Simple Robust Hedging with Nearby Contracts

Transcription

1 Simple Robust Heging with Nearby Contracts Liuren Wu Zicklin School of Business, Baruch College Jingyi Zhu University of Utah Abstract Most existing heging approaches are base on neutralizing risk exposures efine uner a pre-specifie moel. This paper proposes a new, simple, an robust heging approach base on the affinity of the erivative contracts. As a result, the strategy oes not epen on assumptions on the unerlying risk ynamics. Simulation analysis uner commonly propose security price ynamics shows that the heging performance of our methoology base on a static position of three options compares favorably against the ynamic elta heging strategy with aily rebalancing. A historical heging exercise on S&P 5 inex option further highlights the superior performance of our strategy. JEL Classification: E43, E47, G1, G12, C51. Keywors: options, static heging, forwar partial ifferential equation, local volatility. We thank Peter Carr, Tom Hur, an seminar participants at McMaster University an Worcester Polytechnic Institute for iscussions an comments. We welcome comments, incluing references we have inavertently overlooke. liuren.wu@baruch.cuny.eu (Wu) an zhu@math.utah.eu (Zhu).

2 Contents 1 Introuction 1 2 A New Theoretical Framework for Heging with Nearby Contracts Assumptions an notations Heging with a maturity-strike triangle triangle Symmetric triangles A egenerating line of three strikes at one maturity A egenerating line of two maturities at one strike From expansion errors to heging errors Inepenence of portfolio weights on the expansion reference point Leaing-term expansion errors in target an hege options Numerical Experiments Base on Commonly Specifie Dynamics Data-generating processes Monte Carlo proceures Optimal strike spacing choice Heging performance comparisons A Historical Heging Exercise on S&P 5 Inex Options 32 5 Concluing Remarks 34 2

3 Give me a lever long enough an a fulcrum on which to place it, an I shall move the worl. Archimees, Mathematician an inventor of ancient Greece, BC 1. Introuction In heging erivatives risk, many think like Archimees, by making strong, iealistic assumptions on both the security ynamics an the traing environment. For example, Black an Scholes (1973) an Merton (1973) introuce the concept of ynamic heging by assuming that the unerlying security follows a onefactor iffusion process an that one can rebalance the hege portfolio continuously without incurring any transaction cost. Since then, the iea of ynamic heging has been extene to multi-factor continuous ynamics base on continuous rebalancing of a portfolio that inclues both the unerlying security an a finite number of erivative securities. The heging ratios epen on the particular assumptions on the unerlying ynamics. Carr an Wu (22) propose a static heging strategy on vanilla options by assuming that the unerlying security follows a one-factor Markovian process an that one can eploy an infinite number of short-term options across the whole continuum of strikes. In reality, transaction cost is a fact of life, uner which both continuous rebalancing an transacting on a continuum of options lea to immeiate financial ruin. One must therefore iscretize the rebalancing frequency uner the ynamic heging approach an iscretize the strike continuum uner the static heging application to balance heging errors with transaction costs. 1 Even more problematic, however, is the fact that one oes not know the exact ynamics of the unerlying security an thus cannot fully quantify the exposures to each risk source. To the extent that the moel is misspecifie, heging errors can result from either mis-calculating the heging ratios or missing some risk sources all together. For example, most ynamic heging approaches leave the risk of iscontinuous price movements of ranom sizes unhege. Furthermore, both the ynamic elta heging uner the Black-Merton-Scholes (BMS) moel an the static heging strategy propose by Carr an Wu (22) leave volatility risks unhege. 1 Uner the Black-Merton-Scholes moel environment, the epenence of the elta heging error on the iscretization step has been stuie extensively in, for example, Boyle an Emanuel (198), Bhattacharya (198), Figlewski (1989), Galai (1983), Lelan (1985), an Toft (1996). 1

4 Practitioners often perform various a hoc ajustments to brige the gap between iealistic theories an reality. For example, ue to transaction cost, continuous rebalancing is often reuce to rebalancing aily for stock price exposures an opportunistically for volatility exposures. Furthermore, ue to uncertainty in the unerlying ynamics, risk exposures are often compute using the Black-Merton-Scholes (BMS) moel, but risks are manage far beyon the elta risk, even though elta risk is the only risk source uner the BMS moel. The vega of the erivative portfolio, i.e., the portfolio s exposure to volatility risk, is also closely monitore an manage. In aition, vega risks at ifferent segments of the implie volatility surface are often manage separately, implicitly recognizing that volatility risk may have multiple imensions of variation that can affect ifferent segments of the implie volatility surface ifferently. 2 Such practices may seem inconsistent with the unerlying moels use for computing the risk exposures, but they provie a simple, albeit primitive, mechanism to efen against moel uncertainty an against shocks from possibly multiple, unmoele risk sources. In the presence of such uncertainties, the safest way to hege the risk of a erivative position is to use nearby, similar contracts, which share similar risk characteristics regarless of the unerlying ynamics, rather than using vastly ifferent contracts while relying on a moel to compute the risk exposure an the heging ratio. In this paper, we formalize this intuitive iea an erive a heging strategy not base on risk exposures efine in a moel, but base on similarities in observable contract characteristics. To make our iea operational, we focus on European options on the same unerlying security an efine contract similarities base on their istance in strike an time to maturity. We start with a short position in a target option contract, an propose to hege the target position with three nearby option contracts. One can in theory choose more option contracts to form a more accurate hege portfolio, but transaction cost concerns motivate us to focus on a small number. The strikes an maturities of the three heging options can be flexibly chosen to balance between contract availability, transaction cost, an heging effectiveness. We focus on a strike-maturity triangle formulation in which a center strike is place at one maturity an two outsie strikes are place at another maturity. We perform Taylor series 2 As a concrete example, a long five-year strale might be neutralize by a short four-year strale, but may not be neutralize by a one-month strale position. Vegas at ifferent segments of the implie volatility surface are often regare as exposures to ifferent risk sources. 2

