A Cautious Note on the Design of Volatility Derivatives

Transcription

1 A Cautious Note on the Design of Volatility Derivatives arxiv: v1 [q-fin.pr] 18 Jul 2010 Abstract Eckhard Platen and Leunglung Chan July 20, 2010 This cautious note aims to point at the potential risks for the financial system caused by various increasingly popular volatility derivatives including variance swaps on futures of equity indices. It investigates the pricing of variance swaps under the 3/2 volatility model. Carr with Itkin and Sun have discussed the pricing of variance swaps under this type of model. This paper studies a special case of this model and observes an explosion of prices for squared volatility and variance swaps. It argues that such a price explosion may have deeper economic reasons, which should be taken into account when designing volatility derivatives Mathematics Subject Classification: Primary 62P05; secondary 60G35, 62P20 JEL Classification: G10, G13 Key words: 3/2 volatility model, variance swap, numéraire portfolio, squared Bessel process 1 Introduction A significant source of risk in a financial market is the uncertainty of volatility. During the recent financial turmoil, volatility risk was of extreme importance to investors and derivative traders. Additionally, due to the large and frequent shifts in volatility of various assets in the recent past, there is a growing practical need to develop appropriate models for valuing volatility based financial instruments and hedging volatility risk. In 1993, the School of Finance and Economics and Department of Mathematical Sciences, University of Technology, Sydney, PO Box 123, Broadway, NSW 2007, Australia, Eckhard.Platen@uts.edu.au School of Finance and Economics, University of Technology, Sydney, Australia; E- mail: Leunglung.Chan@uts.edu.au

2 E. Platen, L. Chan 2 Chicago Board Options Exchange (CBOE) introduced a volatility index, the VIX, see Figure 1, based on the implied volatilities of options on the S&P 500 index, see Figure 2. The volatility of this diversified index has attracted the most attention in the current literature on volatility derivatives. As such, we take as the security of interest a similarly diversified equity index and study its volatility and related derivatives. Some of the pioneering papers related to volatility indices is Whaley (1993) Figure 1: VIX from 1990 to Figure 2: S&P 500 from 1990 to Variance and volatility swaps have been traded in over-the-counter markets since the collapse of Long Term Capital Management in late In particular, variance swaps on stock indices have been quoted and traded actively, and have been used as a hedge for volatility risk. Investors and fund managers alike, have developed an interest in volatility derivatives since these instruments may substantially increase the value of their holdings, even if the equity market experiences a major crash. At least theoretically, these derivatives can provide some protection against severe market downturns. How effective such portfolio insurance is, from a macro-economic view point, remains an open question. Particularly, when a large and increasing number of pension funds, insurance companies and other investors continue to request this type of insurance.

3 E. Platen, L. Chan 3 There exists substantial literature on volatility modeling, and one may refer to Cont and Tankov (2004) as one of the many references. Several papers, which explore stochastic volatility modeling, have pointed at the seemingly undesirable property of some squared volatility models, where their moments of higher order than one may become infinite in finite time. Examples are given in Andersen and Piterbarg (2007), Lion and Musiela (2007) and Glasserman and Kim (2010). Furthermore, there exist various papers discussing the problem of pricing and hedging variance swaps, including Brenner et al. (1993), Grünbuchler and Longstaff (1996), Carr and Madan (1999), Chriss and Morokoff (1999), Demeterfi et al. (1999), Brockhaus and Long (2000), Matytsin (2000), Javaheri et al. (2002), Swishchuk (2004), Howison et al. (2004), Carr et al. (2005), Windcliff et al. (2006), Zhang and Zhu (2006), Zhu and Zhang (2007), Sepp (2007) and Elliott, Siu and Chan (2007). Recently, CarrandSun(2007),aswellas,ItkinandCarr(2009)discussedin two interesting papers the pricing of variance swaps under the well-known 3/2 volatility model. Versions of 3/2 volatility models were studied, for instance, in Cox et al. (1980, 1985), Platen (1997), Ahn and Gao (1999), Heston (1999), Lewis (2000), Andreasen (2001), Platen (2001) and Spencer (2003). The current paper studies a special case of the 3/2 volatility model, and shows that the price of a variance swap explodes under this realistic model. This raises important questions on the design of variance swaps and other volatility index derivatives. The total face value of these derivatives can easily reach, in a few years, values on the macro-economic scale. This paper emphasizes that the consequences of such massive exposure of the economy to this strongly nonlinear type of leveraged position has to be prudently analyzed. It argues that the infinite price obtained for squared volatility as payoff, may have deep economic reasons. More precisely, in the case of a major market downturn there may not be enough liquid capital available in the financial system to deliver the payoffs demanded by the holders of squared volatility contracts and variance swaps on equity indices. Regulators and politicians have to be made aware of this looming problem within the financial system. For instance, in the case of a similarly severe market downturn, as experienced in the recent financial crisis, issuers of contracts paying uncapped squared volatility, as variance swaps on equity indices, could loose significantly more capital than liquidly available to them. Bailouts of financial institutions of similar magnitudes to those recently experienced may again be necessary to save or restore the functionality of the financial system. As will be proposed in this paper, the introduction of sufficiently high knock-out barriers would make, from a macro-economic perspective, derivative contracts more viable instruments and could prevent another looming financial crisis. The paper is organized as follows: Section 2 introduces variance swaps and

