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Mathematics in Finance Working Paper Series

Robust Replication of Volatility Derivatives Peter Carr and Roger Lee

Working Paper #2008-3 May 21, 2008

The authors retain the copyright. All rights reserved. Courant Institute of Mathematical Sciences, New York University 251 Mercer Street, New York, NY 10012 http://www.cims.nyu.edu/working_paper_series/

Robust Replication of Volatility Derivatives Peter Carr∗ and Roger Lee† This version: April 14, 2008

Abstract We show that the information in European option prices reveals, robustly and nonparametrically, the no-arbitrage prices of general volatility derivatives – contracts on the realized variance of an underlying price process. Our explicit formulas are exact and valid across all dynamics satisfying an independence assumption on the instantaneous volatility. Our methods are moreover immunized, to first order, against the presence of correlation. We solve for not just valuation but also replication, via robust trading strategies which perfectly hedge volatility derivatives. Additionally, these results have relevance to the forecasting of realized volatility and the inference of volatility risk premia.

∗ †

Bloomberg LP and Courant Institute, NYU. 731 Lexington Avenue, New York, NY 10022. [email protected]. University of Chicago. 5734 S University Ave, Chicago IL 60637. [email protected].

1

1

Introduction

The tradeoff between risk and return is a central theme of finance, and volatility and variance of returns are standard measures of risk. The volatility of a stock is revealed by the market price of an option on the stock, if one accepts the model of Black-Scholes [9], which does not require any assumptions on equity risk premium nor expected return. However, that model does assume constant volatility, which contradicts the empirical observation that, for a given expiry, option contracts of different strikes typically have market prices which imply different Black-Scholes volatility parameters. In the presence of these non-constant implied volatilities across strikes – a phenomenon which practitioners describe as the smile or skew – the question arises: what information content, regarding the risk-neutral distribution of the pathdependent realized volatility and variance, is carried by the profile of European option prices at a given expiry? We answer this question with the help of an observation due to Hull-White [32]. Under some assumptions including an independence condition, the distribution of realized variance determines the value of a stock option. We invert this relationship in a more general setting. Analogously to how Latane-Rendleman [34] take as given the market price of a single option and invert the BlackScholes equation to infer a constant volatility, we take as given the market prices of all options at a given expiry and invert a Hull-White-type relationship to infer the entire risk-neutral distribution of the random realized volatility, including its mean. The information in the profile of T -expiry option prices will, therefore, nonparametrically reveal the no-arbitrage prices of volatility derivatives – claims on payoffs contingent on realized volatility. This information will, moreover, allow us to replicate volatility derivatives, by dynamic trading in standard options and the underlying shares. Our valuations and our replication strategies will have explicit formulas in terms of observables, not the parameters of any model. Our inference does not rely on any specification of the market price of volatility risk. Just as knowledge of the stock price sufficiently reflects the equity risk premium in the Black-Scholes framework, knowledge of option prices sufficiently reflects the volatility risk premium in our framework.

1.1

Variance

We define the realized variance of the returns on a positive underlying price S from time 0 to time T to be the quadratic variation of log S at time T . If S has an instantaneous volatility process σt , then

2

realized variance equals integrated variance, meaning the time integral of σt2 . In practice, contracts written on realized variance typically define it discretely as the sample variance of daily or weekly log returns. Following the custom in the derivatives literature, we study the (continuously-sampled) quadratic variation / integrated variance, leaving tests of discrete sampling for future research. Realized variance can be traded by means of a variance swap, a contract which pays at time T the difference between realized variance and an agreed fixed leg. The variance swap has become a leading tool – perhaps the leading tool – for portfolio managers to trade variance. As reported in the Financial Times [31] in 2006, Volatility is becoming an asset class in its own right. A range of structured derivative products, particularly those known as variance swaps, are now the preferred route for many hedge fund managers and proprietary traders to make bets on market volatility. According to some estimates [1], the daily trading volume in equity index variance swaps reached USD 4–5 million vega notional in 2006. On an annual basis, this corresponds to payments of more than USD 1 billion, per percentage point of volatility. From a dealer’s perspective, the variance swap admits replication by a T -expiry log contract (which decomposes into static positions in calls and puts on S), together with dynamic trading in S, as shown in Neuberger [36], Dupire [27], Carr-Madan [21], Derman et al [26], and BrittenJones/Neuberger [15]. Perfect replication requires frictionless markets and continuity of the price process, but does not require the dynamics of instantaneous volatility to be specified. In that sense, the result is model-free. The variance swap’s model-free replicating portfolio became in 2003 the basis for how the Chicago Board Option Exchange (CBOE) calculates the VIX index, an indicator of the risk-neutral market expectation of short-term volatility. Implementation issues arising from data limitations (particularly the need to interpolate and extrapolate from a limited number of strikes) are documented and addressed in Jiang-Tian [33].

1.2

Volatility derivatives

More generally, volatility derivatives, which pay functions of realized variance, are of interest to portfolio managers who desire non-linear exposure to variance. Important examples include calls and puts on realized variance; and volatility swaps (popular especially in foreign exchange markets) which pay realized volatility, defined as the square root of realized variance. In contrast to the variance swap’s replicability by a log contract, general functions of variance 3

present greater hedging difficulties to the dealer. In theory, if one specifies the dynamics of instantaneous volatility as a one-dimensional diffusion, then one can replicate a volatility derivative by trading the underlying shares and one option. Such simple stochastic volatility models are, however, misspecified according to empirical evidence, such as difficulties in fitting the observed cross-section of option prices, and pricing errors out-of-sample, as documented in Bakshi-Cao-Chen [4] and Bates [7]. Moreover, even if one could find a well-specified model, further error can arise in trying to calibrate or estimate the model’s parameters, not directly observable from options prices. Equity derivatives dealers have struggled with these issues. According to a 2003 article [37] in RiskNews, While variance swaps - where the underlying is volatility squared - can be perfectly replicated under classical derivatives pricing theory, this has not generally been thought to be possible with volatility swaps. So while a few equity derivatives desks are comfortable with taking on the risk associated with dealing volatility swaps, many are not. A 2006 Financial Times article [31] quotes a derivatives trader: Variance is easier to hedge. Volatility can be a nightmare. We challenge this conventional wisdom, by developing strategies to price and to replicate volatility derivatives – without specifying the dynamics of instantaneous volatility, hence without bearing the types of misspecification and misestimation risk discussed above. The volatility derivatives studied in this paper (and referenced in the block quotations) are realized volatility contracts, which pay functions of underlying price paths – as opposed to the various types of options-implied volatility contracts, which pay functions of option prices prevailing at a specified time. For example, we do not explicitly study options on VIX (itself a function of vanilla option prices) nor options on straddles (Brenner-Ou-Zhang [13]); rather, we do study, for example, options and swaps on the variance and volatility actually realized by the underlying.

