The Implied of Forward Starting Options: ATM Short-Time Level, Skew and Curvature Elisa Alòs Antoine Jacquier Jorge A. León May 2017

Barcelona GSE Working Paper Series Working Paper nº 988

The of forward starting options: ATM short-time level, skew and curvature

Elisa Al`os∗ Antoine Jacquier Dpt. d’Economia i Empresa Department of Mathematics Universitat Pompeu Fabra Imperial College London and Barcelona GSE London SW7 2AZ, UK c/Ramon Trias Fargas, 25-27 08005 Barcelona, Spain Jorge A. Le´on† Control Autom´atico CINVESTAV-IPN Apartado Postal 14-740 07000 M´exico,D.F., Mexico

Abstract For models, we study the short-time behaviour of the at-the-money implied volatility level, skew and curvature for forward- starting options. Our analysis is based on Malliavin Calculus techniques. Keywords: Forward starting options, implied volatility, Malliavin cal- culus, stochastic volatility models JEL code: C02

1 Introduction

Consider two moment times s > t. A forward-start call with maturity T > s allows the holder to receive, at time s and with no additional cost, a expirying at T , with strike set equal to KSs, for some K > 0. So, the option life starts at s, but the holder pays at time t the price of the option. Some classical applications of forward starting options include employee stock options and cliquet options, among others (see for example Rubinstein (1991)). Under the Black-Scholes formula, a conditional expectation argument leads to show that the price of a forward starting option is the price of a plain vanilla

∗Supported by grants ECO2014-59885-P and MTM2016-76420-P (MINECO/FEDER, UE). †Partially supported by the CONACyT grant 220303

1 option with time to maturity T −t. In the stochastic volatility case, a change-of- measure links the price of the forward option with the price of a classical vanilla (see, for example, Rubinstein (1991), Musiela and Rutkowski (1997), Wilmott (1998), or Zhang (1998)). For stochastic volatility models, change of numeraire techniques can be applied to obtain a closed-form pricing formula in the context of the Heston model (see Kruse and N¨ogel(2005)). The implied volatility surface for forward starting options exhibits substan- tial differences to the classical vanilla case (see for example Jacquier and Roome (2015)). This paper is devoted to the study of the at-the-money (ATM) short time limit of the implied volatility for forward starting options. More precisely, we will use Malliavin Calculus techniques to compute the ATM short-time limit of the implied volatility level, skew and curvature. In particular, we will see that -contrary to the classical vanilla case- the ATM short-time level depends on the correlation parameter (see Lemma 6 and Theorem 7). We will also prove that the skew depends of the Malliavin of the volatility process in a similar way as for vanilla options, while the curvature (see Theorems 14 and 15) is of order O(T − s). The paper is organized as follows. Section 2 is devoted to introduce forward starting options and the main notation used troghout the paper. In Section 3 we obtain a decomposition of the option price that will allow us to compute, in Sections 4, 5 and 6, the limits for the ATM implied volatility level, skew and curvature.

2 Forward start option

We will consider the Heston model for stock price on a time interval [0,T ] under a risk neutral probability P ∗ :   ∗ p 2 ∗ dSt =rS ˆ tdt + σtSt ρdWt + 1 − ρ Bt , t ∈ [0,T ], (1) wherer ˆ is the instantaneous interest rate (supposed to be constant), W ∗ and B∗ are independent standard Brownian motions defined on a probability space (Ω, F,P ∗) and σ is a positive and square-integrable process adapted to the ∗ ∗ filtration generated by W ∗. In the following we will denote by F W and F B the filtrations generated by W ∗ and B∗, respectively. Moreover we define F := ∗ ∗ F W ∨ F B . It will be convenient in the following sections to make the change of variable Xt = log (St) , t ∈ [0,T ]. Now, we consider a point s ∈ [0,T ] . We want to evaluate the following option price:

−rˆ(T −t) ∗ XT α Xs  Vt = e Et e − e e + , (2)

∗ where E denotes the conditional expectation given Ft and α is a real constant. Notice that if t ≥ s this is the payoff of a call option, while in the case t < s this defines a forward start option. We will make use of the following notation

2 • BS(t, x, K, σ) will represent the price of a European call option under the classical Black-Scholes model with constant volatility σ, current log stock price x, time to maturity T − t, K and interest rater. ˆ Remember that in this case

x −rˆ(T −t) BS(t, x, K, σ) = e N(d+) − e KN(d−), where N is the cumulative probability function of the standard normal law and

x − ln K +r ˆ(T − t) σ √ d± := √ ± T − t. σ T − t 2

•L BS (σ) will denote the Black-Scholes differential operator (in the log vari- able) with volatility σ :

∂ 1 ∂2  1  ∂ L (σ) = + σ2 + rˆ − σ2 − r.ˆ BS ∂t 2 ∂x2 2 ∂x

It is well known that LBS (σ) BS (·, ·; K, σ) = 0.

2 • G(t, x, K, σ) := (∂xx − ∂x)BS(t, x, K, σ). 2 • H(t, x, K, σ) := ∂x(∂xx − ∂x)BS(t, x, K, σ). • BS−1(a) := BS−1 (t, x, K, a) denotes the inverse of BS as a function of the volatility parameter.