4 E. Platen, L. Chan 4 provides some remarks on the theoretical underpinning of the popular logcontract hedging. Section 3 studies the risk neutral 3/2 volatility model. Section 4 considers the real world 3/2 volatility model. Section 5 introduces real world pricing. The pricing of squared volatility as payoff is discussed in Section 6. Finally, Section 7 concludes. 2 Variance Swaps A variance swap is a forward contract on annualized variance, which is the square of the realized annualized volatility. Let σ0,t 2 denote the realized annualized variance of the log-returns of a diversified equity index or related futures over the life of the contract, such that σ 2 0,T := 1 T T 0 σ 2 udu. (2.1) Assume that one can trade the futures or index price at discrete times t i = i for i {0,1,...} with time step size > 0. The period between two successive potential trading times is typically the length of one day. If F ti denotes the equity index price at time t i, then the return G ti for the period before this time is defined as G ti = F t i F ti 1 F ti 1 (2.2) for i {1, 2,...}. In practice, variance swaps are often written on the realized variance evaluated from daily closing prices of the form ˆσ 2 0,t n := N n n i=1 ( G ti ) 2 N n n i=1 { ( )} 2 Fti ln := σ 0,t 2 F n, (2.3) ti 1 where T = t n = n i=1 (t i t i 1 ) = n, and N is the number of trading days N per year. Hence, variance swaps with idealized payoffs, depending on the realized variance, as defined in (2.1), are only approximations to those of the actual contracts. As indicated in (2.3), returns or log-returns may both be used to define payoffs on realized variance. Furthermore, by (2.1) and (2.3) it becomes clear that the basic elements of these types of payoffs are some form of squared volatility to be paid at a given time. This is why the paper focuses on the pricing of a payoff that delivers squared volatility. As is common in most of the literature, let (Ω,A T,A,Q) denote some assumed risk neutral, filtered probability space. Here Q is denoting the assumed risk neutral probability measure. The filtration A = (A t ) t [0,T] models the evolution of the market information over time, where A t describes the information available at time t [0,T]. Let K v denote the delivery price for realized variance and L the notional amount of the swap

5 E. Platen, L. Chan 5 in dollars per annualized variance point. Then, the payoff of the variance swap at expiration time t n is given by L( σ 2 0,t n K v ). Intuitively, the buyer of the variance swap will receive L dollars for each point by which the realized annual variance σ 2 0,t n has exceeded K v. Under the risk neutral approach, the value of the variance swap can be evaluated as the expectation of its discounted payoff with respect to the assumed risk neutral measure Q. This value is equivalent to the value of a forward contract on future realized variance with strike price K v. For simplicity, assume throughout the paper that the interest rate is zero. The risk neutral value of the variance swap ˆV at time t = 0 over the period [0,t n ] is then given by the expression ˆV = E Q [L( σ 2 0,t n K v )] = LE Q ( σ 2 0,t n ) LK v, (2.4) where E Q denotes expectation under the assumed risk neutral measure Q. Hence, the valuation of the variance swap relies on calculating the risk neutral expectation of the realized variance E Q ( σ 2 0,t n ). Note by (2.3) and (2.1) that this involves the computation of a sum of risk neutral expectations of squared volatility. Therefore, the problem can be reduced to evaluating the risk neutral expectation of squared volatility, as was mentioned earlier in the introduction. To highlight some of the critical issues with the current methodology, recall the popular log-contract hedging approach to realized annualized variance, for instance described in Neuberger (1994), Dupire (1993), Demeterfi et al. (1999) and Carr and Lee (2009). In the above literature the payoff σ 2 0,t n appearing in (2.3) is usually approximated by the expansion σ 2 0,t n = + n i=1 2N n ( 1 1 ) [ Ft0 (F ti F ti 1 )+ F ti 1 F t0 0 ] n n F t0 2N nk 2(F t n K) + dk N 3n i=1 G 3 t i + i=1 2N nk 2(K F t n ) + dk O(G 4 t i ), (2.5) wherethereturng ti << 1isassumedtobe small. HereO(G q t i )estimates a term that is smaller in absolute value than some constant multiplied by G ti q when G ti asymptotically vanishes. The first two terms on the righthand side of (2.5) are the profit and losses from a dynamic position in the equity index and a static position ( in options ) on the index. For the dynamic component, one holds 2N 1 n F ti 1 1 F t0 futures contracts on the equity index from day t i 1 to day t i. For the static component, one holds 2N dk European puts having a strike K dk below the initial index nk 2 value F t0. One also holds 2N dk European calls having a strike K dk nk 2 greater than the initial index value F t0. As such, one ends up holding more puts than calls. In the literature it is argued that the third and fourth