5 expansions on both the target an the heging options along the strike an maturity imension aroun a common strike an maturity reference point, an we choose the heging portfolio weights to match the ifferent expansion terms in the target option an the heging portfolio. The simple maturity-strike triangle becomes a very effective hege of the target option for several reasons. First, we Taylor expan along the strike imension to the secon orer, an we link the seconorer strike erivative (butterfly spreas) to the first-orer maturity erivative (calenar spreas) via the local volatility efinition of Dupire (1994). While Dupire first proposes the concept of local volatility in a one-factor iffusion setting, the notion of local volatility is well-efine uner a much more general setting. Rather than regaring it as a moel assumption, we use the local volatility to efine the empirically observe relation between butterfly spreas an calenar spreas, without assuming anything about the unerlying price ynamics. Through this linkage, we are able to achieve secon-orer accuracy with merely three options to match coefficients on three terms: the option value, the first strike erivative, an the first maturity erivative at the reference strike an maturity point. Secon, we show that our portfolio formulation allows a partial cancelation of the higher-orer terms in the Taylor expansions of the target an hege options, thus making the heging errors smaller than the expansion errors of each target or hege option. Furthermore, when multiple strikes are available, we can match higher-orer terms between the target option an the triangle hege portfolio through appropriate choice of the strike choice for the hege triangle. Most importantly, the formulation of our heging portfolio is completely inepenent of any assumptions on the unerlying risk ynamics. We choose the heging option maturities an strikes to balance contract availability, transaction cost, an heging effectiveness, an we erive the heging portfolio weights base on observe option prices, from which we estimate a local volatility at the reference strike an maturity point, while making no assumptions on the unerlying risk ynamics an/or the risk exposures of each contract. Through extensive Monte Carlo analysis on commonly use stock price ynamics, we show that our static hege portfolio with three options can perform much better than ynamic hege with the unerlying 3

6 futures in all moel environments. The simulation exercise also illustrates how one can choose the maturity an strike spacing for the heging portfolio to further reuce heging errors. Applying the strategy to S&P 5 inex options in a historical test also shows that we can form many maturity-strike triangles from the available option contracts that outperform elta heging with aily rebalancing. Thus, uner practical scenarios, the triangle is simple an flexible for actual implementation, an robust to ynamics variations. The option pricing literature mostly starts with a funamental backwar partial ifferential equation (PDE), which efines the value of a erivative contract base on the relations between the exposures of the contract to various risk sources. For example, uner the Black-Scholes-Merton moel an assuming zero rates, the theta (time erivative) of an option is linearly relate to the elta (stock price erivative) an ollar gamma (secon stock price erivative) of the option. In the presence of stochastic volatility, vega (volatility erivative), vanna (cross erivative of volatility an price), an volga (secon volatility erivative) also come into the backwar PDE. Such backwar equations efine how the risk sensitivities of the erivative contracts link to each other an form the basis for risk-exposure-base heging approaches. By contrast, our heging result is built on the forwar PDE that relates option erivatives against maturities an strike prices. By exploiting the forwar equation, we can match more terms with fewer options. Even if the ynamics unerlying the original forwar PDE of Dupire (1994) oes not hol, we can still use it as a efinition for local volatility, through which the maturity erivative an secon-strike erivative are linke. To the exten this linkage (an hence the local volatility) is stable over time, better static heging performance can be achieve by matching secon-orer strike erivatives. Since we kept our portfolio weights fixe over the life of the heging exercise, our heging approach is the most closely relate to the static heging propose by Carr an Wu (22), who use a continuum of short-term options to completely span the risk of a long-term option uner the assumption of a onefactor Markovian setting. Since using a continuum of options is not feasible in practice, they also propose a quarature metho to approximate the continuum with a small number of options. We show that their three-strike approximation coincies with a egenerate special case of our maturity-strike triangle, in which all three options in the hege portfolio lie on one maturity. 4

7 In other relate literature, the effective heging of erivative securities has been applie not only for risk management, but also for option valuation an moel verification (Bates (23)). Bakshi, Cao, an Chen (1997), Bakshi an Kapaia (23), an Dumas, Fleming, an Whaley (1998) use heging performance to test ifferent option pricing moels. Kenney, Forsyth, an Vetzal (26) an Kenney, Forsyth, an Vetzal (29) sets up a ynamic programming problem in minimizing the heging errors uner jump iffusion frameworks an in the presence of transaction cost. The iea of static spanning, on the other han, starte with the classic works by Breeen an Litzenberger (1978), Ross (1976), Green an Jarrow (1987), an Nachman (1988). These authors show that a path-inepenent payoff can be hege using a portfolio of stanar options maturing with the claim. More recently, Carr an Chou (1997) consiers the static heging of barrier options an Carr an Maan (1998) proposes a static spanning relation for a general payoff function by a portfolio of bon, forwar, European options maturing at the same maturity with the payoff function. Starting with such a spanning relation, Takahashi an Yamazaki (29a,b) propose a static heging relation for a target instrument that has a known value function. The remainer of the paper is organize as follows. The next section efines the heging proceure, an erives the optimal weights for the maturity-strike triangle heging portfolio. Section 3 provies a numerical stuy on the effectiveness of the heging strategy an how the effectiveness varies across ifferent maturity combinations an strike spacing choices. Section 4 applies the heging strategy to a long history of S&P 5 inex options. Section 5 conclues. 2. A New Theoretical Framework for Heging with Nearby Contracts To make the iea concrete, we start at time t with a unit short position in a European call option with strike K an expiry T, an we consier heging this option position by using a small number (three, to be exact) of European call options at nearby strikes an maturities. When necessary, put-call parity can be applie to switch call options to put options. In the absence of transaction cost, one can in principle form a hege portfolio with more options to achieve better heging performance; nevertheless, practical transaction cost concerns motivate us to limit to three options in forming the hege portfolio. 5