• α∗ :=r ˆ(T − s)

We recall the following result, that can be proved following the same argu- ments as in that of Lemma 4.1 in Al`os,Le´onand Vives (2007)

W Lemma 1 Let 0 ≤ t ≤ s, u < T and Gt := Ft ∨ FT . Then for every n ≥ 0, there exists C = C(n, ρ) such that

a) If u ≤ s

− 1 (n+1) Z T ! 2 n Xu  2 |E (∂x G (u, Xu,Mu, vu)| Gt)| ≤ CE e Gt σs ds . s

b) If u ≥ s

− 1 (n+1) Z T ! 2 n Xs  2 |E (∂x G (u, Xu,Mu, vu)| Gt)| ≤ CE e Gt σs ds . s

3 3 A decomposition formula for forward option prices

We will stand for

∗ α Xs  • Mt := Et e e , t ∈ [0,T ]. We observe that

Z t   α rsˆ X0 α rˆ(s−u) Xu ∗ p 2 ∗ Mt = e e e + σu1[0,s](u)e e e ρdWu + 1 − ρ dBu 0 Z t∧s   α rˆ(s−u) Xu ∗ p 2 ∗ = M0 + e σue e ρdWu + 1 − ρ dBu . 0

1   2 Yt • vt := T −t , with

Z T Z T 2 2 Yt := σu1[s,T ](u)du = σudu t t∨s √ √ Notice that, if t < s, vt T − t = vs T − s.

We will need the following hypotheses: (H1) There exist two constants 0 < c < C such that c < σt < C, for all t∈ [0,T ], with probability one. (H2) σ, σeX ∈ L2,2 ∩ L1,4. Now we are in a position to prove the following theorem, that allows us to identify the impact of correlation in the forward option price

Theorem 2 Consider the model (1) and assume that hypotheses (H1) and (H2) hold. Then, for all 0 ≤ t ≤ s ≤ T ,  ∗ α Vt = Et exp(Xt)BS (s, 0, e , vs) Z T ρ −rˆ(u−t) W ∗ + e H(u, Xu,Mu, vu)σuΛu du 2 s s  ρ Z ∗ α −rˆ(u−t) Xu W + G(s, 0, e , vs) e e σuΛu du . 2 t

W ∗ R T W ∗ 2 where Λu := u∨s Du σθ dθ.

∗ α Xs  Proof. Remember that Mt := Et e e and notice that

−rˆ(T −t) ∗ XT α Xs  −rˆ(T −t) ∗ XT  Vt = e Et e − e e + = e Et e − MT + −rˆ(T −t) = e BS(T,XT ,MT , νT ).

4 Therefore, the anticipating Itˆo’sformula for the Skorohod integral (see for ex- ample Nualart (2006)) allows us to write

∗ −rTˆ  Et e BS(T,XT ,MT , vT ) " Z T ∗ −rtˆ −ruˆ = Et e BS(t, Xt,Mt, vt) − rˆ e BS(u, Xu,Mu, vu)du t Z T −ruˆ ∂BS + e (u, Xu,Mu, vu)du t ∂u Z T  2  −ruˆ ∂BS σu + e (u, Xu,Mu, vu) rˆ − du t ∂x 2 Z T 2 1 −ruˆ ∂ BS 2 + e 2 (u, Xu,Mu, vu)σudu 2 t ∂x Z T 2 −ruˆ ∂ BS + e (u, Xu,Mu, vu)d hX,Miu t ∂x∂K Z T 2 1 −ruˆ ∂ BS + e 2 (u, Xu,Mu, vu)d hM,Miu 2 t ∂K Z T  2  1 −ruˆ ∂ ∂ 2 2  + e 2 − BS(u, Xu,Mu, vu) vu − σu1]s,T ](u) du 2 t ∂x ∂x Z T  2  1 −ruˆ ∂ ∂ ∂ W ∗ + e 2 − BS(u, Xu,Mu, vu)σuρΛu du 2 t ∂x ∂x ∂x T  2  # 1 Z ∂ ∂ ∂ ∗ −ruˆ α rˆ(s−u) Xu W + e 2 − BS(u, Xu,Mu, vu)σue e e ρΛu 1[0,s](u)du . 2 t ∂K ∂x ∂x

