Chapter 9 Asymptotic Analysis of Implied Volatility

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1 Chapter 9 Asymptotic Analysis of Implied Volatility he implied volatility was first introduced in the paper [LR76] of H.A. Latané and R.J. Rendleman under the name the implied standard deviation. Latané and Rendleman studied standard deviations of asset returns, which are implied in actual call option prices when investors price options according to the Black Scholes model. For a general model of call option prices, the implied volatility can be obtained by inverting the Black Scholes call pricing function with respect to the volatility variable and composing the resulting inverse function with the original call pricing function. his chapter mainly concerns the asymptotics of the implied volatility at extreme strikes. In Sect. 9., we define the implied volatility in general models of call option prices and discuss its elementary properties. Implied volatility models free of static arbitrage are characterized in Sect. 9. see heorem 9.6. he rest of the chapter is devoted to sharp asymptotic formulas with error estimates for the implied volatility. We discuss asymptotic formulas of various orders, and show how certain symmetries hidden in stochastic asset price models allow to analyze the asymptotic behavior of the implied volatility for small strikes, by using information about its behavior for large strikes. hese symmetries become more explicit in the so-called symmetric models, which are also discussed in the present chapter. 9. Implied Volatility in General Option Pricing Models Fix >0 and >0. hen the function ρσ = C BS,,σ is increasing on 0,. his follows from the fact that the Greek vega is positive see Sect If 0 <<x 0 e r, then the range of the function ρ coincides with the interval x 0 e r,x 0, while for x 0 e r, the range of ρ is the interval 0,x 0. Definition 9. Let C be a call pricing function. For, 0,, the implied volatility I, associated with C is the value of the volatility σ in the Black Scholes model for which C, = C BS,,σ. he implied volatility I, is defined only if such a number σ exists and is unique. A. Gulisashvili, Analytically ractable Stochastic Stock Price Models, Springer Finance, DOI 0.007/ _9, Springer-Verlag Berlin Heidelberg 0 43

2 44 9 Asymptotic Analysis of Implied Volatility It follows from the discussion above that if 0 <<x 0 e r, then the condition x 0 e r <C,<x 0 is necessary for the existence of the implied volatility I,. Similarly, if x 0 e r, then I, is defined if and only if 0 <C,<x 0. Note that the inequality C, < x 0 holds for all 0 and >0. Moreover, if, [0,, then x0 e r + C,. In the next definitions, we introduce special classes of call pricing functions. Definition 9. he class PF consists of all call pricing functions C, forwhich one of the following equivalent conditions holds:. C, > 0 for all >0and >0with x 0 e r.. P,>e r x 0 for all >0and >0with x 0 e r. 3. For every > 0 and all a>0 the random variable X is such that P [X <a] <. Definition 9.3 he class PF 0 consists of all call pricing functions C, for which one of the following equivalent conditions holds:. P,>0 for all >0and >0with <x 0 e r.. C, > x 0 e r for all >0and >0with <x 0 e r. 3. For every > 0 and all a>0 the random variable X is such that 0 < P [X <a]. Remark 9.4 Suppose the maturity >0 is fixed, and consider the pricing function C and the implied volatility I as functions of the strike price. IfC PF, then the implied volatility I is defined for large values of. his allows to study the asymptotic behavior of the implied volatility as. Similarly, if C PF 0, then Iexists for small values of. Finally, if C PF PF 0, then the implied volatility I,exists for all >0 and >0. 9. Implied Volatility Surfaces and Static Arbitrage Let I, with, 0, be a positive function of two variables, and suppose we would like to model the implied volatility surface by this function. hen the function C defined on [0, by C BS,, I,, if, 0,, C, = x 0 +, if = 0, 0, 9. x 0, if 0, = 0, where x 0 is the initial price of the asset in the Black Scholes model, should be a call pricing function. For the sake of simplicity, we will assume that r = 0. he next definition concerns the implied volatility in a no-arbitrage environment.

3 9. Implied Volatility Surfaces and Static Arbitrage 45 Definition 9.5 It is said that the function I modeling the implied volatility is free of static arbitrage if the model of call prices given by the function C in 9. isfree of static arbitrage see Definition 8.. Our next goal is to provide necessary and sufficient conditions for the absence of static arbitrage in a given implied volatility model. heorem 9.6 Suppose the function I models the implied volatility. Suppose also that for every >0the function I, is twice differentiable on 0,. hen I is free of static arbitrage if and only if the following conditions hold:. For all, 0,, I I + I I x 0 I 4 I + I I For every >0 the function I,is increasing on 0,. 3. For every >0, lim d,,i,=. Proof It suffices to prove that the conditions in heorem 8.3, formulated for the function C given by 9., are equivalent to conditions 3 in heorem 9.6. Fix >0, and differentiate the function C on 0, with respect to. his gives C = C BS C BS I,,I, +,,I, σ. Differentiating again, we obtain C = C BS C BS,,I, + σ + C BS σ,,i, I I,,I, + C BS I,,I, σ. Next, taking into account explicit formulas for the Greeks see Sect. 8.4, we see that for every >0the convexity of the function C, on 0, is equivalent to the following inequality: I + d,,i, I I d,, I, d,,i, + I I + I 0, >0. 9.3

