, and sometimes spectacular, fits to ory [6-9] also shows that the forward price Fˆ(t ) is a Martingale under this the implied volatility curves observed in the marketplace. See Figure 1.1. measure, so the Martingale representation theorem shows that Fˆ(t ) obeys More importantly, the formula shows that the SABR model captures the correct dynamics of the smile, and thus yields stable hedges. dFˆ = C (t, ∗ ) dW, Fˆ(0 ) = f, (2.1c)

for some coefficient C (t, ∗ ), where dW is Brownian motion in this meas- 2 Reprise ure. The coefficient C (t, ∗ ) may be deterministic or random, and may depend on any information that can be resolved by time t. This is as far as

Consider a European call option on an asset A with exercise date tex , settle- the fundamental theory of arbitrage free pricing goes. In particular, one ment date tset , and strike K. If the holder exercises the option on tex , then on cannot determine the coefficient C (t, ∗ ) on purely theoretical grounds. the settlement date tset he receives the underlying asset A and pays the Instead one must postulate a mathematical model for C (t, ∗ ). strike K. To derive the value of the option, define Fˆ(t ) to be the forward European swaptions fit within an indentical framework. Consider a

price of the asset for a forward contract that matures on the settlement European swaption with exercise date tex and fixed rate (strike) Rfix. Let ˆ date tset , and define f = F(0 ) to be today’s forward price. Also let D(t ) be Rs(t ) be the swaption’s forward swap rate as seen at date t, and let ˆ the discount factor for date t; that is, let D(t ) be the value today of $1 to be R0 = Rs(0 ) be the forward swap rate as seen today. In [9] Jamshidean shows delivered on date t. Martingale pricing theory [6-9] asserts that under the that one can choose a measure in which the value of a payer swaption is “usual conditions,” there is a measure, known as the forward measure, ˆ + under which the value of a European option can be written as the expect- Vpay = L0E [Rs(tex ) −Rfix] |F0 , (2.2a) ed value of the payoff. The value of a call options is and the value of a receiver swaption is = ˆ − + | Vcall D(tset ) E [F(tex ) K ] F0 , (2.1a) ˆ + Vrec = L0E [Rfix − Rs(tex )] |F0 and the value of the corresponding European put is (2.2b) ≡ Vpay + L0[Rfix − R0]. ˆ + Here the level L0 is today’s value of the annuity, which is a known quanti- Vput = D(tset ) E [K − F (tex )] |F0 (2.1b) ty, and E is the expectation over the level measure of Jamshidean [9]. In W ≡ Vcall + D(tset )[K − f ]. Appendix A it is also shown that the PV01 of the forward swap; like the

Wilmott magazine 85 wilm003.qxd 7/26/02 7:05 PM Page 86

ˆ 0.28 discount factor rate Rs(t ) is a Martingale in this measure, so once again

ˆ ˆ dRs = C(t, ∗ ) dW, Rs(0 ) = R0, (2.2c) 0.26 where dW is Brownian motion. As before, the coefficient C (t, ∗ ) may be 1m deterministic or random, and cannot be determined from fundamental 0.24 theory. Apart from notation, this is identical to the framework provided 3m

by equations (2.1a–2.1c) for European calls and puts. Caplets and floor- Vol 6m lets can also be included in this picture, since they are just one period 0.22 payer and receiver swaptions. For the remainder of the paper, we adopt 12m the notation of (2.1a–2.1c) for general European options. 2.1 Black’s model and implied volatilities. To go any further 0.20 requires postulating a model for the coefficient C (t, ∗ ). In [10], Black pos- ˆ tulated that the coefficient C (t, ∗ ) is σBF (t ), where the volatilty σB is a ˆ 0.18 constant. The forward price F (t ) is then geometric Brownian motion: 80 90 100 110 120 Strike ˆ ˆ ˆ dF = σBF (t ) dW, F (0 ) = f. (2.3)

Fig. 2.1 Implied volatility σB (K ) as a function of the strike K for 1 month, 3 month, Evaluating the expected values in (2.1a, 2.1b) under this model then 6 month, and 12 month European options on an asset with forward price 100. yields Black’s formula,

Vcall = D (tset ){ f N (d1 ) −KN (d2)}, (2.4a) implied volatility at the barrier K2 , or some combination of the two to price this option? Clearly, this option cannot be priced without a single, self-consistent, model that works for all strikes without “adjustments.”

Vput = Vcall + D(tset )[K − f ], (2.4b) The second problem is hedging. Since different models are being used for different strikes, it is not clear that the delta and vega risks calculated at where one strike are consistent with the same risks calculated at other strikes. ± 1 2 For example, suppose that our 1 month option book is long high strike log f/K 2 σB tex d1,2 = √ , (2.4c) options with a total risk of +$1MM, and is long low strike options with a σ t B ex of −$1MM. Is our is our option book really -neutral, or do we have for the price of European calls and puts, as is well-known [10], [11], [12]. residual delta risk that needs to be hedged? Since different models are All parameters in Black’s formula are easily observed, except for the used at each strike, it is not clear that the risks offset each other.

volatility σB . An option’s implied volatility is the value of σB that needs to Consolidating vega risk raises similar concerns. Should we assume parallel be used in Black’s formula so that this formula matches the market price or proportional shifts in volatility to calculate the total vega risk of our

of the option. Since the call (and put) prices in (2.4a – 2.4c) are increasing book? More explicitly, suppose that σB is 20% at K = 100 and 24% at functions of σB , the volatility σB implied by the market price of an option K = 90, as shown for the 1m options in Figure 2.1 Should we calculate vega is unique. Indeed, in many markets it is standard practice to quote prices by bumping σB by, say, 0.2% for both options? Or by bumping σB by 0.2% for in terms of the implied volatility σB ; the option’s dollar price is then the first option and by 0.24% for the second option? These questions are recovered by substituting the agreed upon σB into Black’s formula. critical to effective book management, since this requires consolidating The derivation of Black’s formula presumes that the volatility σB is a the delta and vega risks of all options on a given asset before hedging, so constant for each underlying asset A. However, the implied volatility that only the net exposure of the book is hedged. Clearly one cannot needed to match market prices nearly always varies with both the strike answer these questions without a model that works for all strikes K.

K and the time-to-exercise tex . See Figure 2.1. Changing the volatility σB The third problem concerns evolution of the implied volatility curve means that a different model is being used for the underlying asset for σB(K ). Since the implied volatility σB depends on the strike K, it is likely to each K and tex . This causes several problems managing large books of also depend on the current value f of the forward price: σB = σB( f, K ). In options. this case there would be systematic changes in σB as the forward price f of The first problem is pricing exotics. Suppose one needs to price a call the underlying changes See Figure 2.1. Some of the vega risks of Black’s

option with strike K1 which has, say, a down-and-out knock-out at model would actually be due to changes in the price of the underlying K2 < K1 . Should we use the implied volatility at the call’s strike K1 , the asset, and should be hedged more properly (and cheaply) as delta risks.

86 Wilmott magazine ^ 87 4 . a (2.7) (2.8) (2.9) (2.10) . +··· . From (2.8), f 2 , the implied ) f K to − f ( ) ) . ] ] f . , the implied volatility f K K = increases + + +···} to ) 0 f f So supppose that thetoday f 1 [ [ 0 +···} { ( . 1 2 1 2 ) f ˆ 1 ( ( F 0 { f )

loc loc to to some new value 0 σ σ f − 0 f f decreases 1 , − 24 0 ˆ + f F of the volatility implied curve, which 2 K + ˆ ( ( FdW TECHNICAL ARTICLE TECHNICAL 0 ; if 1 0 B B direction direction as the price of the underlying asset ) σ ˆ σ F left ( and re-calculating the option prices enables = = ] Local volatility models predict that the market loc f , given that the current value of the forward ) ) K . σ ˆ K F f ( + , = f K loc ˆ opposite [ F ( right σ B d only: 1 2 σ ˆ wrong wrong dynamics and the implied volatility curve seen in the market- F 0 direction as the underlying. f loc σ . Calibrating the model to the market clearly requires ) = same K and the current forward price ) ( 0 f K B , σ . In particular, if the forward price f K ( B σ The behavior of local volatility models can be largely understood by To illustrate the problem, consider the special case in which the local the which in case special the consider problem, the illustrate To curve curve moves to the smile/skew moves in the place is for for an option with strike In [13] and [14] singular perturbationIn and [13] methods[14] used were analyze to this by given are prices put and call thatEuropean found was it There model. with (2.4a-2.4c) volatility Black’s formula the implied On the right hand side, the first term dominates the solution and thesecond term provides a much smaller correction The omitted terms are very less than small, usually 1% of the first term. examining the firstdepends on bothvolatility in term (2.8). The implied the strike be to choosing the volatility local Suppose predictions. its examine us let calibrated, is model thethat Now that the forward value changes from and vega risks for all options, so these risks can be consolidated across perturbing Finally, strikes. curve volatility thatsee thethat predicts implied model the new we (2.9) is price is one to determine how the implied volatilites change with changes in thein withchanges change volatilites theimplied how determine to one underlying asset price. Thus, the local volatility model thus provides smiles market of the presence in options hedging and pricing methodof and skews. It is perhaps the most popular method of managing exotic equity and foreign exchange options. the Unfortunately, local volatility model predicts the volatility curve moves to the leads to inaccruate and often unstable hedges. and often inaccruate unstable leads to is a function of vol forward price is This is opposite to typical market behavior, in which smiles and skews in the move ex as t . In ) , for , (2.6) ˆ ex F 2 (2.5c) (2.5a) (2.5b) ex t , t for for all t ( 1 ex C t J . Instead one Instead . ˆ ,... F B 3 σ , 2 to = ) , f j , ∗ . Re-writing Re-writing . f , one should obtain and exercise dates ) , = ) ˆ t F ( and exercise date ˆ K = ) F , t C ) , 0 K j ( ez t ) ( 0 ( t ˆ C ( F K | ˆ loc F < + = σ − ] are taken to be piecewise con- t f K , C ) ( J 1 ez ez < − ) ˆ t t F ) 1 , , set t − < t > ex j ( ex ( t t t t ( ˆ D loc ˆ FdW F to reproduce the option prices at theprices option reproduce to σ [ An apparent solution to these problems ) + for ˆ ) for for F ˆ E , F t put ( ) to to reproduce the option prices at ( ) V until these theoretical prices match theactu- match thesetheoreticalprices until 2 is Markovian: Markovian: is ) set loc ( loc ) ) t ) ) ˆ ≡ σ C ˆ F ( σ ˆ ˆ ˆ F F F F ( D ( ) ( ( , ) = ) ) ; for example, for 1m, 2m, for 3m, from 6m, example, ; and for 12m t 1 j J 1 ˆ ( ( loc ( ( = F ( loc loc loc σ d loc σ σ σ ,... σ has been obtained by calibration, the local volatility call 3 ex V = = = ) t ˆ ) ) ) , F ˆ ˆ ˆ 2 F F F ex , magazine t t , , , ( t t t , ( ( ( 1 ex loc then yields the “local volatility model,” where the forward t loc loc loc σ ˆ directly from the marketplace by “calibrating” the local volatili- thelocal “calibrating” by themarketplace from directly σ σ σ , then calibrates then, calibrates , and evaluates F ) ) K ) ˆ ˆ ˆ F F F , , , , and so forth. This calibration process can be greatly simplified by t t t In calibration, one starts with a given local volatility function 2.2 Local models. volatility Once K ( ( ( loc loc loc today. Commonly today. the local vols ty model to market prices of liquid European options. prices of liquid market ty model to is provided by the local volatility model of Dupire [2], which is also attrib- also is which [2], Dupire of model volatility local the by provided is uted to Derman [4], [5]. In an insightful work, Dupire essentially argued coefficientthe setting in bold to was Black that strikes strikes stant in time: stant without “adjustment.” Prices of exotic options can now be calculated from this model without ambiguity. This model yields consistent delta σ σ practice liquid markets usually exist only for options with specific exercise dates σ using the results in [13] and [14]. There we solve to obtain the prices of European options under the local volatility model (2.5a–2.5c), and from volatil- theimplied for formulas algebraic explicit obtain we theseprices models. ity of thevol local al market prices of the option for each strike to to obtain the theoretical prices of the options; one then varies the local function volatility should only assume thatassume only should price of the asset is model is a single, self-consistent model which correctly reproduces the market prices of calls (and puts) for all strikes in the forward measure. Dupire argued that of instead theorizing about the unknown local volatility function all One first calibrates Wilmott wilm003.qxd 7/26/02 7:05 PM Page 87 wilm003.qxd 7/26/02 wilm003.qxd 7/26/02 7:05 PM Page 88

To demonstrate the problem concretely, suppose that today’s implied 0.28 volatility is a perfect smile 0.26 0 = + − 2 σB (K ) α β[K f0] (2.11a) 0.24

around today’s forward price f0 . Then equation (2.8) implies that the 0.22

local volatility is Implied Vol 0.20 2 σ (Fˆ ) = α + 3β(Fˆ− f ) +···. (2.11b) loc 0 0.18

f f0 K As the forward price f evolves away from f0 due to normal market fluctu-

ations, equation (2.8) predicts that the implied volatility is Fig. 2.3 Implied volatility σB (K, f ) if the forward price decreases from f0 to f (solid line). = + − 3 − 1 2 + 3 − 2 +··· σB(K, f ) α β K 2 f0 2 f 4 β ( f f0 ) . (2.11c)

The implied volatility curve not only moves in the opposite direction as predicted by the local volatility model. The first term is clearly the risk the underlying, but the curve also shifts upward regardless of whether f one would calculate from Black’s model using the implied volatility from increases or decreases. Exact results are illustrated in Figures 2.2 – 2.4. the market. The second term is the local volatility model’s correction to ˆ There we assumed that the local volatility σloc (F ) was given by (2.11b), the risk, which consists of the Black vega risk multiplied by the predict- and used finite difference methods to obtain essentially exact values for ed change in σB due to changes in the underlying forward price f. In real the option prices, and thus implied volatilites. markets the implied volatily moves in the opposite direction as the direc- Hedges calculated from the local volatility model are wrong. To see tion predicted by the model. Therefore, the correction term needed for

this, let BS( f, K,σB, tex ) be Black’s formula (2.4a–2.4c) for, say, a call real markets should have the opposite sign as the correction predicted by option. Under the local volatility model, the value of a call option is the local volatility model. The original Black model yields more accurate hedges given by Black’s formula than the local volatility model, even though the local vol model is self-consistent across strikes and Black’s model is inconsistent.