8 2.1. Assumptions an notations We use C(K,T) to enote the time-t value of a call option at strike K an expiry T. To avoi notational clustering, we assume zero rates an suppress the epenence of the call option value on calenar time t, the spot price S, an other potential risk sources, as long as no confusion shall occur. Given observe option prices across ifferent strikes an maturities, we efine the local volatility surface, σ(k,t), via the Dupire (1994) equation in terms of the partial erivatives of the option values against strike an maturity, σ 2 (K,T) 2C T(K,T) K 2 C KK (K,T), (1) where C T enotes the first partial erivative of the call option value with respect to maturity an C KK enotes the secon partial erivative with respect to strike. Dupire first erives the forwar PDE in a one-factor iffusion setting; however, the notion of local volatility as efine in equation (1) is well-pose uner a much more general setting. In particular, the existence of a positive an finite local volatility surface can be use as a conition to exclue arbitrage opportunities. In practice, only a finite number of option prices are observable across a iscrete number of strikes an maturities. Thus, one nees to perform interpolation an extrapolation over the finite observations to evaluate the maturity an strike erivatives to arrive at the local volatility estimates. When options are quote in BMS implie volatilities, one can also compute the local volatility irectly from the interpolate implie volatility surface, e.g., Coleman, Li, an Verma (1998), Lee (25), an Gatheral (26). We assume that one can perform reasonably stable interpolation an extrapolation on observe option prices or implie volatilities to obtain finite an positive estimates of local volatilities at strikes an maturities of interest. We make no assumptions on the unerlying security price or volatility ynamics Heging with a maturity-strike triangle triangle We propose a strategy to hege the risk of the target option C(K,T) with three nearby options. In principle, the three option contracts can all have ifferent strikes an maturities. Since often fewer maturities are 6

9 available in practice, we focus on a maturity-strike triangle formulation, where the three options have three ifferent strikes K < K c < K u but two ifferent maturities, with the center strike K c at one maturity T c an the two outsie strikes (K,K u ) at another maturity T o. There is no particular restriction on the orer of the three maturities T c,t o,t, but practically it is likely that one chooses more liqui shorter-term options to hege the possibly less liqui longer-term option, that is, T c,t o < T. Furthermore, it is natural to choose the hege option strikes aroun the target option strike K < K < K u, with possibly K c = K when K is available at T c. The heging strategy that we propose oes not rely on matching the risk exposures of the heging portfolio with that of the target option because risk exposure calculations epen on the particular specification of the unerlying security price an volatility ynamics. Instea, our strategy is base on the affinity of the triangle hege portfolio to the target option in terms of their strikes an maturities. Given the layout of the maturity-strike triangle, we erive the hege portfolio weights through the following proceure. First, we perform Taylor expansions on both the target option an the hege portfolio along the maturity an strike imensions aroun a common reference point, (K,T o ), C(K,T) C(K,T o )+C T (K,T o )(T T o ), (2) C(K,T o ) C(K,T o )+C K (K,T o )(K K)+ 1 2 C KK(K,T o )(K K) 2, (3) C(K u,t o ) C(K,T o )+C K (K,T o )(K u K)+ 1 2 C KK(K,T o )(K u K) 2, (4) C(K c,t c ) C(K,T o )+C K (K,T o )(K c K)+C T (K,T o )(T c T o )+ 1 2 C KK(K,T o )(K c K) 2. (5) We expan the options along the maturity imension to the first orer an along the strike imension to the secon orer. The expansion generate four terms C(K,T o ), C K (K,T o ), C KK (K,T o ), an C T (K,T o ). Unfortunately, we cannot use a portfolio of three options to match four expansion terms. Fortunately, we can replace the secon strike erivative C KK (K,T o ) with the first maturity erivative C T (K,T o ) via the local volatility efinition in equation (1), C KK (K,T o ) = 2 σ(k,t o ) 2 K 2C T(K,T o ). (6) 7

10 With this replacement, we can choose the portfolio weights (w,w c,w u ) for the three options at strikes (K,K c,k u ) to match the coefficients on the three terms between the target option expansion an the hege portfolio expansion: the option value C(K,T o ), the first-orer strike erivative C K (K,T o ), an the first-orer maturity erivative C(T(K,T o ). Matching the coefficients on C(K,T o ), we have 1 = w + w u + w c, (7) which says that the sum of the hege portfolio weights is equal to the target option weight. Matching the coefficients on C K (K,T o ), we have, = w (K K)+w u (K u K)+w c (K c K). (8) Plugging the weight conition in (7) to (8), we have, K = w K + w u K u + w c K c, (9) which says that the weighte average of the chosen strikes in the hege portfolio shoul be equal to the target strike. Finally, matching the coefficients on C T (K,T o ) an normalizing both sies by (T T o ), we have (K j K) 1 = 2 w j j σ 2 (K,T o )K 2 (T T o ) w T o T c c, j =,u,c. (1) T T o We can solve for the three portfolio weights from the three conitions (7), (9), an (1). Now, we introuce a stanarize measure of strike spacing aroun the target strike point, j (K j K) Kσ(K,T o ) T T o, j =,c,u. (11) Intuitively, the stanarize spacing measure j approximates the number of stanar eviations that the 8