6 E. Platen, L. Chan 6 terms on the right hand side of the expression (2.5) represent the most significant source of error when approximating the payoff σ 2 0,t n. Note that the fourth power of the return appears in the remainder term. This paper emphasizes that the expansion (2.5) is of a pathwise nature and relies on the assumption that G ti is sufficiently small for all scenarios. This is a rather delicate assumption, as will be discussed below. It was shown in Platen and Rendek (2008), that the Student-t distribution with approximately four degrees of freedom is the typical estimated log-return distribution of diversified world stock indices when such indices are denominated in different currencies. Similar empirical evidence is provided in Markowitz and Usmen (1996a, 1996b). It is well-known that the fourth moment of the Student-t distribution with four degrees of freedom is infinite. In view of the expansion (2.5) this raises the important question whether the assumption of a vanishing remainder term in (2.5) is sensible in reality. Additionally, it will turn out that the real world 3/2 volatility model generates Student-t distributed log-returns for the equity index with four degrees of freedom. It is a fact that extreme log-returns of diversified equity indices occur in reality and under the 3/2 volatility model with a real world probability that is not negligible. One may question whether it is prudent to assume in (2.5) that G ti is always sufficiently small. More precisely, when pricing, one has to deal with expectations. Hence, the final termof(2.5)createsaprobleminthisrespect asitwillbeinfinitewhen taking expectation under realistic model assumptions that generate Student-t distributed log-returns with four degrees of freedom. Even when the degrees of freedom are slightly higher than four, thus generating theoretically a finite expectation, the corresponding expected value of annualized variance is still enormous. Therefore, extreme caution should be taken when pricing and hedging volatility derivatives using the type of expansion given in (2.5). To visualize this problem, Figure 3 exhibits the cumulative fourth term on the right hand side of (2.5), using S&P 500 daily log-return data, from 1990 until For instance, during the period September to October 2008 the term G 4 t i increased by approximately 300% in just a few days. This demonstrates, at least visually, that the fourth term in (2.5) cannot be simply neglected as a higher order remainder term. Some severe approximation error has to be expected when following the methodology described in (2.5).

7 E. Platen, L. Chan x Figure 3: The cumulative fourth term, n i=1 O(G4 t i ), using daily S&P 500 data from 1990 to Risk Neutral 3/2 Volatility Model The authors of the two interesting papers: Carr and Sun (2007) and Itkin and Carr (2009), derived under some assumed risk neutral probability measure a 3/2 volatility model as the generic stochastic volatility model that naturally emerges under two plausible assumptions. These assumptions are the, so called, stationary volatility ratio hypothesis and the maturity independent diffusion hypothesis. The first assumes that the ratio of the volatility of the variance swap rate to the instantaneous volatility of the underlying asset only depends on the variance swap rate and maturity, but does not depend on calendar time. The second hypothesis requires that the risk-neutral process for the instantaneous variance is a diffusion process whose coefficients are independent of the variance swap maturity date. For a detailed description of these assumptions and the derivation of the risk neutral 3/2 volatility model, the reader is referred to Carr and Sun (2007). This model, which will be described in detail further below, reflects well empirical evidence for short term volatility derivatives and realized volatility. Some of this evidence is provided in Poteshman (1998), Chacko and Viceira (1999), Ishida and Engle (2002), Jones (2003), Javaheri (2004), Bakshi et al. (2004) and Platen and Rendek (2008). Because of the strong empirical support that the 3/2 volatility model attracts this paper places it in a broader setting, and discusses it from a more general perspective. In particular, it studies its dynamics under the real world probability measure P. More precisely, it links it to the Minimal Market Model (MMM), described in Platen (2001) and Platen and Heath (2006). The MMM, being derived theoretically from macro-economic considerations, represents a 3/2 volatility model of the above type when formulated formally under some assumed risk neutral measure, as will be shown below.