4 46 9 Asymptotic Analysis of Implied Volatility It is not hard to see, using the definition of d and d, that 9.3 is equivalent to 9.. Hence the convexity of the function C0, on 0, is equivalent to the validity of 9.. We will next turn our attention to the convexity conditions for the function C, on [0,. Let us assume that condition in heorem 9.6 holds, and put ϕ = C,. hen the function ϕ is twice differentiable and convex on 0,. Moreover, the function ϕ is increasing on 0,, and it follows from 8.7 that for all 0 <x<y<, ϕ x ϕy ϕx y x ϕ y. 9.4 Using the definition of the Black Scholes call pricing function, we see that for all >0 and >0, It will be shown next that ϕ = x d,,i, 0 π e y d,,i, e y dy. 9.5 π lim d,,i, = Indeed, for small values of we have d,,i= x 0 + I x 0 I, 9.7 and 9.6 follows. Using 9.5, we obtain the following equality: dy lim C, = x herefore, the function C, is continuous on [0,. Our next goal is to prove the differentiability of this function from the right at = 0. It follows from 9.4 that there exists the limit M = lim 0 ϕ. In addition, 9.4 and 9.8 give M ϕs ϕ0 S = ϕs x 0 S ϕ S for all 0 <<S<. herefore M = ϕ + 0. Moreover, 9.5 and 9.7imply ϕ = x 0 x 0 π + π e y d,,i, e y d,,i, dy dy

5 9. Implied Volatility Surfaces and Static Arbitrage 47 = x 0 x 0 π e y d,,i, = x 0 x 0 e y π x 0 = x 0 + o dy + o dy + o as 0. herefore, M =, and it follows that for every >0 the function C, is convex on [0, use affine extrapolation. he next step in the proof deals with condition 3 in heorem 9.6. Our goal is to show that lim d,,i, = lim C, = We will first prove the following equality: lim d,,i, =. 9.0 Suppose d,,i, does not tend to as. hen there exists a sequence n such that d, n,i, n e y dy c>0 for all n. It follows from 9.5 that C, n <0forn>n 0, which is impossible. herefore, 9.0 holds. It will be shown next that we always have d,,i, e y dy 0 9. as. Reasoning as in 9.7, we see that for large values of, d,,i x Using formula 8.5, we obtain F d,,i, exp { d,,i, as, where F denotes the function on the left-hand side of 9.. It follows from 9.0 and 9. that 9. holds. Nowit isclearthat 9.5 impliestheequivalence in 9.9. Note that the condition on the right-hand side of 9.9 also holds for = 0. his follows from the definition of the function C. },

6 48 9 Asymptotic Analysis of Implied Volatility Next, we turn our attention to condition in heorem 9.6. It is not hard to see that this condition is equivalent to the following: On the other hand, + I 0, > C = C BS,,I+ C BS σ,,i I, and using the formulas for the Greeks in Sect. 8.4, we obtain C = exp π { d,,i, }[ + I ]. Now 9.3 implies that condition in heorem 9.6 is equivalent to the following condition. For all >0, the function C, is non-decreasing on 0,. For = 0, the same conclusion follows from the definition of the function C. In addition, the function C, is also non-decreasing on [0,. Indeed, for any volatility parameter σ in the Black Scholes model, we have x 0 + C BS,,σ, 0, 0. herefore, C0, C,. Let us denote by μ the second distributional derivative of the function C,, and suppose that the conditions in the formulation of heorem 9.6 hold for the function C. Recall that ϕ + 0 = x 0, ϕ + 0 =, and lim ϕ = 0. he function ϕ is non-decreasing see 9.4 and integrable on [0,. herefore, ϕ is non-positive. Our next goal is to prove that lim ϕ = Using 9.4, we see that ϕ ϕ ϕ, and it is clear that the previous estimate implies 9.4. Next, taking into account 9.4, we obtain μ [0, = lim ϕ ϕ + 0 =. Moreover, the integration by parts formula for Stieltjes integrals implies the following equality: xdμ x = ϕ + 0 lim ϕ + lim ϕ = x 0. [0, It follows that condition in heorem 8.3 is valid for the function C, provided that the conditions in the formulation of heorem 9.6 hold. Finally, it is not hard to see, taking into account what was said above and applying heorem 8.3, that heorem 9.6 holds.