11 security price nees to move from (T o,k j ) to (T,K). We also efine a relative maturity spacing measure, α T o T c T T o, (12) which measures the relative istance between the two maturities in the hege triangle to the istance between the target option maturity an the reference hege maturity T o. The thir conition in (1) can be written in terms of the stanarize strike spacing j an maturity spacing α, 1 = w 2 + w u 2 u + w c ( 2 c α). (13) The following proposal summarizes the results on the maturity-strike triangle hege portfolio. Proposition 1 To hege the risk of a target option at (K,T), we propose to form a hege portfolio with three options forming a maturity-strike triangle, in which the three options are place at three strikes K < K c < K u ) an two maturities with (K,K u ) at T o an K c at T c. The portfolio weights can be chosen to match the maturity an strike expansions of the triangle with that of the target option, w w c w u = K K c K u 2 2 c α 2 u 1 1 K 1. (14) An important observation from the proposition is that the portfolio weights only epen on the relative strike an maturity spacing of the hege an target options, but o not explicitly epen on the calenar time or the spot price level. In this sense, the hege portfolio is static. One caveat is that we use the local volatility σ(k,t o ) to stanarize the strike spacing in (11). To the extent that the local volatility is varying over time, so is the stanarize strike spacing for a fixe set of option contracts. The portfolio weights can vary as a result. In application, we assume that the relation between C T an C KK is stable over time, an treat the hege portfolio as an approximate, static portfolio. 9

12 The proposal imposes little constraints on the strike an maturity choice in the triangle. In what follows, we consier several interesting special cases of the general proposal Symmetric triangles If we place the center strike at the target option strike K c = K an choose equal spacing for the two outer strikes, K u K = K K, we obtain a symmetric (isosceles) triangle. In this case, c = an we let = u = enotes the stanarize equal istance from the two outer strikes to the center. The result becomes particularly simple. Proposition 2 When the maturity-strike triangle is symmetric aroun the target strike, with K c = K an K u K = K K, the portfolio weights are given as a function of the stanarize strike spacing = u = an relative maturity istance α, w c = α, w = w u = 1 2 (1 w c). (15) Proof. From the first an secon conitions in (7) an (9), we can infer that symmetric strike choice leas to symmetric portfolio weights w = w u = 1 w c. Plugging in the symmetric weight conition into the thir conition in (13), we can solve for the center strike weight as in (15). The isosceles triangle has its peak at maturity T c an its base at maturity T o. Depening on the ranking of the three maturities (T c,t o,t), the triangle can be forme in a number of ways. For practical consierations, we focus on the cases in which the maturity of the target option T is longer than the maturities (T o,t c ) of the heging options in the triangle. With this constraint, the triangle can be forme with either (i) T c < T o < T an thus α >, where the triangle points to the shorter maturity, or (ii) T o < T c < T an thus 1 < α <, where the triangle points to the longer maturity. As the relative maturity istance α takes on ifferent ranges of values in the two cases, the portfolio weights also show ifferent behaviors as a function of the stanarize strike spacing. With positive α in the first case, the portfolio weight on the center strike (w c ) increases monotonically with the strike spacing from 1/α at = to 1 as approaches infinity. When the 1

13 three maturities are equally space an hence α = 1, the portfolio weight on the center strike varies from 1% to 1% as the strike spacing increases from zero to infinity. In the secon case in which the center strike maturity is longer than the maturity of the outer strikes (T o < T c < T ) an thus the relative maturity istance α becomes negative, the portfolio weight w c has a singularity at 2 + α =. The center strike weight w c approaches positive infinity as 2 α an negative infinity as 2 α. In both cases, as long as the strikes are space one stanar eviation away ( > 1), the portfolio weights on all three points of the triangle are positive, an the weight on the center strike increases with increasing strike spacing for the two outsie strikes. Figure 1 plots the center strike weight w c as a function of the stanarize strike spacing in both cases with the assumption of equal spacing between the three maturities, an thus α = 1 for the first case an α = 1/2 for the secon case. The soli line shows the monotonic an slow increase of the center strike weight from 1% to 1% as a function of the strike spacing. The ashe line reveals the singularity at = 1/2. When > 1, both cases generate positive weights on the center strike an the two outsie strikes. [Figure 1 about here.] A egenerating line of three strikes at one maturity The hege remains well-efine when the maturity-strike triangle egenerates into a line of three strikes as the two maturities shrink to one T o = T c. In this case, we label the maturity of the options in the heging portfolio as T h. With symmetric strike placement, the portfolio weight on the center strike option increases with strike spacing. Proposition 3 When the symmetric maturity-strike triangle egenerates into a line of three strikes symmetrically place aroun the target strike, the hege portfolio weights are reuce to be a function of the 11