Vcall = BS( f, K,σB(K, f ), tex ) (2.12a) Local volatility models are also peculiar theoretically. Using any func- ˆ tion for the local volatility σloc (t, F ) except for a power law, with the volatility σB(K, f ) given by (2.8). Differentiating with respect to f yields the risk C(t, ∗ ) = α(t ) Fˆβ , (2.13) ∂V ∂BS ∂BS ∂σ (K, f ) ≡ call = + B . (2.12b) ∂f ∂f ∂σB ∂f ˆ ˆβ ˆ ˆ1−β σloc (t, F ) = α(t ) F /F = α(t ) /F , (2.14)

0.28 0.28

0.26 0.26

0.24 0.24

0.22 0.22 Implied Vol Implied Vol 0.20 0.20

0.18 0.18 f K 0 f0 f K

Fig. 2.2 Exact implied volatility σB (K, f0 ) (solid line) obtained from Fig. 2.4 Implied volatility σB (K, f ) if the forward prices increases from ˆ the local volatility σloc (F ) (dashed line): f0 to f (solid line).

88 Wilmott magazine ^ 89 4 . ) , this (2.18) f z (2.17c) (2.16c) z (2.17a) (2.16a) (2.16a) (2.17b) is then is ( x = ) +··· f K , · ex t . K (

B 2 σ ν 2 +··· β +··· ρ 2 . 2 ex 3 − K t α 2 / 24 ,

f f − } ρ 2 4 ) 2 2 ν ) 2 , − ] 2 , d , f β log + z ( ρ K . 4 / 3 ex 24 2 ) f − t + N / − 24 β 2 ) B K − 2 1 − K ρ β z ( σ 1920 [ 1 − 2 log ( 1 2 ) − ex

) 1 t 2 − + ( / 1 + set ) z + ) ± + 1 √ t ρβνα d ) ( B ρ β ( K K 1 is given by is given β fK σ 2 D − / /

1 f f ) − ( N ( 1 ) f − + 2 ) α ( K 1 4 β ρβαν { TECHNICAL ARTICLE TECHNICAL f , 1 ) f log fK − α call ( log 1 + 4 1 ( ( B V set 2 t = f ) σ β + ν α ( − β = 2 , D 1 24 = − 2 1 ) = 1 log d ) ( α put = z f V fK = , + ( f call ) 1 ( 2 V z ) B ( σ 2 x β / ) 24 = − β − 1 1 ( ( ATM )

σ + is defined by fK ) ) ( f 1 z ,

( K = · x ( The SABR model (2.15a–2.15c) is analyzed in Appendix B. There sin- B σ where the implied volatility volatility where the implied For For the special case of at-the-money options, options struck at with Here and hedged hedged by buying and selling options as whether a the market would be if complete these risks matter were not hedged of routine, so is a moot question. gular perturbationare used obtain techniques to the prices of European volatility implied options’ the prices, these From options. obtained. The upshot of this analysis is that under the SABR model, the Black’s formula, by price of European options is given formula reduces to formula . In . We ˆ α (2.15c) (2.15a) (2.15b) volatility into into the ˆ and F ˆ F SABR model now now yields the ˆ α dynamic SABR model -hedging is used to is itself a stochastic

ˆ α f α = = . In practice, vega risks are vega practice, In . ) ˆ ) F 0 . 0 ( ˆ F dt ˆ α( ρ = 2 , , 1 2 dW dW 1 dW β ˆ ˆ α F dW ν α The failure of the model means local volatility = =ˆ , where the “volatility” ˆ ˆ β α F ˆ d F d ˆ α model,” model,” which has become known as the to be (the vega risk) cannot be hedged by buying or selling the magazine ) ˆ ∗ α αβρ , t ( C of an asset which has European options at several different exer- 2.3 The model. SABR It has been authorsmod- claimed many by volatility that stochastic As written, the SABR model may or may not fit the observed . More importantly, it predicts the correct dynamics of the implied . Although intrinsic length scales are theoretically possible, it is diffi- ex ˆ this model, the are forward price and volatility model. That is, the model becomes in inhomogeneous the forward price process. Choosing the simplest reasonable process for under the by: forward processes are correlated measure, where the two shall find that the SABR model can be used fitto accurately the implied date single in exercise theany observed for curves marketplace volatility neutralize the risks to changes in thein price thechanges risksneutralize to t that we cannot use a Markovian model based on a single Brownian motion to manage our smile risk. Instead of making the a develop to model choose we motion, non-Brownian non- on it basing or Markovian, that note markets most we select the model. To second factor factor, two experience both relatively quiescent and relatively chaotic periods. This that is but not is constant, itself suggests volatility function a of random time. Respecting the preceding discusion, we choose the unknown coefficient introduces an intrinsic “length scale” for the forward price F cult understand to the financial origin and meaning of these scales [15], and one naturally wonders whether such scales should be introduced a model without specificinto theoretical justification. underlying asset. However, vega risk can be hedged by buying or selling or buying by hedged be can risk vega However, asset. underlying options on the asset in exactly the same thatway “stochastic- surface cise dates; such markets include foreign exchange options and most the surfaces requires volatility options. Fitting equity which is discussed in an Appendix. els are models of markets, because incomplete the volatility stochastic risk cannot be hedged. This is not true. It is true changes in that the risk to volatility curves. This curves. volatility makes the SABR model an effective means man- to theage smile risk in where markets each asset only has a single exercise these include the markets swaption and caplet/floorletdate; markets. Many other stochastic volatility models have been proposed, for example for proposed, been have models volatility other stochastic Many the [19]. [18], SABR However, [17], [16], model has the virtue of being the in model which is homogenous volatility stochastic simplest Wilmott wilm003.qxd 7/26/02 7:05 PM Page 89 wilm003.qxd 7/26/02 wilm003.qxd 7/26/02 7:05 PM Page 90

These formulas are the main result of this paper. Although it appears 22% formidable, the formula is explicit and only involves elementary trigno- 20% β = 0 metric functions. Implementing the SABR model for vanilla options is very 18% easy, since once this formula is programmed, we just need to send the 16% options to a Black pricer. In the next section we examine the qualitative 14% behavior of this formula, and how it can be used to managing smile risk. The complexity of the formula is needed for accurate pricing. Implied vol 12% Omitting the last line of (2.17a), for example, can result in a relative error 10% that exceeds three per cent in extreme cases. Although this error term 8% seems small, it is large enough to be required for accurate pricing. The 4% 6% 8% 10% 12% omitted terms “+···” are much, much smaller. Indeed, even though we Fig. 3.1 Backbone and smiles for β = 0. As the forward f varies, the implied have derived more accurate expressions by continuing the perturbation volatiliity σB ( f, f ) of ATM options traverses the backbone (dashed curve). Shown are expansion to higher order, (2.17a – 2.17c) is the formula we use to value the smiles σB (K, f ) for three different values of the forward. Volatility data from 1 and hedge our vanilla swaptions, caps, and floors. We have not imple- into 1 swaption on 4/28/00, courtesy of Cantor-Fitzgerald. mented the higher order results, believing that the increased precision of the higher order results is superfluous. 22% There are two special cases of note: β = 1, representing a stochastic 20% log normal model), and β = 0, representing a stochastic normal model. β = 1 The implied volatility for these special cases is obtained in the last sec- 18% tion of Appendix B. 16% 14%

Implied vol 12% 3 Managing Smile Risk 10% 8% The complexity of the above formula for σB(K, f ) obscures the qualita- 4% 6% 8% 10% 12% tive behavior of the SABR model. To make the model’s phenomenology and dynamics more transparent, note that formula (2.17a – 2.17c) can Fig. 3.2 Backbone and smiles as above, but for β = 1. be approximated as

= α − 1 − − σB(K, f ) − 1 (1 β ρλ) log K/f Let us now consider the implied volatility σB(K, f ) in detail. The first f 1 β 2 − (3.1a) factor α/f 1 β in (3.1a( is the implied volatility for at-the-money (ATM) 1 + (1 − β)2 + (2 − 3ρ2 ) λ2 log2 K/f +···, options, options whose strike K equals the current forward f. So the back- 12 1−β bone traversed by ATM options is essentially σB( f, f ) = α/f for the provided that the strike K is not too far from the current forward f. Here SABR model. The backbone is almost entirely determined by the expo- the ratio nent β, with the exponent β = 0 (a stochastic Gaussian model) giving a ν − steeply downward sloping backbone, and the exponent β = 1 giving a λ = f 1 β (3.1b) α nearly flat backbone. − 1 − − The second term 2 (1 β ρλ) log K/f represents the skew, the measures the strength ν of the volatility of volatility (the “volvol”) com- slope of the implied volatility with respect to the strike K. The 1−β − 1 − pared to the local volatility α/f at the current forward. Although equa- 2 (1 β ) log K/f part is the beta skew, which is downward sloping since tions (3.1a–3.1b) should not be used to price real deals, they are accurate 0 ≤ β ≤ 1. It arises because the “local volatility” αˆ Fˆβ /Fˆ1 =ˆα/Fˆ1−β is a 1 enough to depict the qualitative behavior of the SABR model faithfully. decreasing function of the forward price. The second part 2 ρλlog K/f is As f varies during normal trading, the curve that the ATM volatility the vanna skew, the skew caused by the correlation between the volatility

σB(f, f ) traces is known as the backbone, while the smile and skew refer to and the asset price. Typically the volatility and asset price are negatively the implied volatility σB(K, f ) as a function of strike K for a fixed f. That correlated, so on average, the volatility α would decrease (increase) when is, the market smile/skew gives a snapshot of the market prices for dif- the forward f increases (decreases). It thus seems unsurprising that a ferent strikes K at a given instance, when the forward f has a specific negative correlation ρ causes a downward sloping vanna skew. price. Figures 3.1 and 3.2. show the dynamics of the smile/skew predicted It is interesting to compare the skew to the slope of the backbone. As f by the SABR model. changes to f the ATM vol changes to

90 Wilmott magazine ^ 91 4 β ex t ] (3.3) ··· 1 [ and β = 2 2 ATM β − α 2 σ f 2 ) . β β 24 and with − can be negative or 0 f 1 term is typically less is term typically . ( cite log cite normal mod- =

ex considerations considerations usually is essential for trading 1 t β ] (stochastic (stochastic normal), or + 0 plot of historical observa- historical of plot = usually believe that a nor- a that believe usually 0 +··· 1 ··· [ = β 0

ex = t β =

β a priori log 2 β are stochastic variables, this fit- log log ν + 2 α f ρ 3 24 log 1y into 1y and − ) are usually US interest rate desks that f TECHNICAL ARTICLE TECHNICAL 2 β 1 2 is then needed invert- by whenever found + − α = ) 1 β ( β − 1 to to the implied volatility curve, with − ( ρβαν f ν α 4 1 can be determined from historical observations of the of observations historical from determined be can + log and β (stochastic (stochastic lognormal), as the SABR of instead parameters the original parame- = can be extracted from a from extracted be can ρ from “aesthetic” or other 1 ) ν pairs. Since both β f β Implied Implied volatilities as a function of strike. Shown are the curves , = . The parameter . The parameter f 4% 6% 8% 10% 12% ATM ( β and B

σ ,σ 22% 20% 18% 16% 14% 12%

f Implied vol Implied CIR) models. Proponents (stochastic of β,ρ,ν , log Fig. 3.3 obtained by fitting the SABR model with exponent to to the 1y into 1y swaption vol observed on 4/28/00. As usual, both fits are of Cantor-Fitzgerald. good. Data courtesy equally 1 2 α ,β,ρ, Selecting The exponent The It is usually more convenient to use the at-the-money volatility = ATM ters ters than one or two per cent, it is usually ignored in fitting per cent, it is usually than one or two near zero. Proponents of The exponent The mal model, with its symmetric break-even points, is a more effective tool effective more a is points, break-even symmetric withits model, mal for managing risks, and would claim that term term is small. With this parameterization, fitting the requires fitting SABR model given. In many markets, the ATM volatilities need to be updated fre- say quently, once or twice a while day, the smiles and skews need to be tions of results in ing (2.18) on the fly; this inversion is numerically easy since the markets markets like interest Yen rates, where the forwards “backbone” or selected from “aesthetic considerations.” Equation (2.18) options is of ATM that volatility theshows implied that thebest backbone horizontal believe or natural.” “more being as els represents their market. These proponents often include desks trading of Proponents options. exchange foreign ting procedure can be quite noisy, and as the noisy, ting procedure can be quite trust in CIR models. developed have β σ ) ν 2 , it , ρ K 3 (3.2a) (3.2b) affect . With f − increas- ρ 2 α ( 1 12 moves moves away K . , and then re-fit the 0 +··· is known or has been +··· f = f β far from the money rep- , it is difficult to distin- and correlation − f β f f − cannot be determined by