8 E. Platen, L. Chan 8 As before, denote by F t the price at time t [0,T],T <, of the underlying asset, which one can interpret as a diversified equity index, for instance, the S&P 500 total return index. The instantaneous variance of this index shall be denoted by v t, which represents its squared volatility. To remain close to the work by Carr and his co-authors this section models the stochastic processes under some assumed risk neutral probability measure Q. It assumes, as in Carr and Sun (2007), that the equity index F t or its futures satisfies the following stochastic differential equation (SDE): df t = F t vt d Z t, (3.1) and its squared volatility v t the SDE: ( ) dv t = p(t)v t +qvt 2 dt+ǫv 3/2 t d W t, (3.2) t [0,T]. Here p(t), t [0,T], and q are assumed to be given real valued, deterministic quantities. Furthermore, Z = { Zt,t [0,T]} and W = { W t,t [0,T]} represent dependent Q-standard Wiener processes. The dependence between Z and W is modeled by the covariation [ Z, W] t = ρt, (3.3) for t [0,T], with given constant correlation parameter ρ [ 1,1]. One notes the power 3/2 for the squared volatility v t in the diffusion coefficient of the SDE (3.2) from which the model derives its name. It is well-known that the movements of the squared volatility of a diversified equity index are in reality strongly negatively correlated to those of the index itself. This stylized empirical fact has been known as the leverage effect, see Black (1976). Realistic calibrations identify a negative correlation parameter ρ relatively close to 1. Therefore, before going into the analysis of any multi-factor model, the authors of this paper consider it to be crucially important to clarify first the key properties of its most likely onefactor version. Therefore, the paper studies below a 3/2 volatility model where ρ = 1, and thus, a single Wiener process is driving the dynamics of the index, as well as that of its volatility. To be even more precise and avoid any ambiguity, it considers the one-factor model which follows from (3.1) by setting v t = αexp{ηt} F t. (3.4) Here α > 0 is a scaling parameter, and η > 0 will play the role of the long term average growth rate of the index F t under the real world probability measure P. This growth rate becomes important when one requests that the squared volatility process v = {v t,t 0} should have some stationary density under P.

9 E. Platen, L. Chan 9 The relation (3.4) yields, by (3.1), the SDE Ft αexp{ η df t = F t} 2 t d Z t = αexp{ η 2 t} F t d Z t (3.5) for t 0. Under the assumed risk neutral measure Q this is the SDE of a time transformed squared Bessel process of dimension zero, see Revuz and Yor (1999). One confirms by application of the Itô formula to (3.4) that by using (3.5) one obtains for v t a 3/2 volatility model with SDE ( dv t = d α exp{ηt} ) F t = αexp{ηt} F 2 t (F t vt d Z t ) + αexp{ηt} F t v t dt+η αexp{ηt} dt F t = v t ( η +v t )dt v 3 2 t d Z t (3.6) for t [0,T] with v t > 0. In particular, the squared volatility satisfies the SDE (3.2) for the parameter choice ρ = 1, p(t) = η, q = 1 and ǫ = 1. The squared volatility process characterized by the SDE (3.6) has nonlinear drift and diffusion coefficients. At first glance, this seems to make it difficult to obtain rather clear statements about its probabilistic properties. However, due to the fact that F t in (3.5) is a squared Bessel process, which is an extremely well-studied diffusion process, one knows analytically the probability law of v t and many other properties. To exploit this one may study the expression R t = α v t = F t exp{ ηt}, (3.7) as in Carr and Sun (2007), and obtains by the Itô formula the SDE dr t = α ) ] [v vt 2 t (η +v t dt v 3 2 t d Z t + α v 3 vt 3 t dt = ( αη α+α)dt+αv 1/2 t d Z t v t = ηr t dt+ αr t d Z t. (3.8) Under the assumed risk neutral measure Q, not only the squared Bessel process F but also thesquare rootprocess R = {R t,t [0, )}isofdimension δ = 0, see Revuz and Yor (1999). It is well-known that both processes hit the level zero with strictly positive Q-probability in any nonzero finite time period. In such an event the volatility explodes, see (3.7). This gives an indication that the expectation of the squared volatility v T = α R T may become infinite under Q. Indeed, the first negative risk neutral moment of the squared Bessel process R of dimension δ = 0 is infinite, see Revuz and

10 E. Platen, L. Chan 10 Yor (1999). Consequently, the risk neutral expectation of squared volatility has under the above 3/2 volatility model an infinite value, that is, E Q (v T ) =. (3.9) Thus, the typical payoffs that constitute the building blocks of a variance swap onadiversified equity indexdo notappear tohave afiniterisk neutral price under the above 3/2 volatility model. As mentioned in the introduction, other volatility models, in particular those that model the leverage effect, may create similar volatility explosions and infinite prices under an assumed risk neutral measure, see Andersen and Piterbarg(2007), Lion and Musiela (2007) and Glasserman and Kim(2010). The current paper focuses in the following on the aforementioned 3/2 volatility model to highlight the type of problems that can emerge under risk neutral volatility explosions. At this point it is appropriate to mention that the assumptions leading to the SDE (3.6) in relation to the parameter choice for the risk neutral 3/2 volatility model are slightly different from those imposed in Carr and Sun (2007) and Itkin and Carr (2009). In particular, these authors require in the SDE (3.2) that the relation q < ǫ2 holds. As a consequence, 2 the dimension δ of their square root process R under the assumed risk neutral measure Q is greater than two. Their condition aims to secure a zero risk neutral probability for the event that the volatility will explode in finite time. The current paper argues that there is no need to restrict the volatility dynamics to avoid a volatility explosion under some assumed risk neutral measure. Since the reasoning of Carr and his co-authors, who favour a 3/2 type volatility model, is rather compelling, it considers below a real world 3/2 model with the same structural properties that the risk neutral one suggests, however, adjusted to a description under the real world probability measure P. What really matters are the dynamics of the volatility process under the real world probability measure P. Under this probability measure the volatility should remain finite, as observed in reality. Still it should reach occasionally very extreme levels with nonnegligible probabilities. It has been shown in Platen and Heath (2006), and will be discussed in Section 5, that the formal risk neutral price of a nonnegative contingent claim can be significantly higher than the price to be identified in Section 5 under the real world probability measure, this price being the minimal possible price for a replicable contingent claim. This means, if there would be no volatility explosion under the real world probability measure, then there may still be a chance that the price of squared volatility, and thus that of a variance swap could remain finite for a 3/2 volatility model.

