Model-independent lower bound on variance

swaps

Nabil Kahale ∗

First version: October 29, 2009This version: April 1, 2011

Abstract

It is well known that, under a continuity assumption on the price of astock S, the realized variance on S for maturity T can be replicated by aportfolio of calls and puts maturing at T . This paper assumes call priceson S maturing at T are known for all strikes but makes no continuityassumptions on S. We derive semi-explicit expressions for the supremumlower bound Vinf on the hedged payoff, at maturity T , of a long positionin the realized variance of S. Equivalently, Vinf is the supremum strike Ksuch that an investor with a long position in a variance swap with strikeK can ensure to have a non-negative payoff at T . We study examples withconstant implied volatilities and with a volatility skew. In our examples,Vinf is rather close to the fair variance strike obtained under the continuityassumption.

Keywords: model risk, hedging, sub-replication, realized variance, varianceswap

1 Introduction

A variance swap with maturity T and strike K on a stock S is a contract thatpays the realized variance minus K at time T . If the value of such a contractis null at inception, we say that K is the fair variance strike of the swap.Variance swaps can be used to speculate on future volatility levels and to hedgethe volatility exposure of other positions. As noted by Demeterfi et al. (1999),whereas stock options provide exposure to both the direction of the stock priceand its volatility, variance swaps provide pure exposure to the volatility level.Carr and Lee (2009) give a detailed history of the variance swaps and othervolatility derivatives market.Unless otherwise specified, we assume that call prices on S for maturity T

are known for all strikes. Under a continuity assumption on the stock price, therealized variance can be replicated (Dupire, 1993, Neuberger, 1994) by hedging

∗The author thanks Lorenzo Bergomi, Bruno Dupire, and other participants at the FrenchFinance Association spring meeting, May 2010, Saint-Malo, and the Modeling and manag-ing financial risks conference, January 2011, Paris. Address correspondence to Nabil Ka-hale, ESCP Europe, 79 avenue de la Republique, 75543 Paris Cedex, France; e-mail: nka-hale@escpeurope.eu.

1

a position in a log contract whose payoff at time T is 2T−1 ln(F0/ST ), whereST is the price of S at T and F0 is the forward price of the stock. Thus thefair variance strike is equal to the forward price Vlog of the log contract. Onthe other hand, the log contract can itself be statically replicated (Carr andMadan, 1998, Demeterfi et al., 1999, Britten-Jones and A. Neuberger, 2000) bya portfolio of calls and puts maturing at T . Furthermore,

(1.1) Vlog =2

T

∫

(0,∞)

C(K)−max(0, F0 −K)K2

,

where C(K) is the forward price of a call with maturity T and strike K. Eq. 1.1has provided the basis for an analytic approximation (Demeterfi et al., 1999) ofVlog in the presence of a volatility skew and for the calculation of the VIX indexby the Chicago Board Options Exchange (CBOE).Consider an investor long in the floating leg of a variance swap. The investor

is allowed to buy or sell at time 0 call options maturing at T and to take dynamicpositions in S during the period [0, T ]. The hedged payoff of the investor at T isdefined as the combined result of his positions. This paper answers the followingquestion: what is the supremum lower bound Vinf on the hedged payoff, at T ,of a long position in the realized variance? Equivalently, Vinf is the supremumstrike K such that an investor with a long position in a variance swap withstrike K can ensure to have a non-negative payoff at T independently of thestock behavior. The answer to the above question is Vlog under the continuityassumption since the price of a variance swap with strike Vlog is null in this case.This paper, however, makes no assumptions on the stock price process otherthan being strictly positive. Several authors (Demeterfi et al., 1999, Broadieand Jain, 2008, Carr and Wu, 2009, Carr et al., 2009) have shown that, in thepresence of jumps, the price of a variance swap with strike Vlog depends on thejumps size, their frequency, and can be significantly positive or negative.In related work, model-independent bounds on lookback and barrier options

have been derived (Hobson, 1998, Brown et al., 2001) in terms of a continuumset of calls with the same maturity. Bounds on prices of basket options (Hobsonet al., 2005a,b, Laurence and Wang, 2005, d’Aspremont and El Ghaoui, 2006)and spread options (Laurence and Wang, 2008, 2009) have been established interms of prices of vanilla options with the same maturity. Kahale (2010) hasshown that the calculation of extremum prices of a claim in multiperiod marketsconsistently with other options prices is reducible to finding an appropriatemodel of bounded size. Hobson and Neuberger (2010) have recently derived anoptimal upper bound for forward start options in terms of vanilla options withcorresponding maturities.The rest of the paper is organized as follows. Section 2 states the remaining

definitions and assumptions. Section 3 shows how to obtain lower bounds onvariance swaps from a new class of functions, the V-convex functions. By build-ing upon the results of section 3, section 4 gives a semi-explicit lower bound onvariance swaps via jump processes. Section 5 shows this lower bound is optimalwhenever Vinf is finite by constructing martingale measures that asymptoticallyattain it. It also gives a necessary and sufficient condition for the finiteness ofVinf and, under a regularity condition on the call function, a simplified expres-sion for Vinf . Section 6 illustrates our results on two examples. In 6.1, numericalcalculations and a simple approximation for Vinf are given if the implied volatil-ities for maturity T are the same for all strikes. An example with a discrete set

2

of strikes and a volatility skew is treated in 6.2. Section 7 contains concludingremarks. Proofs of some of the results are included in the appendix.

2 Definitions and assumptions

We assume the short risk-free rate r(t) is deterministic and bounded by κ fort ∈ [0, T ]. To simplify the notation, we assume that S pays no dividends. It iseasy to show that our results still hold in the case where S pays a continuous,deterministic and bounded dividend rate. The call options on S with maturityT are assumed to be arbitrage free and liquid for any non-negative strike K.Thus C(K) is a convex and decreasing function of K (e.g., Davis and Hobson2007), with C(0) = F0. We further assume that C

′+(0) = −1 and that C(y)

goes to 0 as y goes to ∞. A standard result (Follmer and Schied, 2004, Lemma7.23) then implies the existence of a probability measure μ on [0,∞) such thatC(K) =

∫(K,∞)(z − K) dμ(z) for K ≥ 0, and so C

′+(K) = −μ((K,∞)). Let

P (K) be the forward price of a put with maturity T and strike K. By the call-put parity relation, C(K)−P (K) = F0−K. For simplicity of presentation, weexclude the trivial case where μ is the Dirac measure δF0 . Given a subdivisionof [0, T ] with time-steps 0 = t0 < ∙ ∙ ∙ < tn+1 = T , denote by Fi the forwardprice of the stock at time ti for maturity T . The realized variance of S over thetime-period [0, T ] is defined as

V [t] =1

T

n∑

i=0

ln2(Si+1

Si),

where Si is the value of the stock at time ti. Let Vinf [t] be the supremum lowerbound on the hedged payoff, at maturity T , of a long position in V [t]. Thus,Vinf = inftVinf [t].