K β K f β ) ) (vol-gamma) effect. Physically effect. (vol-gamma) trade trade less frequently than at- β with β f − − volga 1 1 ( ( is twice as steep as the slope of rtheis twice as steep varies. From a single market snap- at the current forward price at a given 1 2 ) α, ρ, ν K − K f K − , 1 f

1 have have different effects on the curve: the ( . Note that there is no substantial differ- The exponent B

β 1 ν σ . In particular, − α β 1 β − f = -component of the skew. -component α 1 decreases, so strikes β f β and = α happen more often when the volatility ) ˆ = F

as the strike ) f f ) with appears to appears be to a smile term, but (quadratic) it is domi- , α, ρ, ,

f f controls the curve’s skew, and changing the vol of vol vol of thevol changing and skew, thecurve’s controls K component of skew expands as of skew component , f / ( as a function of ( K ρ B K β B ( ) σ due to the due to 2 B σ magazine f mainly mainly controls the overall height of the curve, changing σ ) , α, ρ, ν f α K log is the smile induced by the by induced thesmile is , ( , the 2 f f B away away from the current forward K ) / as a function of the strike ( σ K B = β K ) , the volatility quotes become more suspect because they become more suspect because are more quotes , the volatility σ 2 f f K − , The last term in (3.1a) also contains two parts. The first part Suppose for the moment that the exponent 3.1 Fitting market data. Figure Figure 3.3 also exhibits a common data quality issue. Options with 1 K log given, given, fitting the SABR model is a straightforward procedure. The ( ( 2 B 1 12 this smile arises because of “adverse selection”: unusually large move- ments of the forward nated by the downward sloping beta skew, and, at reasonable strikes strikes reasonable at and, skew, beta sloping thedownward by nated the volatility smile in similar ways. They both cause a downward sloping skew in ence in the quality of the fits, despite the presence of market noise. This noise. market of presence the fits,despite the of quality the in ence matches our general experience: market smiles can be fit equally well with any specific value of guish between the two parameters. This is demonstrated by figure 3.3. There fitwe the SABR parameters smile Near Near likely likely to be out-of-date and not represent bona fide offers to buy or sell options. so the slope of the backbone just just modifies this skew somewhat. The second part parameter parameter selected. selected. Taking a snapshot of the market yields the implied volatility resent, on average, high volatility environments. high volatility resent, on average, σ β seper- widely the of Because exhibits. curve the smile much how controls ated roles these parameters the play, fitted parameter values tend to be noise. in the amounts of market presence of large very even stable, parameters parameters thecorrelation es, and less often when shot of three parameters the-money and near-the-money options. as Consequently, λ fitting a market smile since this would clearly amount to “fitting the noise.” strikes from Wilmott wilm003.qxd 7/26/02 7:05 PM Page 91 wilm003.qxd 7/26/02 wilm003.qxd 7/26/02 7:05 PM Page 92

U99 Eurodollar option Z99 Eurodollar option 30 20

25 18

20 Vol (%) Vol Vol (%) Vol 16 15

10 14 93.0 94.0 95.0 96.0 97.0 92.0 93.0 94.0 95.0 96.0 Strike Strike

Fig.3.4 Volatility of the Sep 99 EDF options Fig. 3.5 Volatility of the Dec 99 EDF options

H00 Eurodollar option updated infrequently, say once or twice a month. With the new parame- 24 terization, σATM can be updated as often as needed, with ρ, ν (and β) updated only as needed. 22 Let us apply SABR to options on US dollar interest rates. There are three key groups of European options on US rates: Eurodollar future 20 options, caps/floors, and European swaptions. Eurodollar future options

are exchange-traded options on the 3 month Libor rate; like interest rate (%) Vol 18

futures, EDF options are quoted on 100(1 − rLibor ). Figure 1.1 fits the 16 SABR model (with β = 1) to the implied volatility for the June 99 con- β = 1 tracts, and figures 3.4–3.7 fit the model (also with ) to the implied 14 volatility for the September 99, December 99, and March 00 contracts. 92.0 93.0 94.0 95.0 96.0 97.0 All prices were obtained from Bloomberg Information Services on March Strike 23, 1999. Two points are shown for the same strike where there are quotes for both puts and calls. Note that market liquidity dries up for the Fig. 3.6 Volatility of the Mar 00 EDF options later contracts, and for strikes that are too far from the money. Consequently, more market noise is seen for these options. the skew is overwhelmed by the smile for short-dated options, but is Caps and floors are sums of caplets and floorlets; each caplet and important for long-dated options. This picture is confirmed tables 3.1 floorlet is a European option on the 3 month Libor rate. We do not con- and 3.2. These tables show the values of the vol of vol ν and correlation ρ sider the cap/floor market here because the broker-quoted cap prices obtained by fitting the smile and skew of each “m into n” swaption, must be “stripped” to obtain the caplet volatilities before SABR can be again using the data from April 28, 2000. Note that the vol of vol ν is very applied. high for short dated options, and decreases as the time-to-exercise A m year into n year swaption is a European option with m years to the increases, while the correlations starts near zero and becomes substan- exercise date (the maturity); if it is exercised, then one receives an n year tially negative. Also note that there is little dependence of the market swap (the tenor, or underlying) on the 3 month Libor rate. See Appendix skew/smile on the length of the underlying swap; both ν and ρ are fairly A. For almost all maturities and tenors, the US swaption market is liquid constant across each row. This matches our general experience: in most for at-the-money swaptions, but is ill-liquid for swaptions struck away markets there is a strong smile for short-dated options which relaxes as from the money. Hence, market data is somewhat suspect for swaptions the time-to-expiry increases; consequently the volatility of volatility ν is that are not struck near the money. Figures 3.8–3.11 fits the SABR model large for short dated options and smaller for long-dated options, regard- (with β = 1) to the prices of m into5Y swaptions observed on April 28, less of the particular underlying. Our experience with correlations is less 2000. Data supplied courtesy of Cantor-Fitzgerald. clear: in some markets a nearly flat skew for short maturity options We observe that the smile and skew depend heavily on the time-to- develops into a strongly downward sloping skew for longer maturities. In exercise for Eurodollar future options and swaptions. The smile is pro- other markets there is a strong downward skew for all option maturities, nounced for short-dated options and flattens for longer dated options; and in still other markets the skew is close to zero for all maturities.

92 Wilmott magazine ^ 93 4 , K (3.5) are very sta- ν 12% and ρ is given by equations ) ) ex t , , re-valuing the option after ) α f , K α, β, ρ, ν ( B ; f , ,σ may may need to be updated every few K K ( , α B f 1Y into 5Y 5Y into 5Y TECHNICAL ARTICLE TECHNICAL σ ( BS ≡ ) , the “backbone” is downward sloping, so theso sloping, downward theis , “backbone” = f risk for the moment, and calculate the other 1 , be Black’s formula (2.4a–2.4c) for, say, a call K call

) ( V . This predicted dynamics of the smile matches B f ex β< σ t , Volatilities of 5 year into 5 year swaptions 5 year into of 5 year Volatilities B Volatilities of 1 year into 1 year swaptions 1 year into of 1 year Volatilities ,σ K 4% 6% 8% 10% 4% 6% 8% 10% 12% , or, equivalently, , f ( Fig. 3.9 Fig. 3.10 ATM 17% 16% 15% 14% 13% 12% 18% 17% 16% 15% 14% 13% BS σ is assumed to is be assumed and constant), a to be need given re-fitto everyonly β Let us set aside the Since the SABR model is a single model self-consistent for all strikes shift in the implied volatility curve is not purely horizontal. Instead, this Instead, horizontal. purely not is curve volatility shifttheimplied in curve shifts up and down as the at-the-money point traverses the backbone. Our experience suggests that the parameters market experience. If experience. market (2.17a–2.17c). Differentiating@(2.17a–2.17c). practice footnote:{In risks are calculated by finite differences: valuing the option at amount as the price risks. Let the model, the SABR to value of a call is option. According few weeks. This stability may be because the reproduces the SABR model may This stability weeks. few usual dynamics of smiles and skews. In contrast, thevolatility at-the-money ble ( the risks calculated at one strike are consistent with the risks calculated at withtheriskscalculated consistent are strike one at theriskscalculated be can asset same the on options the all of risksthe Therefore strikes. other be hedged. the and only residual risk needs to added together, where the volatility hours in fast-paced markets. hours in fast-paced same (3.4c) (3.4a) , and (3.4b) ν , ρ changes, the changes, f and fitting without “adjustment,” f α β K = direction and by the = ) ) 0 0 ( ˆ F dt same ˆ α( ρ = 2 , Strike 1 , 2 dW 3M into 5Y After choosing dW 1 dW β ˆ ˆ α F dW ν α M00 Eurodollar option M00 Eurodollar = =ˆ ˆ ˆ F α d d Volatilities of 3 month into 5 year swaption 5 year of 3 month into Volatilities 4% 6% 8% 10% 12% , the SABR model Volatility of the Jun 00 EDF options Volatility magazine 92.0 93.0 94.0 95.0 96.0 97.0 ATM Fig. 3.8

20% 18% 16% 14% 12% σ

24 22 20 18 16 14 Vol (%) Vol or Fig, 3.7 α 3.2. Managing smile risk. so we so can we use this model price to options exotic The without ambiguity. theprice forward thatwhenever predicts also model SABR the implied volatility curve shifts in the fits the smiles and skews observed in the take market us Let . quite money the from well, away especially quotes price of quality the considering self-consistent single, a have we Then enough. fitswell it that granted for model that fits the option prices for all strikes either with Wilmott wilm003.qxd 7/26/02 7:05 PM Page 93 wilm003.qxd 7/26/02 wilm003.qxd 7/26/02 7:05 PM Page 94

10Y into 5Y bumping the forward to α + δ, and then subtracting to determine the 13% risk This saves differentiating complex formulas such as (2.17a–2.17c). with respect to α yields the vega risk, the risk to overall changes in 12% volatility: 11% ∂V ∂BS ∂σ (K, f ; α, β, ρ, ν ) 10% call = · B . (3.6) ∂α ∂σB ∂α 9% 4% 6% 8% 10% 12% This risk is the change in value when α changes by a unit amount. It is Fig. 3.11 Volatilities of 10 year into 5 year options traditional to scale vega so that it represents the change in value when

the ATM volatility changes by a unit amount. Since δσATM = (∂σATM /∂α ) δα, the vega risk is

; ∂V ∂BS ∂σB (K,f α,β,ρ,ν ) TABLE 3.1 vega ≡ call = · ∂α (3.7a) ∂σ (f;α,β,ρ,ν ) ∂α ∂σB ATM VOLATILITY OF VOLATILITY ν FOR EUROPEAN ∂α SWAPTIONS. ROWS ARE TIME–TO–EXERCISE; = COLUMNS ARE TENOR OF THE UNDERLYING SWAP. where σATM ( f ) σB( f, f ) is given by (2.18). Note that to leading order, ∂σB/∂α ≈ σB/α and ∂σATM /∂α ≈ σATM /α, so the vega risk is roughly given by 1Y 2Y 3Y 4Y 5Y 7Y 10Y 1M 76.2% 75.4% 74.6% 74.1% 75.2% 73.7% 74.1% ∂BS σ (K, f ) ∂BS σ (K, f ) 3M 65.1% 62.0% 60.7% 60.1% 62.9% 59.7% 59.5% vega ≈ · B = · B . (3.7b) ∂σ σ ( f ) ∂σ σ ( f, f ) 6M 57.1% 52.6% 51.4% 50.8% 49.4% 50.4% 50.0% B ATM B B 1Y 59.8% 49.3% 47.1% 46.7% 46.0% 45.6% 44.7% 3Y 42.1% 39.1% 38.4% 38.4% 36.9% 38.0% 37.6% Qualitatively, then, vega risks at different strikes are calculated by bump- 5Y 33.4% 33.2% 33.1% 32.6% 31.3% 32.3% 32.2% ing the implied volatility at each strike K by an amount that is propor-

7Y 30.2% 29.2% 29.0% 28.2% 26.2% 27.2% 27.0% tional to the implied volatiity σB(K, f ) at that strike. That is, in using 10Y 26.7% 26.3% 26.0% 25.6% 24.8% 24.7% 24.5% equation (3.7a), we are essentially using proportional, and not parallel, shifts of the volatility curve to calculate the total vega risk of a book of options. Since ρ and ν are determined by fitting the implied volatility curve observed in the marketplace, the SABR model has risks to ρ and ν changing. Borrowing terminology from foreign exchange desks, TABLE 3.2 vanna is the risk to ρ changing and volga (vol gamma) is the risk MATRIX OF CORRELATIONS ρ BETWEEN THE UNDERLY- to ν changing: ING AND THE VOLATILITY FOR EUROPEAN SWAPTONS. ∂V ∂BS ∂σ (K, f ; α, β, ρ, ν ) vanna = call = · B , (3.8a) ∂ρ ∂σB ∂ρ 1Y 2Y 3Y 4Y 5Y 7Y 10Y ∂V ∂BS ∂σ (K, f ; α, β, ρ, ν ) 1M 4.2% –0.2% –0.7% –1.0% –2.5% –1.8% –2.3% volga = call = · B . (3.8b) ∂ν ∂σ ∂ν 3M 2.5% –4.9% –5.9% –6.5% –6.9% –7.6% –8.5% B 6M 5.0% –3.6% –4.9% –5.6% –7.1% –7.0% –8.0% 1Y –4.4% –8.1% –8.8% –9.3% –9.8% –10.2% –10.9% Vanna basically expresses the risk to the skew increasing, and 3Y –7.3% –14.3% –17.1% –17.1% –16.6% –17.9% –18.9% volga expresses the risk to the smile becoming more pronounced. 5Y –11.1% –17.3% –18.5% –18.8% –19.0% –20.0% –21.6% These risks are easily calculated by using finite differences on the

7Y –13.7% –22.0% –23.6% –24.0% –25.0% –26.1% –28.7% formula for σB in equations (2.17a–2.17c). If desired, these risks 10Y –14.8% –25.5% –27.7% –29.2% –31.7% –32.3% –33.7% can be hedged by buying or selling away-from-the-money options.