14 stanarize strike spacing only, w c = 1 1 2, w = w u = 1 2 (1 w c). (16) When we approximate the target option with three strikes at one maturity T h, the approximation is analogous to a trinomial tree, an the weight on the center strike increases with the strike spacing. When the outer strikes are about one stanar eviation away from the center = 1, the center weight is zero an the trinomial tree egenerates into a binomial tree. When the strikes are space more than one stanar eviation away, the weights on all three strikes become positive. For example, at two stanar eviation strike spacing = 2, the center strike takes a weight of 3 4, an the weights on the two outer strikes are 1 8 each. The three strikes take on equal weight of 1 3 each when the strike spacing is = 3/2. Uner a one-factor Markovian setting, Carr an Wu (23) (henceforth CW) erives a static heging strategy for a vanilla option C(K,T) using a continuum of options at at a shorter maturity T h < T. Different from our approximations base on Taylor series expansions, the CW static hege is an exact relation if (i) the unerlying security price ynamics is known, (ii) the security price ynamics is one-factor Markovian, an (iii) a continuum of options are available at a shorter maturity to form the heging portfolio. However, none of the three conitions are likely to hol in reality. Investors o not know the true unerlying price ynamics. The ynamics are unlikely to be one-factor Markovian because stochastic volatilities for most securities, with inepenent variations, are well-ocumente. Finally, option contracts are available only at a finite number of strikes. Furthermore, to minimize transaction costs, one can only use a small number of options to form the hege portfolio. CW propose a iscrete-strike implementation proceure in which the strikes an portfolio weights are chosen base on a Gauss-Hermite quarature approximation of the integral in the theoretical relation. In particular, given the quarature points an weights (x j,ω j ) an with zero rates, the strikes an portfolio weights are given as, K j = Ke 2x j σ (T T h ) 1 2 σ2 (T T h ), w j = ω j / π, (17) 12

15 where σ enotes a volatility estimate for the unerlying security return. If we ignore the convexity term an the ifference between percentage returns an log returns, our stanarize strike spacing measure j relates approximately to the quarature point by j 2x j. In the three-strike case, the quarature points are given as (,± 3/2), corresponing to a stanarize strike spacing of 3. The quarature weight for the center point is 2 3 π, corresponing to a portfolio weight for the center strike of 2/3, exactly the same as implie by equation (16) in our Proposition 3, w c = 1 1/ 2 = 2/3. Therefore, the CW three-strike iscrete implementation coincies with a very special example of our egenerate case of a line of three strikes, with the strike spacing being pre-set accoring to the quarature rule. Our approach is much more general. It allows the allocation of the three strikes at two arbitrary maturities; the strike spacing is not pre-etermine, but can be chosen with flexibility to balance contract availability, transaction cost, an heging performance; an finally, the hege portfolio formulation is inepenent of any ynamics assumptions A egenerating line of two maturities at one strike When all three strikes in the hege portfolio coincie with the target strike K u = K = K c = K, the maturitystrike triangle further egenerates into a line of two contracts at two maturities. If we retain the notation of T c an T o, with no particular ranking, the portfolio weights are etermine purely by the relative maturity istance α. Proposition 4 When the symmetric maturity-strike triangle egenerates into a line of two option contracts at two maturities (T c,t o ) an the same strike K, the hege portfolio weights are reuce to a function of the relative maturity spacing α only, w c = 1 α, w o = 1+ 1 α. (18) When the target option maturity T is either longer or shorter than both maturities in the hege portfolio, the portfolio always contains a short position in the shorter maturity an a long, levere position in the longer maturity. On the other han, if the target option maturity is sanwiche by the two maturities in the hege portfolio, the portfolio weights are positive for both options. 13

16 While our focus is on the maturity-strike triangle, the two egenerate lines illustrate the generality of our proposal as it inclues the CW static heging as a very special case, an it allows investors to trae both the implie volatility smile an the term structure, either together or separately, while managing their risk exposures From expansion errors to heging errors Our heging portfolio weights are erive by matching the corresponing terms in the Taylor expansions of the target option an the hege portfolio. The expansion error on each option contract increases with the istance between the contract s strike an maturity an the reference strike an maturity expansion point. In practice, strikes are often available at a fine gri, but maturities ten to be more sparse. Thus, the expansion errors along the maturity imension can be large. However, we show in this section that the expansion error of the hege portfolio can be much smaller than the average expansion error of the iniviual contracts ue to cancelation. We illustrate this point through two angles. First, we show that although the expansion error on each option contract epens on the reference point aroun which the expansion is performe, the portfolio weights for the hege triangle o not explicitly epen on the particular choice of the expansion reference point. Secon, we perform the Taylor expansion to a higher orer an show how the leaing-term expansion error on the target options cancels with the leaing-term expansion error on the hege portfolio. We further show how one can maximize the cancelation via appropriate choice of the strike spacing in the hege portfolio Inepenence of portfolio weights on the expansion reference point In eriving our portfolio weights, we expan both the target option an the hege options aroun a common strike an maturity reference point (K,T o ). This reference point is a convenient choice because with this reference point, we only nee to perform maturity expansion for the target option an the hege option at the center strike when K c = K, an we only nee to perform strike expansion on the two hege options at the outsie strikes. Choosing other expansion points woul lea to more terms. However, the following 14