3 Variance swaps and V-convex functions

We introduce here the class of V-convex functions and use them to derive lowerbounds on Vinf . The definition and many of the properties of V-convex functionsin this section and in section 4 are inspired from those of convex functions.

Definition 3.1. A real-valued function g defined on a sub-interval J of (0,∞)is V-convex if, for x, y, z ∈ J with x < z < y,

(3.1)g(x) + ln2(x/z)− g(z)

x− z≤g(y) + ln2(y/z)− g(z)

y − z.

Lemma 3.2. Let g be a real-valued function on an open sub-interval J of (0,∞).The following conditions are equivalent:

1. g is V-convex.

2. g is continuous, has a right derivative g′+ and, for x, y ∈ J with x < y,

(3.2) g′+(x)(y − x) ≤ g(y) + ln2(y/x)− g(x).

3

3. For x > 0 and y ∈ J ,

δx,y;g = infv∈J,v>y

g(v) + ln2(v/x)− g(y)v − y

is finite and, for z ∈ J ,

(3.3) g(y)− g(z) ≤ ln2(z/x) + δx,y;g(y − z).

Proof. See the appendix.

Note that if we omit the ln2 terms in Definition 3.1 and Lemma 3.2, weobtain classical properties of convex functions.

Remark 3.3. The proof of Lemma 3.2 shows that Condition 2 can be replacedby the following one: g is continuous and, for x ∈ J except for a finite numberof points, g′+(x) exists and Eq. 3.2 holds for y ∈ J with x < y.

The following variant of results proven in (Breeden and Litzenberger, 1978,Carr and Madan, 1998, Demeterfi et al., 1999) shows how to approximatelysuper-replicate a European derivative meeting certain conditions with standardcall options. The idea behind the proof is to super-replicate the derivative payoffwith a piece-wise affine function and to observe that such a function is, up to aconstant, equal to the payoff of a portfolios of calls.

Lemma 3.4 (Breeden Litzenberger). Consider a Lipschitz function f on [0,∞).Then

∫(0,∞) |f | dμ <∞ and, for any ε > 0, a European option that pays f(ST )

at maturity T can be super-replicated by a portfolio of a zero-coupon bond andof call options maturing at T with forward price at most ε+

∫(0,∞) f dμ.

Proof. Let k > 0 be such that f is k-Lipschitz. Since |f(z)| ≤ |f(0)| + kz, theintegral

∫(0,∞) |f | dμ is finite. Consider a strikeK > 0 such that C(K) < ε/k and

let m = dkK/εe. Define fm as the unique function on [0,∞) that coincides withf at the points jK/m, 0 ≤ j ≤ m, is affine on the intervals [jK/m, (j+1)K/m],0 ≤ j ≤ m− 1, and on [K,∞), with slope k on the latter interval. Then

(3.4) fm(z)− ε− 2kmax(0, z −K) ≤ f(z) ≤ fm(z) + ε

for z ≥ 0. Since fm + ε is piece-wise affine, it can be statically replicated bya portfolio P of a zero-coupon bond and of m + 1 call options (e.g., Demeterfiet al. 1999) with maturity T . Using Eq. 3.4, we conclude that the forward priceε +

∫(0,∞) fm dμ of P is at most 4ε +

∫(0,∞) f dμ. Replacing ε with ε/4 finishes

the proof.

Theorem 3.5. Let g be a Lipschitz function on [0,∞) which is V-convexon (0,∞) and such that g(F0) = 0. Then

∫(0,∞) |g| dμ < ∞ and Vinf ≥

−T−1∫(0,∞) g dμ.

Proof. Consider a subdivision of [0, T ] with time-steps 0 = t0 < ∙ ∙ ∙ < tn+1 = T .

For 0 ≤ i ≤ n, let Zi = exp−∫ Tti+1

r(u) du. By Condition 3, ξi = δSi/Zi,Fi;g is

finite. By Eq. 3.3 and the relation ZiFi+1 = Si+1,

(3.5) g(Fi)− g(Fi+1) ≤ ln2(Si+1/Si) + ξi(Fi − Fi+1),

4

and so

(3.6) 0 ≤ g(ST ) + T V [t] +

n∑

i=0

ξi(Fi − Fi+1).

Furthermore, the value of ξi is known at time ti. Fix ε > 0. By Lemma 3.4,there is a portfolio P of a zero-coupon bond and of call options maturing at Twith forward price at most ε +

∫(0,∞) g dμ that super-replicates g. Consider a

long position in the realized variance V [t]. We hedge the position by taking along position in T−1P at time 0 together with a short position of T−1ξi stocksat time ti and unwinding the latter position at time ti+1. By Eq. 3.6, thepayoff at time T of the position in V [t] combined with the hedges is at least−T−1(ε+

∫(0,∞) g dμ). Thus Vinf ≥ −T

−1∫(0,∞) g dμ.

Example 3.6. Given a ∈ (0, F0], the function ga(x) = − ln2(x/a)1x≥a is Lips-

chitz on [0,∞) and, by Condition 2 of Lemma 3.2, is V-convex on (0,∞). Weconclude that Vinf ≥ T−1(

∫(a,∞) ln

2(x/a) dμ(x)− ln2(F0/a)).

Remark 3.7. It is easy to show that ξi = g′+(Si) if interest rates are null.

4 A lower bound via jump processes

Throughout the rest of the paper, let a = F0, b = max{x ≥ 0 : P (x) = 0} andI = {y ≥ a : C ′−(y) < 0}. For 0 < x ≤ y and any function g with a rightderivative at x, define

g(x, y) = g(x) + g′+(x)(y − x)− ln2(y/x).

This section establishes a lower bound on Vinf which will be shown to be optimalin Section 5 in the case where Vinf is finite. The main idea behind our approachis to obtain the best lower bound on Vinf achievable through Theorem 3.5. Thisis a convex optimization problem since the set of V-convex functions is convex.Assume first that g is known on [0, a]. We want to find the smallest V-convex

extension (if it exists) of g to (a,∞). Since g(x, y) ≤ g(y) for 0 < x < a < y byEq. 3.2, a natural such extension for g is given by Eq. 4.1 below. Lemma 4.1shows that, under certain assumptions on g, the proposed extension of g to(a,∞) satisfies the conditions of Theorem 3.5.