94 Wilmott magazine ^ 95 4 at α (A.4) (A.2) (A.3a) (A.1c) (A.1b) (A.3b) = ) ) t T , , when the when , t ˆ t α( ˆ . α( . , > f < , which factors α T ) satisfies the for- = A dF dA = ) p α set , ) t for t ) ( by ( t A = dF ˆ , and then specialize then and , F ) D , ) , ) ˆ α( ˆ t F + A t F , ( , , F f = ˆ F C ex [21], [21], [22] in the language α( t , T , ; < = T f AA ; ) ) , (A.1a) ] be the option’s exercise date, exercise option’s the be at ,α t the density = 1 T , p f ( , ( ,α ex 2 , , ˆ ˆ ) f 2 t F dt F t A A t ) , ( [ dW ρ ( t , 2 ˆ < ( p α ) F F dW ν ˆ Let ) , = | . F p ˆ dA F 2 α ( − T α ε 2 + K C ] + 1 2 A εν ˆ α = ˝kker-Planck equation) ˝kker-Planck K − A ε δ( dW = + TECHNICAL ARTICLE TECHNICAL ) F 1 − ) t ( prob < ˆ ) = distinguished limit . This is notationally simpler than working than simpler notationally is This . α f FA ] ˆ β ∞ ˆ d α( F ex dW = ˆ p K FF t F − d ] ) , ( f p F ˆ F F ) [ ( ∞ to to get the answer in terms of the original vari- δ( = F C −∞ ( dF dA 2 ) ν E = 2 ) t A [ ( C p forward variables forward = = [ A ˆ F 2 , ) ρν . This is the −→ F A 2 1 2 , ε be the value of a European call option at date date at option call European a of value the be ,α ε T εν f ) 1 2 ; + , t ( ε ,α = ,α f f V and T , , t t be its strike. Omitting be theits strike. discount factor p ( ( K α, p V . Define the probability density t We first We use thethe simplify forwardto equation option Kolmogorov Let −→ ˆ ˆ α As a function of the of singular perturbation theory. After obtaining the results we replace with in the limit ables. We first analyze the model with a general general a with model the analyze first We ables. with and let ward Kolmogorov equation (the Fo equation Kolmogorov ward out exactly, the value of the option is out exactly, the results to the power law law power the to results the date date with the law power throughout, and the more general result may prove valuable in some future application. pricing problem. Suppose the economy is in state economy is in state state in is economy as is [24], well-known [25], [26]. Here, and throughout, we use subscripts partial derivatives. denote to ε . f . )

is a and ) (3.9) f , one (3.10) . f α f , ˆ risk. It α, ε f

ATM ∂ α, β, ρ, ν ; −→ . f . Then ∂α(σ ˆ , ν 1 α to be small. To small. be to more effective . K ) held fixed. The , ν ( ) = constant constant while ρ B , σ β ATM β σ much ATM ≡ , . Suppose first use we σ ) ν f AMT , , α, β, ρ, ν f ∂α σ α, β, ρ, ν ρ K ; ∂ with ( f ; , B f f , β σ , . As discussed theabove, pre- K change in the implied volatil- implied the in change , f ( K B ( B ATM and the “volvol” “volvol” the and risk from equation (3.9) to risk man- ∂σ σ ˆ ∂σ α

+ B or from (3.10) is a BS ) predicted ν ∂σ ∂ needed to keep , so that from equation (3.9). Then, when the risk one would calculate from Black’s , the volatility curve rises and falls as the curve the volatility

ν α ρ + 1

equilibrated equilibrated over the next few days, one , , to compensate, and to thiscompensate, re-marking would f ρ β BS ∂ , , α ∂ yields the ATM f β< α β α, β, ρ, ν σ ∂ f ; = , f α defined implicitly by (2.18). Differentiating (3.5) , f f call K ( ∂ V B ∂ ∂σ risk and ≡

back to its original value, which would change the

hedges. hedges. As to to change following significant changes in the forward did not immediately change following a change in

B α ATM magazine BS B σ

∂σ ∂ . In addition, if σ f ATM σ + , and analyze f . In these markets, using εν BS caused by the change in the forward f ∂ ∂ B Theoretically Theoretically one should use the The delta risk theexpressed by SABR model depends on whether one Now Now suppose we use the parameterization σ ≡ −→ change change the last last term is just the change in uses the parameterization the parameterization now yields the now would would mark dicted dicted change consists of a sideways movement of the volatility curve in the same direction (and by the same amount) as the change in price ward the for- price function of The delta risk is now the risk to changes in hedge. hedge. For suppose one used The first term is the ordinary at-the-money point up traverses and down the backbone. There may also in changes the due to shape of the to skew/smile be minor changes ν Appendix A. Analysis of the SABR Model A. Appendix options price European to use singular perturbation techniques Here we under the SABR model. Our is analysis based on a expan- small volatility volatility theboth take we where sion, hedges hedges back to their original value. This “hedging chatter” caused by be costly. to can prove delays market changes. changes. Clearly this last term must just cancel out the vertical component of the backbone, leaving only the that this sideways is zero for Note term curve. volatilty implied movement of the volatility model. The second term is the SABR model’s correction to the Differentiating respect to Differentiating respect to age option books. In many markets, however, it may take several days for days several take it may however, markets, option books.age In many volatilities carry out this analysis in a systematic fashion, we re-write re-write carry fashion, we out this in a systematic analysis would be would forced to re-mark consists of the Black vega times the times vega Black the of consists ity Wilmott wilm003.qxd 7/26/02 7:05 PM Page 95 wilm003.qxd 7/26/02 wilm003.qxd 7/26/02 7:05 PM Page 96

See (2.1a). Since 1 2 2 2 2 2 1 2 2 2 Pτ = ε α C ( f ) Pff + ε ρνα C( f ) Pf α + ε ν α Pαα, for τ>0, (A.13a) tex 2 2 p (t, f,α; tex , F, A ) = δ(F − f ) δ(A − α ) + pT(t, f,α; T, F, A ) dT, (A.5) t we can re-write V(t, f,α) as P = α2δ(f − K ), for τ = 0. (A.13b)

; tex ∞ ∞ In this appendix we solve (A.13a), (A.13b) to obtain P (τ, f,α K ), and + V (t, f,α) = [ f − K ] + (F − K ) pT(t, f,α; T, F, A ) dA dF dT. then substitute this solution into (A.12) to obtain the option value t K −∞ (A.6) V (t, f,α). This yields the option price under the SABR model, but the resulting formulas are awkward and not very useful. To cast the results

We substitute (A.3a) for pT into (A.6). Integrating the A derivatives in a more usable form, we re-compute the option price under the normal 2 2 1 2 2 2 ε ρν[A C(F ) p]FA and 2 ε ν [A p]AA over all A yields zero. Therefore our model option price reduces to d Fˆ = σ dW, (A.14a) N tex ∞ ∞ + 1 2 2 2 V (t, f,α) = [ f − K ] + ε A (F − K ) [C (F ) p]FF dF dA dT, 2 t −∞ K (A.7) and then equate the two prices to determine which normal volatility σN needs to be used to reproduce the option’s price under the SABR model. That is, we find the “implied normal volatility” of the option under the where we have switched the order of integration. Integrating by parts SABR model. By doing a second comparison between option prices under twice with respect to F now yields the log normal model

ˆ ˆ ∞ dF = σB FdW (A.14b) 1 tex V (t, f,α) = [ f − K ]+ + ε2C 2(K ) A2p(t, f,α; T, K, A ) dA dT. (A.8) 2 −∞ t and the normal model, we then convert the implied normal volatility to the usual implied log-normal (Black-Scholes) volatility. That is, we quote The problem can be simplified further by defining the option price predicted by the SABR model in terms of the option’s ∞ implied volatility. P (t, f,α; T, K ) = A2p(t, f,α; T, K, A ) dA. (A.9) A.1 Singular perturbation expansion. Using a straightforward per- −∞ turbation expansion would yield a Gaussian density to leading order, Then P satisfies the backward’s Kolmogorov equation [24], [25], [26]

1 1 α − (f−K )2 2 2 2 2 2 2 2 2 2 2 2 Pt + ε α C ( f ) Pff + ε ρνα C( f ) Pf α + ε ν α Pαα = 0, for t < T P = e 2ε α C (K ) τ {1 +···}. (A.15a) 2 2 2 2 (A.10a) 2πε C (K ) τ

Since the “+···” involves powers of ( f − K ) /εαC (K ), this expansion P = α2δ(f − K ), for t = T. (A.10b) would become inaccurate as soon as ( f − K ) C (K ) /C (K ) becomes a significant fraction of 1; i.e., as soon as C ( f ) and C (K ) are significantly dif- Since t does not appear explicitly in this equation, P depends only on the ferent. Stated differently, small changes in the exponent cause much combination T − t, and not on t and T separately. So define greater changes in the probability density. A better approach is to re-cast the series as τ = T − t,τ= t − t. (A.11) ex ex − 2 α − (f K ) {1+···} P = e 2ε2 α2 C 2 (K ) τ (A.15b) 2 2 Then our pricing formula becomes 2πε C (K ) τ τex + 1 and expand the exponent, since one expects that only small changes to V (t, f,α) = [ f − K ] + ε2C 2(K ) P (τ, f,α; K ) dτ (A.12) 2 0 the exponent will be needed to effect the much larger changes in the density. This expansion also describes the basic physics better —- P is where P (τ, f,α; K ) is the solution of the problem essentially a Gaussian probability density which tails off faster or slower

96 Wilmott magazine ^ 97 4 ) (A.20) (A.22) (A.23) z (A.19e) (A.21a) (A.24a) (A.21b) (A.19d) (A.24b) P − α α z z ˆ P P τ, α τ, (α d ) d . ) z ) z ) z 2 ∂ 2 ∂ . ν ν ) ,α 2 2 z ,α 2 z z ε z 2 α ε δ( − . − (τ, + (τ, ) . This is the near identi- 0 ˆ P P 2 K 2 . z ( ex ex = ∂ 1 τ 0 ... τ ∂ C (ερν 0 (ερν 0 , τ 2 2 ) = εα , + z α + ) 2 ) . 0 at z τ ˆ z P K P K = (ε + ( )

P (

at 2 B O C K B B )) τ> C ( B a 2 ∂α 2 C a K ) εα ε by ∂ ε z ( ε z ε α 1 2 1 2 ) ∂ for 1 2 1 2 zC δ( z ) TECHNICAL ARTICLE TECHNICAL + . So it is a Gaussian to leading order. The leading order. . So it is a Gaussian to = α − ) 2 z + ,α − ˆ 0 0 z P + K zz ] + δ( ( ˆ zz α ] − δ(εα P = P C K )

(τ, K 2 = ε ˆ

τ> 2 τ = 2 α P ˆ ˆ 2 − P ∂ P z − ) z ∂α f = 2 f 2 α at [ K for ν [ is the solution of the standard diffusion problem ν 2 P ) 2 2 ˆ = z P − ε ε = + f −→ ) δ( ) + , and then through + a 2 αα ) a a δ( is the solution of ˆ z 2 , z = P α , f is the solution of ∂ f 2 ˆ P , (ε ) ∂α P ) , t 2 t ( ερν O (α ερν α ( , we obtain , we 2 2 ,α ˆ 2 2 V ,α V P z ν z ν 2 − 2 − with ε ε 1 (τ, 1 (τ, 1 2 zz ˆ 1 2 ˆ P P P 1 2 1 2 1 2 + + To leading order To = = = τ τ τ ˆ ˆ P fy transform method [28]. systems Hamiltonian which has proven so powerful in near- where of In terms where Therefore, (A.12) through (A.13b) become through (A.13b) (A.12) Therefore, next stage is to transform the problem to the standard diffusion problem through Also, Accordingly, let us define Accordingly, P P (A.17) (A.16) (A.19c) (A.18a) (A.19a) (A.18b) (A.19b) . , decreases or z , , ) ∂ ∂ +···} f z ( z z α 1 ∂ ∂ { ∂ ∂ C 2 ) − 1 ) α z z ) ∂ − ∂α

(εα f 2

(εα 2 ( df z B ∂ B ∂ C −→ f K . z , α . εα ) )

)

∂ − f ∂α f − f . To leading order, the density is leading order, . To ( 1 ( df df ( 2 εα 2 / 2 C C C 2 z 2 ∂α ∂ z f K f K ∂ ∂ z − = τ ∂ e ,