5 That is, using t ≤ s,  ∗ Vt = Et BS(t, Xt,Mt, vt) Z T −rˆ(u−t) + e LBS (vu) BS(u, Xu,Mu, vu)du t Z T  2  1 −rˆ(u−t) ∂ ∂ 2 2 + e 2 − BS(u, Xu,Mu, vu) σu − vu du 2 t ∂x ∂x Z T 2 −rˆ(u−t) ∂ BS + e (u, Xu,Mu, vu)d hX,Miu t ∂x∂K Z T 2 1 −rˆ(u−t) ∂ BS + e 2 (u, Xu,Mu, vu)d hM,Miu 2 t ∂K Z T  2  1 −rˆ(u−t) ∂ ∂ 2 2  + e 2 − BS(u, Xu,Mu, vu) vu − σu1]s,T ](u) du 2 t ∂x ∂x Z T  2  1 −rˆ(u−t) ∂ ∂ ∂ W ∗ + e 2 − BS(u, Xu,Mu, vu)σuρΛu du 2 t ∂x ∂x ∂x s  2   1 Z ∂ ∂ ∂ ∗ −rˆ(u−t) α rˆ(s−u) Xu W + e 2 − BS(u, Xu,Mu, vu)σue e e ρΛu du . 2 t ∂K ∂x ∂x Thus, we get, for t ≤ s,  ∗ Vt = Et BS(t, Xt,Mt, vt) Z T −rˆ(u−t) + e LBS (vu) BS(u, Xu,Mu, vu)du t Z T  2  1 −rˆ(u−t) ∂ ∂ 2 2  + e 2 − BS(u, Xu,Mu, vu) σu − σu1]s,T ](u) du 2 t ∂x ∂x Z T 2 −rˆ(u−t) ∂ BS + e (u, Xu,Mu, vu)d hX,Miu t ∂x∂K Z T 2 1 −rˆ(u−t) ∂ BS + e 2 (u, Xu,Mu, vu)d hM,Miu 2 t ∂K Z T  2  1 −rˆ(u−t) ∂ ∂ ∂ W ∗ + e 2 − BS(u, Xu,Mu, vu)σuρΛu du 2 t ∂x ∂x ∂x s  2   1 Z ∂ ∂ ∂ ∗ −rˆ(u−t) α rˆ(s−u) Xu W + e 2 − BS(u, Xu,Mu, vu)σue e e ρΛu du . 2 t ∂K ∂x ∂x Now, taking into account the facts that

LBS (vu)(BS)(u, Xu,Mu, vu) = 0,

2 α rˆ(s−u) Xu d hM,Xiu = σue e e 1[0,s](u)du,

6 2 2α 2ˆr(s−u) 2Xu d hM,Miu = σue e e 1[0,s](u)du, ∂2BS 1 ∂2BS ∂BS  (t, x, K, σ) = − − (t, x, K, σ) ∂x∂K K ∂x2 ∂x and ∂2BS 1 ∂2BS ∂BS  (t, x, K, σ) = − (t, x, K, σ) , ∂K2 K2 ∂x2 ∂x it follows that  ∗ Vt = Et BS(t, Xt,Mt, vt)

Z T  2  1 −rˆ(u−t) ∂ ∂ ∂ W ∗ + e 2 − BS(u, Xu,Mu, vu)σuρΛu du 2 t ∂x ∂x ∂x s  2   1 Z ∂ ∂ ∂ ∗ −rˆ(u−t) α rˆ(s−u) Xu W + e 2 − BS(u, Xu,Mu, vu)σue e e ρΛu du 2 t ∂K ∂x ∂x  ∗ = Et BS(t, Xt,Mt, vt)

Z T  2  1 −rˆ(u−t) ∂ ∂ ∂ W ∗ + e 2 − BS(u, Xu,Mu, vu)σuρΛu du 2 s ∂x ∂x ∂x Z s  2  1 −rˆ(u−t) ∂ ∂ ∂ W ∗ + e 2 − BS(u, Xu,Mu, vu)σuρΛu du 2 t ∂x ∂x ∂x Z s  2   1 −rˆ(u−t) ∂ ∂ ∂ W ∗ + e 2 − BS(u, Xu,Mu, vu)σuMuρΛu du . 2 t ∂K ∂x ∂x Since we have ∂ ∂2BS ∂BS  exN 0(d )  d  − (t, x, k, σ) = √ + 1 − √ + ∂x ∂x2 ∂x σ T − t σ T − t and ∂ ∂2BS ∂BS  exN 0(d )  d  − (t, x, k, σ) = √ + √ + , ∂K ∂x2 ∂x Kσ T − t σ T − t then it is easy to see that  ∗ Vt = Et BS(t, Xt,Mt, vt) Z T ρ −rˆ(u−t) W ∗ + e H(u, Xu,Mu, vu)σuΛu du 2 s Z s  ρ −rˆ(u−t) W ∗ + e G(u, Xu,Mu, vu)σuΛu du . 2 t

7 Hence, due to

BS(t, Xt,Mt, vt)  √  −α +r ˆ(T − s) vs T − s = exp(Xt)N √ + vs T − s 2  √  α −rˆ(T −s) −α +r ˆ(T − s) vs T − s −e exp(Xt)e N √ − vs T − s 2 α = exp(Xt)BS (s, 0, e , vs) and, for all u < s,

√  −α+ˆr(T −s)  eXu N 0 √ + vs T −s vs T −s 2 G(u, Xu,Mu, vu) = √ vs T − s Xu α = e G(s, 0, e , vs), the proof is complete.

Remark 3 We have chosen Hypotheses (H1) and (H2) for the sake of simplic- ity, which can be substituted by adequate integrability conditions.

Remark 4 Notice that, if t = s we recover the decomposition formula for call option prices presented in Al`os,Le´onand Vives (2007).

Remark 5 If the volatility process is constant (σu = σ, for some positive con- W stant σ), then vs = σ and Λ ≡ 0, which implies that, for t ≤ s,

α Vt = exp(Xt)BS (s, 0, e , σ) . (3)

Consequently, we recover the well-known option pricing formula for forward start options under the Black-Scholes model (see for example Rubinstein (1991), Wilmott (1998), or Zhang (1998)).