Lemma 4.1. Let g be a V-convex and Lipschitz function on [0, a] which is affineon [0, b′], for some b′ ∈ (0, a), with g(a) = 0. Define

(4.1) g(y) = supx∈(0,a)

g(x, y)

for y > a. Then g is Lipschitz and V-convex on [0,∞).

Proof. Since g is V-convex and Lipschitz on [0, a], it follows from Eq. 3.2 thatEq. 4.1 holds for y = a. Thus, for y ≥ a,

g(y) = supx∈(b′/2,a)

g(x, y).

5

This implies that the restriction of g to [a,∞) is Lipschitz since it is the supre-mum of functions with uniformly bounded derivatives. Thus g is Lipschitz on[0,∞). We show that g is V-convex on (0,∞) by applying Remark 3.3. Since gis V-convex on [0, a], Eq. 3.2 holds for 0 < x < y ≤ a. By Eq. 4.1, it follows thatEq. 3.2 holds for 0 < x < a and x < y. On the other hand, Example 3.6 showsthat the function y 7→ − ln2(y/x) is V-convex on [a,∞) if 0 < x ≤ a. Sincethe set of V-convex functions on I is invariant by adding an affine function andby taking the supremum, it follows from Eq. 4.1 that g is V-convex on (a,∞).Thus Eq. 3.2 holds for a < x < y. We have thus shown that Eq. 3.2 holdsfor 0 < x < y with x 6= a. By Remark 3.3, it follows that g is V-convex on[0,∞).

Consider now a decreasing function φ from (b, a] to [a,∞) such that I is theinterval spanned by φ((b, a]). Let

(4.2) g(x) = 2

∫

(x,a)

(x− u)ln(φ(u)/u)

u(φ(u)− u)du

for b ≤ x ≤ a and

(4.3) g(y) = supx∈(b,a)

g(x, y)

for y ∈ I − {a}.

Lemma 4.2. The function g is finite and V-convex on [b, a) ∪ I and, for x ∈(b, a],

(4.4) g(φ(x)) = g(x, φ(x)).

For y ∈ I,

(4.5) − ln2(y/a) ≤ g(y).

If∫[b,a)∪I |g| dμ <∞ then Vinf ≥ −T

−1∫[b,a)∪I g dμ.

Proof. See the appendix.

To motivate Eq. 4.2, assume that a function g has a continuous secondderivative on (b, a), is V-convex on [b, a) ∪ I and satisfies Eq. 4.4. Thus thefunction u 7→ g(u, φ(x)) attains its maximum at x. Taking the derivative withrespect to u, it follows that g′′(x) = 2 ln(φ(x)/x)/(x(x − φ(x)), which can besolved via Eq. 4.2. We now justify Eq. 4.4 by assuming that b > 0, interestrates are null, and that a function g satisfies Eq. 4.4 and the conditions inTheorem 3.5. The assumptions made in this paragraph are for intuition and arenot be used elsewhere in the paper. Let xi = a(b/a)

i/(n+1). By Remark 3.7, if Fifollows a jump process such that Fi+1 = Fi or Fi = xi and Fi+1 ∈ {xi+1, φ(xi)},the difference between the right-hand side and the left-hand side of Eq. 3.5 is atmost ln2(xi+1/xi). Thus the right-hand side of Eq. 3.6 is bounded by a constantdivided by n and Vinf equals −T−1

∫[b,a)∪I g dμ. Thus, g would provide the best

lower bound on Vinf .The process described above is not, in general consistent with the forward

call function C. Under martingale measures consistent with C that will bedescribed in section 5, the forward price follows a process similar to the above-described one for an appropriate φ. The following lemma motivates the choiceof φ.

6

Lemma 4.3. Given i ∈ [0, n] and 0 < x < y, assume the forward price is aQ-martingale and

(4.6)

{Fi = x and ST /∈ (x, y) or

Fi ∈ (x, y] and Fj = Fi for j ≥ i,

Q almost surely. Assume Q is consistent with C and let Ci(K) = EQ(max(Fi−K, 0)). Then

(4.7) Ci(K) = C(K)− C(y) +P (x)− C(y)

y − x(K − y)

and

(4.8) C ′−(y) ≤C(y)− P (x)

y − x≤ C ′+(y).

Furthermore, for any Borel subset A of [0,∞),

(4.9) Q(Fi+1 ∈ A ∧ Fi = x) = Q(Fi+1 ∈ A)−Q(Fi ∈ A) +Q(Fi = x)δx(A).

Proof. Since Fi and ST have the same distribution on (x, y) and Ci(y) = 0, forK ∈ (x, y),

Ci(K) = C(K)− C(y) + α(K − y),

where α is a constant. Since Ci(x) = F0 − x we conclude, using the call-putparity relation, that

α =P (x)− C(y)

y − x,

which yields Eq. 4.7. Denote by C ′i− (resp. C′i+) the left (resp. right) derivative

of Ci. The lower bound in Eq. 4.8 follows by observing that C′i−(y) = C

′−(y)+α.

On the other hand, the relation Q(Fi > x) ≤ Q(ST ∈ (x, y]) implies thatC ′i+(x) ≥ C ′+(x) − C

′+(y). Since C

′i+(x) = C ′+(x) + α, this yields the upper

bound in Eq. 4.8. Finally, Eq. 4.9 follows by observing that

Q(Fi+1 ∈ A ∧ Fi = x) = Q(Fi+1 ∈ A)−Q(Fi+1 ∈ A ∧ Fi 6= x)

= Q(Fi+1 ∈ A)−Q(Fi ∈ A ∧ Fi 6= x).

Note that Eq. 4.8 implies that the line (x, P (x)) and (y, C(y)) is tangentat y to the call functions curve. For (x, y) ∈ [0,∞)2, define H(x, y) = C(y) +(x− y)C ′+(y)−P (x). Since C is convex, the function y 7→ H(x, y) is decreasingand right-continuous on [x,∞). Lemma 4.3 motivates the following choice of φ,which will be used throughout the rest of the paper:

(4.10) φ(x) = min{y ≥ a : H(x, y) ≤ 0}

for x ∈ (b, a]. It is easy to see that Eq. 4.8 holds if y = φ(x). The function φ isdecreasing on (b, a] and, by the call-put parity relation, φ(a) = a. Finally, set

(4.11) G(x) = supy>a

P (x)− C(y)y − x

for x ∈ [0, a].

7

Lemma 4.4. The interval I is the interval spanned by φ((b, a]). The functionG is increasing and continuous on [0, a]. For x ∈ (b, a),

(4.12) G(x) =P (x)− C(φ(x))

φ(x)− x.