1.3

Our approach

We prove that general functions of variance, including volatility swaps, do admit valuation and replication using portfolios of the underlying shares and European options, dynamically traded according to strategies valid across all underlying dynamics specified in Section 2. Our approach has the following benefits.

4

First, in contrast to analyses of particular models (such as Matytsin’s [35] analysis of Heston and related dynamics), we take a nonparametric approach, both robust and parameter-free, in the sense that we do not specify the dynamics nor estimate the parameters of instantaneous volatility. Our robust pricing and hedging strategies remain valid across a whole class of models – including non-Markovian and discontinuous volatility processes as well as diffusive volatility – so we avoid the risk of misspecification and miscalibration present in any one model. Specifically, by robust, we mean that our strategies are valid across all underlying continuous price processes whose instantaneous volatility satisfies an independence assumption (and some technical conditions, designated (B, W, I)). Moreover, in case the independence condition does not hold, we immunize our schemes, to first order, against the presence of correlation; thus we can price approximately under dynamics which generate implied volatility skews – without relying on any particular model of volatility. Our parameter-free pricing formulas typically take the form of an equality of risk-neutral expectations of functions of realized variance hXiT and price ST respectively: Eh(hXiT ) = EG(ST ),

(1)

where we find formulas for G, given various classes of payoff functions h, including the square root function which defines the volatility swap. The left-hand side is the value of the desired volatility or variance contract. The right-hand side is the value of a claim on a function of price, and is therefore model-independently given by the values of European options. Thus our formula for the volatility contract value is expressed not in terms of the parameters of any model, but rather in terms of prices directly observable, in principle, in the vanilla options market. Second, in contrast to approximate methods (such as Carr-Lee’s [19] use of a displaced lognormal to approximate the distribution of realized volatility) we find exact formulas for prices and hedges of volatility contracts. For example, the typical result (1) is exact under the independence condition. Third, in contrast to studies of valuation without hedging (such as Carr-Geman-Madan-Yor’s [18] model-dependent variance option valuations under pure jump dynamics), we cover not just valuation but also replication, by proving explicit option trading strategies which enforce the valuation results. The holdings in our replicating portfolios are rebalanced dynamically, but the quantity to hold, at each time, depends only on contemporaneously observable prices, not on the parameters of any model; this result arises because the observable prices already incorporate all quantities of possible relevance, such as instantaneous volatility, volatility-of-volatility, and market price of volatility risk. Indeed, to our knowledge, this paper is the first one to study nonparametrically the 5

pricing restrictions induced by, and the volatility payoffs attainable by, the ability to trade options dynamically. Moreover, because perfectly hedging against a short (long) holding of some realized volatility payoff is equivalent to perfectly replicating a long (short) position in that volatility payoff, our replication strategies therefore provide explicit robust hedges of volatility risk. Fourth, in contrast to conventional approaches narrowly focused on particular contracts and models, we contribute toward a broad program to link together two fundamental families of risks: path-dependent and path-independent. This program nonparametrically utilizes European options – which pay functions of ST alone – to extract information about risks dependent on the entire path of S, and to hedge those risks robustly. As Breeden-Litzenberger [12] showed, the information in the collection of T -expiry option prices at all strikes, fully and model-independently reveals the risk-neutral distribution of price ST . We show that the same option price information, under our assumptions, fully and robustly reveals the risk-neutral distribution of volatility, a path-dependent variable. (Another type of path-dependent risk encompassed in this program is barrier-contingent risk, treated in [20].) This paper, moreover, breaks ground for ongoing research into jointly price/variance-dependent payoffs (including options on CPPI, constant proportion portfolio insurance [42]), and into alternative dynamics with local volatility and jumps.

2

Assumptions

Fix an arbitrary time horizon T > 0. Assume for simplicity zero interest rates on a risk-free asset B with price 1 at all times, which we will call the bond or cash. On a filtered probability space (Ω, F, {Ft }, P ) satisfying the usual conditions, assume there exists an equivalent probability measure P such that the underlying share price S solves dSt = σt St dWt ,

S0 > 0

(Assumption W)

for some (Ft , P)-Brownian motion Wt and some measurable Ft -adapted process σt which satisfy Z

T

σt2 dt is bounded by some m ∈ R

(Assumption B)

0

and σ and W are independent

(Assumption I)

and such that P is a risk-neutral pricing measure in the following sense: for all p ∈ C and t ≤ T , a claim paying the real part of STp at time T has time-t price equal to the real part of Et STp , where Et 6

denotes Ft -conditional P-expectation; and likewise for the imaginary parts. Denote the logarithmic returns process by Xt := log(St /S0 )

(2)

and write hXi for the quadratic variation of X or equivalently the realized variance of the returns on S. Under assumption (W), Z hXit =

t

σu2 du.

(3)

0

Unless otherwise stated, the assumptions (B, W, I) are in effect throughout this paper. These assumptions are sufficient for the validity of our methodology, but not necessary. Indeed each of the three assumptions can be relaxed: Remark 1. In this paper we will relax our reliance on assumption (I), by finding results robust – in a sense to be defined in Section 4 – to correlation between σ and W . Moreover, in ongoing research, we drop assumption (I), by allowing an S-dependent “local volatility” multiplier. Assumptions (I) and (W) taken together imply that implied volatility skews are symmetric [6] – contrary to typical implied volatility skews in equity markets, which slope downward. Therefore our robustness to correlation has practical importance. Remark 2. We drop assumption (B) in Section 8. Remark 3. We drop assumption (W) in ongoing research, by introducing jumps in the price process. In particular, we allow asymmetries in the jump distribution which can generate asymmetric volatility skews. Remark 4. We need not and will not work under the actual physical probability measure P . All expectations are with respect to risk-neutral measure P. Our typical result, of the form Eh(hXiT ) = EG(ST ),

(4)

states nothing directly about the the physical expectation of h(hXiT ). Rather, our conclusion is that the value of the contract that pays h(hXiT ) equals the price of the contract that pays G(ST ), by the following reasoning: the G(ST ) claim, plus dynamic selffinancing trading, replicates the h(hXiT ) payoff with risk-neutral probability 1, hence with physical probability 1, because P and P agree on all events of probability 1. Thus, given the availability of the appropriate European-style contracts as hedging instruments, the variance payoff h(hXiT ) is dynamically spanned, and valuation result (4) follows, by absence of arbitrage.