7 9.3 Asymptotic Behavior of Implied Volatility Near Infinity Asymptotic Behavior of Implied Volatility Near Infinity In this section, we find sharp asymptotic formulas for the implied volatility I associated with a general call pricing function C. It is assumed that the maturity is fixed and the implied volatility is considered as a function of the strike price. We also assume that C PF. his guarantees the existence of the implied volatility for large values of the strike price. he next theorem provides an asymptotic formula for the implied volatility associated with a general call pricing function. heorem 9.7 Let C PF. hen I= + + O 9.5 as. heorem 9.7 and the mean value theorem imply the following statement: Corollary 9.8 For any call pricing function C PF, I= [ + ] + O 9.6 as. Proof of heorem9.7 he next lemma will be needed in the proof of heorem 9.7. Lemma 9.9 Let C be a call pricing function, and fix a positive continuous increasing function ψ, satisfying ψ as. Suppose φ is a positive function such that φ as and ψ φ exp { φ }. 9.7

8 50 9 Asymptotic Analysis of Implied Volatility hen the following asymptotic formula holds: I= x 0 e r + φ φ + O ψ φ 9.8 as. Remark 9.0 It is easy to see that if 9.7 holds, then C PF. Proof of Lemma 9.9 Let us compare the implied volatility I with a function Ĩ such that Our goal is to prove that I= Ĩ+ O as, where d, σ is defined in 8.9. It is not hard to see that the function ρ given by satisfies the equalities and 0 < Ĩ < I, > { exp d },Ĩ ρ = x 0 e r 9.0 d,ρ = 0 9. d,ρ = ρ. 9. Plugging 9. and 9. into the Black Scholes formula formula 8., we obtain x 0 C BS,ρ = } e r exp { y dy x π ρ as. Next, taking into account 9.3 and the fact that C BS,I = 0 as, we see that C BS, I < C BS, ρ for all > 0. herefore, I < ρ, > Here we use the fact that for every fixed >0 and >0 the vega is a strictly increasing function of σ.

9 9.3 Asymptotic Behavior of Implied Volatility Near Infinity 5 It is easy to see that for sufficiently large values of, the function increases. It follows from 9. and 9.4 that σ d, σ 9.5 d,i < 0, >. 9.6 Moreover, using the explicit expression for the vega see Sect. 8.4 and the mean value theorem, we get C BS,I CBS,Ĩ = x 0 { I Ĩ exp π d, λ }, >, 9.7 where Ĩ<λ<I. Since the function in 9.5 increases and 9.6 holds, d, I <d, λ < d,i < 0, >. 9.8 Now, using 9.7 and 9.8, we establish the validity of formula 9.0. Let us continue the proof of Lemma 9.9. Suppose Ĩ is a function satisfying the equality d, I = φ, > Such a function exists, since for large values of the function σ d, σ increases from to. It follows from 9.9 and from the definition of d that Ĩ= x 0 e r + φ φ Our next goal is to use formula 9.0 with Ĩ defined in However, we have to first prove inequality 9.9. Using 8., 8.5, and 9.9, we see that there exist constants c > 0 and c > 0 such that C BS,I CBS, I = C BS, I c ψ φ exp = c ψ φ exp { φ { φ { d, Ĩ exp d, Ĩ } { φ } c } c φ exp Since ψ as and 9.3 holds, we get C BS,I >CBS, I }, >. 9.3

10 5 9 Asymptotic Analysis of Implied Volatility for sufficiently large values of. Using the fact that the vega is an increasing function of σ, we obtain inequality 9.9. Now it is clear that 9.8 follows from 9.7, 9.0, and 9.9. he proof of Lemma 9.9 is thus completed. Let us return to the proof of heorem 9.7.Letψ be a positive increasing function such that ψ as. We also assume that the function ψ tends to infinity slower than the function. Put hen we have [ ] φ= + ψ. φ as. It follows that } ψexp { φ φ as. Using formula 9.8, we obtain I= x 0 e r + φ φ + O ψ 9.3 as. Now, it is not hard to see that 9.5 can be derived from 9.3, the mean value theorem, and Lemma 3.. his completes the proof of heorem Corollaries Our objective in this section is to replace the function C in formula 9.5 by another function C. Corollary 9. Let C PF, and suppose C is a positive function such that as. hen I= +