Furthermore, for any nonnegative Borel-measurable function γ on (a,∞),

(4.13)

∫

(b,a)

(γ ◦ φ) dG =∫

(a,∞)γ dμ,

where the integral in left-hand side is the Stieltjes integral with respect to G. Forx ∈ (b, a],

(4.14) −C ′+(φ(x)) ≤ G(x) ≤ −C′−(φ(x)).

Proof. See the appendix.

Note that, by Eq. 4.14, if C and φ are sufficiently smooth, Eq. 4.13 reducesto the change of variables theorem. Using Eq. 4.10, we conclude the section byan alternative expression for the lower bound in Lemma 4.2.

Lemma 4.5. The integral∫[b,a)∪I |g| dμ is finite and Vinf is lower-bounded by

T−1∫

(b,a)

ln2(φ(x)/x) dG(x) = −T−1∫

[b,a)∪Ig dμ.

Proof. See the appendix.

5 Martingales attaining the lower bound

Consider the subdivision of [0, T ] with time-steps ti = iT/(n+1), 0 ≤ i ≤ n+1,and let Ω = {F0} × (0,∞)n+1 be the set of possible values for the sequence(F0, F1, . . . , Fn+1). We construct martingales consistent with C and that attain,if Vinf is finite, the lower bound in Lemma 4.5 as n goes to infinity. Considera strictly decreasing sequence xi, 0 ≤ i ≤ n + 1, with x0 = a and xn+1 = b.We want Fi to be a Q-martingale such that, Q almost surely, for 0 ≤ i ≤ n,Fi+1 = Fi or Fi = xi and Fi+1 ∈ [xi+1, xi]∪[φ(xi), φ(xi+1)]. Thus, condition 4.6holds for x = xi and y = φ(xi). We construct Q by reverse-engineering Eq. 4.7.For 0 ≤ i ≤ n, define

(5.1) Ci(K) =

F0 −K if 0 ≤ K < xi

C(K)− C(φ(xi)) +G(xi)(K − φ(xi)) if xi ≤ K < φ(xi)

0 if φ(xi) ≤ K,

and Cn+1 = C. Using Eq. 4.14, it can be shown that Ci is convex and decreasingon [0,∞), with C ′i+(0) = −1. For 0 ≤ i ≤ n+1, let μi be the probability measureinduced by Ci. In other words, μi is the unique probability on [0,∞) such thatCi(K) =

∫(K,∞)(z − K) dμi(z) for K ≥ 0. Motivated by Eq. 4.9, define the

signed measure

(5.2) νi = δxi +1

μi({xi})(μi+1 − μi)

8

for 0 ≤ i ≤ n. Using Eq. 4.14 once again, it can be easily verified that μi({xi}) >0 and νi(A) ≥ 0 for any Borel subset A of [0,∞), and so νi is a probabilitymeasure on [0,∞). Let Q be the unique probability on Ω such that (Fi), 0 ≤i ≤ n + 1, is a Q-Markov chain with initial probability δF0 and transitionprobabilities πi, 0 ≤ i ≤ n, defined as follows:

(5.3) πi(z) =

{νi if z = xi

δz otherwise.

Lemma 5.1. For 0 ≤ i ≤ n+ 1 and any Borel subset A of [0,∞),

(5.4) Q(Fi ∈ A) = μi(A).

Furthermore, (Fi), 0 ≤ i ≤ n+ 1, is a Q-martingale.

Proof. We show Eq. 5.4 by induction on i. The base case clearly holds. IfEq. 5.4 is true for i then Q(Fi = xi) = μi({xi}) and, for any Borel subset A of[0,∞),

Q(Fi+1 ∈ A) = Q(Fi = xi)νi(A) +Q(Fi 6= xi ∧ Fi+1 ∈ A)

= μi({xi})νi(A) +Q(Fi 6= xi ∧ Fi ∈ A)

= μi+1(A)− μi(A− {xi}) + μi(A− {xi})

= μi+1(A).

Thus Eq. 5.4 holds for i+ 1.On the other hand, by Eqs. 5.3 and 5.4,

EQ((Fi+1 − Fi)1Fi=xi) = EQ(Fi+1 − Fi)

= Ci+1(0)− Ci(0)

= 0.

This shows that Fi is a Q-martingale.

Lemma 5.2. For 0 ≤ i ≤ n+1 and any nonnegative Borel-measurable functionγ on (a,∞),

(5.5)

∫

(xi,a)

(γ ◦ φ) dG =∫

(a,∞)γ dμi.

Proof. See the appendix.

Theorem 5.3. If

(5.6)

∫

(0,∞)ln2(

x

S0) dμ(x) =∞

then Vinf =∞. Otherwise, Vinf is finite and equal to

(5.7) −T−1∫

[b,a)∪Ig dμ = T−1

∫

(b,a)

ln2(φ(x)

x) dG(x).

9

Proof. Assume first that Eq. 5.6 holds. For ε > 0, let

fε(z) =

{0 if z ∈ [0, ε)

ln2(z/S0) if z ∈ [ε,∞).

By Lemma 3.4, a European option that pays f(Sn+1) at maturity T can be sub–replicated by a portfolio of a zero-coupon bond and of call options maturing at Twith forward price at least

∫(0,∞) fε dμ− ε. By the Cauchy-Schwartz inequality

ln2Sn+1

S0≤ (n+ 1)

n∑

i=0

ln2Si+1

Si,

we conclude that T (n+ 1)Vinf [t] ≥∫(0,∞) fε dμ− ε. Letting ε go to 0 and using

the monotone convergence theorem implies that Vinf =∞.Assume now that

∫(0,∞) ln

2(x/S0) dμ(x) is finite. By lemma 4.5, the two

sides of Eq. 5.7 are finite, equal and lower bound Vinf . We show this lowerbound is optimal using the probability Q. By Eq. 5.3, for 0 ≤ i ≤ n,

EQ

(

ln2Fi+1

Fi

)

= Q(Fi = xi)EQ

(

ln2Fi+1

xi|Fi = xi

)

= μi({xi})∫

(0,∞)ln2

z

xidνi(z).

Since νi is null on (0, xi+1) ∪ (xi, a], we conclude using Lemma 5.2 that

EQ

(

ln2Fi+1

Fi

)

≤ ln2xi+1

xi+ μi({xi})

∫

(a,∞)ln2

z

xidνi(z)

= ln2xi+1

xi+

∫

(xi+1,xi]

ln2φ(x)

xidG(x)

for 0 ≤ i ≤ n− 1 and

EQ

(

ln2Fn+1

Fn

)

=

∫

(0,xn)

ln2z

xndμ(z) +

∫

(b,xn]

ln2φ(x)

xndG(x).