7

The irrelevance of physical expectations (for this paper’s valuation and replication purposes) renders also irrelevant the mapping between risk-neutral expectations and physical expectations. Thus we have no need of any assumptions about the volatility risk premia (nor indeed any other type of risk premia) which mediate between the risk-neutral and the physical probability measures. In particular, our results are valid regardless of the market’s risk preferences, and regardless of whether volatility risk is priced or unpriced. Any effects of risk premia are already impounded in the prices of our hedging instruments. Remark 5. This paper’s purpose is distinct from, but complementary to, the forecasting of realized volatility or variance, a topic surveyed in Andersen-Bollerslev-Christoffersen-Diebold [2] and Engle [28]. The main distinction is that we explore here the replication and arbitrage-free valuation – not the forecasting – of functions of future realized variance. This distinction may be expressed in terms of probability distributions of realized variance: In contrast to forecasting/prediction methodologies which extract information about the distributions under physical measure, our valuation methodology instead extracts information about (and indeed fully determines) realized variance distributions under risk-neutral measure, as discussed in Remark 4. A second important distinction is that forecasting methodologies do not seek to create payoffs contingent on the quantities they predict, whereas our replication methodology indeed creates the types of variance-contingent payoffs desired by volatility traders. This paper is nonetheless complementary to the forecasting/prediction literature in the sense that our robust valuations of variance payoffs could serve as regressors to incorporate into a forecasting framework. For example, our synthetic volatility swap (SVS), constructed in Section 6.3 from vanilla options, has an observable value which robustly matches the value of a contract paying realized volatility. Our SVS therefore provides a natural alternative to two other option-based regressors which have appeared in empirical studies of volatility forecasts; first is the Black-Scholes implied volatility (in, for example, Canina-Figlewski [17], Christensen-Prabhala [24], Blair-PoonTaylor [10]), and second is the VIX or some revision thereof (in Andersen-Frederiksen-Staal [3] and Jiang-Tian [33]). The Black-Scholes implied volatility lacks the SVS’s robustness; while the VIX lacks the SVS’s property of robustly valuing volatility (the square root of variance), because the VIX is the square root of a model-free valuation of variance – which is not equivalent, due to the square root’s concavity.

8

In other words, a natural direction for future research is to forecast volatility using our SVS volatility valuation, instead of (or in addition to) using VIX, the square root of a variance valuation. Of course, in the presence of variance risk premia (empirically nonzero and negative according to Coval-Shumway [25], Buraschi-Jackwerth [16], Bakshi-Kapadia [5], Carr-Wu [23, 22], and Bondarenko [11]), correct valuation does not imply unbiased prediction, but this fact is not particular to our SVS; indeed all forecasting methodologies which use options-implied indicators (including SVS, VIX, and Black-Scholes implied volatility) must contend with risk premia, as done in Andersen-Frederiksen-Staal [3] and references cited therein. A second, related, application of our work is to conduct inference on the volatility risk premium, by comparing risk-neutral expectations of volatility (as revealed by our results) against physical expectations of volatility (as inferred from time-series data). Such a study would parallel the inference conducted on the variance risk premium by Carr-Wu [23, 22]. Here we will not pursue further these possible applications, as our purpose is to develop the robust valuation/replication theory for the volatility payoff, and for other functions of variance. Remark 6. Our replication strategy assumes frictionless trading in options. Of course, options trading incurs transaction costs in practice, but our results maintain relevance. First, transactions costs have decreased, and continue to decrease, as options markets become more liquid. Second, in practice a dealer typically manages a portfolio of volatility contracts, which mitigates trading costs, because offsetting trades (buying an option to hedge one volatility contract, selling that option to hedge another contract) need not actually be conducted. Third, our frictionless valuation can be regarded, in the presence of frictions, as a “central” valuation, relative to which a dealer planning to bid (offer) should make a downward (upward) adjustment dependent on transaction costs. Fourth, regardless of trading costs, our results are still implementable in non-trading contexts, such as the development of VIX-like indicators of expected volatility, and applications to volatility forecasting, as discussed in Remarks 5 and 42.

3

Variance swap

A variance swap pays hXiT minus an agreed fixed amount, which we take to be zero unless otherwise specified. Replication of a variance swap does not require assumption (I). As shown in Neuberger [36], Dupire [27], Carr-Madan [21], Derman et al [26], and Britten-Jones/Neuberger [15], Itˆo’s rule

9

implies Z XT = log(ST /S0 ) = 0

T

1 1 dSu − Su 2

Z

T



 −1 2 2 σu Su du. Su2

0

so Z

T

2 dSu . Su

hXiT = −2XT + 0

(5)

Hence the following self-financing strategy replicates the hXiT payoff. At each time t ≤ T hold 1

t

Z 0

log contract, which pays −2 log(ST /S0 )

2 St

shares

2 dSu − 2 Su

bonds

which is a static position in the log contract, plus a dynamically traded share position which “deltahedges” the log contract, plus a bond position that finances the shares and accumulates the trading gains or losses. By replication, therefore, the variance swap’s time-0 value equals the price of the log contract. Alternatively, this may be derived by taking expectations of (5) to obtain E0 hXiT = E0 [−2 log(ST /S0 )],

(6)

or of (5) together with the initial delta hedge, to obtain E0 hXiT = E0 [−2 log(ST /S0 ) + 2(ST /S0 ) − 2].

(7)

The hedged log contract in (7) may be regarded as the time-0 synthetic variance swap. At general times t ∈ [0, T ], Et hXiT = hXit + Et [−2 log(ST /St ) + 2(ST /St ) − 2],

(8)

by similar reasoning. Remark 7. By Breeden-Litzenberger [12] and Carr-Madan [21], the log contract, and indeed a claim on a general function G(ST ), can be synthesized if we have bonds and T -expiry puts and calls at all strikes. Specifically, if G : R+ → R is a difference of convex functions, then for any κ ∈ R+ we have for all x ∈ R+ the representation Z 0 G(x) = G(κ) + G (κ)(x − κ) +

00

+

Z

G (K)(x − K) dK +

G00 (K)(K − x)+ dK

(9)

0 S0 puts at strikes K < S0

(36)

Corollary 39 (Put/call decomposition of seasoned synthetic volatility swap). The seasoned (hXit > 0) correlation-robust synthetic volatility swap decomposes into the payoffs of Z  dK ∞ e−zhXit  √ θ+ (K/St )p+ + θ− (K/St )p− dz calls at strikes K > St , puts at K < St 1/2 2 π 0 K z 1/2 hXit

(37)

bonds

together with a zero-cost delta-hedge. Remark 40. By simplifying the basic volatility valuation formula (30) that we introduced, FrizGatheral [30] find one Bessel representation of the Carr-Lee basic synthetic volatility swap. In contrast, in this section, we simplify our correlation-robust volatility valuation formula (32); and thereby we find two Bessel representations of the the Carr-Lee correlation-robust synthetic volatility swap (SVS), in Corollaries 37 (Bessel formula for payoff) and 38 (Bessel formula for put/call decomposition). Our SVS provides not only valuation, but also replication of the volatility swap. Indeed, holding at each time t a delta-hedged claim on Gsvs (ST , St , hXit ) replicates the volatility swap. Proposition 41 (Synthetic volatility swap replicates the volatility swap). Holding at each time t a delta-hedged claim on Gsvs (ST , St , hXit ) replicates the volatility swap. In other words: Choose an arbitrary constant κ > 0 as a put/call separator. For K ∈ (0, κ) let Pt (K) be the time-t value of a K-strike T -expiry binary put. For K ≥ κ let Pt (K) be the time-t value of a K-strike T -expiry binary call. Let the time-t binary option holdings be given by the signed measure ϕt defined by the density function K 7→ ±∂Gsvs /∂ S (K; St , hXit ) on the domain K ∈ (0, ∞), where the + and − correspond to K > κ and K < κ respectively. Then the self-financing strategy of holding at each time t ϕt

options

Gsvs (κ, St , hXit ) + δt St

bonds

−δt

shares

p hXiT , where ∂ ∂Gsvs Gsvs (s ST /St , St , hXit ) = −Et (ST , St , hXit ) δt := Et ∂ s s =St ∂u