11 9.4 Corollaries 53 as. herefore, I= + O [ + ] + O as. Formula 9.33 can be established exactly as 9.5. Formula 9.34 follows from 9.33 and the mean value theorem. We can also replace a call pricing function C in 9.5 byafunction C under more general conditions. However, this may lead to a weaker error estimate. For instance, put τ= hen the following theorem holds: heorem 9. Let C PF, and suppose C is a positive function satisfying the following condition. here exist > 0 and c with 0 <c< such that τ<c 9.36 for all >, where τ is defined by hen as. I= + [ ] + O + τ 9.37

12 54 9 Asymptotic Analysis of Implied Volatility Proof It is not hard to check that 9.36impliestheformula as. Now using 9.5, 9.35, and the mean value theorem, we obtain he next statement follows from heorem 9. and the mean value theorem. Corollary 9.3 Let C PF, and suppose C is a positive function satisfying the following condition. here exist ν>0and 0 > 0 such that ν 9.38 for all > 0. hen as. I= [ + ] + O Remark 9.4 It is not hard to see that if as,orif9.38 holds, then as. Corollary 9.5 Let C PF, and suppose C is a positive function satisfying the condition 9.39 as. hen as. I Proof It follows from 9.6 that I [ + [ + ] 9.40 ] Λ 9.4

13 9.5 Extra erms: First-Order Asymptotic Formulas for Implied Volatility 55 where + + Λ =. + + We will next prove that Λ as.wehave Λ + Λ + Λ Λ = Λ + + where Λ = and Λ = It is not hard to show that for all positive numbers a and b, a + b a + b. herefore, Λ = Λ + Λ Λ + + Λ Λ + + Λ 9.4 for > 0. It follows from 9.39 and 9.4 that Λ as.next using 9.4 we see that 9.40 holds. his completes the proof of Corollary Extra erms: First-Order Asymptotic Formulas for Implied Volatility Formula 9.5 characterizes the asymptotic behavior of the implied volatility in terms of the call pricing function C, while in formula 9.33, the function C is replaced by a function C, equivalent to C in a certain sense. We call these formulas zero-order asymptotic formulas for the implied volatility. In an important recent paper [GL],. Gao and R. Lee obtained a hierarchy of higher-order asymptotic formulas generalizing formula 9.5. Note that formula 9.33 cannot be generalized in a similar way. In the present section we establish a first-order asymptotic formula, which is different from similar first-order formulas obtained in [GL]. Higher-order asymptotic formulas from [GL] are discussed in Sect Our proofs of abovementioned formulas are refinements of the proof of heorem 9.7 given in Sect. 9.3, and they differ from the proofs given in [GL]. For the sake of simplicity, we assume x 0 = and r = 0.

14 56 9 Asymptotic Analysis of Implied Volatility heorem 9.6 Let C PF, and suppose there exist a number λ>0 and a continuous function Λ satisfying the following conditions: Λ = o and as. hen = λ + O Λ 9.43 I= + λ + λ + π λ + as. λ + λ + π λ O Λ + O Proof Suppose ψ is a positive slowly increasing function such that ψ and as. Put ϕ = Λ + ψ 0 + A Λ + + [ + ψ Here A>0 is a constant that will be chosen later. We have { ϕ } exp = A [ + Aψ Λ + ] ]

15 9.5 Extra erms: First-Order Asymptotic Formulas for Implied Volatility 57 Lemma 9.7 Let Ĩ be the function, for which 9.9 holds with ϕ given by Set λ + λ A = π λ hen Ĩ I. Proof It follows from 8. and 8.5 that { ϕ } exp C BS,Ĩ = π ϕ π + ϕ as.using9.43, 9.45, and the formula we obtain + O ϕ h = + Oh, h 0, as. Moreover, we have ϕ = + O 3 + ϕ = λ λ O Λ + O as. Our next goal is to combine formulas It is not hard to see that there exists 0 > 0 such that C BS,I CBS,Ĩ = C BS,Ĩ > 0 for all > 0. Now Lemma 9.7 follows from the fact that the vega is an increasing function of σ.

16 58 9 Asymptotic Analysis of Implied Volatility Let us return to the proof of heorem 9.6. Since formula 9.7 holds, we have { ϕ } [ I Ĩ= O exp CBS ],Ĩ as. Now using formulas again, we obtain I= Ĩ [ + O ψ Λ + ] as. It follows from 9.30 and 9.45 that Ĩ= + + A + V + A + V, 9.5 where Λ + V= + ψ Λ + = O ψ 9.53 as. Applying the mean value theorem to 9.5 and taking into account 9.47, 9.5, and 9.53, we obtain 9.44 with an extra factor ψ in the error term. Finally, using Lemma 3., we get rid of the extra factor. his completes the proof of heorem 9.6. Formula 9.44 will be used in Sect. 0.5 to study the asymptotic behavior of the implied volatility in the correlated Heston model. 9.6 Extra erms: Higher-Order Asymptotic Formulas for Implied Volatility In this section, we discuss higher-order asymptotic formulas for the implied volatility obtained in [GL]. We restrict ourselves to second- and third-order formulas, since the higher-order cases can be treated similarly. Note that when the order grows, the formulas become more and more complicated. hat is why we decided to use simpler formulas from Sects. 9.3 and 9.4 in the rest of the present book.