We now set xi = a(1/n+ (1− 1/n)b/a)i/n for 0 ≤ i ≤ n. It follows that

n∑

i=0

EQ

(

ln2Fi+1

Fi

)

≤α

n+

∫

(0,xn)

ln2z

xndμ(z) +

∫

(b,a)

ln2φ(x)

xdG(x),

where α is a constant independent of n. Since∫(0,∞) ln

2(x/S0) dμ(x) is finite,

so is∫(0,∞) ln

2(x/a) dμ(x). By Lebesgue’s dominated convergence theorem we

conclude that

(5.8) lim supn→∞

n∑

i=0

EQ

(

ln2Fi+1

Fi

)

≤∫

(b,a)

ln2φ(x)

xdG(x).

Since ln(Si+1/Si) ≤ κT/(n + 1) + ln(Fi+1/Fi), it follows from the Cauchy-Schwarz inequality that

(5.9)n∑

i=0

ln2Si+1

Si≤(κT )2

n+

n∑

i=0

ln2Fi+1

Fi+2κT√n

√√√√

n∑

i=0

ln2Fi+1

Fi.

10

Combining Eqs. 5.8 and 5.9 and using Jensen’s inequality shows that

lim supn→∞

n∑

i=0

EQ

(

ln2Si+1

Si

)

≤∫

(b,a)

ln2φ(x)

xdG(x).

Since Q is risk-neutral, this implies that Vinf ≤ T−1∫(b,a)ln2(φ(x)/x) dG(x).

Remark 5.4. It follows from Eqs. 4.2 and 4.5 that g(z) ≥ glog(z) for z ∈[b, a) ∪ I, where glog(z) = 2(ln(z/a) − z/a + 1). Hence, when Vinf is finite,Vinf ≤ Vlog since Vlog = −T−1

∫[b,a)∪I glog dμ.

Remark 5.5. For y ∈ I, let ψ(y) be the unique solution to the equation C(y)+(x− y)C ′−(y) = P (x). It is easy to show that ψ(y) ∈ (b, a]. Furthermore,

φ(u) ≤ y for u ∈ (ψ(y), a] and φ(u) ≥ y for u ∈ (b, ψ(y)).

By a similar proof to that of Eq. 4.4, we conclude that g(y) = g(ψ(y), y).

Remark 5.6. Assume Vinf is finite. Using the proofs and notation of Theo-rem 3.5 and of Lemma 4.2, it follows that, for any ε > 0, there is b′ > b suchthat an investor with a long position in the floating leg of a variance swap canrealize a payoff at least Vinf − ε at T by buying or selling at time 0 a finitenumber of calls maturing at T and dynamically maintaining a short position ofT−1δSi/Zi,Fi;gb′ stocks at ti.

Theorem 5.7. Assume the function C is differentiable on (a,∞) and Vinf isfinite. Then,

(5.10) Vinf = T−1∫

I

ln2y

ψ(y)dμ(y).

Proof. Consider x ∈ (b, a). Eq. 4.10 shows that H(x, φ(x)) ≤ 0 and thatH(x, y) > 0 for y ∈ (a, φ(x)). Since C is convex and differentiable on (a,∞),C ′ is continuous on (a,∞). We conclude that H(x, φ(x)) = 0. Equivalently,ψ(φ(x)) = x for x ∈ (b, a). By applying Eq. 4.13 to the function γ defined on(a,∞) with

γ(y) =

{ln2(y/ψ(y)) if y ∈ I − {a}

0 otherwise,

it follows that∫

(b,a)

ln2φ(x)

xdG(x) =

∫

I−{a}ln2

y

ψ(y)dμ(y).

We conclude the proof by noting that ψ(a) = a and using Theorem 5.3.

Eq. 5.10 has a simple intuitive interpretation: when n goes to infinity, theforward price follows a process where it may jump from ψ(y) to y.

6 Examples

We study an example with constant implied volatilities and an example with avolatility skew.

11

σ 10% 15% 20% 25% 30% 35%√Vinf 9.782% 14.510% 19.129% 23.641% 28.044% 32.340%

σ − c2σ2√T 9.782% 14.509% 19.128% 23.637% 28.038% 32.329%

Table 1:√Vinf and its approximation obtained via Theorems 5.7 and 6.1 when

T = 0.25.

6.1 A constant implied volatilities example

This subsection assumes that the implied volatilities for maturity T and allstrikes are equal to a constant σ. For u ≥ 0, denote by θ(u) the unique solutionto the equation N ′(u)−N ′(θ) = θ(N(θ) +N(−u)) in the interval [−u, 0].

Theorem 6.1. If the call prices for all strikes and maturity T are equal to theBlack-Scholes prices with volatility σ, then Vinf = σ

2−cσ3√T+O(σ4) as σ → 0,

where

c =1

2

∫ ∞

0

u(u− θ(u))2N ′(u) du ≈ 0.8721.

Proof. See the appendix.

Table 1 shows values of√Vinf calculated numerically via Theorem 5.7 when

T = 0.25 and compares them with the approximation σ − cσ2√T/2.

6.2 A discrete set of strikes example

In practice, European call (or, equivalently, put) options prices with a givenmaturity T are known only for a finite set of positive strikes K1 < ∙ ∙ ∙ < Kl.Demeterfi et al. (1999) have calculated Vlog in this case by approximating thefunction glog with a piecewise affine function. This is computationally equiv-alent to extending the call prices to all strikes via a piecewise affine interpo-lation. We use a similar approach to calculate Vinf . Assume the sequenceC(0), C(K1), . . . , C(Kl) is strictly convex and strictly decreasing with respectto the strike, where C(0) = F0, and that C(K1) ≥ F0 −K1. By the necessaryand sufficient conditions for the absence of arbitrage among the call prices es-tablished by Davis and Hobson (2007), it can be shown there are two strikes0 < K0 < K1 and Kl+1 > Kl such that the following extension of C to allstrikes is arbitrage-free: C(K) = F0 −K for K ≤ K0, C(K) = 0 for K ≥ Kl+1,and

C(K) = C(Ki) +C(Ki+1)− C(Ki)

Ki+1 −Ki(K −Ki)

for Ki < K < Ki+1. Furthermore, the support of μ is included in the set{K0, . . . ,Kl+1}, with

μ(K0) = 1−C(K1)− C(K0)

K1 −K0,

μ(Ki) =C(Ki+1)− C(Ki)

Ki+1 −Ki−C(Ki)− C(Ki−1)

Ki −Ki−1for 1 ≤ i ≤ l, and

μ(Kl+1) =C(Kl)

Kl+1 −Kl.