1 ) 2 z to

) ∂ z ∂ f 1 z 1 ) εα εα ) z z (εα = (εα = B 2 1 1 (εα z z 1 (εα B B , which is determined by “easy” how or “hard” 2 B +···}= z , which closely matches the physics. underlying matches , which closely α f εα 1 2 εα { ε is essentially is essentially to τ = P ) K , truncating terms. all higher order −→ 2 ) z K ) ∂ ∂ ( −→ 2 2 K 2 ) ∂α 2 (ε 2 f C ∂ f f − 2 ∂ O ( ∂ magazine ∂ f 1 α C ( 2 ε εα 2 −→ f We We can refine this approach by noting that the exponent is the Let us change variables from Let us change increases should be largely irrelevent to the quality of the the expansion. quality to irrelevent increases should be largely ∂ ∂ 2 so that the solution increases. integral it is to diffuseit is to from This approach is directly related to the geometric optics technique that is so successful in wave propagation and quantum electronics [27], [22]. be To more specific, we shall use the near identity transform method to carry out the geometric optics expansion. This method, pioneered in [28], transforms the problem order-by-order into a simple canonical obtain the Here we solution problem, which can thentrivially. be solved throughonly The fact that the Gaussian changes by orders of magnitude as z depending on whether the “diffusion coefficient” and to avoid confusion, we define we confusion, avoid and to Then Gaussian in the variable and Suppose we defineSuppose we the variable new Wilmott wilm003.qxd 7/26/02 7:05 PM Page 97 wilm003.qxd 7/26/02 wilm003.qxd 7/26/02 7:05 PM Page 98

Note that the variable α does not enter the problem for Pˆ until O(ε ), so of α until O(ε2 ) and the α derivatives are actually no larger than O(ε2 ). Thus, the last term is actually only O(ε3 ), and can be neglected since we ˆ ˆ ˆ 2 P (τ, z,α) = P0(τ, z ) +P1(τ, z,α) +··· (A.25) are only working through O(ε ). So, 1 1 B Consequently, the derivatives Pˆ , Pˆ , and Pˆ are all O (ε ). Recall that we H = 1 − 2ερνz + ε2ν2z2 H − ε2ρνα (zH − H ) zα αα α τ 2 zz 2 B z are only solving for Pˆ through O (ε2 ). So, through this order, we can re- (A.31a) 1 B 3 B 2 write our problem as + ε2α2 − H for τ>0 4 B 8 B 2

1 1 B Pˆ = 1 − 2ερνz + ε2ν2z2 Pˆ − εa Pˆ + ερναPˆ for τ>0 H = δ(z ) at τ = 0. (A.31b) τ 2 zz 2 B z zα (A.26a) There are no longer any α derivatives, so we can now treat α as a parameter instead of as an independent variable. That is, we have succeeded in Pˆ " = "δ(z ) at τ = 0. (A.26b) effectively reducing the problem to one dimension. 1 ˆ 2 Let us now eliminate the 2 εa(B /B ) Pz term. Define H(τ, z,α) by Let us now remove the Hz term through O(ε ). To leading order,

B (εαz ) /B(εαz ) and B (εαz ) /B(εαz ) are constant. We can replace these Pˆ = C ( f ) /C(K )H ≡ B(εαz ) /B(0 )H. (A.27) ratios by

Then b1 = B (εαz0 ) /B(εαz0 ), b2 = B (εαz0 ) /B (εαz0 ), (A.32) 1 B Pˆ = B(εαz ) /B(0 ) H + εα H , (A.28a) commiting only an O(ε ) error, where the constant z will be chosen later. z z 2 B 0 We now define Hˆ by 2 ˆ B 2 2 B B ε2 ρναb z 2 /4 ˆ P = B(εαz ) /B(0 ) H + εα H + ε α − H , (A.28b) H = e 1 H. (A.33) zz zz B z 2B 4B 2 Then our option price becomes 1 B 1 B 1 B ˆ = + + + + 2 P zα B(εαz ) /B(0 ) H zα εz Hz εα Hα ε H O(ε ) τex 2 B 2 B 2 B + 1 ε2 ρναb z 2 /4 ˆ V (t, f, a ) = [ f − K ] + εα B(0 ) B(εαz )e 1 H(τ, z ) dτ, 2 (A.28c) 0 (A.34)

The option price now becomes where Hˆ is the solution of τex + 1 1 1 3 V (t, f, a ) = [ f − K ] + εα B(0 ) B(εαz ) H(τ, z,α) dτ, (A.29) Hˆ = 1 − 2ερνz + ε2ν2z2 Hˆ + ε2α2 b − b2 Hˆ 2 τ 2 zz 4 2 8 1 0 (A.35a) 3 where + ε2ρναb Hˆ for τ>0 4 1 1 2 2 2 1 2 B ˆ Hτ = 1 − 2ερνz + ε ν z Hzz − ε ρνα (zHz − H ) H = δ(z ) at τ = 0. (A.35b) 2 2 B 1 B 3 B 2 1 B + ε2α2 − H + ερνα(H + εα H ) for τ>0 We’ve almost beaten the equation into shape. We now define 4 B 8 B 2 zα 2 B α (A.30a) 1 ενz dζ x = εν 0 1 − 2ρζ + ζ 2 H = δ(z ) at τ = 0. (A.30b) (A.36a) − + 2 2 2 − + = 1 1 2ερνz ε ν z ρ ενz log − , Equations (A.30a), (A.30b) are independent of α to leading order, and at εν 1 ρ + 1 B ; O(ε ) they depend on α only through the last term ερνα(Hzα 2 εα B Hα ). As above, since A.30a is independent of α to leading order, we can con- which can be written implicitly as

clude that the α derivatives Hα and Hz α are no larger than O(ε ), and so the last term is actually no larger than O(ε2 ). Therefore H is independent ενz = sinh ενx − ρ(cosh ενx − 1 ) . (A.36b)

98 Wilmott magazine ^ 99 4 2 , and (A.51) z (A.46) (A.47) (A.49) (A.50) ) 1 (A.48c) (A.45a) (A.48a) (A.45b) (A.48b) 2 b , 2 2 τ, (ε z / 2 1 d / 3 O b 3

ρνα ,

2 κτ . ) 2 θ ε ρνα ε 4 2 2 dq 4 1 ε e ε τ 1 4 2 θ (ε +··· 2 τ d / 2 e x 2 O + ε 3 τ 1 ) θ

e 2 2 through z + τ θ − ε 2 ) 2 , κε θ + / , we can simplify our 2 1 ε z q 2 / 2 ) 0 2 3 x (εν 3 − ε 2 2 θ

− − e / 2 e 2 − (εν z ε 1 θ (ε x q . I 2 e τ> 1 O ) / 0 τ τ 1 z 2 2 πτ 1 ex 2 τ / ∞ 1 τ ε x = 2 xI 2 2 2 τ. for x / 2 d 2 τ (εα √ − x e | B − , − κτ ex ) at 2 τ e K 1 π 2 τ ε log 0 0 x 2 Q e πτ ( √ − 1 τ κ πτ + 2 B f 4 2 1 = 2 K 2 / = | 2 ε √ ) x x 2 ) q TECHNICAL ARTICLE TECHNICAL √ − / ex . Therefore our option price is . Therefore x − + z + f 3 τ εα e 3

0 = δ( + 1 2 1 2 xx ] τ κ/ (εα 2 Q πτ K = κτ K 1 + + B 2 1 2 2 1 ε κε ) Q x − + + e − 2 3 √ ] ] f 0 = τ f [ ( K K 2 ex τ / B τ − 1 2 2 0 − = Q x − 1 f f − ) matches matches + [ · K e [ z a 2 + ] , x = = − = f εα K πτ f ) , . This can be shown by expanding ) 1 . θ/ t 2 κτ ) ) a ( 2 2 − a ε 2 , 2 ε √ V f , f e f [ , (ε (ε , t log will be chosen Then later. through t = O O ( = ( 0 V Q = V ) z a θ , 2 f This solution yields the option price ε , t ( equation to equation The solution of A.45a, A.45b is clearly through through noting that where where Moreover, quite amazingly, quite Moreover, Observe that as thisObserve can be written variables to and changing integration reduces this to V τ, Q 1 d b ) (A.44) (A.40) (A.42) (A.39) (A.37) x (A.41a) (A.43a) (A.38a) (A.41b) (A.43b) (A.38b) ρνα τ, 2 (τ, , d ε Q

) 3 4 ex ˆ Q τ x H . 0 + Q

2 (τ, I z 4 2 1 Q

ˆ 1 / b H I b by by the constants 1 2

3 8

1 4 ex 2 1 , τ ) ) 2 b ρνα 0 z

z 0 2 − + 3 8 2 ε z Q I errors. Define the con- 1 4 2 ν 4 )

b e / 2 (εν , ) − I z 2 (εν

ε ) 1 1 4 z

2 . . . I 1 2 1 b z I (ε 2 b b 0 , 0 + (εν O ) ζ 1 4

2 z I z 2 8 1 = (εν = 0 ρνα α ρνα + 2 ν 2 2 ε / − τ 2 τ 2 εν 4 3 ε 1 ερν e (εν ε ) α

I 1 2 ) ρζ 2 2 I at ) τ> 0 at + + z 2 + z ε , z + x − ) x ˆ − x z H + 2 1 for 1 Q (εα (εν

b ) (εα Q I 1 Q I B ) z B

) ) 3 8 (εν ) = ) ) x x 0 I by

εν z 0 z I − 0 ˆ Q (εν

= δ( ( δ( Q

( H + I 2 I 1 ) B B b )) (εν 1 8 (εν = b = xx

2 1 4 x εν ˆ (ζ / I , and commit only Q ( Q H I − 1 1 2 )

z 1 4 I εα εα I ρνα 0 2 )

2 1 2 z 1 2 − I α z = ε (εν 2 1 4 xx x 2 + 3 4 + ν + ˆ ˆ / (εν (εν H H 1

+ + 2 I ] ] = I + 2 2 1 / , 1 ν K K κ ) = I = 2 0 ε ˆ magazine − − τ z H = ˆ f f H + [ [ xx (εν ˆ xx

= , with = H is theof solution I Q 0 ) by ) , Q 1 2 ) a a κ 0 , , f f z = As above, we can replace The final step is to defineThe final is to step τ> , , t t τ ( ( Q (εν V Here stant stant with and so I In terms of x, our problem is In terms for for Then where Wilmott V wilm003.qxd 7/26/02 7:05 PM Page 99 wilm003.qxd 7/26/02 wilm003.qxd 7/26/02 7:05 PM Page 100

That is, the value of a European call option is given by f − K θ σ = 1 + ε2 τ +··· (A.56) N x x2 ex ∞ −q | f − K | e 2 V(t, f, a ) = [ f − K ]+ + √ dq, (A.52a) through O(ε ). 2 3/2 4 π x − ε2 θ q 2τex Before continuing to the implied log normal volatility, let us seek the simplest possible way to re-write this answer which is correct through 2 = + with O(ε ). Since x z [1 O(ε )], we can re-write the answer as − f K z 2 1/2 = + + + +··· εαz xI (ενz ) 1 σN 1 ε (φ1 φ2 φ3) τex , (A.57a) ε2θ = log B(0 ) B(εαz ) + log + ε2ρναb z 2, z x(z ) f − K z 4 1 (A.52b) where 2 through O(ε ). −1 f − K εα( f − K ) 1 f df A.2. Equivalent normal volatility. Equations (A.52a) and (A.52a) are a = = . f df − z f K K εαC(f ) formula for the dollar price of the call option under the SABR model. K C( f ) The utility and beauty of this formula is not overwhelmingly apparent. To obtain a useful formula, we convert this dollar price into the equiva- This factor represents the average difficulty in diffusing from today’s for- lent implied volatilities. We first obtain the implied normal volatility, and ward f to the strike K, and would be present even if the volatility were then the standard log normal (Black) volatility. not stochastic. Suppose we repeated the above analysis for the ordinary normal model The next factor is z ζ ˆ ˆ = √ dF = σN dW, F(0 ) = f. (A.53a) , (A.57b) x(z ) 1−2ρζ+ζ 2 −ρ+ζ log 1−ρ where the normal volatily σN is constant, not stochastic. (This model is where also called the absolute or Gaussian model). We would find that the option f value for the normal model is exactly ν df ν f − K ζ = ενz = = 1 + O(ε2 ) . (A.57c) ∞ −q α K C(f ) α C( faν ) + | f − K | e V(t, f ) = [ f − K ] + √ dq (A.53b) 4 π (f−K )2 q3/2 2 = 2σ τex Here fav fK is the geometric average of f and K. (The arithmetic aver- N age could have been used equally well at this order of accuracy). This fac-

This can be seen by setting C (f ) to 1, setting εα to σN and setting ν to 0 in tor represents the main effect of the stochastic volatility. A.52a, A.52b. Working out this integral then yields the exact European The coefficients φ1 , φ2 , and φ3 provide relatively minor corrections. option price Through O(ε2 ) these corrections are f − K √ f − K = − N √ + G √ V(t, f ) ( f K ) σN τex , (A.54a) 1 εαz σN τex σN τex ε2φ = log C( f ) C(K ) 1 z2 f − K for the normal model, where N is the normal distribution and G is the (A.57d) 2γ − γ 2 Gaussian density = 2 1 ε2α2C 2 ( f ) +··· 24 av