17 9.6 Extra erms: Higher-Order Asymptotic Formulas for Implied Volatility 59 Let us begin with a second-order formula see [GL], formula 6. in Corollary 6.. Our presentation of this result of Gao and Lee is different from that in [GL]. he main idea is to replace the constant λ in heorem 9.6 by the function λ = and put Λ = 0. hen formula 9.47 takes the following form: + A = π + his choice of the function A leads to the cancellation of all the terms in the upper estimate for the function C BS, Ĩ, except for the higher-order error terms see the proof of heorem 9.6. o justify the previous statement, we will need the estimate 0 A = π = O aking into account the previous remarks, we see that the following assertion holds. heorem 9.8 Let C PF. hen I= + + A + A + O as, where the function A is defined by Our next goal is to establish a third-order asymptotic formula for the implied volatility see formula 9.68 below. he proof of this formula is similar to that of

18 60 9 Asymptotic Analysis of Implied Volatility formula 9.56, but is more involved. Put ϕ = + + A [ + B + ψ In 9.57, ψ is a positive continuous function such that ψ and ] ψ as. he function B, appearing in 9.57, will be chosen later. his function should satisfy the following condition: as.wehave { ϕ exp B = O } = A [ + B + ψ On the other hand, using 8. and 8.5, we obtain { ϕ } exp C BS,Ĩ = [ π ϕ + ϕ 5 + O as. Set h = + + A [ + B + ψ 9.59 ] ] ϕ ϕ 3 ]. 9.6

19 9.6 Extra erms: Higher-Order Asymptotic Formulas for Implied Volatility 6 Using 9.55, 9.58, and 9.59, we obtain h = O as. herefore, = πϕ + h π = π h + O h = A + π 8 π 3 + O as. Similarly, π + ϕ = π + O + 5 A + 8 π as. Moreover, = πϕ π 5 + O 9.64 and π + ϕ 3 = 4 π + 3 3

20 6 9 Asymptotic Analysis of Implied Volatility 5 + O 9.65 as. Our next goal is to combine formulas Recalling the cancellation properties of the function A, we see that the correct choice of the function B is as follows: B = 8 π A [ ] A Indeed, it is not hard to see that with this choice of B all the terms in the estimate for the difference C BS, Ĩ, containing the factor 3, cancel out. It follows that formula 9.66 can be rewritten in the following form: B = A Here we take into account It remains to prove that the function B satisfies condition It is not hard to see that this condition follows from formulas 9.55 and Analyzing the proof sketched above, we see that the following assertion holds. heorem 9.9 Let C PF. hen I = + + O [ + A + [ + A B ] + B ] 9.68

21 9.7 Symmetries and Asymptotic Behavior of Implied Volatility Near Zero 63 as, where the function A is defined by Formula 9.68 is a third-order asymptotic formula for the implied volatility in a general model of call prices. 9.7 Symmetries and Asymptotic Behavior of Implied Volatility Near Zero In this section, we turn our attention to the asymptotic behavior of the implied volatility as 0. It is interesting to mention that one can derive asymptotic formulas for the implied volatility at small strikes from similar results at large strikes, by taking into account certain symmetries existing in the world of stochastic asset price models. We will next describe those symmetries and explain what follows from them. Let C be a general call pricing function, and let X be the corresponding stock price process. his process is defined on a filtered probability space Ω, F, {F t }, P, where P is a risk-neutral probability measure. We assume that the interest rate r and the initial condition x 0 are fixed, and denote by μ the distribution of the random variable X. Put η = x 0 e r. We call η a symmetry transformation. It is easy to see that the Black Scholes pricing function C BS satisfies the following condition: C BS,,σ= x 0 e r + e r x 0 C BS,η, σ On the other hand, the put call parity formula implies that where G is given by C, = x 0 e r + e r x 0 G,η, 9.70 G, = It follows from 8.4 and 9.7 that η G, = x 0 dμ x x 0 e r 0 x 0 e r P,η. 9.7 η 0 xdμ x. 9.7 Define a family of Borel measures { μ } 0 on 0, as follows. For every Borel subset A of 0, put μ A = x 0 e r xdμ x η A