12

We now can apply Theorem 5.3 to calculate Vinf . The function φ can be calcu-lated by inspection via Eq. 4.10, which implies that Im(φ) ⊆ {F0} ∪ Supp(μ).On the other hand, Eq. 4.13 shows that, for y ∈ Supp(μ) ∩ (F0,∞), the setφ−1({y}) is a sub-interval of (K0, F0] with a non-empty interior. Thus, an el-ement x of φ−1({y}) can be found using the bisection method. We can thencalculate g(y) via Eq. 4.4.In our numerical example in table 2, we assume that Ki = 35 + 5i for

1 ≤ i ≤ l, with l = 22. We chose K0 = 35 and Kl+1 = 150. The function ghas been calculated on Supp(μ) = {K0, . . . ,Kl+1} via Eqs. 4.2 and 4.3, and Ghas been evaluated via Eq. 4.12. The integrals in Eqs. 4.2, 4.3 and 5.7 havebeen calculated using the midpoint method with 100000 points. The differencebetween the right-hand and left-hand sides of Eq. 5.7 is of order 10−7. Eq. 5.7shows that Vinf = (24.263%)

2. In comparison, we have calculated Vlog via the

equation Vlog =∑l+1i=0 μ(Ki)glog(Ki) and found that Vlog = (25.608%)

2.Table 3 compares the values of Vinf and of Vlog using the same implied

volatility function σimp(K) = 0.45 − 0.002K as in table 2 and different valuesfor the strikes spacing. We do the same comparison in table 4 using a constantimplied volatility function σimp(K) = 0.25. In the limit case ΔK = 0, Vinf hasbeen extracted from table 1.

7 Concluding remarks

Alternative semi-explicit expressions for the optimal lower bound Vinf on thehedged payoff of a long position in the realized variance have been given. Wehave assumed that call prices on S with maturity T are known for all strikesbut did not make any continuity assumptions on S.When Vinf is finite, for any ε > 0, an explicit hedging scheme allows an

investor with a long position in the floating leg of a variance swap to realize apayoff at least Vinf − ε at T . Furthermore, the expected realized variance is atmost Vinf + ε under an explicitly constructed martingale.Numerical values and an approximation for Vinf have been given when the

implied volatilities for maturity T are constant, and a numerical example with adiscrete set of strikes and a volatility skew has been treated. Quite surprisingly,even though the models we have used to attain Vinf are jump processes, thevalues of Vinf in our examples are rather close to the fair variance strikes obtainedunder the continuity assumption.

A Proof of Lemma 3.2

1 ⇒ 2. Assume that g is V-convex. Since the function u 7→ ln2 u + 4 lnu isconcave on an open interval containing 1,

(A.1)ln2(y/z) + 4 ln(y/z)

y − z≤ln2(x/z) + 4 ln(x/z)

x− z

for 0 < (1 + ε)−1z < x < z < y < (1 + ε)z, where ε is a sufficiently smallpositive constant. Combining Eqs. 3.1 and A.1 shows that, for any z ∈ J, thefunction g(v) − 4 ln v is convex on the interval (z, (1 + ε)z) ∩ J . Consequently,

13

Strike Implied Volatility Call price μ g glog35 0.000000 -0.5770 -0.806240 37% 60.199502 0.000002 -0.4725 -0.638645 36% 55.224448 0.000015 -0.3832 -0.502550 35% 50.249468 0.000088 -0.3066 -0.391355 34% 45.274923 0.000394 -0.2411 -0.300260 33% 40.302340 0.001411 -0.1855 -0.225765 32% 35.336777 0.004169 -0.1387 -0.165170 31% 30.391954 0.010436 -0.0998 -0.116475 30% 25.499053 0.022568 -0.0681 -0.077980 29% 20.718432 0.042685 -0.0431 -0.048385 28% 16.150173 0.071128 -0.0242 -0.026690 27% 11.935778 0.104687 -0.0109 -0.011795 26% 8.242208 0.135842 -0.0030 -0.0031100 25% 5.224458 0.154436 0.0000 0.0000105 24% 2.975040 0.152160 -0.0019 -0.0019110 23% 1.482628 0.127838 -0.0081 -0.0084115 22% 0.626216 0.089568 -0.0179 -0.0190120 21% 0.215409 0.050812 -0.0304 -0.0334125 20% 0.057392 0.022458 -0.0447 -0.0512130 19% 0.011107 0.007359 -0.0597 -0.0723135 18% 0.001437 0.001677 -0.0743 -0.0963140 17% 0.000111 0.000245 -0.0865 -0.1231145 16% 0.000004 0.000021 -0.0947 -0.1524150 0.000001 -0.0970 -0.1841

Table 2: The spot price is $100, implied volatilities are given for European calloptions with maturity T = 0.25 and strikes ranging from $40 to $145, spaced$5 apart. There are no dividends and the continuously compounded risk-freeinterest rate with maturity T is 2%.

ΔK 0.1 1 5Vinf (23.955%)2 (23.967%)2 (24.263%)2

Vlog (25.267%)2 (25.280%)2 (25.608%)2

Table 3: Variance rates in the presence of skew. Strikes range in the interval[40, 200].

ΔK 0 0.1 1 5Vinf (23.641%)2 (23.641%)2 (23.653%)2 (23.951%)2

Vlog (25.000%)2 (25.000%)2 (25.014%)2 (25.344%)2

Table 4: Variance rates with flat volatility. Strikes range in the interval [40, 200].

14

g is continuous and has a right derivative. Consider now x, y ∈ J , with x < y.Eq. 3.2 follows by taking limits as z → x+ in Eq. 3.1.2⇒ 3. Assume 2. holds. We first show that, for x, y, z ∈ J with x < z < y,

(A.2)g(z)− g(x)z − x

≤g(y)− g(z)y − z

+ln2(y/x)

y − x.

Fix x, y ∈ J , with x < y. For z ∈ [x, y], let

h(z) = g(z)− g(x)−g(y)− g(x)y − x

(z − x) +ln2(y/x)

(y − x)2(z − y)(z − x).