1 2 G = √ −q /2 (q ) e . (A.54b) 2 1 x 1/4 2 − 3ρ 2π ε2φ = log 1 − 2ερνz + ε2ν2z2 = ε2ν2 +··· (A.57e) 2 z2 z 24 From A.53b it is clear that the option price under the normal model matches the option price under the SABR model A.52a, A.52a if and only 2 1 2 B (ενz0 ) 1 2 if we choose the normal volatility σN to be ε φ3 = ε ραν = ε ρναγ1C( fav ) +··· (A.57f) 4 B(ενz0 ) 4 1 x2 θ = 1 − 2ε2 τ . (A.55) where 2 − 2 2 ex σN (f K ) x

C (fav ) C ( fav ) Taking the square root now shows the option’s implied normal (absolute) γ1 = ,γ2 = . (A.57g) volatility is given by C(fav ) C( fav )

100 Wilmott magazine ^ 4 101 N σ and this f (A.61) (A.65) ) (A.64) (A.63) (A.62c) (A.62a) (A.62b) 2 = +··· (ε ) ) 2 ex f O τ av ( f 4 B ( C σ C 4 . 1 ε . +··· 1 K 1 to / 5760 f ρναγ ε 4 4 1 + +··· , ex log ex f + , , τ , ] as before. τ ) 2 ) f B 2 ex ) B = 1 2 σ av τ to to stay consistent with the σ f 1920 ) 2 − d 2 set B 2 ( ( ε t B 0 ε σ K 2 ( [

( + 2 N ) ex ˆ C εσ D ε 1 F K τ K 2 24 K 1 2 / set / f α f t √ − ( − 2 B 2 2 ± av ) f D 1 1 K / εσ +··· , ·

log / d in (A.59a – A.59c) shows that the nor- log 1 + f ( ex ) 0 τ TECHNICAL ARTICLE TECHNICAL 1 1 ) + 24 ˆ K N 2 120 call FdW K 2 f log ε ζ 1 B = V / (ζ 24 + − ˆ f γ is −

x ν f = εσ = 2 1 = 1 ( ex − ν 2 B τ · log , = 2 2 call 1 put ˆ 1 γ V ρ 24 F and εσ K d V ) 2 3 d

f f f

+ 24 ( = df − C = ) 1 + f K 2 log ) K f 1 ( α + fK (

N . Indeed, in [14] it is shown that the implied normal σ C . We . can We find the implied Black vol for the SABR model by ) = B ) 2 4 ) εσ obtained from Black’s model in equation (A.63) equal to (ε K (ε ( O B N O = and exercise date date and exercise σ σ ) K K . Setting ( 0 N We We can obtain the implied normal volatility for Black’s model by σ = repeating the preceding analysis for the SABR model with preceding analysis. For Black’s model, the value of a European call with strike through setting yields This is the main result of this article. As before, the implied log normal calls, and this puts is the can be re-cast formula same as for for volatility setting of the simpley in terms original variables by where we are omitting the overall factor factor are omitting the overall where we obtained from the SABR model in (A.59a – B.59c). Through mal volatility is mal volatility volatility for Black’s model is for volatility with through where we have written the volatility as ν , ν is ex to 1 τ (A.60) −→ ε (A.58c) (A.58a) (A.58b) (A.59b) (A. 59c) . εν , ex α f ζ τ normal model (A.59a) − , + √ ) ) N ) K −→ ρ σ av av av f f f − ( ( εα (

2 G C ρ C ζ C 1 ex f τ and exercise date − + α = = √ K 1 2 N ) = ρναγ ρζ σ 0 ) 2 1 4 ( ˆ + 0 F , − . + ˆ is given by the same formulas dt α( 1 ) ,γ ρ ex f N av ) ) f τ σ , = ( av − av √ 1 f f 2 2 N , +··· K ( (

C 2 σ log 2 C ex dW C ) dW τ ) α 1 With the exception of JPY traders, most 2 ˆ dW = 2 F ζ = 1 ε ˆ (ζ ( N α ) ˆ γ x 1

C ) dW 2 ˆ εν α f (ζ − ˆ ν 24 ε x · 2 2 = − γ ) ρ = ˆ K 2 α ) K ˆ 3 (

F ,γ d

f 24 d ( df − , − = C fK + f ) usually provideing corrections of around 1% or so. provideing usually f K 2 ) K 1 av ex f K

+ τ − εα( = ( · 2 f , provided we use the implied normal volatility use the implied we , provided C ε av ] magazine ) = ,α, f f ν α 2 ) ( ··· [ (ε K = ( put O N + V ζ σ 1 One can repeat the a analysis for European put use option, or simply We We can revert to the original units by replacing Let us briefly summarize before continuing. Under the A. 3. Equivalent Black vol. call/put parity. This shows that the value of the put option SABR model is under the (A.59a – A.59c) as the call. the value of a European call option with strike Here everywhere. traders traders prefer to quote prices in terms of Black (log normal) volatilities, rather than normal volatilities. To derive the implied Black volatility, consider Black’s model the value of the call option is given by through the same formula, at least withthe remaining behavior, thedominant provide factors firstThe two factor everywhere in the above formulas; this is equivalent to setting given by (A.54a), (A.54b). For the (A.54a), (A.54b). For SABR model, by given where the implied normal volatility Wilmott wilm003.qxd 7/26/02 7:05 PM Page 101 wilm003.qxd 7/26/02 wilm003.qxd 7/26/02 7:05 PM Page 102

A.4. Stochastic β model. As originally stated, the SABR model con- where sists of the special case C ( f ) = f β : 2 ν − 1 − 2ρζ + ζ − ρ + ζ dFˆ = εαˆ Fˆβ dW , Fˆ(0 ) = f (A.66a) ζ = ( fK)(1 β ) /2 log f/K, xˆ(ζ ) = log . 1 α 1 − ρ (A.69b) dαˆ = εναˆ dW2, α(ˆ 0 ) = α (A.66b) This is the formula we use in pricing European calls and puts.

dW1dW2 = ρdt. (A.66c) To obtain the implied Black volatility, we equate the implied normal

volatility σN (K ) for the SABR model obtained in (A.69a – A.69b) to the Making this substitution in (A.58a–A.58b) shows that the implied normal implied normal volatility for Black’s model obtained in (A.63). This volatility for this model is shows that the implied Black volatility for the SABR model is εα(1 − β )( f − K ) ζ = εα 1 · ζ σ (K ) = · σB(K ) (1−β /2 − 2 − 4 N 1−β − 1−β ˆ ( fK) ) + (1 β ) 2 + (1 β ) 4 +··· xˆ(ζ ) f K x(ζ ) 1 24 log f/K 1920 log f/K − − 2 − 2 (1 − β )2 α2 ρανβ 2 − 3ρ2 β(2 β ) α ρανβ 2 3ρ 2 2 2 2 · 1 + + + ν ε τ +··· · 1 + + + ν ε τex +··· , 2−2β 1−β ex 1−β (1−β ) /2 24 24fav 4fav 24 24( fK) 4( fK) (A.67a) (A.69c)

2 through O(ε ), where ζ and xˆ(ζ ) are given by (A.69b) as before. Apart 2 through O(ε ), where fav = fKas before and from setting ε to 1 to recover the original units, this is the formula quoted in section 2, and fitted to the market in section 3. A.5. Special cases Two special cases are worthy of special treatment: ν f − K 1 − 2ρζ + ζ 2 − ρ + ζ = ζ = , xˆ(ζ ) = log . (A.67b) the stochastic normal model (β 0) and the stochastic log normal β − α fav 1 ρ model (β = 1). Both these models are simple enough that the expansion can be continued through O(ε4). For the stochastic normal model (β = 0) the implied volatilities of European calls and puts are We can simplify this formula by expanding − 2 2 3ρ 2 2 σN (K ) = εα 1 + ε ν τex +··· (A.70a) 24 1 1 f − K = fK log f/K 1 + log2 f/K + log4 f/K +···, (A68a) 24 1920 2 − 2 log f/K ζ α 2 3ρ 2 2 σB(K ) = εα · · 1 + + ν ε τex +··· (1−β ) /2 f − K xˆ(ζ ) 24fK 24 f 1−β − K 1−β = (1 − β )(fK ) log f/K (A.68b) (A.70b) (1 − β )2 (1 − β )4 · 1 + log2 f/K + log4 f/K +··· 24 1920 through O(ε4), where ν 1 − 2ρζ + ζ 2 − ρ + ζ and neglecting terms higher than fourth order. This expansion reduces ζ = fKlog f/K, xˆ(ζ ) = log . (A.70c) α 1 − ρ the implied normal volatility to β/2 + 1 2 + 1 4 +··· = 1 24 log f/K 1920 log f/K ζ For the stochastic log normal model (β 1 ) the implied volatilities are σN (K ) = εα( fK ) · (1−β )2 2 (1−β )4 4 ˆ 1 + log f/K + log f/K +··· x(ζ ) f − K ζ 1 1 24 1920 σ (K ) = εα · · 1 + − α2 + ραν N log f/K xˆ(ζ ) 24 4 −β(2 − β ) α2 ρανβ 2 − 3ρ2 · + + + 2 2 +··· (A.71a) 1 1−β − ν ε τex , 24( fK) 4( fK)(1 β ) /2 24 1 2 2 2 + (2 − 3ρ ) ν ε τex +··· (A.69a) 24

102 Wilmott magazine ^ 4 103 ¯ ) υ s

s η( (B.5) (B.6) (B.8) (B.9c) (B.7a) (B.9a) (B.4b) (B.7b) (B.9b) < and s . ¯

η : dt ,

H )

time-independ- ds t ( 0for ) . 2 ) K γ = t ,

ex t s 0 strategy [23]. strategy In this ; /γ ( αα P ) . 2 ,α t

= f α s , and then to , and then to , ν( ˆ . ) s ex P ( T s s = ( = P s to 2 = ) , ex υ , s P . s ) t

, ) for + f f P dt ) df ( ( ) so that all terms arising from the for α ) effective media f C

C , K K ¯ P υ ) t ( f K ( ) ( 2 = , C K f 2 ) ,υ( C ) ( ε cancel out. As we shall see, this simul- ) α γ 2 − K z t and T C ) ε f 1 TECHNICAL ARTICLE TECHNICAL 2 0 εα t ¯ = η − 1 2 δ( α ˆ f (εα P /γ ( 2 υ( ) = B + ) = α s . The resulting problem is relatively complex, relatively is problem resulting The . δ( z t

¯ 2 + υ ] s η( = α ν( As in Appendix B, we change independent As vari- in change Appendix B, we and 2 K ) P ) t = , + t − and

P f ff ¯ ρ( η solves the forward problem solves η( [ P dt ) ) ) by definingby the time variable new = =

f K ) ) t ) , ( t ( s

2 2 a s C , γ( η( ; γ f 2 t , 0 α t ,α ( f , V 2 . If we could find these constant values, this would reduce the s ε = ) ( s P s 1 2 υ( + We solve thisWe problem by using an We carry the applying same series of by We out this strategy B.1 Transformation. transformations transformations that was used to solve the non-dynamic SABR model s P problem to the non-dynamic SABR model solved in Appendix B. the non-dynamic SABR model solved problem to ent in Appendix B, defining the transformations in terms of the constants unknown) (as yet time dependence of We eliminate eliminate We yield the same option price as the the time dependent coefficients and taneously determines the “effective” and parameters allows us to use the of the option. volatility obtain the implied in Appendix B to analysis ables to Then the option price becomes where more complex than the canonical problem obtained in Appedix B. We use We B. Appedix in obtained problem canonical thethan complex more have we once and problem, this solve to expansion perturbation regular a this choose solved problem, we and define Here then dependent variables from change We strategy strategy our objective is to determine which constant values T (B.3) . Let < (B.1c) t (B.4a) (B.1a) (B.2a) (B.1b) (B.2b) by t (A.71c) (A.71b) ) , . ) and exer- and A for T +··· to to get the dT , K F ) ) ˆ ex ζ α( and “volvol” and , t τ K T ˆ 2 at date , and analyze α + α , < ν( ) ; ε ) , T t ρ . t α A

0 ; = ,α 2 f , − ) dA γ( ν = = εν( , −→ ,α t 2 t f ) ) ) dF ) ( ρ , ζ 2 t ˆ α( t t p A ( + ρ αα − , ˆ + , −→ α( P P 3 f F K 1 2 ) εν( ) , , f t − α , T = ρζ T < ( 2 1 2 ; ) 2 , ) 2 ν( = ν , ( t T 2 γ ) ( dt − .and ,α ˆ ( dW + ˆ F f t 1 ) ) ex F 24 ) t ( 1 , t t | α ˆ ˆ and t dW t f F F ˆ ( < P ( , + α ρ( γ( p ) C dA ) ) ) F 2 t ˆ t α f = K A + ( ) ( 2 ραν 2 t C −→ log A ∞ εγ ( εν( 2 1 4 C −∞ ) 2 = t < dW

prob = ε εγ ( 1 ) [23] to extend [23] theextend to preceding analysis theto ˆ 1 2 −→ + = α ≡ = εγ ( d ) ργνα (ζ ) 1 ˆ dW ˆ + F x t 2