17 proposition shows that the particular choice of the reference point is not important for computing the hege portfolio weights. Proposition 5 When the local volatility is flat across strikes an maturities, the portfolio weights o not epen on the reference maturity an strike point, aroun which the Taylor expansion is performe. Proof. Let (K m,t m ) be an arbitrary strike-maturity reference point, with which we perform the Taylor expansion on the target an hege options: C(K,T) C +C K (K K m )+C T (T T m )+ 1 2 C KK (K K m ) 2, C(K u,t o ) C +C K (K u K m )+C T (T o T m )+ 1 2 C KK (K o K m ) 2, C(K u,t o ) C +C K (K K m )+C T (T o T m )+ 1 2 C KK (K o K m ) 2, C(K c,t c ) C +C K (K c K m )+C T (T c T m )+ 1 2 C KK (K c K m ) 2, where the term C, C K, C T, an C KK are all evaluate at the reference point (K m,t m ) an we hie the epenence to reuce notation clustering. Matching the option level C term, we have 1 = w u + w + w c as before. Matching the C K term, we have (K K m ) = w u (K u K m )+w (K K m )+w c (K c K m ), (19) which in combination with the first conition leas to, K = w u K u + w K + w c K c. (2) Thus, neither the first nor the secon conition epens on the reference point choice (K m,t m ). Matching the C T an C KK term, we have C T (T T m )+ 1 2 C KK (K K m ) 2 = w j C T (T j T m )+ 1 j 2 C KK (K j K m ) 2, (21) 15

18 with j = u,,c. We see that the T m terms cancel out. Furthermore, if we write K j K m = (K j K)+ (K K m ) an expan the (K j K m ) 2 terms, the conition in (21) simplifies to C T T = w j C T T j + 1 j 2 C KK (K j K) 2, (22) which oes not has any explicit epenence on the reference point (K m,t m ). Therefore, the portfolio weights o not have explicit epenence on the reference point for the Taylor expansion. An implicit epenence arise when we convert C KK into C T via the local volatility efinition. Since the local volatility is evaluate at the reference point (K m,t m ), portfolio weights epen on the reference point to the extent that the local volatility is strike an maturity epenent. When the local volatility surface is flat, the portfolio weights are completely inepenent of the reference point choice. The expansion error on each option contract epens obviously on the reference point. The closer the reference point is to the strike an maturity of the contract, the smaller the expansion error is. Yet, the above proposition shows that the hege portfolio weights are quite robust with respect to the reference point choice. In particular, with a flat local volatility surface, the portfolio weights an hence the heging errors are inepenent of the reference point that we choose for the expansion Leaing-term expansion errors in target an hege options To analyze how the expansion errors cancel between target an hege options, we expan each option contract to a higher orer an analyze the behavior of the leaing-term expansion error. To reuce notation clustering, we use (K,T o ) as the reference point for the expansion, we hie the explicit epenence on the reference point in the notation, an we focus on the symmetric maturity-strike triangle for the analysis, with 16

19 K = K u K = K K an T = T T o. The expansions become, C(K,T) C +C T T C TT ( T) 2, (23) C(K u,t o ) C +C K K C KK ( K) C KKK ( K) C KKKK ( K) 4, (24) C(K,T o ) C C K K C KK ( K) C KKK ( K) C KKKK ( K) 4, (25) C(K,T c ) C C T α T C TT α 2 ( T) 2. (26) From the above expansions, we can see that when K =, the leaing-term expansion error in the target option is 1 2 C TT ( T) 2, which partially cancels with the leaing-term expansion error in the hege portfolio, w c α C TT ( T) 2, when the portfolio weight on the center strike is positive. When K >, aitional expansion errors are introuce in the hege portfolio in terms of the C KKKK term. These aitional expansion errors can be use to further cancel out the errors on the C TT terms. To link these higher-orer terms, we further ifferentiate the forwar PDE with respective to T, C TT = 1 2 σ2 K 2 (C T ) KK σ2 T K2 C KK, = 1 4 σ4 K 4 C KKKK + ( 1 2 σ2 σ 2 K K4 + σ 4 K 3) C KKK + ( 1 2 ( σ 4 + σ 2 T) K 2 + σ 2 σ 2 K K σ2 σ KK K 4) C KK, (27) where σ 2 T, σ2 K, an σ2 KK enote the partial erivatives of the local variance σ2, which are all zero in the case of a flat local volatility surface. To remove the C KKK term in equation (27), we assume that S = K an link C KKK to C KK accoring to the BMS moel, C KKK 3 2 C KK K. Then, we have C TT a 2 C KK σ4 K 4 C KKKK, (28) with a 2 = ( 1 2 σ2 T σ4) K σ2 σ 2 K K σ2 σ 2 KK K4. Now, we can use the forwar PDE an equation (28) to convert the C T an C T T terms in the expansions 17

20 (23) to (26) to C KK an C KKKK terms, C(K,T) C σ2 K 2 C KK T a 2C KK σ4 K 4 C KKKK ( T) 2, (29) C(K u,t o ) C +C K K C KK ( K) C KKK ( K) C KKKK ( K) 4, (3) C(K,T o ) C C K K C KK ( K) C KKK ( K) C KKKK ( K) 4, (31) C(K,T c ) C α 1 2 σ2 K 2 C KK ( T)+ 1 2 a 2C KK + α σ4 K 4 C KKKK ( T) 2. (32) The terms on C an C K remain the same as before, from which we obtain w u + w + w c = 1 an w u = w = (1 w c )/2. Matching the C KK terms, we have, 1+ a 2 T σ 2 K 2 = (1 w c) 2 αw c ( 1 α a 2 σ 2 K 2 T ), from which we can solve for w c, w c = 2 1+h 2 + α+σ 2 h, (33) where h = a ( 2 σ 2 K 2 T = σ 4 1 σ 2 T 2 σ σ2 K K 1 ) 4 σ2 KK K2 T, (34) which is a function of the local volatility level, its slope along the term structure an strike imension, its curvature along the strike imension, an the maturity-istance between the target option an the hege options for the two outsie strikes. Thus, by matching higher-orer terms, the portfolio weights are moifie by the higher-orer term h. When the local volatility is flat across strike an maturity, h = σ 4 T becomes a very small term an can be safely ignore. When the local volatility surface is heavily skewe across strike or is having a steep term structure, the ajustment can become significant. Finally, if we are free to choose the strike spacing, we can also match the higher-orer term C KKKK term by setting, w c = α2. (35) 18