Since the map u 7→ (y − u)−1ln(y/u) is decreasing on (0, y),

h(z) + h′+(z)(y − z) = g(z) + g′+(z)(y − z)− g(y)−ln2(y/x)

(y − x)2(y − z)2

≤ ln2(y/z)−ln2(y/x)

(y − x)2(y − z)2

≤ 0.(A.3)

Let v be the first point in the interval [x, y] where h attains its maximum. Ifh(v) > 0 then h is positive on an interval [u, v], where u ∈ (x, v). By Eq. A.3,the function h is decreasing on [u, v], leading to a contradiction. We concludethat h(z) ≤ 0 for x ≤ z ≤ y which, after some simplifications, is equivalent toEq. A.2.Now, for x > 0 and u, y, v ∈ J with u < y < v, the Cauchy-Schwarz

inequality

(√v − y

ln(v/x)√v − y

+√y − u

ln(x/u)√y − u

)2≤ (v − u)

(ln2(v/x)

v − y+ln2(x/u)

y − u

)

can be rewritten as

(A.4)ln2(v/u)

v − u≤ln2(v/x)

v − y+ln2(u/x)

y − u.

Combining Eq. A.2, applied to (u, y, v), and Eq. A.4 yields

g(y)− g(u)− ln2(u/x)y − u

≤g(v)− g(y) + ln2(v/x)

v − y.

This implies that δx,y;g is finite and that Eq. 3.3 holds.3 ⇒ 1. Assume 3. holds. Consider x, z, y ∈ J with x < z < y. It follows

from Eq. 3.3 thatg(x) + ln2(x/z)− g(z)

x− z≤ δz,z;g.

Since

δz,z;g ≤g(y) + ln2(y/z)− g(z)

y − z,

we conclude that Eq. 3.1 holds.

15

B Proof of Lemma 4.2

We first observe that φ(a) = a. Eq. 4.2 shows that g is continuous on [b, a] andthat, for x ∈ (b, a),

(B.1) g′(x) = 2

∫

(x,a)

ln(φ(u)/u)

u(φ(u)− u)du.

It follows from Eqs. 4.2 and B.1 that, for b < x < z < a and y ≥ z,

(B.2) g(z, y)− g(x, y) = 2∫

(x,z)

y − uu

(ln(y/u)

y − u−ln(φ(u)/u)

φ(u)− u

)

du.

Since the map v 7→ ln vv−1 is decreasing on (1,∞), Eq. B.2 implies that g(x, y) ≤

g(y, y) for b < x < y < a and so Eq. 3.2 holds for b < x < y < a. Thus g isV-convex on [b, a]. Since φ is decreasing, Eq. B.2 implies that g(x, y) ≤ g(z, y)for b < x < z < a ≤ y ≤ φ(z), and so g(y) is finite. Eq. 4.4 follows from Eq. B.2by a similar argument.Given b′ ∈ (b, a), let

gb′(z) =

g(b′) + g′(b′)(z − b′) for z ∈ [0, b′)

g(z) for z ∈ [b′, a]

supx∈[b′,a) g(x, z) for z ∈ (a,∞).

By Eq. B.1, the function gb′ is Lipschitz on [0, a]. On the other hand, byCondition 2 of Lemma 3.2 and since g is V-convex on [b, a], the function gb′ isV-convex on [0, a]. By Lemma 4.1, it follows that gb′ is Lipschitz and V-convexon [0,∞) and so, by Theorem 3.5,

(B.3) Vinf ≥ −T−1∫

[b,a)∪Igb′ dμ.

On the other hand, by Eq. B.1, g is concave and increasing on (b, a) and so

(B.4) g(x) ≤ gb′(x) ≤ 0

for b ≤ x ≤ a. Also, since g′(x) goes to 0 as x goes to a, x < a,

− ln2(y/a) ≤ gb′(y) ≤ g(y)

for y ∈ I. Thus, the functions gb′ are dominated by a μ-integrable function on[b, a)∪I. Furthermore, from Eq. B.4 and the bound gb′(b) ≤ g(b′), the functiongb′ converges pointwise to g on [b, a)∪ I as b′ goes to b. The lemma thus followsfrom Eq. B.3 and Lebesgue’s dominated convergence theorem.

C Proof of Lemma 4.4

The function G is increasing and continuous on [0, a] since it is the supre-mum of increasing and continuous functions on [0, a] with uniformly boundedright derivatives on any closed sub-interval of [0, a). Fix x ∈ (b, a). Eq. 4.12follows from Eq. 4.10 by observing that the right derivative of the function

y 7→ P (x)−C(y)y−x is positive for a < y < φ(x) and at most 0 for y > φ(x).

16

We first show Eq. 4.13 in the case where γ = 1(v,∞), with v ≥ a. Let

(C.1) u = min{x ≥ 0 : H(x, v) = 0}.

By the convexity of C, the function y 7→ C(y)+(u−y)C ′+(v) attains its minimumat v, and u ≤ a. Thus, by Eq. C.1,

miny≥a

C(y) + (u− y)C ′+(v) = P (u),

and so

(C.2) G(u) = −C ′+(v).

On the other hand, it follows from Eqs. 4.10 and C.1 that, for x ∈ (b, a),

v < φ(x) ⇔ H(x, v) > 0

⇔ x < u.

Thus, by Eq. C.2 and since G(b) = 0, Eq. 4.13 holds when γ = 1(v,∞). Conse-quently, Eq. 4.13 holds when γ is the indicator function of any Borel subset of(a,∞). Since any nonnegative Borel-measurable function on (a,∞) is a point-wise limit of an increasing sequence of non-negative simple functions on (a,∞),this concludes the proof of Eq. 4.13.Eq. 4.14 follows by applying Eq. 4.13 in the case where γ = 1(φ(x),∞) or

γ = 1[φ(x),∞).We now prove the first assertion in the lemma. Consider y ∈ I − {a}.

Since the right-hand side of Eq. 4.13 is −C ′−(y) when γ = 1[y,∞), there isx ∈ (b, a) with y ≤ φ(x). Thus, y belongs to the interval spanned by φ((b, a]).Conversely, consider x ∈ (b, a) and assume for contradiction that C ′−(φ(x)) = 0.By Eqs. 4.14 and 4.12, we conclude that P (x) = C(φ(x)) = 0, leading to acontradiction. Thus, I contains the interval spanned by φ((b, a]).

D Proof of Lemma 4.5

By Lemma 4.4, for u ∈ (b, a),∫

(b,u)

(φ(x)− φ(u)) dG(x) =∫

(b,a)

max(0, φ(x)− φ(u)) dG(x)

=

∫

(a,∞)max(0, y − φ(u)) dμ(y)

= C(φ(u)).

Since∫

(b,u)

(φ(x)− u) dG(x) =∫

(b,u)

(φ(x)− φ(u)) dG(x) + (φ(u)− u)G(u),

we conclude, using Eq. 4.12, that

∫

(b,u)

(φ(x)− u) dG(x) = P (u).