11 E. Platen, L. Chan 11 4 Real World 3/2 Volatility Model To study realistic dynamics of the underlying diversified equity index and its volatility under the real world probability measure, this paper adopts the benchmark approach described in Platen and Heath (2006). This approach is rather general and does not require the restrictive assumption on the existence of an equivalent risk neutral probability measure. It only assumes the existence of the numéraire portfolio, see Long (1990), Becherer (2001), Karatzas and Kardaras (2007) and Kardaras and Platen (2008). If there exists an equivalent risk neutral probability measure for a given model, then the benchmark approach recovers fully the results of the risk neutral approach. Otherwise, it still provides a sound pricing methodology, as will be explained in Section 5. Under the real world probability measure P one needs additionally to model the drift in the SDE for the real world dynamics of the diversified index F t. It has been shown in Platen (2005) that such a well diversified index of a large equity market is, under weak assumptions, well approximated by the numéraire portfolio. The numéraire portfolio is known to be the growth optimal portfolio of the market, which maximizes expected logarithmic utility, see Kelly (1956). This fact allows one to determine its SDE under the real world probability measure P by log-utility maximization. Assuming from now on that F t models the value of a continuous numéraire portfolio at time t, it follows that its volatility v t becomes the market price of risk, and its risk premium the squared volatility v t. Details on these statements can be found in Platen and Heath (2006). The SDE for F t under the real world probability measure P has the form: df t = F t ( v t dt+ v t dz t ). (4.1) Here Z t = Z t t 0 vs ds, (4.2) forms a Wiener process under the real world probability measure P, and vt is the market price of risk, which provides the link to the process Z. The squared volatility for the above 3/2 volatility model satisfies by (3.6) and (4.2) the SDE dv t = ηv t dt v 3/2 t dz t (4.3) for t [0,T], where Z is the Wiener process as given in (4.2) under P. The real world dynamics of the process Y = {Y t = 1 v t,t 0} of the inverse squared volatility is given by Y t = 1 v t = R t α = F t αexp{ηt}. (4.4)

12 E. Platen, L. Chan 12 Using the Itô formula one obtains by (4.4) and (4.3) for Y t the SDE ) dy t = (1 ηy t dt+ Y t dz t, (4.5) being a square root process of dimension δ = 4, see Revuz and Yor (1999). The only parameter in the SDE (4.5) is the net growth rate η. Note, by using (3.4) and (4.4) it follows that the index F t can be expressed as the product of an exponential function and a square root process of dimension δ = 4, that is, F t = αexp{ηt}y t (4.6) for t [0,T]. Obviously, the square root process Y has a stationary density. Furthermore, it is known that it never hits zero. Therefore, the above 3/2 volatility model has no volatility explosion because Y t is never hitting zero. Additionally, the parameter η can be interpreted as the long term growth rate of the discounted diversified equity index. This growth rate is the key macro-economic variable. To elaborate on the interpretation of the real world dynamics of the diversified index one notes from (4.6) and (4.5) that the index F t satisfies the SDE df t = αexp{ηt}dt+ αexp{ηt}f t dz t, (4.7) where Z is a P-Wiener process. Essentially, the deterministic drift in (4.7) models the increase per unit of time in the underlying fundamental value oftheequityindexf t. Ontheotherhand, theremainingmartingaletermin (4.7) reflects the speculative fluctuations of the index. This parsimonious model makes good economic sense, in particular, in the very long term. Remarkably, it has only two parameters, α and η. Furthermore, it coincides with the stylized MMM proposed in Platen (2001), see also Platen and Heath (2006) Figure 4: Logarithm of the discounted S&P 500 from 1920 to 2010.