4 The ATM short-time limit of the implied volatil- ity.

For t ∈ [0, s], we define the implied volatility I(t, s; α) as the Ft-adapted process satisfying α Vt = exp(Xt)BS (s, 0, e ,I(t, s; α)) . (4)

Notice that, from (3), in the constant volatility case σu = σ, I(t, s; α) = σ. For the sake of simplicity, we will consider first the uncorrelated case ρ = 0. In this case, we have the following result

8 Lemma 6 Consider the model (1) with ρ = 0 and assume that hypotheses in Theorem 2 hold. Then, for all t < s,

∗ ∗ lim I(t, s; α ) = Et (σs) . T →s Proof. We know that

I(t, s; α∗) −1 ∗ ∗ = BS (Et (BS (s, 0, exp(α ), vs))) ∗  −1 ∗ = Et BS (BS (s, 0, exp(α ), vs)) −1 ∗ ∗ −1 ∗  +BS (Et (BS (s, 0, exp(α ), vs))) − BS (BS (s, 0, exp(α ), vs)) ∗  −1 ∗ ∗ = Et vs + BS (Et (BS (s, 0, exp(α ), vs))) −1 ∗  −BS (BS (s, 0, exp(α ), vs)) .

Notice that a direct application of Clark-Ocone’s formula gives us that Z T ∗ ∗ ∗ BS(s, 0, exp(α ), vs) = Et (BS(s, 0, exp(α ), vs)) + UrdWr, t with   W ∗ R T 2 ! ∗ ∂BS ∗ Du s σr dr Uu = Eu (s, 0, exp(α ), vs) . (5) ∂σ 2(T − s)vs

−1 Then, applying Itˆo’s formula to the process BS (Au), where Z u ∗ ∗ Au := Et (BS(s, 0, exp(α ), vs)) + UrdWr t and taking expectations, we get

∗  −1 ∗ ∗ −1 ∗  Et BS (Et (BS (s, 0, exp(α ), vs))) − BS (BS (s, 0, exp(α ), vs)) Z T 1 ∗ −1 00 ∗ 2 = − Et (BS ) (Er(BS (s, 0, exp(α ), vs))Ur dr. 2 t Hence

lim I(t, s, α∗) T →s ∗ = Et (σs) Z T 1 ∗ −1 00 ∗ 2 − lim Et (BS ) (Er(BS (s, 0, exp(α ), vs))Ur dr. T →s 2 t Now, considering that

−1 00 ∗ (BS ) (Er(BS (s, 0, exp(α ), vs))  −1 ∗  1 BS (Er(BS (s, 0, exp(α ), vs))(T − s) = √ , 0 2 4 N (d1) T − s

9 −1 ∗ √ BS (Er (BS(s,0,exp(α ),vs)) T −s where d1 := 2 , we get that Z T 1 ∗ −1 00 ∗ 2 lim Et (BS ) (Er(BS (s, 0, exp(α ), vs))Ur dr = 0. T →s 2 t Therefore, the proof is complete. As a consequence of the previous result, we have the following limit in the correlated case. Theorem 7 Consider the model (1) and assume that hypotheses in Theorem 2 hold. Then, for all t < s,

 s  ρ 1 Z ∗ ∗ ∗ −Xt ∗ −rˆ(u−t) Xu W 2 lim I(t, s; α ) = Et (σs) + e Et e e σuDu σs du . T →s 2 σs t Proof. We know, by Theorem 2, that

I(t, s; α∗) −1 ∗ ∗ = BS [Et (BS (s, 0, exp(α ), vs)) Z T ρ ∗ −rˆ(u−t) W ∗ + exp(−Xt)Et e H(u, Xu,Mu, vu)σuΛu du 2 s  s  ρ Z ∗ ∗ ∗ −rˆ(u−t) Xu W + exp(−Xt)Et G(s, 0, exp(α ), vs) e e σuΛu du . 2 t

∗ ∗ By the mean value theorem, we can find θ between Et (BS(s, 0, exp(α ), vs)) and  ∗ ∗ Et BS (s, 0, exp(α ), vs) Z T ρ −rˆ(u−t) W ∗ + exp(−Xt e H(u, Xu,Mu, vu)σuΛu du 2 s s  ρ Z ∗ ∗ −rˆ(u−t) Xu W + exp(−XtG(s, 0, exp(α ), vs) e e σuΛu du 2 t such that

∗ −1 ∗ ∗ I(t, s; α ) − BS (Et (BS(s, 0, exp(α ), vs))) √ BS−1(θ) 2π exp( 8 (T − s)) hρ = √ exp(−Xt) T − s 2 Z T ! ∗ −rˆ(u−t) W ∗ ×Et e H(u, Xu,Mu, vu)σuΛu du s  s  ρ Z ∗ ∗ ∗ −rˆ(u−t) Xu W + Et G(s, 0, exp(α ), vs) e e σuΛu du 2 t = I1 + I2. (6)

10 From Lemma 1, we have

Z T exp(−Xt) ∗ √ ∗ Xs  −1 W lim |I1| ≤ C lim E e |Gt (T − s) Λu du T →s T →s T − s s 1/2 1/2 ∗ 2Xs  ≤ C exp(−Xt) lim (T − s) E e |Gt = 0. T →s

Finally, (6) and Lemma 6 imply

lim I(t, s; α∗) T →s 2 vs (T −s) s ! ρ exp(X ) exp( ) Z ∗ ∗ t ∗ 8 −rˆ(u−t) Xu W = Et (σs) + lim Et e e σuΛu du 2 T →s vs(T − s) t  s  ρ exp(X ) 1 Z ∗ ∗ t ∗ −rˆ(u−t) Xu W 2 = Et (σs) + Et e e σuDu σs du . 2 σs t Therefore, the proof is complete.