17

Equivalently,

(D.1)

∫

(b,a)

(pu + (φ− I)pu′+) dG = −

∫

[b,a)

pu dμ,

where pu(x) = max(0, u−x), pu′+ is the right derivative of pu with respect to xand I is the identity function. Since, for x ∈ (b, a),

g(x) =

∫

(b,a)

ω(u)pu(x) du

and

g′(x) =

∫

(b,a)

ω(u)pu′+(x) du,

where ω(u) = −2 ln(φ(u)/u)u(φ(u)−u) , it follows from Eq. D.1 and Fubini’s theorem that

(D.2)

∫

(b,a)

(g + (φ− I)g′) dG =∫

[b,a)

−g dμ.

Both integrands in Eq. D.2 are positive and, since g is continuous,∫[b,a)−g dμ

is finite. Eqs. D.2 and 4.4 show that the integral∫(b,a)max(0, g ◦φ) dG is finite.

Also, by setting g(y) = 0 for y ∈ (a,∞) − I, it follows from Eq. 4.5 that theintegral

∫(a,∞)max(0,−g) dμ is finite. Thus, by Lemma 4.4,

(D.3)

∫

(b,a)

(g ◦ φ) dG =∫

(a,∞)g dμ.

Combining Eqs. D.2, 4.4 and D.3 concludes the proof.

E Proof of Lemma 5.2

The proof is similar to that of Eq. 4.13, which is identical to Eq. 5.5 wheni = n + 1. Assume now that 0 ≤ i ≤ n. We first observe that both sides ofEq. 5.5 are null in the case where γ = 1(v,∞), with v ≥ φ(xi). Similarly, whenγ = 1(a,v], with v < φ(xi), then by Eq. 4.13

∫

(xi,a)

(γ ◦ φ) dG =

∫

(b,a)

(γ ◦ φ) dG

= μ((a, v])

= μi((a, v])

=

∫

(a,∞)γ dμi.

Thus Eq. 5.5 also holds in this case. Finally, since C(a) = P (a) and C isconvex, it follows from Eq. 4.11 that G(a) = −C ′+(a), and so Eq. 5.5 holdswhen γ is constant and equal to 1 on (a,∞). Consequently, Eq. 5.5 holds whenγ is the indicator function of any Borel subset of (a,∞), and so it holds for anynonnegative Borel-measurable function γ on (a,∞).

18

F Proof of Theorem 6.1

Assume for simplicity that F0 = T = 1. Then, for v ≤ 0 ≤ u and σ ≥ 0,

(F.1) H(eσv, eσu) = N(−u+σ

2) +N(v −

σ

2)− eσv(N(−u−

σ

2) +N(v +

σ

2)),

where N(z) = (2π)−1/2∫ z−∞ exp(−w

2/2) dw. It follows that H(1, eσu) ≤ 0 ≤H(e−σu, eσu) and so

(F.2) ψ(eσu) ∈ [e−σu, 1].

Leth(v, u) = N ′(u)−N ′(v)− vN(−u)− vN(v)

andη(σ; v, u) = e−σv/2H(eσv, eσu).

Eq. F.1 shows that the definitions of H and η can be extended to any realnumber σ and the function η is odd with respect to σ. On the other hand,∂η(σ; v, u)/∂σ = h(v, u) at σ = 0. Furthermore, an easy calculation shows thatthere is a constant γ > 0 such |∂3η(σ; v, u)/∂σ3| ≤ γe2uN ′(v) for −u ≤ v ≤ 0 ≤u and 0 ≤ σ ≤ 1 and so, by Taylor-Lagrange theorem,

(F.3) |η(σ; v, u)− σh(v, u)| ≤ γσ3e2uN ′(v)/6.

Let us now assume that H(eσv, eσu) = 0 and γσ2e4u ≤ 1, with v ≤ 0 ≤ uand 0 ≤ σ ≤ min(1, 1/γ). By Eq. F.2, −u ≤ v ≤ 0 and so, by Eq. F.3,|h(v, u)| ≤ γσ2e2uN ′(v)/6. We show that

(F.4) |v − θ(u)| ≤ γσ2e3u.

Since the function N(z) − ezN ′(z) has a positive derivative and goes to 0 as zgoes to −∞,

(F.5) evN ′(v) ≤ N(v).

We now distinguish two cases:

• v ≤ θ(u): since ∂h(v, u)/∂v = −N(v) − N(−u) and h is concave withrespect to v,

h(θ(u), u)− h(v, u) ≤ −(θ(u)− v)(N(v) +N(−u)).

Thus (θ(u) − v)N(v) ≤ γσ2e2uN ′(v)/6. By Eq. F.5, we conclude thatEq. F.4 holds in this case.

• v > θ(u): Let z = v − γσ2e3u. By Taylor-Lagrange theorem and Eq. F.5,

h(z, u) ≥ h(v, u) + (v − z)(N(v) +N(−u))−(v − z)2

2N ′(v)

≥ N ′(v)(−γσ2e2u/6 + (v − z)e−u −(v − z)2

2)

≥ 0

≥ h(v, u).

Thus Eq. F.4 holds again in this case.

19

Eq. F.4 implies that, for 0 ≤ σ ≤ min(1, 1/γ) and 0 ≤ u ≤ − ln(γσ2)/4,

(F.6) | ln2eσu

ψ(eσu)− σ2(u− θ(u))2| ≤ 4γσ4ue3u.

By Theorem 5.7, Eqs. F.2 and F.6,

Vinf =

∫ ∞

1

ln2y

ψ(y)C ′′(y) dy

=

∫ ∞

0

ln2eσu

ψ(eσu)N ′(u+ σ/2) du

=

∫ − ln(γσ2)/4

0

ln2eσu

ψ(eσu)N ′(u+ σ/2) du+O(σ4)

= σ2∫ ∞

0

(u− θ(u))2N ′(u+ σ/2) du+O(σ4)

= σ2∫ ∞

0

(u− θ(u))2N ′(u) du− cσ3 +O(σ4).

On the other hand, by the implicit function theorem the function θ has a con-tinuous derivative for u > 0. Thus∫ ∞

0

(u− θ(u))2N ′(u) du =1

2+ 2

∫ ∞

0

θ(u)N ′′(u) du+

∫ ∞

0

θ(u)2N ′(−u) du

=1

2− 2

∫ ∞

0

θ′(u)(N ′(u)− θ(u)N(−u)) du

=1

2− 2

∫ ∞

0

θ′(u)(N ′(θ(u)) + θ(u)N(θ(u))) du

=1

2+ 2

∫ 0

−∞(N ′(z) + zN(z)) dz

= 1.

This concludes the proof of the theorem.

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