13 E. Platen, L. Chan 13 Figure 4 shows the logarithm of the discounted S&P 500 total return index, observed inusdollarsfrom1920until March Onenotesits longterm growth with seemingly stationary fluctuations around the average growth. By taking the logarithm on both sides of (4.6) it follows that ln(f t ) = ηt+ln(y t )+ln(α). The slope of the fitted line in Figure 4 equals the net growth rate η, which turns out to be approximately 5% for the US market, see also Dimson, Marsh andstaunton (2002). The value ln(y t ) isthe logarithmof the square root process at time t and is modeled in (4.5) as a diffusion process with a stationary probability density. This fits well what one observes in Figure 4. The scaling parameter α can be fitted by ensuring that the average of Y t equals 1 η. Under the real world probability measure the above model appears to be a reasonable model for the dynamics of a well diversified equity index, e.g. the discounted S&P 500 total return index. Carr and Sun (2007) and Itkin and Carr (2009) use very different arguments, compared to the macro-economic argument presented above, to obtain a 3/2 volatility model. Recall that under the real world probability measure P the above square root process Y = {Y t,t [0,T]} has dimension δ = 4, and is a scalar diffusion process with stationary density that never reaches zero under P. Under the assumed risk neutral measure Q the dimension δ of R, and thus the dimension of the square root process Y, is zero. Consequently, there is a strictly positive risk neutral probability for the event that this process hits zero in a finite time period, see Revuz and Yor (1999). As a result, the measures Q and P do not have the same events of measure zero. This leads to the conclusion that under the above 3/2 volatility model the putative risk neutral measure Q is not equivalent to the real world probability measure P. In a complete market one has for the corresponding Radon-Nikodym derivative Λ Q t at time t the expression Λ Q t = ( Ft F 0 ) 1, (4.8) which forms under the above 3/2 volatility model the inverse of a squared Bessel process of dimension δ = 4 under P. It is well-known that in a continuous market this process is a local (A,P)-martingale but not a true (A,P)-martingale, see Revuz and Yor (1999). More precisely, Λ Q is a non-negative strict (A, P)-local martingale, and thus, a strict (A, P)- supermartingale. The above 3/2 volatility model is an example of a viable financial market model, as discussed in Loewenstein and Willard (2000), where the traditional notion of no-arbitrage, see Delbaen and Schachermayer (1998), cannot be verified. Therefore, a more general pricing method than the classical risk neutral one is needed to price derivatives.

14 E. Platen, L. Chan 14 5 Real World Pricing It is the aim of this section to price derivatives under the real world probability measure. Since an equivalent risk neutral probability measure does not exist under the above model, one can follow Platen and Heath (2006) and can use the numéraire portfolio as numéraire. In this context the following notion turns out to be crucial: [ ] U Definition 5.1 A price process U = {U t,t [0,T]}, with E T F T <, is called fair if the corresponding benchmarked price process Û = {Ût = U t F t,t [0,T]} forms an (A,P)-martingale, that is, ] Û t = E [Û T A t (5.1) for all 0 t T T. AsshowninPlatenandHeath(2006),inthegivengeneralsetting, whichassumes the existence of the numéraire portfolio process F = {F t,t [0,T]}, any benchmarked nonnegative portfolio is a supermartingale. The minimal supermartingale, which replicates a given benchmarked contingent claim, is the corresponding martingale. Since the minimal price is economically the reasonable price for a replicable claim, it is the fair price that should determine the value of a derivative if no other constraint exists. For a replicable contingent claim H T, payable at time T [0,T] with E( H T ) <, F T this yields the real world pricing formula ( ) V t = F t E H T A t (5.2) for all t [0, T], T [0,T]. Now, one can discuss the link between real world pricing and classical risk neutral pricing. As shown in Platen and Heath (2006), and as already indicated in (4.8), in a complete market the Radon-Nikodym derivative process Λ Q = {Λ Q t,t [0,T]} for the putative risk-neutral probability measure Q is given by the ratio F T Λ Q t = dq dp A t = F 0 F t. (5.3) Using this notion, one obtains from the real world pricing formula (5.1) the equivalent expression ( Λ Q ) T V t = E H T A t (5.4) Λ Q t for all t [0, T], T [0,T]. Recall that zero interest rates are assumed, for simplicity.