Remark 8 Notice that, in the case s = t, we recover the results by Durrleman (2008) for classical vanilla options. Also, contrary to the classical vanilla case, the ATM short-time limit depends on the correlation parameter.

We will need the following lemma later on.

  R T 2  0 σudu Lemma 9 Consider the model (1) and assume that E exp 2 < ∞.Then √ I(t, s; α∗) T − s → 0 as T → s.

Proof. We know that, in this case,

   −rˆ(T −t) ∗ XT rˆ(T −s) Xs Vt = e Et e − e e , + which converges to zero as T → s. Indeed, we have   eXT − erˆ(T −s)eXs → 0, + as T → s with probability 1. Moreover, by Novikov theorem (see for example Karatzas and Shreve (1991)), we have   eXT − erˆ(T −s)eXs ≤ eXT + erˆ(T −s)eXs + and   E∗ eXT + erˆ(T −s)eXs = 2erTˆ → 2ersˆ

11 as T → s. Thus, the dominated convergence theorem implies that Vt → 0 as T → s, with probability one. On the other hand, √ √  I(t, s; α∗) T − s  I(t, s; α∗) T − s V = exp(X ) N − N − t t 2 2 √   I(t, s; α∗) T − s = exp(X ) 1 − 2N − , t 2 which allows us to complete the proof.

5 An expression for the derivative of the implied volatility

Equation (4) leads us to get

∂V ∂BS t = exp(X ) (s, 0, eα,I(t, s; α)) ∂α t ∂k ∂BS ∂I + exp(X ) (s, 0, eα,I(t, s; α)) (t, s; α), t ∂σ ∂α

∂BS ∂BS where ∂k := ∂(ln K) . Then, from Theorem 2 we are able to write ∂I (t, s; α) ∂α ∂Vt − exp(X ) ∂BS (s, 0, eα,I(t, s; α)) = ∂α t ∂k ∂BS α exp(Xt) ∂σ (s, 0, e ,I(t, s; α)) E∗  ∂BS (s, 0, eα, v ) − ∂BS (s, 0, eα,I(t, s; α)) = t ∂k s ∂k ∂BS α ∂σ (s, 0, e ,I(t, s; α)) h T ∗ i ∗ R −rˆ(u−t) ∂H α Xs W ρ Et s e ∂k (u, Xu, e e , vu)σuΛu du + 2 ∂BS α exp(Xt) ∂σ (s, 0, e ,I(t, s; α)) s ∗ ∗  ∂G α R Xu −rˆ(u−t) W  ρ E (s, 0, e , vs) e e σuΛ du + t ∂k t u . (7) 2 ∂BS α exp(Xt) ∂σ (s, 0, e ,I(t, s; α)) Remark 10 Note that in the uncorrelated case (i.e., ρ = 0), equality (7) implies

∂I E∗  ∂BS (s, 0, eα, v ) − ∂BS (s, 0, eα,I(t, s; α)) (t, s; α) = t ∂k s ∂k . ∂α ∂BS α ∂σ (s, 0, e ,I(t, s; α))

12 In this case, for α = α∗, we get, by Theorem 2, ∂BS  E∗ (s, 0, exp(α∗), v ) t ∂k s √   v T − s = E∗ −N − s t 2  √   √   vs T −s vs T −s  N 2 − N − 2 − 1 = E∗ t  2 

BS (s, 0, exp(α∗), v ) − 1 = E∗ s t 2 V exp (−X ) − 1 = t t 2 and ∂BS (s, 0, exp(α∗),I(t, s; α∗)) ∂k √  I(t, s; α∗) T − s = −N − 2 V exp (−X ) − 1 = t t . 2 So, we conclude that ∂BS  ∂BS E∗ (s, 0, exp(α∗), v ) − (s, 0, exp(α∗),I(t, s; α∗)) = 0 t ∂k s ∂k

∂I ∗ and, consequently, ∂α (t, s; α ) = 0. We will need the following hypothesis (H3) There exist two constants C > 0 and δ ≥ 0 such that, for all t ≤ θ < u < r  2 ∗  W ∗  2δ Et Du σr ≤ C(r − u) (8) and  2 ∗  W ∗ W ∗  2δ −2δ Et Dθ Ds σr ≤ C (r − s) (r − θ) . (9)

Theorem 11 Consider the model (1), and assume that hypotheses (H1), (H2) + and (H3) hold and that here exists a Fs-measurable random variable Ds σs such that  4 ∗ ∗  W ∗ 2 + 2 Es sup Eu Du σθ − Ds σs → 0 as T → s . (10) s≤u≤θ≤T Then, for all s < t, ∂I ρ exp (−rˆ(s − t))  D+σ2  ∗ ∗ Xs s s lim (t, s; α ) = Et e 2 . T →s ∂α 4 exp(Xt) σs