K d + · , ] γ( + T dF dA K ff ; ) P ) , ) − is the solution of the problem backwards A ,α ζ K f f f (ζ , ) ˆ ( / . We obtain the prices of European options, and from [ , x F f 2 t 1 K , ( C , = T P 2 · T ; , where log ) α ; magazine ) a 2 ε ν α 4 ,α εα , f γ ,α . As before, define. As before, the density transition f f , (ε , = ex t = , t O t be the value of, say, a European call option with strike withstrike option call European a say, of, thevalue be ( 2 t ( ) ( effective medium theory ζ ε ) p V P K 1 2 ( ,α B to be small, writng to f + σ Suppose the economy is in state , ) t t t ( P where in the limit through with V ν( Appendix B. Analysis of the Dynamic Appendix B. SABR Model. use We dynamic SABR model. As before, we take the volatility volatility the take we before, As model. SABR dynamic answer in terms of the of in terms original variables. answer cise date cise date and define these prices we obtain the implied volatity of these options. After obtain- After theseoptions. of volatity theimplied obtain we theseprices ing the results, we replace Repeating Repeating the analysis in Appendix B through equation (A.10a), (A.10b) that by the shows option price is given now Wilmott wilm003.qxd 7/26/02 7:05 PM Page 103 wilm003.qxd 7/26/02 wilm003.qxd 7/26/02 7:05 PM Page 104

= ˆ ≡ ˆ 1 2 ˆ H C(K ) /C( f )P B(0 ) /B(εαz )P. (B.9d) change in price due to 2 ε αb1δzHz term. In this way to the transformation cancels out the zHz term “on average.” Following the reasoning in Appendix B, we obtain In a similar vein we define sex + 1 V (t, f, a ) = [ f − K ] + εα B(0 ) B(εαz ) H(s, z,α; s ) ds , (B.10) I(ευ¯ z ) = 1 − 2εη¯z + ε2υ¯ 2z 2, (B.16a) 2 s where H(s, z,α; s ) is the solution of and 1 1 B 2 2 2 2 ευ¯ z 2 2 2 Hs + 1 − 2εηz + ε υ z Hzz − ε ηα (zHz − H ) 1 dζ 1 1 − 2εη¯z + ε υ¯ z −¯η/υ¯ + ευ¯ z 2 2 B x = = log , (B.11a) ¯ ¯ −¯ ¯ 1 B 3 B 2 ευ 0 I(ζ ) ευ 1 η/υ + ε2α2 − H = 0 (B.16b) 4 B 8 B 2 for s < s , and where the constants η¯ and υ¯ will be chosen later. This yields

H = δ(z ) at s = s (B.11b) 1 1 − 2εηz + ε2υ2z 2 1 Hˆ + (Hˆ − ευ¯I (ευ¯z ) Hˆ ) − ε2αb (η − δ ) xHˆ s 2 1 − 2εη¯z + ε2υ¯2z 2 xx x 2 1 x through O(ε2 ). See (A.29), (A.31a), and (A.31b). There are no α derivatives 1 1 3 + 2 + ˆ + 2 2 − 2 ˆ = in equations (B.11a), (B.11b), so we can treat α as a parameter instead of a ε αb1(2η δ ) H ε α b2 b1 H 0fors < s , 4 4 8 variable. Through O(ε2 ) we can also treat B /B and B /B as constants: (B.17a)

B (εαz0 ) B (εαz0 ) b1 ≡ , b2 ≡ , (B.12) B(εαz ) B(εαz ) 0 0 Hˆ = δ(x ) at s = s , (B.17b)

2 ˆ ˆ where z0 will be chosen later. Thus we must solve through O(ε ). Here we used z = x +··· and zHz = xHx +··· to leading order to simplify the results. Finally, we define 1 1 H + 1 − 2εηz + ε2υ2z 2 H − ε2ηαb (zH − H ) s 2 zz 2 1 z (B.13a) Hˆ = I 1/2(ευ¯z ) Q. (B.18) 1 3 + ε2α2 b − b2 H = 0fors < s , 4 2 8 1 Then the price of our call option is H = δ(z ) at s = s . (B.13b) 1 V(t, f, a ) = [ f − K ]+ + εα B(0 ) B(εαz )I 1/2 2 At this point we would like to use a time-independent transformation sex (B.19) 1 ε2 αb δz 2 · ¯ 4 1 ; to remove the zHz term from equation (B.13a). It is not possible to cancel (ευz ) e Q(s, x s ) ds , this term exactly, since the coefficient η(s ) is time dependent. Instead we s use the transformation where Q(s, x; s ) is the solution of

1 ε2 αb δz2 ˆ H = e 4 1 H, (B.14) 1 1 − 2εηz + ε2υ2z 2 1 1 Q + Q − ε2αb (η − δ ) xQ + ε2αb (2η + δ ) Q s 2 1 − 2εη¯z + ε2υ¯2z 2 xx 2 1 x 4 1 where the constant δ will be chosen later. This transformation yields 1 1 1 3 + 2 ¯2 − + 2 2 − 2 = ε υ I I I I Q ε α b2 b1 Q 0fors < s , ˆ 1 2 2 2 ˆ 1 2 ˆ 4 8 4 8 Hs + 1 − 2εηz + ε υ z Hzz − ε αb1(η − δ ) zHz (B.20a) 2 2 1 2 ˆ 2 2 1 3 2 ˆ + ε αb (2η + δ ) H + ε α b − b H = 0fors < s , 4 1 4 2 8 1 Q = δ(x ) at s = s , (B.20b)

Hˆ = δ(z ) at s = s , (B.15a) Using through O(ε2 ). Later the constant δ will be selected so that the change in 1 2 the option price caused by the term 1 ε2αb ηzHˆ is exactly offset by the z = x − εη¯x +···, (B.21) 2 1 z 2

104 Wilmott magazine ^ 4 105 ,

) b ds 2 ( ) Q

(B.31c) (B.31a) (B.28a) (B.29a) (B.30a) s (B.31b) (B.28b) (B.29b) (B.30b) ; , J x , and )

, , 2 s s ( ) s z δ ( a Q 1 ) 2 < s ( , , . )

ab 2 s 2 (

z Q s s ε I , Q

1 4 ) (ε I < < ex e 2 2 1 s for ( / ) 1 8 s s +··· s υ 1 ,

Q z

I ) − −¯

2 (ε ds were constant in time, were for for ) z ds I ε 2 2 0

) ) / ( ) υ I 1

s +

) ) so that the last threethatinte- theso last I Q s (εα . 1 4

s s 0 )

¯ ( ; ( xx B . υ( , 1 2 υ ; s

− Q ds ) x Q s x s ) 2 , 2 − , ) 0 = s we can we carry out this strategy, 0 ¯ x s υ (

( x ) (εα ( = = 2 1 s ( ] and 2 and ) s B if 1 ) b B ) s s ( xx ε b ) s , and the solution would be just ; xQ ) ¯ ( 2 3 8 at η s η, ( 0 x Q − 0 xQ , ) , Q ( ) η( −¯ − εα s δ ex ) ≡ s ex s δ B ( ) s ( 2 1 2 ) s arise from the time-dependence of the ) η 0at s 0at x b 1 b Q ) − ( ¯ 2 η(η δ b 1 4 TECHNICAL ARTICLE TECHNICAL −¯ + ( δ( 2 = = 2 2 Q 1 3 z εα ( (η Q δ ε ) + ) ex ] 1 s 1 (η 1 2 − = Q b ab a 2 b s 2 2 2 ab ≡ + K ) ( ( α 2 α s = ε

+ ) ( ε 2 Q Q ε ) − 1 4 1 2 a 4 3 ε . Indeed, if Q ds + and b 2 f e ] would be the solution of the problem would static ) ( 2 [ + , ) · ( xx − ) s = K ) Q

s a Q ( ) s = . In this case, the option price woud be given by . In this be given case, the price woud option =− 2 υ( a − ds ( ; 1 2 Q ≡ x ) . Therefore, we will first solve for for solve first will we Therefore, . 2 ) f ( xx , ) s ) x ) Q [ ( xx a ) s

1 , + , Q 2 , ( s ) 2 s Q ( f and ) ( = 1 2 1 ; Q b ) , 1 2 ( Q ) (ε t ) 2 a x 2 ( s s ( Q O 2 + a , ε ( + V Q s to choose the constants theconstants choose to ) , η( ( ) f Q a + ) s 2 , ex ( s ( s s 0 ) t try ( ( s Q 0 Q ( Q V ex Q 2 s ε s ≡ The terms The terms + ) s ( = J where and where by thengiven is price thentheoption theseequations, solved have we Once we we would have This is exactly the time-independent problem solved in Appendix B. See (A.42), equations (A.43a), and (A.43b). So we can obtain option prices for the dynamic SABR model by reducing the thethem model. prices for static previously-obtained to where and, through and thenand coefficients grals are zero for all grals are zero for Q )

s ; x (B.26) (B.23) , (B.24a) (B.24a) (B.25a) (B.27a) (B.22a) (B.25b) (B.27b) (B.22b) s xx ( . Q Q ) 2 x 0 ( ,

+··· ) Q s ) η

s < , I

−¯

; s s I Q x 1 8 , , < ¯ s

η(η for

( s s I 3 − )

, ,

2 I I ) ( ) < s −

b 1 8 0 for I . Q s 2 , ( 2

( 2 < 1 4 , s . . s υ Q

ε −

Q ) s ) ) s s I for + = 2 0 −¯ =

Q ( xx 2 + 2 1 ) I s = 2 s = b ) (ε υ

− 1 4 s a s υ s Q O 3 8 xQ can be replaced by the constants 2 at x at −¯ ; ( ) 0for 2 ) x Q − 2 2 1

η ε xQ 0 , ¯ b ( I ) υ 2 = ( s 1 2 Suppose we were Suppose to expand were we + ) 2 ( b 3 8 −¯ ) Q x 0at 0at ) ) ε δ δ 0 ) 1 4 s 1 − ( xx δ( 1 ( 2 − (η = x = ( − and − Q through Q xx ) 2 ) ab = 2 Q δ( , s Q ε 1 2 = ) b 1

2 ) 3 4 ( α (η δ ( I ) 0 xQ 1 4 + 1 1 = 0 = z , Q 1 + ( ) Q ) b b I ( xx − ¯ ) υ )

Q Q η α α 2 Q 2 0 s ( =− 2 2 ( s (ε ; 1 2 α ε ε Q

−¯ ) Q 2 x s I : 3 1 2 4 ε 2 , + ( xx s ε ) ( ε(η Q − + − ) 1 ( s 1 2 0 ( = , and Q ) Q + . Note that xx 0 ) ) magazine z Q = s 2 ¯ 2 υ 1 2 ( ) s (ε

Q (ε s

O + we can break the solution into we I ; we have we s ) , x ) Q ) 2 , 0 s (ε (ε z ( B. 2. Perturbation expansion. ¯ O O Q υ (ε Substituting this expansion into (B.22a), (B.22b) yields the following hier- following the yields (B.22b) (B.22a), into expansion this Substituting have we leading order To of equations. archy through as a power series in as a power I we can simplify this to can simplify we At At where Wilmott wilm003.qxd 7/26/02 7:05 PM Page 105 wilm003.qxd 7/26/02 wilm003.qxd 7/26/02 7:05 PM Page 106

1 1 x2 + B.2.1. Leading order analysis The solution of (B.24a), (B.24b) is (2a ) + (2a ) =− − Q s Q xx αb1(η δ ) G Gaussian: 2 2 (B.38) √ 1 =− αb1(η − δ ) Gxx − αb1(η − δ ) G Q (0 ) = G(x/ ) (B.32a) 2 for s < s , with Q (2a ) = 0 at s = s . Solving then yields where ∂ s √ (2a ) = − ˜ ˜ − ˜ − √ Q αb1 (s s )[η(s ) δ]dsG(x/ s s ) . (B.39) 1 − 2 ∂s G(x/ ) = √ e x /2 , = s − s. (B.32b) s 2π This term makes a contribution of For future reference, note that sex sex √ x x2 − x3 − 3 x Q (2a )(s, x; s ) ds = αb (s − s˜)[η(s˜) −δ]ds˜ G(x/ s − s ) (B.40) G =− G ; G = G ; G =− G ; (B.33a) 1 ex ex x xx 2 xxx 3 s s

4 − 2 + 2 5 − 3 + 2 to the option price, so we choose = x 6 x 3 ; =−x 10 x 15 x G xxxx G G xxxxx G, (B.33b) sex − ˜ ˜ − ˜ 4 5 (sex s )[η(s ) δ]ds δ =¯η = s . (B.41) 1 − ˜ 2 2 (sex s ) x6 − 15 x4 + 45 2x2 − 15 3 G = G. (B.33c) to eliminate this contribution. xxxxxx 6 B.2.4. The ε2Q (2b ) term Substituting Q (1 ) and Q (0 ) into equation B.2.2. Order ε Substituting Q (0 ) into the equation for Q (1 ) and using (B.29a), we obtain (B.33a) yields (2b ) 1 (2b ) 1 2 + = −¯ + −¯ − Q s Qxx (η η ) AxGxxxxx 2(η η ) As xGxxx κx Gxx, (B.42a) 1 x3 − x 2 2 Q (1 ) + Q (1 ) = (η −¯η ) G s xx 2 2 for s < s , where

=−(s − s )(η −¯η ) G xxx − 2(η −¯η ) Gx for s < s , (B.34) κ = υ2(s ) −¯υ2 − 3η¯[η(s ) −¯η]. (B.42b)