22 64 9 Asymptotic Analysis of Implied Volatility It is not hard to see that { μ } 0 is a family of probability measures. Moreover, for all >0 and 0, we have and d μ x = x 0 e r η 0 xdμ x 9.74 η xd μ x = x 0 e r dμ x It follows from 9.7, 9.74, and 9.75 that G, = e r xd μ x e r d μ x Remark 9.0 Suppose for every >0the measure μ is absolutely continuous with respect to the Lebesgue measure on 0,. Denote the Radon Nikodym derivative of μ with respect to the Lebesgue measure by D. hen, for every >0the measure μ admits a density D Y given by D x = x 0e r 3 x 3 D 0 x0 e r x, x >0. he next theorem has important consequences. For example, it will allow us to establish a link between the asymptotic behavior of the implied volatility at large and small strikes. heorem 9. Let C be a call pricing function and let P be the corresponding put pricing function. hen the function G defined by 9.7 is a call pricing function with the same interest rate r and the initial condition x 0 as the pricing function C. Moreover, if X is the stock price process associated with G, then for every >0 the measure μ defined by 9.73 is the distribution of the random variable X. Proof According to heorem 8.3, it suffices to prove that conditions 5 in the formulation of this theorem are valid for the function G. We have already shown that for every 0, μ is a probability measure. In addition, equality 8.8 holds for μ,by9.75. Put V,= xd μ x d μ x. hen G, = e r V,. Moreover, the function V, is convex on [0,, since its second distributional derivative coincides with the measure μ. his establishes conditions and in heorem 8.3. he equality G0,= x 0 + can be obtained using 9.7. hus condition 4 holds. Next, we see that 9.76

23 9.8 Symmetric Models 65 implies G, e r xd μ x, and hence lim G, = 0. his establishes condition 5. In order to prove the validity of condition 3 for G, we notice that 9.70 gives the following: G,e r = C,e r x 0 + x x 0 Now it is clear that condition 3 for G follows from the same condition for C. herefore, G is a call pricing function. his completes the proof of heorem 9.. Remark 9. It is not hard to see that if the call pricing function C in heorem 9. satisfies C PF, then G PF 0. Similarly, if C PF 0, then G PF. Let C be a call pricing function such that C PF PF 0. hen G PF PF 0, and hence the implied volatilities I C and I G associated with the pricing functions C and G, respectively, exist for all >0 and >0. Replacing σ by I C in 9.69 and taking into account 9.70 and the equality we see that herefore, the following lemma holds. C BS,,IC, = C,, C BS,η, I C, = G,η. Lemma 9.3 Let C PF PF 0, and let G be defined by hen for all >0 and >0. I C, = I G,η 9.78 Lemma 9.3 shows that the implied volatility associated with C can be obtained from the implied volatility associated with G by applying the symmetry transformation. 9.8 Symmetric Models he notion of a symmetric model is based on the symmetry properties of stochastic models discussed in the previous section.

24 66 9 Asymptotic Analysis of Implied Volatility Definition 9.4 A stochastic asset price model is called symmetric if, for every >0 the distributions μ and μ coincide. Lemma 9.5 he following statements hold:. Suppose for every >0 the measure μ admits a density D. hen the model is symmetric if and only if for all >0, D x = x 0 e r 3 x 3 D x0 e r x 9.79 almost everywhere with respect to the Lebesgue measure on 0,.. Suppose the asset price process X is strictly positive and for every >0the measure μ admits a density D. Define the -price process by X = X and denote by D the distribution density of X, >0. hen the model is symmetric if and only if D x = x 0e r e x D x + x0 e r almost everywhere with respect to the Lebesgue measure on R. 3. he model is symmetric if and only if for all >0 and >0, G, = C,. 4. he model is symmetric if and only if for all >0 and >0, C, = x 0 e r C, x 0 e r + x 0 e r. 5. Let C PF PF 0. hen the model is symmetric if and only if for all >0 and >0, I,= I, x 0 e r. Proof Part 3 of Lemma 9.5 follows from 8.3, 9.76, and from the fact that the measures μ and μ are the second distributional derivatives of the functions C, and G,, respectfully. Part 4 can be easily derived from As for part 5 of Lemma 9.5, it can be established using part 3 and Lemma 9.3. In addition, part follows from Definition 9.4 and Remark 9.0. Finally, the equivalence follows from the standard equalities D x = ex D e x and D y = y D y. his completes the proof of Lemma 9.5. Special examples of symmetric models are uncorrelated stochastic volatility models in a risk-neutral setting. Let us consider a stochastic model defined by { dxt = rx t dt + fy t X t dw t, 9.80 dy t = by t dt + σy t dz t, where W and Z are independent Brownian motions on Ω, F, {F t }, P, and suppose that the measure P is risk-neutral. Suppose also that the solvability conditions