15 E. Platen, L. Chan 15 In a complete market the Radon-Nikodym derivative process Λ Q equals the normalized benchmarked savings account. If the savings account would be a fair price process, that is, a martingale, then the candidate Radon- Nikodym derivative process Λ Q would bean(a,p)-martingale. This would guarantee that the risk-neutral probability measure Q exists. In this case, one would obtain from (5.3) by the Bayes rule the standard risk neutral pricing formula with V t equal to E Q (H T A t ). However, when looking at discounted long term S&P 500 total return index data, one observes that Λ Q t exhibits in the long run much smaller values than at the beginning. This is a reflection of the existence of the equity premium. It suggests that it may not be realistic to interpret Λ Q as an (A,P)-martingale when modeling over long periods of time. Note that in the above 3/2 volatility model the Radon-Nikodym derivative is not a martingale because it is a strict supermartingale. Moreover, it follows from (5.3) by the supermartingale property of the Radon-Nikodym derivative that Λ Q t E t (Λ Q T) (5.5) for 0 t T T. Therefore, as indicated earlier, for any nonnegative contingent claim H T the application of formal risk neutral pricing would produce a price greater than or equal to the corresponding real world price. This can be seen when re-expressing (5.4) using (5.5) as the inequality V t E ( Λ Q T E Λ Q t ( Λ Q T Λ Q t H T A t ) A t ). (5.6) Theright handside oftheaboveinequality represents theformal riskneutral price, which can be substantially greater than the real world price. The concept of real-world pricing generalizes classical risk neutral pricing. It does not impose the restrictive condition that Λ Q has to form an (A, P)-martingale. As a consequence, real world pricing removes from the assumptions of Carr and Sun (2007) and Itkin and Carr (2009) the necessity to require the condition q < ǫ2, which prevented under the assumed 2 risk neutral probability the volatility from exploding. The freedom gained allows one to focus on the real world dynamics of the 3/2 volatility model, which is a realistic model from the perspective of short term volatility derivatives, as has been discussed in Carr and Sun (2007) and Itkin and Carr (2009). 6 Valuation of Squared Volatility Note that when pricing volatility derivatives one has to determine the expectations of payoffs or higher order moments of the squared volatility v T

16 E. Platen, L. Chan 16 at some maturity date T [0,T]. For the above 3/2 volatility model this can be achieved by using the explicitly known transition density of a time transformed squared Bessel process of dimension δ = 4, see Revuz and Yor (1999). For completeness, the transition density function p(ϕ t,x t ;ϕ T,x T) of a time-transformed squared Bessel process X = {X ϕt, ϕ t [ϕ 0,ϕ T]} of dimension δ = 4 is summarized below. It refers to a move from x t = X ϕt at some transformed time ϕ t to the level x T = X ϕ T at a later transformed time ϕ T. From Revuz and Yor (1999) one has the transition density 1 p(ϕ t,x t ;ϕ T,x T) = 2(ϕ T ϕ t ) ( x T x t )ν { 2 exp x } ( ) t +x T xt x T I ν 2(ϕ T ϕ t ) ϕ T ϕ t (6.1) for ϕ t [ϕ 0,ϕ T], where I ν is the modified Bessel function of the first kind with index ν = 1. For ϕ (0, ) one can show that the moment of order β has the value E(X β ϕ A 0) = { (2ϕ) β exp{ X 0 } 2ϕ k=0 ( ) k X 0 Γ(β+k+ δ 2 ) 2ϕ k!γ(k+ δ 2 ) for β > 2 otherwise. (6.2) Note for β 2 the corresponding moment does not exist because it is infinite. Let the ϕ-time be given in the form ϕ t = ϕ 0 + α ( ) exp{ηt} 1 4η (6.3) for t [0, ), see Delbaen and Shirakawa (1997). Given a squared Bessel process X of dimension δ = 4 in ϕ-time one obtains by setting X ϕt = F t the square root process Y = {Y t = Xϕ t,t [0, )} of dimension δ = 4 αexp{ηt} in t time for t [0, ). Recall that by (3.7) and X ϕt = F t one has the squared volatility in the form v t = αexp{ηt} X ϕt = 1 Y t, (6.4) which leads to the expression ( ) ( ) E v T αexp{η T} = E F T X 2 ϕ T ( 1 = αexp{η T}E X 2 ϕ T ). (6.5) Since β = 2, it follows by relation (6.2) that ( ) E v T = (6.6) F T

17 E. Platen, L. Chan 17 for 0 < T T. This provides the insight that even under real world pricing squared volatility as a payoff still demands an infinite price, which destroys the hope, formulated at the end of Section 5, that real world pricing may allocate a finite price to the squared volatility as payoff. Note that the volatility itself does not explode under P when using the real world 3/2 volatility model. It is the interplay between volatility and the index that creates a probability law which yields the infinite expectation in (6.6). The following discusses the question whether the above price explosion for squared volatility as a payoff is just a mathematical curiosity under the discussed model or whether it has deeper economic reasons, which should be taken into account in financial modeling and the design of volatility derivatives. Recall from (3.4) that the benchmarked squared volatility at time T is of the form v T F T = αexp{η T}. (6.7) F 2 T On the other hand, the benchmarked index remains simply a constant. If theindexf T falls, thenthebenchmarkedsquaredvolatility v T increases significantly and vice versa. More precisely, one observes an inverse quadratic F T relationship. One may wonder if such a strong nonlinearity in the payoff structure is sustainable in reality, in particular, from a macro-economic point of view. Potentially, in the coming years large pension funds and many other investors may seek protection against a downturn in the market by holding the floating part of a variance swap or a similar volatility derivative. To illustrate the looming problem, Figure 5 exhibits the normalized benchmarked squared VIX, from 1990 until One notes the enormous increase in September and October If one would simulate a corresponding scenario under the 3/2 volatility model, then the trajectory would have similar extreme spikes but also similar longer periods with relatively low benchmarked squared volatility. The benchmarked squared Figure 5: Normalized benchmarked squared VIX from 1990 to 2010.