(38)

replicates the payoff

is observable from the time-t prices of T -expiry options. If (I) holds, then δt = 0. Regardless of whether (I) holds, the strategy is (ρ, ∆)-neutral. 25

(39)

6.4

Evolution of the synthetic volatility swap

As variance accumulates during the life of the synthetic volatility swap, its payoff profile evolves. Proposition 34 makes this precise, but here let us give some intuition. The initial payoff resembles a straddle struck at-the-money. The dynamics of the payoff depend on two factors. First, as the spot moves, the “strike” of the “straddle” floats to stay at-the-money. Second, as quadratic variation (an increasing process) accumulates, the “straddle” smooths out, losing its kink; indeed, only when hXit = 0 does the kink literally exist. We can, moreover, understand intuitively the shape which the payoff approaches as it smooths out. At time t, consider a decomposition of hXiT into the already-revealed portion hXit > 0, and the random remaining variance Rt,T := hXiT − hXit . The volatility contract pays q p p 1 hXiT = hXit + Rt,T ≤ hXit + p Rt,T . 2 hXit

(40)

The inequality holds because on the domain [hXit , ∞), the square root function is concave and hence lies below its tangent at hXit . By (7), therefore, we have Et

p p   1 hXiT ≤ hXit + p Et − 2 log(ST /St ) + 2(ST /St − 1) . 2 hXit

(41)

Intuitively, as hXit increases, the curvature of the square root function on [hXit , ∞) decreases, hence the difference between the square root and its tangent decreases, and the inequalities (40) and (41) become approximate equalities. It is natural, then, to expect that as time t rolls forward and hXit accumulates, the synthetic volatility swap should evolve toward a synthetic variance swap plus cash, which has total time-T payoff p 1 hXit + p (ST /St − 1 − log(ST /St )). hXit

(42)

This is visually confirmed in the right side of Figure 7, which compares the two time-T payoff functions (contracted at time t): the SVS payoff Gsvs (ST , St , hXit ) and the log-contract-plus-cash payoff (42).

26

Figure 5: At initiation (hXit = 0.0), the volatility swap and synthetic volatility swap (SVS) 0.7

3

0.6

2.5

0.5 2

Payoff

Payoff

0.4 1.5

0.3 1

0.2

0.5

0.1

0

0

0

0.05

0.1

0.15

0.2 0.25 Realized Variance

0.3

0.35

0.4

0.45

0

0.2

0.4

0.6

0.8

1 ST/St

1.2

1.4

1.6

1.8

2

Figure 6: Seasoned (hXit = 0.1) volatility swap and synthetic volatility swap (SVS) 0.7

3

0.6

2.5

0.5 2

Payoff

Payoff

0.4 1.5

0.3 1

0.2

0.5

0.1

0

0

0

0.05

0.1

0.15

0.2 0.25 Realized Variance

0.3

0.35

0.4

0.45

0

0.2

0.4

0.6

0.8

1 ST/St

1.2

1.4

1.6

1.8

2

Figure 7: Seasoned (hXit = 0.25) volatility swap and SVS, compared to variance swaps 3

0.7

Synthetic volatility swap Synthetic variance swap (plus cash)

Volatility swap Variance swap (plus cash)

0.6

2.5

0.5 2

Payoff

Payoff

0.4 1.5

0.3 1

0.2

0.5

0.1

0

0

0

0.05

0.1

0.15

0.2 0.25 Realized Variance

0.3

0.35

0.4

0.45

27

0

0.2

0.4

0.6

0.8

1 ST/St

1.2

1.4

1.6

1.8

2

6.5

Accuracy of the ρ-neutral synthetic volatility swap

Figure 8 shows how closely the time-0 ρ-neutral synthetic volatility swap (SVS) price approximates the true volatility swap fair value, under Heston dynamics with parameters from Bakshi-Cao-Chen [4]. For comparison, we plot also the ATM implied volatility, and the basic (correlation-sensitive) synthetic volatility swap price. As approximations of the true volatility swap value, our correlation-robust SVS outperforms ATM implied volatility and outperforms our basic (correlation-sensitive) replication – across essentially all correlation assumptions. In the case ρ = 0, both of our methods are (as promised) exact and the implied volatility approximation is nearly exact; but more importantly, in the empirically relevant case of ρ 6= 0, our correlation-robust SVS’s relative “flatness” with respect to ρ results in its greater accuracy. This illustrates why, in equity markets, we do not recommend any method or approximation which relies on assumption (I), unless it has the additional correlation-robustness present in our SVS. Figure 8: Heston dynamics: Volatility swap valuations as functions of correlation 19.4 Volatility swap fair value (VOL0) 19.3

ATM implied volatility (IV0) Basic synthetic vol swap Correlation−robust synthetic vol swap (SVS)

Value (in percentage points)

19.2

19.1

19

18.9

18.8

18.7 dV=1.15(0.04−V)dt + 0.39V1/2dW, 18.6

18.5 −1

V0=0.04

T=0.5

−0.8

−0.6

−0.4

−0.2

0 ρ

28

0.2

0.4

0.6

0.8

1

We comment on each curve in greater detail. p hXiT as in Section 6.1) equals the R expectation of realized volatility. It is determined by the distribution of realized variance Vt dt, The volatility swap fair value (denoted by VOL0 := E0

which is determined entirely by the given dynamics 1/2

dVt = 1.15(0.04 − Vt )dt + 0.39Vt

dWt ,

V0 = 0.04

(43)

of instantaneous variance Vt = σt2 . So the correlation ρ is irrelevant to VOL0 , which therefore plots R as a horizontal line. Its level 0.1902 is computable via the known distribution of Vt dt given (43). The basic (correlation-sensitive) synthetic volatility swap payoff is approximately the payoff of √