25 9.8 Symmetric Models 67 discussed in Sect.. hold. It is clear that for such a model, formula 9.79 follows from formula 3.6. herefore, part of Lemma 9.5 shows that the model in 9.80 is symmetric. Remark 9.6 he symmetry condition for the implied volatility in part 4 of Lemma 9.5 becomes especially simple if the strike is replaced by the moneyness k defined by k =, >0. x 0 er In terms of the -moneyness, the symmetry condition can be rewritten as follows: Ik= I k for all <k<. For uncorrelated stochastic volatility models, the previous equality was first obtained in [R96]. In [CL09], P. Carr andr. Leeestablishedthatundercertainrestrictions, stochastic volatility models are symmetric if and only if ρ = 0. We will next prove this result of Carr and Lee. We restrict ourselves to models with time-homogeneous volatility equation. However, heorem 9.7 also holds when volatility equations are inhomogeneous see [CL09]. Let us consider the stochastic model given by { dxt = rx t dt + Y t X t dw t, 9.8 dy t = by t dt + σy t dz t. Itisassumedin9.8 that Z = ρ Z + ρw, where Z is a standard Brownian motion independent of W, and the correlation coefficient ρ is such that ρ. It is also assumed that the functions b and σ in 9.8 satisfy the linear growth condition and the Lipschitz condition, the function σ is positive, and for every ρ and every positive initial condition y 0 the solution Y to the second equation in 9.8 is a positive process. heorem 9.7 Suppose the model in 9.8 satisfies the conditions formulated above. In addition, suppose the discounted price process is a martingale. hen the model is symmetric if and only if ρ = 0. Remark 9.8 It is worth mentioning that the conditions in heorem 9.7 are rather restrictive. For example, this theorem is not applicable to the Stein Stein model, or the Heston model. Indeed, in the Stein Stein model the volatility process is not positive, while in the Heston model the function σ does not satisfy the Lipschitz condition. On the other hand, heorem 9.7 can be used to prove that a negatively correlated Hull White model cannot be symmetric. Indeed, in such a model the volatility process is a geometric Brownian motion, and hence it is a positive process. Moreover, the stock price process is a martingale use heorem.33. Note also that if a geometric Brownian motion Y is the solution to the equation dy t = νy t dt + ξy t dz t

26 68 9 Asymptotic Analysis of Implied Volatility with the initial condition y 0 > 0, then the process Ỹ = Y is also a geometric Brownian motion satisfying the equation ν dỹ t = ξ Ỹ t dt + ξ 8 Ỹt dz t with the initial condition y 0. Summarizing what was said above, we see that heorem 9.7 can be applied to the negatively correlated Hull White model. If the Hull White model is positively correlated, then heorem.33 implies that the stock price process is not a martingale. herefore, heorem 9.7 cannot be applied to such a model. It would be interesting to extend heorem 9.7 to a larger class of stochastic volatility models. Proof It has already been established that for ρ = 0, the model is symmetric. We will next prove the converse statement. With no loss of generality, we can assume r = 0. Fix ρ>0, and suppose the symmetry condition holds for the model given by { dxt = Y t X t dw t, dy t = by t dt + ρ σy t d Z t + ρσy t dw t. Using the Itô formula, we can rewrite the model above in terms of the -price process defined by X = X and X 0 = x 0. his gives dx t = Y t dt + Y t dw t, 9.8 dy t = by t dt + ρ σy t d Z t + ρσy t dw t. Let us fix >0. Since the process X is a martingale, the measure P determined from d P = x 0 X dp is a probability measure. Define a new process by Ŵ t = W t t 0 Ys ds, 0 t. It follows from Girsanov s theorem that the process Ŵ t, Z t, t [0,], isatwodimensional standard Brownian motion under the measure P. herefore, the same is true for the process W t, Z t,0 t, where W t = Ŵ t for all t [0,]. It is easy to see that under the measure P, the system in 9.8 can be rewritten as follows: d X t = Y t dt + Y t d W t, 9.83 dy t = ΦY t dt + ρ σy t d Z t + ρσy t d W t where Φu = bu + ρσu u.