13 Proof. From (7) we obtain ∂I (t, s; α∗) ∂α E∗  ∂BS (s, 0, exp(α∗), v ) − ∂BS (s, 0, exp(α∗),I(t, s; α∗)) = t ∂k s ∂k ∂BS ∗ ∗ ∂σ (s, 0, exp(α ),I(t, s; α )) ρ ∗ hR T −rˆ(u−t) ∂H W ∗ i 2 Et s e ∂k (u, Xu,Mu, vu)σuΛu du + ∂BS ∗ ∗ exp(Xt) ∂σ (s, 0, exp(α ),I(t, s; α )) s ∗ ρ ∗  ∂G ∗ R Xu −rˆ(u−t) W  E (s, 0, exp(α ), vs) e e σuΛ du + 2 t ∂k t u ∂BS ∗ ∗ exp(Xt) ∂σ (s, 0, exp(α ),I(t, s; α )) = T1 + T2 + T3. Proceeding as in Remark 10 and using Theorem 2,we get ∂BS  ∂BS E∗ (s, 0, exp(α∗), v ) − (s, 0, exp(α∗),I(t, s; α∗)) t ∂k s ∂k BS (s, 0, exp(α∗), v ) − 1 V exp (−X ) − 1 = E∗ s − t t t 2 2 ( Z T exp (−Xt) ∗ ρ −rˆ(u−t) W ∗ = − Et e H(u, Xu,Mu, vu)σuΛu du|α=α∗ 2 2 s s  exp (−X ) ρ Z ∗ t ∗ −rˆ(u−t) Xu W − G(s, 0, exp(α ), vs) e σue Λu du . 4 t Then, lim T1 = lim I1 + lim I2, T →s T →s T →s where

∗ R T −rˆ(u−t) W ∗  exp (−Xt) ρEt s e H(u, Xu,Mu, vu)σuΛu du |α=α∗ I = − 1 ∂BS ∗ ∗ 4 ∂σ (s, 0, exp(α ),I(t, s; α )) and s ∗ ∗ ∗ R −rˆ(u−t) Xu W  exp (−Xt) ρE G(s, 0, exp(α ), vs) e σue Λ du I = − t t u . 2 ∂BS ∗ ∗ 4 ∂σ (s, 0, exp(α ),I(t, s; α ))

It is easy to see that, under (H3), limT →s I1 = 0 due to Lemma 1. On the other hand, ∂G G(s, 0, exp(α∗), v ) = 2 (s, 0, exp(α∗), v ), s ∂k s which yields

I2 s ∗ ∗ ∂G ∗ R −rˆ(u−t) Xu W  ρ exp (−Xt) E (s, 0, exp(α ), vs) e e σuΛ du = − t ∂k t u 2 ∂BS ∗ ∗ ∂σ (s, 0, exp(α ),I(t, s; α )) = −T3.

14 This gives us that I2 + T3 = 0. On the other hand, using Lemmas 1 and 9, and the anticipating Itˆo’sformula again, it follows that

ρ ∗ hR T −rˆ(u−t) ∂H W ∗ i 2 Et s e ∂k (u, Xu,Mu, vu)σuΛu du lim T2 = lim T →s T →s ∂BS ∗ ∗ exp(Xt) ∂σ (s, 0, exp(α ),I(t, s; α )) ∗ h ∂H −rˆ(s−t) R T W ∗ i ρ Et ∂k (s, Xs,Ms, vs)e s σuΛu du = lim 2 T →s ∂BS ∗ ∗ exp(Xt) ∂σ (s, 0, exp(α ),I(t, s; α )) " Z T # ρ ∗ exp (Xs) exp (−rˆ(s − t)) W ∗ = lim Et 3 2 σuΛu du 2 T →s exp(Xt)vs (T − s) s " Z T # ρ ∗ exp (Xs) exp (−rˆ(s − t)) W ∗ = lim Et 3 2 σuΛu du . 2 exp(Xt) T →s σs (T − s) s

Now, (10) allows us to write

 + 2  ρ exp (−rˆ(s − t)) ∗ Ds σs lim T2 = Et 2 , T →s 4 exp(Xt) σs and now the proof is complete.

Remark 12 If t = s, this formula agrees with the at-the-money short-time limit skew proved in Al`os,Le`onand Vives (2007).

6 The second derivative of the implied volatility

∂2I ∗ In this section we figure out ∂α2 (t, s; α ). Towards this end, note that ∂V ∂BS t = exp(X ) (s, 0, exp(α),I(t, s; α)) ∂α t ∂k ∂BS ∂I + exp(X ) (s, 0, exp(α),I(t, s; α)) (t, s; α) t ∂σ ∂α implies

∂2V ∂2BS t = exp(X ) (s, 0, exp(α),I(t, s; α)) ∂α2 t ∂k2 ∂2BS ∂I +2 exp(X ) (s, 0, exp(α),I(t, s; α)) (t, s; α) t ∂σ∂k ∂α ∂2BS  ∂I 2 + exp(X ) (s, 0, exp(α),I(t, s; α)) (t, s; α) t ∂σ2 ∂α ∂BS ∂2I + exp(X ) (s, 0, exp(α),I(t, s; α)) (t, s; α). (11) t ∂σ ∂α2