2π/S0 calls, as noted in Remark 33. Therefore its value and the ATM Black-Scholes implied

volatility IV0 are nearly equal, due to (28). The plots confirm this across the full range of ρ. More importantly, the plots confirm that VOL0 is well-approximated by these two values if ρ = 0, but due to the correlation-sensitivity of IV0 and of the basic synthetic volatility swap, both values underestimate VOL0 by more than 40 basis points, for certain values of ρ. Our correlation-robust SVS, as promised, has value SVS0 which exactly matches VOL0 if ρ = 0. Furthermore, SVS0 is, as intended by its design, ρ-invariant to first-order, at ρ = 0. There is no guarantee that this flatness will extend to ρ far from 0, but for these parameters the ρ-neutrality does indeed result in accuracy gains across the entire range of ρ, as confirmed in the plot. Finally we comment on a benchmark not plotted in the figure. The variance swap value (which p equals the log-contract value) is 0.04; and its square root (which we denote by VAR0 = E0 hXiT as in Section 6.1, and which the VIX seeks to approximate) is 0.20, regardless of ρ. Therefore, a plot of VAR0 would be a horizontal line far above the upper boundary of Figure 8, and would not be a competitive approximation to VOL0 = 0.1902. To summarize, in this example the best approximation of VOL0 , for essentially all ρ ∈ [−1, 1], is given by our correlation-robust SVS value (SVS0 ), and the worst is given by the VIX-style quantity VAR0 . The other approximations – ATM implied volatility IV0 and our basic (correlation-sensitive) volatility swap value – are accurate for the ρ = 0 case in which (I) holds. Remark 42. Figure 8 can be regarded as a numerical comparison of two notions of “model-free implied volatility” (MFIV). When defined in the “VIX-style,” MFIV is understood to mean VAR0 , the square root of the variance swap (or log contract) value. Here we have introduced the correlationrobust synthetic volatility swap, whose observable value we regard as an alternative notion of MFIV. Indeed, let us define “SVS-style” MFIV to be SVS0 , the time-0 value of our SVS. 29

Our SVS-style MFIV is truly an implied volatility, in the sense that it does indeed equal VOL0 , the expected realized volatility, according to Proposition 34 – in contrast to the VIX-style definition of MFIV as VAR0 , the square root of expected realized variance. Moreover, although Proposition 34 assumes (I), we observe that even in the (I)-violating ρ 6= 0 dynamics of Figure 8, the expected volatility VOL0 is still approximated much more accurately by our SVS-style MFIV (with errors of only 9 basis points even in the worst cases near ρ = −1) than by the VIX-style MFIV (with errors of 98 basis points).

7

Pricing other volatility derivatives

Using exponential variance payoffs, we can price general variance payoffs.

7.1

Fractional or negative power payoffs

Our volatility swap formula is the r = 1/2 case of the following generalization to powers in (0, 1). Proposition 43. For 0 < r < 1, Et hXirT = Et Gpow(r) (ST , St , hXit ) where

1 − e−zq (S /u )p+ 1 − e−zq (S /u )p− + θ dz − z r+1 z r+1 0 1 1 1√ 1 p± := p± (−z) := ± 1 − 8z θ± := θ± (−z) := ∓ √ 2 2 1 − 8z 2 2

Gpow(r) (S , u , q) :=

r Γ(1 − r)

For each t, the payoff function

Z

∞

θ+

(44) (45)

S 7→ Gpow(r) (S , St , hXit ) is ρ-neutral.

For arbitrary negative powers, we have the following formula for “inverse variance” claims. Proposition 44. For any r > 0 and any ε such that hXit + ε > 0, Et (hXiT + ε)−r = Et Gpow(−r) (ST , St , hXit + ε) where Gpow(−r) (S , u , q) := θ± := θ± (−z 1/r ) :=

1 rΓ(r)

Z

∞

0

(θ+ (S /u )p+ + θ− (S /u )p− )e−z

1 1 ∓ p 2 2 1 − 8z 1/r

p± := p± (−z 1/r ) :=

1 ± 2

For each t, the payoff function F (S ) := Gpow(−r) (S , St , hXit ) is ρ-neutral. 30

1/r q

dz

q 1/4 − 2z 1/r .

Figure 9: Polynomial variance claims hXinT on the left, and their European-style synthetic counterparts Gpow(n) (ST , S0 , hXi0 ) on the right, for n = 1, 2, 3 and hXi0 = 0 0.7

1

n=1 n=2 n=3

n=1 n=2 n=3

0.9

0.6

0.8 0.5 0.7

0.4 Payoff

Payoff

0.6

0.5

0.3 0.4

0.3

0.2

0.2 0.1 0.1

0

0

7.2

0.1

0.2

0.3

0.4 0.5 0.6 Realized variance

0.7

0.8

0.9

1

0 0.5

1

1.5

2

ST/S0

Polynomial payoffs

We obtain polynomials in variance by differentiating, in λ, the exponential of λhXiT . Proposition 45. For each positive integer n, Et hXinT = Et Gpow(n) (ST , St , hXit ) where Gpow(n) (S , u , q) := ∂λn Gexp (S , u , q, λ) λ=0

(46)

with Gexp defined in (23). In particular, for n = 1, 2, 3: E0 hXiT = E0 (−2XT + 2eXT − 2) E0 hXi2T = E0 (4XT2 + 16XT + 8XT eXT − 24eXT + 24) E0 hXi3T = E0 (−8XT3 + 24XT2 eXT − 72XT2 − 192XT eXT − 288XT + 480eXT − 480). For each t, the payoff function F (S ) := Gpow(n) (S , St , hXit ) is ρ-neutral. Note that n = 1 recovers the usual valuation of the variance swap using a hedged log contract. Figure 9 plots Gpow(n) for n = 1, 2, 3.

31

7.3

Payoffs whose transforms decay exponentially

In Sections 7.3 to 7.5 we make use of exponential variance payoffs as basis functions, to span a space of general variance payoffs. Proposition 46. Assume the continuous payoff function h : R → R has bilateral Laplace transform Z

∞

H(z) :=

e−zq h(q)dq,

Re(z) > A,

(47)

−∞

such that |H(α + βi)| = O(e−|β|µ ) as |β| → ∞ for some α > A and some µ > m/2. Then Et h(hXiT ) = Et Gh (ST , St , hXit ) where 1 Gh (S , u , q) := 2πi θ± := θ± (z) :=

Z

α+∞i

α−∞i

H(z)ezq [θ+ (S /u )p+ + θ− (S /u )p− ]dz

1 1 ∓ √ 2 2 1 + 8z

p± := p± (z) :=

(48)

1 p ± 1/4 + 2z. 2

In particular, we prove the convergence and finite expectation of Gh . For each t, the payoff function

S 7→ Gh (S , St , hXit ) is ρ-neutral.