27 9.8 Symmetric Models 69 Recall that, by our assumption, the model described by 9.8 issymmetric.using part of Lemma 9.5, we obtain E [ X ] = xd x dx = ue u D x u du + x 0 e u D u du 0 x 0 = E [ X X ] x 0 + E[X ]. x 0 x 0 Next, using the fact that the process X is a martingale, we see that E [ X ] [ = Ẽ X ] + x he next step in the proof is to take the expectation E in the first stochastic differential equation in 9.8, written in the integral form. his gives E [ X ] = Similarly, applying Ẽ to the first equation in 9.83, we obtain Ẽ [ X ] = 0 It follows from 9.84, 9.85, and 9.86 that 0 0 E[Y t ] dt = E[Y t ] dt + x Ẽ[Y t ] dt + x Ẽ[Y t ] dt We will next use a coupling argument. Consider the following processes: X, Y, W, Z under the measure P and X, Y, W, Z under the measure P. Applying the lemma formulated on p. 4 of [IW77], we see that there exist a filtered measure space Ω,F, F t, P and adapted stochastic processes X, Y, X, Y, W, and Z on Ω such that the following conditions hold: he processes X,Y,W, Z and X,Y,W,Z have the same law under the measures P and P, respectively. he processes X,Y, W, Z and X,Y,W,Z have the same law under the measures P and P, respectively. he process W,Z is a two-dimensional F t -Brownian motion under the measure P. It follows 9.8, 9.83, and the previous statements that under the measure P, dx t = Y t dt + Y t dw t, dy t = b Y t dt + ρ σ Y t dz t + ρσ Y t 9.88 dw t

28 70 9 Asymptotic Analysis of Implied Volatility and dx t = Y dy t = Φ Y t t dt + Y t dw t, dt + ρ σ Y t dz t + ρσy t dw t Moreover, 9.87 implies that 0 Ê [ Y t ] dt = 0 Ê [ Y t ] dt Now we are ready to finish the proof. Applying the strong comparison theorem for stochastic differential equations heorem 54 in [Pro04] to 9.88 and 9.89, we see that Y t >Y t for all 0 <t<. 9.9 Here we take into account that bu < Φu and the initial condition x 0,y 0 is the same for the processes X,Y and X,Y. However, 9.9 contradicts It follows that if ρ>0, then the model cannot be symmetric. he case where ρ<0is similar. his completes the proof of heorem Asymptotic Behavior of Implied Volatility for Small Strikes Lemma 9.3 and the results obtained in Sect. 9.3 imply sharp asymptotic formulas for the implied volatility as 0. heorem 9.9 Let C PF 0, and let P be the corresponding put pricing function. Suppose P P as 0, 9.9 where P is a positive function. hen the following asymptotic formula holds: I= P P P + O P P 9.93 as 0.

29 9.9 Asymptotic Behavior of Implied Volatility for Small Strikes 7 Corollary 9.30 he following asymptotic formula holds: [ I= as 0. P ] P + O P P An important special case of heorem 9.9 is as follows: Corollary 9.3 Let C PF 0, and let P be the corresponding put pricing function. hen [ ] I= P P as 0. + O P P Proof of heorem 9.9 Formulas 9.7 and 9.9 imply that G G as where G = P η Next, applying Corollary 9. to G and G, we get I G = + G G G G + O G 9.95 as. It follows from 9.78, 9.94, 9.95, and from the mean value theorem that I= x 0e r + x 0 e r P x 0 e r P

30 7 9 Asymptotic Analysis of Implied Volatility = + O x 0 e r P x 0 e r P P P + O P P P P as 0. his completes the proof of heorem Notes and References he books [FPS00, Reb04, Haf04, Fen05, Gat06, H-L09], the dissertations [Dur04, Rop09], the surveys [Ski0, CL0], and the papers [SP99, SH99, Lee0, CdF0, Lee04a, CGLS09, Fri0] are useful sources of information on the implied volatility. Section 9. is mostly adapted from [Rop0]. However, the conditions in heorem 9.6 are not exactly the same as in the similar result heorem.9 in [Rop0]. Moreover, heorem 9.6 is formulated in terms of the strike price, while the moneyness is used in [Rop0]. he asymptotic formulas for the implied volatility included in Sects. 9.3, 9.4, and 9.9 are taken from [Gul0]. he material in Sects. 9.7 and 9.8 symmetries and symmetric models comes mostly from [Gul0]. We send the interested reader to [CL09, eh09a, DM0, DMM0] for more information on symmetric models. he paper [GL] of. Gao and R. Lee is an important recent work on smile asymptotics. In Sects. 9.5 and 9.6 of this chapter, several theorems from [GL] are presented. hese theorems provide higher-order approximations for the implied volatility at extreme strikes. However, we have not touched upon the results in [GL] characterizing the asymptotic behavior of the implied volatility with respect to the maturity, or in certain combined regimes.

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