15 6.1 The uncorrelated case In this subsection we assume that ρ = 0. We will need the following result, similar to Theorem 5 in Al`osand Le´on (2015). Lemma 13 Consider the model (1) with ρ = 0 and assume that hypotheses in Theorem 11 hold. Then ∂BS ∂2I (s, 0, exp(α∗),I(t, s; α∗)) (t, s; α∗) ∂σ ∂α2 "Z T 2 # 1 ∗ ∂ Ψ ∗ ∗ 2 = Et 2 (Eu (BS (s, 0, exp(α ), vs))) Uudu , 2 t ∂a where ∂2BS Ψ(a) := s, 0, exp(α∗),BS−1 (a) ∂k2 and U is given in (5). Proof. This proof is similar to that in Al`osand Le´on(2015) (Theorem 5), so we only skecth it. Notice that (11) and Remark 10 give us

2 ∂ Vt | ∗ ∂α2 α=α ∂2BS = exp(X ) (s, 0, exp(α∗),I(t, s; α∗)) t ∂k2 ∂BS ∂2I + exp(X ) (s, 0, exp(α∗),I(t, s; α∗)) (t, s; α∗). t ∂σ ∂α2 ∗ −1 Then, taking into account Theorem 2 and the fact that I(t, s; α ) = BS (exp(−Xt)Vt), we are able to write ∂BS ∂2I exp(X ) (s, 0, exp(α∗),I(t, s; α∗)) (t, s; α∗) t ∂σ ∂α2 2 2 ∂ Vt ∂ BS ∗ ∗ = | ∗ − exp(X ) (s, 0, exp(α ),I(t, s; α )) ∂α2 α=α t ∂k2 ∂2BS ∂2BS  = exp(X )E∗ (s, 0, exp(α∗), v ) − (s, 0, exp(α∗),I(t, s; α∗)) t t ∂k2 s ∂k2 ∂2BS = exp(X )E∗ s, 0, exp(α∗),BS−1 (BS(s, 0, exp(α∗), v )) t t ∂k2 s ∂2BS  − s, 0, exp(α∗),BS−1 (E∗ (BS(s, 0, exp(α∗), v ))) . ∂k2 t s Now, using (4), applying Itˆo’sformula to the process ∂2BS A := s, 0, exp(α∗),BS−1 (E∗ (BS(s, 0, exp(α∗), v )) u ∂k2 u s and taking expectations, the result follows.

16 Theorem 14 Consider the model (1) with ρ = 0 and assume that hypotheses in Theorem 2 are satisfied. Then

2 Z s   W ∗ 2 2 ! ∂ I ∗ 1 ∗ ∗ Du σs ∗ −3 lim (T − s) 2 (t, s; α ) = Et Eu (Eu (σs)) du . T →s ∂α 4 t σs Proof. Lemma 13 yields ∂2I lim (T − s) (t, s; α∗) T →s ∂α2  ∗ hR s ∂2Ψ ∗ ∗ 2 i Et t ∂a2 (Eu (BS (s, 0, exp(α ), vs))) Uudu = lim (T − s)  T →s ∂BS ∗ ∗ 2( ∂σ (s, 0, exp(α ),I(t, s; α ))

 ∗ hR T ∂2Ψ ∗ ∗ 2 i Et s ∂a2 (Eu (BS (s, 0, α , vs))) Uudu + lim (T − s)   T →s ∂BS ∗ ∗ 2( ∂σ (s, 0, exp(α ),I(t, s; α ))

= lim (T1 + T2). T →s Remember that we are assuming that there exist two positive constants c, C > 0 such that c ≤ σ ≤ C. Thus, the fact that BS(s, 0, erˆ(T −s), ·) is an increasing function, together with (H3) and

∂2Ψ (E∗ (BS (s, 0, α, v ))) ∂a2 u s  −1 rˆ(T −s) 2  √ (BS (E∗u(BS(s,0,e ,vs)))) (T −s) 2 2π exp 8 = , −1 rˆ(T −s) 3 3/2 BS (E ∗u (BS(s, 0, e , vs))) (T − s) allows us to deduce that, considering that C is a constant that may change from line to line,

Z T C(T −s) ! Z T ∗ exp ( 8 ) 2 C ∗ 2 0 < T2 ≤ CEt 3 Uudu ≤ Et Uu du s c (T − s) (T − s) s ≤ C(T − s)1/2. which implies that limT →s T2 = 0. Finally, the dominated convergence theorem and Lemma 6 give us that Z s I(u,s;α∗)2(T −s) exp( 8 ) ∗ 2 lim T1 = π lim ∗ 3 Et Uu du T →s T →s t I(u, s; α ) (T − s) T ∗ !!2 1 Z s 1 R (DW σ2)dr ∗ ∗ s √u r = lim Et ∗ 3 Eu du 4 T →s t I(u, s; α ) (T − s) vs T − s Z s   W ∗ 2 2 ! 1 ∗ ∗ Ds σr ∗ −3 = Et Eu (Eu (σs)) du 4 t σs

17 and this allows us to complete the proof.