Remark 47. Recall the heuristic that the smoother a function, the more rapid the decay of its transform. For payoff functions h which are not sufficiently smooth (including call and put payoffs), the transform H will not decay rapidly enough to satisfy the assumption of Proposition 46. For payoff functions h well-behaved enough to satisfy the stated assumptions, Proposition 46 guarantees that the volatility contract can be priced identically to our “synthetic” volatility contract with payoff Gh (ST , St , hXit ), defined by the convergent integral in (48). Although this payoff Gh may be oscillatory in ST , Proposition 46 guarantees that the payoff has a well-defined price, in the sense that the payoff’s positive and negative components each have finite expectation. Nonetheless, implementation difficulties can arise if these finite-priced components are very large and/or concentrated at illiquid strikes, which can occur for volatility contracts h whose replicating price-contracts Gh have payoff profiles with too much variation. In such cases, regularization of the payoff profile can be accomplished by using a finite set of basis functions (such as in Section 7.5); alternatively, following Friz-Gatheral [30], distributional inference can be conducted using a finite set of pricing benchmarks.

32

7.4

Payoffs whose transforms are integrable

If instead of having exponential decay, the payoff’s transform is merely integrable, then our usual pricing formulas of the form Eh(hXiT ) = EG(ST ) may not be available by the Laplace transform method. Nonetheless, the prices of claims on ST do still determine the price of the h(hXiT ) contract. Proposition 48. Assume the continuous payoff function h : R → R has bilateral Laplace transform H, defined in (47), and integrable along Re(z) = α for some α > A. Then Et h(hXiT ) =

1 2πi

Z

α+∞i

H(z)ezhXit Et [θ+ (ST /St )p+ + θ− (ST /St )p− ]dz

(49)

α−∞i

where θ± := θ± (z) :=

1 1 ∓ √ 2 2 1 + 8z

p± := p± (z) :=

1 p ± 1/4 + 2z. 2

In particular, we prove the convergence of the integral. In the case of a variance call, defined by h(q) = (q − K)+ , we have H(z) = e−zK /z 2 for all Re(z) > 0. For all α > 0, therefore, (49) exists and gives the variance call price. Remark 49. Despite its generality, Proposition 48 has a practical drawback, relative to the earlier results. To price a variance contract exactly using Proposition 48 requires the valuation of infinitely many different functions of ST (one for each z). In contrast, using Propositions 34, 43, 44, 45, 46, to price one variance contract exactly requires the valuation of only one individual function of ST . If, instead of demanding closed form formulas, we accept [a sequence of] approximate prices which converge to the exact price, then a general class of variance contracts can be priced using [a sequence of] individual functions of ST . That is the subject of the next section.

7.5

General payoffs continuous on [0, ∞]

Let C[0, ∞] denote the set of continuous h : [0, ∞) → R such that h(∞) := limq→∞ h(q) exists in R. For example, the variance put payoff h(q) = (K − q)+ belongs to C[0, ∞]. This section gives two ways to determine prices of general payoffs in C[0, ∞]. The first will take limits of uniform approximations, and the second will take limits of mean-square approximations. Although variance call payoffs do not belong to C[0, ∞], they can still be priced by the methods of this section, using put-call parity (in the sense that a variance call equals a variance put plus a variance swap). In this section let h ∈ C[0, ∞] and let c > 0 be an arbitrary constant.

33

Proposition 50 (Prices as limits of uniform approximations’ prices). Define h∗ : [0, 1] → R by h∗ (0) := h(∞) and h∗ (x) := h(−(1/c) log x) for x > 0. For integers n ≥ k ≥ 0, let bn,k :=

k X j=0

Then Et h(hXiT ) = lim Et n→∞

n X

   n k h (j/n) (−1)k−j . k j ∗

bn,k e−ckhXit [θ+ (ST /St )p+ + θ− (ST /St )p− ],

(50)

(51)

k=0

where θ± :=

1 1 ∓ √ 2 2 1 − 8ck

p± :=

1 p ± 1/4 − 2ck. 2

(52)

In particular, we prove the existence of the limit. Proposition 51 (Prices as limits of L2 projections’ prices). Let µ be a finite measure on [0, ∞). Let an,n e−cnq + an,n−1 e−c(n−1)q + · · · + an,0 =: An (q) be the L2 (µ) projection of h onto span{1, e−cq , . . . , e−cnq }. Let conditional on Ft . Assume

P

P

denote the P-distribution of hXiT ,

is absolutely continuous with respect to µ and dP /dµ ∈ L2 (µ). Then

Et h(hXiT ) = lim Et n→∞

n X

an,k e−ckhXit [θ+ (ST /St )p+ + θ− (ST /St )p− ]

(53)

k=0

where θ± :=

1 1 ∓ √ 2 2 1 − 8ck

p± :=

1 p ± 1/4 − 2ck. 2

(54)

In particular, we prove the existence of the limit. Remark 52. For each n, the an,k (k = 0, . . . , n) are given by the solution to the linear system n X

an,k he−cjq , e−ckq i = hh(q), e−cjq i,

j = 0, . . . , n

(55)

k=0

of normal equations, where hα(q), β(q)i :=

R∞ 0

α(q)β(q) dµ(q). In practice, one can compute an,k

as the coefficients in a weighted least squares regression of the h(q) function on the regressors {q 7→ e−ckq : k = 0, . . . , n}, with weights given by the measure µ. For example, consider the variance put payoff h(hXiT ) = (0.04 − hXiT )+ with expiry T = 1. Under the Heston variance dynamics specified in Figure 8 with ρ = 0, let us compare the put’s true time-0 value Eh(hXiT ) against the sequence of European prices in the right-hand side of (53). For example, let c = 0.5, and let µ be the lognormal distribution whose parameters are consistent with 34

the values of T -expiry variance and volatility swaps (which are observable from European options, by Propositions 34 and 45). We compute: EA3 (hXiT )

EA4 (hXiT )

EA5 (hXiT )

···

Eh(hXiT )

0.01108

0.01133

0.01147

···

0.01149

(56)

Here small values of n have sufficed to produce an accurate approximation of Eh(hXiT ). Remark 53. In principle, each An and Bn function admits perfect pricing by European options, via (51) and (53) respectively; in practice, the convergence benefits of incrementing n must be considered in the context of whether the available European options data (which may have noisy or missing observations) can provide sufficient resolution. Remark 54. Each An and Bn function is a linear combination of exponentials, hence admits perfect replication by European options, according to Proposition 26. Consequently, by the explicit uniform approximation (68), any variance payoff continuous on [0, ∞] can be replicated to within an arbitrarily small uniform error.