21 Combining (33) an (35), we can solve the stanarize strike spacing that matches the higher-orer term, 2 = ( ) 3 1 α+ (1 α) (1 h(1 α)) α(1 h(1 α)), (36) which is a function of the relative maturity spacing α an the term h, which is proportional to the maturity spacing T. When we ignore the higher-orer term h an set α = an hence all three strikes in the hege portfolio fall on the same maturity, we have 2 = 3, the same as the result from the Hermite-Gauss quarature approximation in Carr an Wu (22). Therefore, in the sense of matching leaing-term expansion errors, the quarature strike choice is optimal. On the other han, when the three maturities are equally space T T o = T o T c an hence α = 1, we have 2 = 3. It is important to realize that the optimal strike spacing in (36) is erive uner strong assumptions to remove the C KKK term, an thus shall not be taken literally. Nevertheless, the erivation shows the potential of further reucing the heging error by appropriate strike spacing choice. To show the prospect of the expansion error cancelation, Figure 2 plots the leaing-term expansion error of the hege portfolio as a function of the stanarize strike spacing measure. The plots are compute from the Black-Scholes moel with σ = 5, zero rates, an with the maturity choices (T c,t o,t ) being one, two, an six months, respectively. The three lines represent three ifferent target option strikes at K = $9 (ashe line), $1 (soli line), an $11 (ash-otte line), relative to a normalize spot price level of $1. The plots highlight the prospect of choosing strike spacing juiciously to eliminate the leaing-term expansion errors of the hege portfolio. [Figure 2 about here.] 19

22 3. Numerical Experiments Base on Commonly Specifie Dynamics We gauge the performance of our propose heging strategy uner several commonly specifie security price ynamics. First, we analyze how the strike spacing choice affects the heging performance uner each strategy. Then, we compare the heging performance of the ifferent strategies with one another an with aily elta heging with the unerlying futures Data-generating processes We consier four ata generating processes: the Black-Scholes moel (BS), the Merton (1976) jumpiffusion moel (MJ), the Heston (1993) stochastic volatility moel (HV), an the jump-iffusion stochastic volatility moel of Huang an Wu (24) (HW). The time-series stock price ynamics are governe by the following stochastic ifferential equations, BS: S t /S t = µt+ σw t, MJ: S t /S t = µt+ σw t + R (ex 1)(ν(x,t) λn(x)xt), n(x) = 1 exp ( (x µ j) 2 2πv j 2v j ), HV: S t /S t = µt+ v t W t, HW: S t /S t = µt+ v t W t + R (ν(x,t) v tλ n(x)xt), v t = κ(θ v t )t ω v t Z t, E[Z t W t ] = ρt, (37) where W t enotes a stanar Brownian motion in all four moels. The MJ moel also incorporates a compoun Poisson jump component, where we use ν(x,t) to enote the counting measure for the jumps, R to enote the real line excluing zero, an λn(x)xt to be the compensator, with λ measuring the mean jump intensity or arrival rate, n(x) enotes a normal probability ensity function capturing the jump size istribution in log return conitional on a jump occurring. Uner the Heston (HV) moel, Z t enotes another stanar Brownian motion that governs the ranomness of the instantaneous variance rate. The two Brownian motions have an instantaneous correlation of ρ. The HW moel combines HV with MJ an allows the jump arrival rate to be proportional to the instantaneous variance rate, λ t = λ v t. The HW moel is labele as MJDSV3 in Huang an Wu (24), who show that the moel performs better in pricing S&P 2

23 5 inex options than oes a similar moel with constant jump arrival rate propose by Bates (1996) an Bakshi, Cao, an Chen (1997). The four processes are carefully chosen for our analysis. The BS an MJ moels serve as static pure iffusion an jump-iffusion benchmarks, respectively, whereas the HV an HW moels allow stochastic volatility for the two benchmarks. Option prices uner the BS moel can be reaily compute using the analytical Black-Scholes option pricing formula. Uner the MJ moel, option prices can be compute as a Poisson-probability weighte sum of the Black-Scholes formulae. For HV an HW, option prices are compute numerically through fast Fourier inversion of the analytical return characteristic function. To simulate the ata-generating processes an price options on each simulate path, we nee to choose appropriate values for the moel parameters. To make the analysis comparable to our historical analysis on the S&P 5 inex (SPX) options in the next section, we set the parameter values to those calibrate to the SPX options market. Specifically, we perform aily calibration of the HV moel an the HW moel on SPX options from January 1996 to March 29, an use the sample averages of the aily parameter estimates for the simulation analysis. The parameters for the BS moel an the MJ moel are aopte irectly from the corresponing parameters from the HV an HW moels, respectively, with the constant volatility level set to its long-run mean estimate. Table 1 reports the parameter values use in our analysis. Estimating the HV moel generates an average long-run mean volatility of θ = 22.77%, an average instantaneous volatility rate level of v t = 18.64%. The ifference between the two implies an average upwar sloping implie volatility term structure. The average mean-reversion coefficient is at κ = , corresponing roughly to quarterly frequency (1/κ). The average volatility of volatility coefficient estimate is quite large at ω =.995, which contributes to the curvature of the implie volatility smile. Finally, the average correlation between return an return variance is strongly negative at ρ =.6824, consistent with the strongly negatively skew observe in the implie volatility smile on SPX options. By aing a jump component in the HW moel, the average long-run mean volatility of the iffusion component becomes lower at θ = 18.69% because the jump component also contributes to the total volatility level, which is at θ(1+λ (µ 2 j + σ2 j )) = 22.44%, very close to the HV estimate. The average 21