6.2 General case Now we are ready to analyze the curvature in the correlated case. So we assume ρ 6= 0.

Theorem 15 Under the assumptions of Theorem 2, we have

∂2I lim (T − s) (t, s; α∗) T →s ∂α2 Z s   W ∗ 2 2 ! 1 ∗ ∗ Du σr ∗ −3 = Et Eu (Eu (σs)) du 4 t σs 1 1 + − ∗  s ∗  E (σs) ∗ ρ ∗ 1 R −rˆ(u−t) Xu W 2 t E (σs) + exp(−Xt)E e e σuD σ du t 2 t σs t u s  s  ρ 1 Z  ∗  ∗ −rˆ(u−t) Xu W 2 − exp(−Xt)Et 3 e e σu Du σs du . 2 σs t Proof. By Theorem 2 and (11), we deduce

∂BS ∂2I exp(X ) (s, 0, exp(α∗),I(t, s; α)) (t, s; α) t ∂σ ∂α2 ∂2BS ∂2BS  = exp(X )E∗ (s, 0, exp(α∗), v ) − (s, 0, exp(α∗),I(t, s; α)) t t ∂k2 s ∂k2 ∂2BS ∂I −2 exp(X ) (s, 0, exp(α∗),I(t, s; α)) (t, s; α) t ∂σ∂k ∂α ∂2BS  ∂I 2 − exp(X ) (s, 0, exp(α∗),I(t, s; α∗)) (t, s; α∗) t ∂σ2 ∂α " T 2 ρ Z ∂ H ∗ + E∗ e−rˆ(u−t) (u, X ,M , v )σ ΛW du t 2 u u u u u 2 s ∂k α=α∗ 2 s  ∂ G Z ∗ ∗ −rˆ(u−t) Xu W + 2 (s, 0, exp(α ), vs) e e σuΛu du . (12) ∂k t Thus we can write

18 ∂2I (T − s) (t, s; α∗) ∂α2 ∗  ∂2BS ∗ ∂2BS ∗ 0 ∗  Et ∂k2 (s, 0, exp(α ), vs) − ∂k2 s, 0, exp(α ),I (t, s; α ) = (T − s) ∂BS ∗ ∗ ∂σ (s, 0, exp(α ),I(t, s; α )) ∗  ∂2BS ∗ 0 ∗  ∂2BS ∗ ∗  Et ∂k2 s, 0, exp(α ),I (t, s; α ) − ∂k2 (s, 0, exp(α ),I(t, s; α )) +(T − s) ∂BS ∗ ∗ ∂σ (s, 0, exp(α ),I(t, s; α )) 2 2 ∂ BS (s, 0, exp(α∗),I(t, s; α∗)) ∂I (t, s; α∗) −(T − s) ∂σ∂k ∂α ∂BS ∗ ∗ ∂σ (s, 0, exp(α ),I(t, s; α )) 2 2 ∂ BS (s, 0, exp(α∗),I(t, s; α∗)) ∂I (t, s; α∗) − (T − s) ∂σ2 ∂α ∂BS ∗ ∗ ∂σ (s, 0, exp(α ),I(t, s; α )) ρ ∗ hR T −rˆ(u−t) ∂2H W ∗ 2 exp(−Xt)Et s e ∂k2 (u, Xu,Mu, vu)σuΛu du +(T − s) ∂BS ∗ ∗ ∂σ (s, 0, exp(α ),I(t, s; α )) α=α∗  2 s ∗  ρ ∗ ∂ G ∗ R −rˆ(u−t) Xu W 2 exp(−Xt)Et ∂k2 (s, 0, exp(α ), vs) t e e σuΛu du + (T − s) ∂BS ∗ ∗ ∂σ (s, 0, exp(α ),I(t, s; α )) = T1 + T2 + T3 + T4 + T5 + T6. Here I0(t, s; α) denotes the implied volatility in the uncorrelated case ρ = 0. It is easy to see te definition of BS leads us to limT →s (T3 + T4 + T5) = 0. Moreover, we can easily see that ∂BS (s, 0, exp(α∗),I(t, s; α∗)) lim ∂σ = 1. T →s ∂BS ∗ 0 ∗ ∂σ (s, 0, exp(α ),I (t, s; α )) Then, the proof of Lemma 13 yields ∂2I0(t, s, α∗) lim T1 = lim (T − s) . T →s T →s ∂k2 ∂2BS By computing ∂k2 it is easy to see that  1 1  lim T2 = lim − . T →s T →s I0(t, s; α∗) I(t, s; α∗) Therefore, we can use Lemma 6 and Theorem 7 to figure out this limit. On the other hand

lim T6 T →s  2 s ∗  ∗ ∂ G ∗ R −rˆ(u−t) Xu W ρ Et ∂k2 (s, 0, exp(α ), vs) t e e σuΛu du = exp(−Xt) lim (T − s) 2 T →s ∂BS ∗ ∗ ∂σ (s, 0, exp(α ),I(t, s; α ))  s  ρ 1 Z  ∗  ∗ −rˆ(u−t) Xu W 2 = − exp(−Xt)Et 3 e e σu Du σs du . 2 σs t

19 Finally, the result is a consequence of Lemma 6, and Theorems 7 and 14.

Aknowledgements. Elisa Al`osacknowledges nancial support from the Span- ish Ministry of Economy and Competitiveness, through the Severo Ochoa Pro- gramme for Centres of Excellence in R&D (SEV-2015-0563) Jorge A. Le´on thanks Universitat Barcelona, Universitat Pompeu Fabra and the Centre de Recerca Matem`aticaof Universitat Aut`onomade Barcelona for their hospital- ity and economical support.

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