This can be re-written as with the “initial” condition Q (1 ) = 0 at s = s . The solution is (2b ) 1 (2b ) + =− −¯ + − −¯ Q s Qxx (η η ) A[ Gxxxxxx 5Gxxxx ] 2(η η ) As [ Gxxxx (1 ) 2 Q = A(s, s ) G + 2A (s, s ) G (B.35a) xxx s x 1 + 3G ] − κ[ 2G + 5 G + 2G] (B.43) xx 2 xxxx xx ∂ √ = 2A(s, s ) G (x/ s − s , (B.35b) Solving this with the initial condition Q (2b ) = 0 at s = s yields ∂s x

where (2b ) 1 2 Q = A (s, s ) Gxxxxxx + 2A(s, s ) As (s, s ) Gxxxx 2 s s s A(s, s ) = (s − s˜)[η(s˜) −¯η]ds˜; As (s, s ) = [η(s˜) −¯η]ds˜. (B.35c) 2 + ˜ −¯ ˜ ˜ + s s 3 [η(s ) η]A(s, s ) dsGxxxx 3As (s, s ) Gxx s This term contributes 1 s 5 s s + − ˜2 ˜ ˜ + − ˜ ˜ ˜ + ˜ ˜ [s s] κ(s ) dsGxxxx [s s]κ(s ) dsGxx κ(s ) dsG. sex √ 2 s 2 s s (1 ) Q (s, x; s ) ds = 2A(s, sex ) Gx(x/ sex − s ) (B.36) s to the option price. See equations (B.30a), (B.30b). To eliminate this con- This can be written as tribution, we chose η¯ so that A(s, s ) = 0: ex ∂ s Q(2b ) = 4A2(s, s ) G − 12 [η(s˜) −¯η]A(s˜, s ) dsG˜ s ss s ex (s − s˜) η(s˜) ds˜ ∂s s ¯ = s ex η 1 2 . (B.37) (B.45) − s 2 s 2 (sex s ) −2 (s − s˜) κ(s˜) dsG˜ s + (s − s˜) κ(s˜) dsG˜ s s B.2.3. The ε2Q (2a ) term From equation (B.28a) we obtain

106 Wilmott magazine ^ 4 107 , )

s and ; ¯ (B.54) (B.55) (B.52) η , x

(B.51a) (B.50a) (B.53a) (B.51b) (B.50b) (B.53b) , s , s depends ( 0 ) < ) s s ( 2 ( Q ) τ> Q . The integral s J 2 ( ε 2 Q for s / + τ, 1 +··· d ) I

for )

) I I

) s . The option price is s

z , I ds ds )

I x ; s } } 1 8 − . 1 8 x ex ) ) s (εα , 0

ex = (τ, ) s − ; B s s s ) s − +··· ( ( ( s x I ) ; ; ) = I ( ) +··· ¯

, Q ) υ ¯ 0

κ x x 0 s I s ( Q ) 2 τ I 2 ( ex , so its solution ( ( , , s s s ) s 1 4 Q η/ s s ε B 1 4 2 1 Q − ( ( at s 0 ; 1 2 b ( ) ) ex − ˆ s s s = x 3 8 Q 2 →¯ 2 2 0 2 1 , 2 at ) + ( ( ex ¯ ¯ s b υ s εα s 2 υ s ( ( Q Q 2 . The option price is therefore − ) ) 2 3 8 1 2 2 2 2 s ¯ ε Q s z κ( contribute nothingcontribute to 0 ε 2 ¯ − ε ε ( η 2 b ) 1 − − + + ε − Q + + ex − a ) 1 4

ab 2 s ) 2 1 2 ) )

) + 2 TECHNICAL ARTICLE TECHNICAL s ( 2 s ex b ] s s

ε ( ) s ( υ, ρ 1 4 Q ) s s = s 1 4 K 2 + Q = e Q , − ; ; ¯ x s ¯ κ( . We chose the . We “effective parameters” η α ) δ ) − s 2 ex x x τ − η 1 2 →¯ 1 1 δ( s 2 z ( ε − , , f ε ex δ( s s [ 1 2 α ν ab s Q =¯ ab ( ( (ε ex = = 2 ( 2 ˆ ) ) − 2 s = ε δ s ε = 0 0 ¯ ε ) κ ( ( ex ˆ ) s Q 2 s 3 4 ) ( 3 4 s + Q Q ( → ε Q a { { Q 2 1 · , = ex ex ˆ ex f s s . ) τ s s , =− s t we we can combine ( xx + ( ) s Q is theof solution V ( xx ) = = , where 1 2 2 ) ) Q J 2 x (ε 1 2 − solves the problem static solves O (ε ) ) s + (τ, O s ( τ s ( then contributed then contributed ) s Q Q Q ) ( s b Q 2 ( The option price defined by (B.52), (B.53a), and (B.53b) is identical to Q so that the of integrals ¯ the static model’s option price defined by (A.42), (A.43a), and (A.43b), pro- (A.43b), and (A.43a), (A.42), definedby price option model’s static the the identifications make vided we of and where we have use and where have we only on theonly time difference through Through where where This problem is homogeneous in the time υ )

s ds ˜ s − ) d (B.46) (B.47)

to to set ] ex (B.49a) (B.48a) s (B.48b) (B.49b) , ) η s ¯ ) ; υ s , , −¯ x s √ J , ) / − 2 ˜ s z s − x ( ¯ η ( ) ex 1 , ) s s ex η( G +··· ) [ s 2 s ab (

) 2 2 √ ˜ ex ε ˜ s √ Q / s s − 1 4 . ds / ˜ x d s ; ex e x ( − s ex ) ) . d s ( ) x s ] s

1 G ex , G 2 z ex s s s √ s ds υ ; ( ( 2 . Therefore . theTherefore contri- / (ε , ˜ ) 2 ˜ ˜ x s ex ε s s x 2 0 s −¯ 0 , / ( ( d ( d s ds s ) 1 ] + ) A G ˜ ( ) = Q I s ] ˜ to simplify (B.48b). simplify to ) ˜

η s ) s ( ¯ η 2 2 ) b η to to set the coefficient of 2 z ) ) 0 2 −¯ ds 3 κ( ( s s κ( −¯ ex υ ) ¯ [ ) s ) 2 ) ) Q υ = ˜ ) 2 ˜ ˜ − s , − −

˜ s (εα s ˜ s s s ex s s s ˜ B ( s ex ex d − ; s η( − η( ) s s A [ − d or the coefficientthe or of ) x ( ( ][ ex ) ex 0 , ) ex s η ˜ ex η 2 s ¯ ¯ ( s s κ κ s s s s ( ε ( ( ( B ( −¯ −¯ ) 1 1 2 2 ex 2 ex . s ex s ) − 1 ) ] s + ( s ˜ υ s s η s 1

= = 12 ) 2 s Q ex ˜

εα s −¯ s We can now determine We the volatili- implied ds ex 2 (η( 2 s ) η( 1 2 [ √ ) ds ) ) s − s + ˜ + 2 / s

s ) . We choose s + x

s ) ex η( 1 =− ( ε [ s s b s − ; − s + ( ¯ 1

2 η ] s ; x ( G ex + ex x ex 3 , s ds K Q s s

s , s ( ( ) ( ex s − s ) ex

( 1 2 − ds s s ) a s s

2 f 2 b ) ; [ ( υ 2 3

= ( x 6 ) Q s −¯ ˜ , = Q ¯ s κ ; s ex ) to to zero, for reasons that will become apparent in a − to the price is option to s ( ) x ex ) s s s − ) , magazine ( 1 a b was chosen above so thatwas chosen above ) s s b 2 2 , ( s ex 2 ( ¯ f 2 ) η ( s υ , ε ( Q 0 t Q − ( 1 3 ( ex = s Q + V ex s s κ ex = s s √ 2 / ¯ B.3. Equivalent volatilities. B.3. Equivalent volatilities. We We can choose the remaining “effective media” parameter υ x ( = s J where G to to zero, but cannot set both to zero to completely eliminate the contribution of the term Recall thatRecall either the coefficienttheeither of where where Here we have used have Here we ty for ty thefor dynamic model by mapping the problem back theto static model that from (B.30a), (B.30b) of Appendix B. Recall the value of the option is Then the remaining contribution to theThen the option price is remaining contribution to moment: bution of Wilmott wilm003.qxd 7/26/02 7:05 PM Page 107 wilm003.qxd 7/26/02 wilm003.qxd 7/26/02 7:05 PM Page 108

in Appendix B for the original non-dynamic SABR model, provided we REFERENCES make the identifications D. T. Breeden and R. H. Litzenberger, (1994), Prices of state-contingent claims implicit τ in option prices, J. Business, 51 pp. 621–651. = + 2 ˜ 2 ˜ −¯2 ˜ τex τ ε τ [υ (τ ) υ ]dτ, (B.56a) Bruno Dupire, (1994), Pricing with a smile, Risk, Jan. pp. 18–20. 0 Bruno Dupire, (1997), Pricing and hedging with smiles, in Mathematics of Derivative Securities, M.A. H. Dempster and S. R. Pliska, eds., ν →¯η/υ,¯ ν →¯υ. (B.56b) Cambridge University Press, Cambridge, pp. 103–111. E. Derman and I. Kani, (1994), Riding on a smile, Risk, Feb. pp. 32–39. See equations (A.42 – A.43b). Following the reasoning in the preceding E. Derman and I. Kani, (1998), Stochastic implied trees: Arbitrage pricing with stochas- Appendix now shows that the European call price is given by the formula tic term and strike structure of volatility, Int J. Theor Appl Finance, 1 pp. 61–110. J.M. Harrison and D. Kreps, (1979), Martingales and arbitrage in multiperiod securities f − K √ f − K markets, J. Econ. Theory, 20 pp. 381–408 V(t, f, K ) = ( f − K ) N √ + σN τex G √ , (B.57) σN τex σN τex J.M. Harrison and S. Pliska, (1981), Martingales and stochastic integrals in the theory of continuous trading,Stoch. Proc. And Their Appl., 11 pp. 215–260 with the implied normal volatility I. Karatzas, J.P. Lehoczky, S.E. Shreve, and G.L. XU, (1991), Martingale and duality − methods for utility maximization in an incomplete market, SIAM J. Control Optim, 29 pp. = εα(f K ) · ζ · σN (K ) 702–730 f df xˆ(ζ ) K C(f ) J. Michael Steele, (2001) Stochastic Calculus and Financial Applications, Springer, F. Jamshidean, (1997), Libor and swap market models and measures, Fin. Stoch. 1 pp. 2γ − γ 2 + 2 1 2 2 + 1 ¯ (B.58a) 293–330 1 α C (fav) 4 ηαγ1C (fav) 24 F. Black, (1976), The pricing of commodity contracts, Jour. Pol. Ec., 81 pp. 167–179 J. C. Hull, (1997), Options, Futures and Other Derivatives, Prentice-Hall. 2υ¯2 − 3η¯2 1 ¯ 2 + + θ ε τex +···. Paul Wilmott, (2000), Paul Wilmott on Quantitative Finance, John Wiley & Sons, . 24 2 where Patrick S. Hagan and Diana E. Woodward, (1999),Equivalent Black volatilities, App. Math. Finance, 6 pp. 147–157. P.S. Hagan, A. Lesniewski, and D. Woodward, Geometrical optics in finance, in prepara- ¯ − − ¯ ¯ + 2 −¯ ¯ + tion = υ f K ˆ = 1 2ηζ/υ ζ η/υ ζ ζ , x(ζ ) log , (B.58b) Fred Wan, (1991), A Beginner’s Book of Modeling, Springer-Verlag, Berlin, New York, . α C( fav ) 1 −¯η/υ¯ J. Hull and A. White, (1987), The pricing of options on assets with stochastic volatilities, J. of Finance, 42 pp. 281–300 C ( fav ) C ( fav ) f = fK,γ= ,γ= , (B.58c) S. L. Heston, (1993), A closed-form solution for options with stochastic volatility with av 1 C( f ) 2 C( f ) av av applications to bond and currency options, The Review of Financial Studies, 6 pp. 327–343 τ τ˜[υ2(τ˜ ) −¯υ2]dτ˜ θ¯ = 0 . (B.58d) A. Lewis, (2000), Option Valuation Under Stochastic Volatility, Financial Press, 1 2 2 τ J.P. Fouque, G. Papanicolaou, K.R. Sirclair, (2000), Derivatives in Financial Markets with Equivalently, the option prices are given by Black’s formula with the Stochastic Volatility, Cambridge Univ Press, N. A. Berner, BNP Paribas, (2000), Private communication, effective Black volatility of J.D. Cole, (1968), Perturbation Methods in Applied Mathematics, Ginn-Blaisdell, J. Kevorkian and J.D. Cole, (1985), Perturbation Methods in Applied Mathematics, = α log f /K · ζ σB(K ) Springer-Verlag, f df xˆ(ζ ) K C(f ) J. F. Clouet, (1998), Diffusion Approximation of a Transport Process in Random Media , SIAM J. Appl. Math, 58 pp. 1604–1621 2γ − γ 2 + 1/f 2 · + 2 1 av 2 2 + 1 ¯ I. Karatzas and S. Shreve, (1988), Brownian Motion and Stochastic Calculus, Springer. 1 α C ( fav) 4 ηαγ1C (fav) 24 B. Okdendal, (1998), Stochastic Differential Equations, Springer

¯ 2 − ¯2 M. Musiela and M. Rutkowski, (1998), Martingale Methods in Financial Modelling, 2υ 3η 1 ¯ 2 + + θ ε τ +··· Springer 24 2 ex G.B. Whitham, (1974), Linear and Nonlinear Waves, Wiley J.C. Neu, (1978), Doctoral Thesis, California Institute of Technology

108 Wilmott magazine