24 jump frequency is λ θ = 995, about one jump every two years. Conitional on a jump occurring, the average jump size in return is µ j = 11%, with a stanar eviation of σ j = 14.32%. The large negative jump size contributes to short-term implie volatility skews in the SPX options, an the jump size uncertainty (σ j ) as curvature to the skew. With the jump component, both the mean-reversion coefficient an the volatility of volatility coefficient average lower at κ = an ω =.3811, respectively. The return-volatility correlation remains strongly negative at ρ = The aily calibration on SPX options generates parameter estimates uner the risk-neutral measure. To obtain the corresponing values for the statistical process, we assume zero risk premiums by setting µ = r q, an use the same set of parameters both for simulating the sample paths an for option pricing. During this sample perio, the S&P 5 inex starte at 617.7, went over 15 in year 2 an 27, but ene the sample at The average ex-ivien return on the inex over the sample perio is 2.17%. The interest rates (r) an ivien yiels (q) unerlying the option contracts average at 4.17% an 2.58%, which we use as constants for the simulation an option pricing Monte Carlo proceures In each simulation, we generate a time series of aily unerlying security prices accoring to an Euler approximation of the respective ata generating process. The starting value for the stock price is normalize to $1, an the starting values of the instantaneous variance rates for the HV an HW moels are also fixe to the average values in Table 1. We consier a heging horizon of one month an simulate paths over this perio. We assume that there are 21 business ays in a month. To be consistent with the historical analysis in the next section, we think of the simulation as starting on a Wenesay an ening on a Thursay four weeks later, spanning a total of 21 week ays an 29 actual ays. The security price moves accoring to the ata-generating processes in equation (37) only on week ays. Figure 3 plots the 1, simulate sample paths for the security price uner each of the four moel environments. The pure iffusive moels BS an HV generate mostly small price movements, whereas large iscontinuous movements are apparent uner the MJ an HW moel environments. 22

25 [Figure 3 about here.] The HV an HW moels also generate stochastic volatility. We plot the corresponing simulate sample paths for the instantaneous return volatility, v t, in Figure 4. Given the large volatility of volatility coefficient uner the HV moel, several volatility sample paths hit the lower boun of zero. By incorporating jumps in the security price ynamics, the estimate stochastic volatility ynamics uner HW moel look more well-behave. [Figure 4 about here.] At each week ay, we compute the relevant option prices base on the realizations of the security price an the instantaneous variance rate, as well as the moel ynamics. We monitor the heging error (profit an loss) at each week ay base on the simulate security price an the option prices. The heging error at each ate t, e t, is efine as the ifference between the value of the hege portfolio an the value of the target call option being hege, e t = 3 w j C t (K j,t j ) C t (K,T). (38) j=1 Since the portfolio is erive using Taylor expansion, the initial values of the heging portfolio an the target option may not be exactly the same. We remove this initial value mismatch through a proportional scaling of the three portfolio weights. We hol this portfolio statically for one month an investigate the heging error uring the process an, in particular, at the en of one month. Our portfolio weights are stable over time as they epen mainly on the structural features of the option contracts such as the strike price an the relative expiration istance α. To the extent that the local volatility estimates vary over time, the stanarization () of the strike spacing varies accoringly an so shoul be the portfolio weights. Nevertheless, we regar these variations as small an hol the portfolio weights fixe for the whole month while investigating its heging performance. We assume that option contracts are available at a finite number of strikes an maturities. That is, these 23

26 contracts can be trae at observable market prices. We exclue bi-ask spreas from our analysis. The target option choice an heging portfolio formulation are all from this pool of available option contracts. To compute the portfolio weights, we estimate the local volatility by interpolating the implie volatility surface constructe from the finite number of option observations. At the start of each simulation, we assume that options are available at maturities of one, two, three, six, an 12 months, an that option strikes are centere aroun the normalize spot price of $1, an space at intervals of $1, $1.5, $2, $2.5, an $3 for the five maturities, respectively. The assume strike spacing increase with maturities match the behavior of SPX options market, where the strike spacing averages from $1 to $3 on an unerlying inex level of about $1,. We set the target option strike at the center K = 1, an consier three types of maturity-strike placements for the hege portfolio: (A) Symmetric maturity-strike triangles pointing to short maturity, with T c < T o < T, (B) symmetric maturity-strike triangles pointing to long maturity, with T o < T c < T, an (C) a line of three strikes at the same maturity (T h < T ). Figure 5 plots schematically the maturity-strike placement for each type. Within each type, we form ten istinct target-hege maturity combinations out of the five available maturities. For each maturity combination, we also have many flexible choices on the strike spacing. Through this extensive simulation exercise, we strive to gain a better unerstaning on the epenence of the heging performance on maturity-strike placement patterns, target-hege maturity istances, an strike spacing. [Figure 5 about here.] 3.3. Optimal strike spacing choice For each maturity combination uner each of the three maturity-strike placement types (A), (B), an (C), we analyze the effect of strike spacing on the heging performance, through which we etermine the optimal strike spacing choice. Given the potential instability of the portfolio weight when < 1, we start at an outer strike choice close to = 1 an then progressively move to the next available strike further away from K. We 24