8

Extension to unbounded quadratic variation

Here we show how to drop the assumption (B) that hXiT 6 m for some constant m. It could be argued that this section is mainly of theoretical interest, because in practice a bound of, say, m = 1010 T may be an acceptable assumption for an equity index. Theoretically, however, models such as Heston do violate (B). Proposition 55. Assume the measurable functions h and G satisfy Eh(hXiT ) = EG(ST )

(57)

for all S which satisfy (B, W, I). Assume that h is bounded or that h is nonnegative and increasing. Assume that G has a decomposition G = G1 − G2 , where G1,2 are convex and EG1,2 (ST ) < ∞. Then (57) holds, more generally, for all S which satisfy (W) and (I) and EhXiT < ∞. Remark 56. The finiteness of Eh(hXiT ) is a conclusion, not an assumption. Remark 57. The assumptions on G are very mild, in the following sense: They are satisfied by any payoff function which can be represented as a mixture of calls and puts at all strikes, such that the long and short positions have finite values. 35

Corollary 58. Propositions 16, 24 on exponential variance valuation, Propositions 32, 34 on volatility swap valuation, Propositions 43 44, 45 on valuation of fractional and integer powers of variance, and Proposition 46 on valuation by Laplace transform, all hold without assuming (B) – provided that the long and short positions in call and puts in the replicating portfolios have finite values.

9

Conclusion

Contracts on realized variance allow investors to tailor their exposure to volatility risk, but derivatives dealers have faced difficulties in pricing and hedging such contracts. We find robust solutions by deriving explicit robust formulas to value realized variance contracts in terms of vanilla option prices – not in terms of the parameters of any model. The formulas are exact under an independence condition, and they are first-order robust to the presence of correlation. In this setting, the information in option prices at a single expiry fully reveals the risk-neutral distribution of realized variance. For hedging purposes, we enforce these valuation formulas by replicating the variance payoffs using explicit trading strategies in vanilla options and the underlying shares. Although we focus on pricing and hedging, we have also suggested possible applications to the forecasting of realized volatility and the inference of volatility risk premia. Future research can build on the dynamics we study and the risks we replicate. This paper lays the groundwork for ongoing research to add jumps and local volatility to the price dynamics, and it contributes to a broad program which nonparametrically utilizes European options – which pay functions of ST alone – to extract information about risks dependent on the entire path of S, and to hedge those risks robustly.

36

A

Proofs

Proof of Proposition 9. We have p 1 dXt = − σt2 dt + 1 − ρ2 σt dW1t + ρσt dW2t 2 p ρ2 1 − ρ2 2 =− σt dt + 1 − ρ2 σt dW1t − σt2 + ρσt dW2t 2 2 So conditional on HT ∨ Ft ,   2 p 2 1−ρ XT ∼ Normal Xt + log Mt,T (ρ) − σ ¯t,T , σ ¯t,T 1 − ρ2 . 2 Hence the time-t price of the F (ST ) claim is Et F (ST ) = Et (Et (F (ST )|HT )) = Et F BS (St Mt,T (ρ), σ ¯t,T

p

1 − ρ2 )

as desired. Proof of Proposition 16. We apply a more general version of Hull-White’s [32] conditioning argument. Conditional on FTσ , the W is still a Brownian motion, by independence. So conditional on Ft ∨ FTσ , Z XT − Xt = t

T

1 σu dWu − (hXiT − hXit ) 2

∼

 hXi − hXi  t T Normal − , hXiT − hXit . 2

For each p ∈ C, therefore, h i Et ep(XT −Xt ) = Et Et (ep(XT −Xt ) |FTσ ) i h σ σ = Et eEt (pXT −pXt |FT )+Vart (pXT −pXt |FT )/2 h 2 i = Et e(p /2−p/2)(hXiT −hXit ) = Et eλ(hXiT −hXit ) , where λ = p2 /2 − p/2. Equivalently, p =

1 2

±

q

1 4

+ 2λ.

Proof of Proposition 18. Our trading strategy at each time t has value Nt Pt − (pNt Pt− /St )St + pNt Pt− = Nt Pt , and in particular it has at time T the desired terminal value NT PT = eλhXiT . To prove that it self-finances, we have d(Nt Pt ) = Nt dPt + Pt− dNt + d[P, N ]t   −pNt = Nt dPt + Pt− dSt + dAt St 37

where A has finite variation. The continuity of S implies the continuity of N , hence [P, N ], hence A. Moreover, A is a local martingale because Nt Pt (= Et eλhXiT by Proposition 16) and the stochastic integrals with respect to P and S are all local martingales. Therefore dA vanishes. Moreover, because dB = 0, we have d(Nt Pt ) = Nt dPt − (pNt Pt− /St )dSt + pNt Pt− dBt which proves self-financing. Proof of Proposition 24. The weights θ± have the properties that θ+ +θ− = 1 and θ+ p+ +θ− p− = 0. The first property, together with Remark 23, implies (22). To see that the second property implies ρ-neutrality, let φv be the lognormal density with parameters (−v/2, v). Then  Z ∞ ∂F Pv p− p− λhXit ∂ p+ p+ φv (y)dy (St ) = e θ+ (s /St ) y + θ− (s /St ) y ∂s ∂ s s =St 0  Z ∞ Z p+ θ+ p+ + θ− p− ∞ p+ p− = eλhXit θ+ y p+ + θ− y p− φv (y)dy = eλhXit y φv (y)dy = 0 St St St 0 0 using the equality of integrals of y p+ φv (y) and y p− φv (y). Proof of Proposition 26. The strategy is a linear combination of the two strategies (+, −) specified in Proposition 18, with constant weights θ+ and θ− which sum to 1. Each strategy self-finances and replicates eλhXiT , so the combination does also. Moreover, δt = 0 because Nt+ Pt+ = Nt− Pt− and θ+ p+ + θ− p− = 0. Regardless of (I), we have ∆-neutrality because the time-t holdings have combined payoff function (in the sense of (11)) F (S ) := Gexp (S , St , hXit ; λ) − δt (S − St ), which satisfies    ∂F Ptrue ∂ (St ) = eλhXit Et θ+ (s ST /St2 )p+ + θ− (s ST /St2 )p− − δt = 0. ∂s ∂s s =St Finally, Proposition 24 implies the ρ-neutrality condition. Proof of Proposition 27. The upper bound (c) is known (Britten-Jones/Neuberger [15]) to hold, by Jensen’s inequality.

38

BS , For (b), we have by Proposition 9 and the concavity of Fatmc BS BS BS Fatmc (S0 , IV0 ) = E0 Fatmc (ST ) = E0 Fatmc (S0 , σ ¯0,T ) ≤ Fatmc (S0 , E0 σ ¯0,T ). BS , therefore, IV ≤ E σ By the monotonicity of Fatmc 0 0 ¯0,T . For (a), √ √ √ 2π 2π BS 2π S0 IV0 + √ E0 (ST − S0 ) = Fatmc (S0 , IV0 ) ≤ = IV0 S0 S0 S0 2π

(58)

(59)

BS (S , ·) lies everywhere below its tangent at 0. because concavity implies that Fatmc 0

Proof of Proposition 32. Of the ±, we prove the + equation; the − equation is similar. The square root function has the integral representation √