18 E. Platen, L. Chan 18 volatility increased in Figure 5 by approximately 2, 000% during a short time period in This demonstrates the extreme movements that a benchmarked payoff of squared volatility can exhibit. If those financial institutions, which typically issue on a large scale volatility derivatives with benchmarked payoffs of the type (6.7), were to have most of their capital invested in the equity market, then the authors doubt that they would have enough liquid capital to deliver the demanded payoffs inthe event of a market crash of the kind seen in 2008, or even a more severe one such as the one before the great depression. It is probably reasonable to predict that when most large pension funds, insurers and many investors hold variance swaps on equity indices on a large scale, the market would suffer a liquidity crisis with major bankruptcies and it would probably need larger bailout funding than that provided during the recent financial crisis. The strong nonlinearity of the benchmarked payoffs would require crippling bailouts in the event of a decline in the index, as experienced with the great depression. In summary, the authors of this paper are deeply concerned about the rapidly developing exposure to volatility derivatives in financial markets. In particular, the currently traded standard variance swaps on equity indices could severely de-stabilize the financial system. The consequences of such destabilization should not be underestimated. Pensions could be lost to a large extent and large financial institutions may have to declare bankruptcy. It is certain that the global market will again move substantiallydownward inthelongrun. Thisissimply a consequence ofitsrandom nature. Professionals responsible for designing derivatives, politicians who need to safeguard their financial market, and regulators who oversee the provision of regulatory capital for institutions with derivative portfolios should take action to mitigate the systemic risks that could accumulate over the coming years due to a poor design of volatility derivatives. What is it that could help to reduce significantly the risks that have been pointed out above? Assume for the moment that there would be no bailout of large issuers of volatility derivatives in the case of a severe market downturn. One could then argue that even if the standard payoff structure of a variance swap on an equity index does not declare a knock-out barrier or cap, in reality, such a barrier is implicitly present due to the limited liquid capital and likely default of the issuer as a result of such a rare but extreme event. The introduction of some barrier in the design of the payoff would make volatility derivatives less expensive, and less risky for the issuer. It would also make the risks for both counter parties more transparent. Variance swaps designed for single stocks have often a cap, which acts in the direction indicated above. Such a cap is part of the design to cover the case of bankruptcy of the respective company. However, variance swaps on equity indices, which are of primary interest to institutional and large investors, typically do not have such a cap or knock-out barrier. One does typically not imagine that large parts of the industry that issues these instruments could face bankruptcy. To make these volatility deriva-

19 E. Platen, L. Chan 19 tives more realistic and even calculable, it should become mandatory that variance swaps on equity indices or similar payoff structures must include some knock-out barrier level. The volatility derivative payoff remains then bounded even when the equity index falls significantly below the knockout barrier and pushes the index volatility high up. Mathematically, such a knock-out barrier would make most common volatility derivative prices finite. The real world pricing formula of such a derivative with a knock-out barrier B would be of the form ( H(v T) Ṽ t = F 0 E 0 1 {τ> T} ), (6.8) F T where τ = inf{t 0 : F t < B} is the knock-out time. Here H(v T) is the payoff and B > 0 denotes the knock-out barrier level. One could set B slightly above the minimum level for the index, where the issuer of the derivative can most likely still serve its obligations in the case of a downturn. The price of the above barrier type option can be obtained by calculating the conditional expectation in (6.8) using standard numerical methods, for instance, finite-difference methods or Fourier and Laplace transform methods, described e.g. in Carr and Sun (2007) and Itkin and Carr (2009). 7 Conclusions This study has formally pointed at the potential risk of a liquidity crisis for the financial system that currently has been building up through various increasingly popular volatility derivatives, in particular, variance swaps on equity indices. It studies the case of the 3/2 volatility model and obtains infinite prices for payoffs delivering squared volatility. It has been argued that these price explosions may have deeper economic reasons. Consequently, care should be taken when designing volatility derivatives such as variance swaps on equity indices. The mandatory inclusion of knock-out barriers has been recommended to keep the emerging systemic risk manageable. BIBLIOGRAPHY Ahn, D.,& Gao, B.(1999). A parametric nonlinear model of term structure dynamics. Review of Financial Studies, 12, Andersen, L.B.G., & Piterbarg, V.(2007). Moment explosions in stochastic volatility models. Finance and Stochastics, 11, Andreasen, J. (2001). Credit explosives. working paper, Bank of America.

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