1 q= √ 2 π

Z

∞

0

1 − e−zq dz z 3/2

q ≥ 0;

as shown in sources such as Sch¨ urger [40]. So Z ∞ p 1 1 − e−zhXiT E0 hXiT = √ E0 dz 2 π z 3/2 0 Z ∞ 1 − e−zhXiT 1 √ E0 dz = 2 π 0 z 3/2 √ Z ∞ 1 1 − e(1/2− 1/4−2z)XT = √ E0 dz 2 π 0 z 3/2 √ Z ∞ 1 1 − e(1/2− 1/4−2z)XT dz. = √ E0 2 π z 3/2 0 and take real parts. The first application of Fubini is justified by |1 − e−zhXiT | < 1 − e−zm . The √ second application of Fubini is justified by E0 |1 − e(1/2− 1/4−2z)XT | = O(1) as z → ∞; and on the other hand for z sufficiently small, √ √ (E0 |1 − e(1/2− 1/4−2z)XT |)2 ≤ E0 (|1 − e(1/2− 1/4−2z)XT |2 ) √ √ = E0 (1 − 2e(1/2− 1/4−2z)XT + e(1/2− 1/4−2z)2XT ) = 1 − 2E0 e−zhXiT + E0 e(

√ 1−8z− 1−8z )hXiT 2

= 1 − 2(1 − zf 0 (0) + O(z 2 )) + 1 − 2zf 0 (0) + O(z 2 ) = O(z 2 )

as z → 0

using the analyticity of the moment generating function f (ξ) := eξhXiT , which follows from (B).

39

Proof of Proposition 34. For arbitrary Ft -measurable q ≥ 0 we have Z ∞ p 1 − e−z(hXiT −hXit +q) 1 dz Et hXiT − hXit + q = √ Et 2 π z 3/2 0 Z ∞ 1 1 − Et e−z(hXiT −hXit +q) = √ (θ+ + θ− ) dz 2 π 0 z 3/2 Z ∞X 1 1 − e−zq Et ep± (XT −Xt ) √ θ± dz = 2 π 0 ± z 3/2 Z ∞X 1 1 − e−zq ep± (XT −Xt ) = √ Et dz θ± 3/2 2 π z 0 ±

(60) (61) (62) (63)

Taking q := hXit yields the result. The first application of Fubini (61) is justified by |1 − e−z(hXiT +q) | < 1 − e−z(m+q) . The second application of Fubini (63) is justified by √ √ A± := (Et |1 − e−qz+(1/2± 1/4−2z)(XT −Xt ) |)2 ≤ Et (|(1 − e−qz+(1/2± 1/4−2z)(XT −Xt ) |2 )

(64)

which is O(1) as z → ∞, hence √ |θ± (1 − e−qz+(1/2± 1/4−2z)(XT −Xt ) )| = O(z −3/2 ) Et 3/2 z

z → ∞.

On the other hand, for z sufficiently small, the term in the absolute values in (64) is real, so √ √   A± ≤ Et 1 − 2e−qz+(1/2± 1/4−2z)(XT −Xt ) + e−2qz+2(1/2± 1/4−2z)(XT −Xt ) = 1 − 2e−qz Et e−z(hXiT −hXit ) + e−2qz Et e(

√ 1−8z± 1−8z )(hXiT −hXit ) 2

hence as z → 0, A+ = O(1)

(65)

A− = 1 − 2(1 − zf 0 (0) − qz + O(z 2 )) + 1 − 2zf 0 (0) − 2qz + O(z 2 ) = O(z 2 )

(66)

using analyticity of moment generating function f (ξ) := eξhXiT , which follows from (B). Combining this with θ− = O(1) and θ+ = O(z) as z → 0, we have √ 1/2 |θ± |A± |θ± (1 − e−qz+(1/2± 1/4−2z)(XT −Xt ) )| Et = = O(z −1/2 ) z 3/2 z 3/2

z → 0,

which allows the interchange in (63). To establish ρ-neutrality, let φv be the lognormal density with parameters (−v/2, v). Then  Z ∞Z ∞ ∂F Pv 1 ∂ 1 − e−zhXit (s y/St )p− 1 − e−zhXit (s y/St )p+ (St ) = √ + θ− dz φv (y)dy θ+ ∂s 2 π ∂ s s =St 0 z 3/2 z 3/2 0 Z ∞Z ∞ 1 −e−zhXit (θ+ p+ y p+ + θ− p− y p− ) = √ φv (y)dzdy = 0 2 π 0 St z 3/2 0 using the equality of integrals of y p+ φv (y) and y p− φv (y), and the identity θ+ p+ + θ− p− = 0. 40

Proof of Corollary 37. By a Mathematica computation, r Z ∞ 1 − ep+ XT 1 − ep− XT π XT /2 1 √ θ+ + θ− dz = e |XT I0 (XT /2) − XT I1 (XT /2)|. 3/2 3/2 2 2 π 0 z z The result now follows from Proposition 34. Proof of Corollary 38. From (35), compute ψ 00 (K), and apply Remark 7. Proof of Corollary 39. From (32), compute ∂ 2 Gsvs /∂ S 2 (K, St , hXit ), and apply Remark 7. Proof of Proposition 41. For background in measure-valued trading strategies, see [8]. The trading strategy at each time t has value Z Pt (K)ϕt (dK) + Gsvs (κ, St , hXit )

Vt = Z

∂Gsvs (K; St , hXit )dK + Gsvs (κ, St , hXit ) ∂S p = Et Gsvs (ST , St , hXit ) = Et hXiT

=

Pt (K)(−1)IK 0, EG+ (STm )I(G+ (STm ) > A) ≤ EG+ (ST )I(G+ (ST ) > A). By the convexity of G+ , there exist a, b ∈ [0, ∞] such that I(G+ (S ) > A) = I(S < S0 − a) + I(S > S0 + b)

for all

S >0

Moreover, the function U (S ) := G+ (S )I(G+ (S ) > A) −

A A (S − S0 )I(S > S0 + b) − (S0 − S )I(S < S0 − a) b a

is convex. We have EG+ (STm )I(G+ (STm ) > A)   A A = E (S0 − STm )I(STm < S0 − a) + (STm − S0 )I(STm > S0 + b) + U (STm ) a b   A A m m m m m = E E[(S0 − ST )I(ST < S0 − a)|hXiT ] + E[(ST − S0 )I(ST > S0 + b)|hXiT ] + E[U (ST )|hXiT ] a b   A A ≤ E E[(S0 − ST )I(ST < S0 − a)|hXiT ] + E[(ST − S0 )I(ST > S0 + b)|hXiT ] + E[U (ST )|hXiT ] a b = EG+ (ST )I(G+ (ST ) > A) where the inequality holds because if Z has a mean-S0 lognormal distribution with Var log(Z) = σ 2 , then each of the quantities E[(S0 − Z)I(Z < S0 − a)], E[(Z − S0 )I(Z > S0 + b)], and EU (Z) is an increasing function of σ. For the first two expectations, this comes from direct calculation; for the third expectation, it follows from convexity and Jensen’s inequality.

45

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