The Fundamental Theorem of Derivative Trading - Exposition, Extensions and Experiments

Quantitative Finance

ISSN: 1469-7688 (Print) 1469-7696 (Online) Journal homepage: https://www.tandfonline.com/loi/rquf20

The Fundamental Theorem of Derivative Trading - exposition, extensions and experiments

Simon Ellersgaard, Martin Jönsson & Rolf Poulsen

To cite this article: Simon Ellersgaard, Martin Jönsson & Rolf Poulsen (2017) The Fundamental Theorem of Derivative Trading - exposition, extensions and experiments, Quantitative Finance, 17:4, 515-529, DOI: 10.1080/14697688.2016.1222078 To link to this article: https://doi.org/10.1080/14697688.2016.1222078

Published online: 14 Sep 2016.

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Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=rquf20 Quantitative Finance, 2017 Vol. 17, No. 4, 515–529, http://dx.doi.org/10.1080/14697688.2016.1222078

The Fundamental Theorem of Derivative Trading - exposition, extensions and experiments

SIMON ELLERSGAARD, MARTIN JÖNSSON and ROLF POULSEN∗

Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark

(Received 14 December 2015; accepted 28 July 2016; published online 14 September 2016)

When estimated volatilities are not in perfect agreement with reality, delta-hedged option portfolios will incur a non-zero proﬁt-and-loss over time. However, there is a surprisingly simple formula for the resulting hedge error, which has been known since the late 1990s. We call this The Fundamental Theorem of Derivative Trading. This paper is a survey with twists on that result. We prove a more general version of it and discuss various extensions and applications, from incorporating a multi-dimensional jump framework to deriving the Dupire–Gyöngy–Derman–Kani formula. We also consider its practical consequences, both in simulation experiments and on empirical data, thus demonstrating the beneﬁts of hedging with implied volatility.

Keywords: Delta hedging; Model uncertainty; Volatility arbitrage JEL Classiﬁcation: G13, G17

1. A meditation on the art of derivative hedging geometric Brownian motion, one can proceed to measure the standard deviation of past log returns over time. Yet this pro- 1.1. Introduction cedure raises uncomfortable questions pertaining to statistical measurement: under ordinary circumstances, increasing the Of all possible concepts within the field mathematical finance, sample space should narrow the confidence interval around our that of continuous time derivative hedging indubitably emerges sample parameter. Only here, there is no a priori way of telling as the central pillar. First used in the seminal work by Black if and when a model undergoes a structural change.§ Inevitably, and Scholes (1973),† it has become the cornerstone in the this implies that extending the time series of log returns too determination of no-arbitrage prices for new financial prod- far into the past might lead to a less accurate estimator as we ucts. Yet a disconnect between this body of abstract mathe- might end up sampling from a governing dynamics that is no matical theory and real-world practice prevails. Specifically, longer valid. Of course, we may take some measures against successful hedging relies crucially on us having near-perfect this issue by trying our luck with ever more intricate time series information about the model that drives the underlying asset. analyses, until we stumble upon a model, the parameters of Even if we boldly adopt the standard stochastic differential which satisfy our arbitrary tolerance for statistical significance. equation paradigm of asset pricing, it remains to make exact Nevertheless, in practice this procedure invariably boils down specifications for the degree to which the price process reacts to checking some finite basket of models and selecting the to market fluctuations (i.e. to specify the diffusion term, the best one from the lot. Furthermore, unknown structural breaks volatility). Alas, volatility blatantly transcends direct human continue to pose a problem no matter what. observation, being, as it were, a Kantian Ding an sich‡ of which Alternatively, we might try to extract an implied volatility we only have approximate knowledge. from the market by fitting our model to observed option prices. One such source supervenes upon historical realizations of the underlying price process: for example, assuming that the governing model can at least locally be approximated as a §This scenario is not at all implausible. Unlike the physical sciences where the fundamental laws are assumed to have no sufficient reason ∗ to change (in the Leibnizian or Occamian sense), this philosophical Corresponding author. Email: [email protected] †That Black and Scholes along with Merton were the first is the principle would hardly withstand scrutiny in a social science context. general consensus, although the paper by Haug and Taleb (2011) Asset price processes are fundamentally governed by market agents shows that the view is not universal. and their reactions to various events (be they self-induced or ‡Literally, thing in itself or the noumenon. Kant held that there is a exogenous). There is really no reason to assume that these market distinction between the way things appear to observers (phenomena) players will not drastically change their opinions at some point (for and the way reality actually is construed (noumena). one reason or the other). © 2016 Informa UK Limited, trading as Taylor & Francis Group 516 S. Ellersgaard et al.

Nevertheless, the inadequacy of this methodology quickly quite reached the ‘textbook’ status we reckon it deserves (in becomes apparent: first, implied volatility might be ill-defined particular, it rarely finds its way into the curriculum of aspir- as it is the case for certain exotic products such as barrier ing financial engineers). To end this regrettable situation, we options. Secondly, it is quite clear that the market hysteria here provide a thorough exposition. Taking our vantage point which drives the prices of traded options need not capture in the tripartite philosophy of Henrard and Carr, we endeav- the market hysteria which drives the corresponding market our to provide a simple proof of the Fundamental Theorem, for the underlying asset. Fair pricing ultimately boils down to whilst simultaneously spicing up the result by considering a understanding the true nature of the underlying product: not to more general dynamics in a full-fledged multi-asset frame- mimic the collective madness of option traders. work. Specifically, the structure of this paper is as follows: Whilst volatility at its core remains elusive to us, the situation in section 2, we state and prove a generalized version of the is perhaps not as dire as one might think. Specifically, we can FundamentalTheorem of DerivativeTrading for basket options develop a formal understanding of the profit-&-loss we incur and discuss its various implications for hedging strategies and upon hedging a portfolio with an erroneous volatility—at least applications. Key insights here include the benefit of hedging insofar as we make some moderate assumptions of the dynami- with the implied volatility with regards to attaining ‘smooth’ cal form of the underlying assets. To give a concrete example of P&L paths, and a somewhat surprising connection to Dupire’s this, consider the simple interest rate free framework presented local volatility formula. In section 3, we expose the impli- in Andreasen (2003) where the price process of a single non- cations of adding a multi-dimensional jump process to the dividend paying asset is assumed to follow the real dynamics dynamics, thus emphasizing the relative ease with which the original proof can be adapted. In particular, we argue that dX = X [μ , dt + σ , dW ]. t t t r t r t long option positions with convex pay-out profiles profit from i Let Vt be the value of an option that trades in the market at a discontinuous movement in one of the underlying stocks. a certain implied volatility σi —possibly quite different from Finally, section 4 presents an empirical investigation into what the epistemically inaccessible σt,r . Now if we were to set up actually happens to our portfolio when we hedge using various a hedge of a long position on such an option, using σi as our volatilities. In particular, using actual quotes for stocks and call hedge volatility, an application of It¯o’s formula, coupled with options, we demonstrate that self-financing portfolios hedged the Black–Scholes equation, shows that the infinitesimal value at the implied volatility indeed give rise to a smoother P&Lover = i − ∂ i · change in the hedge portfolio t Vt x Vt Xt ,isofthe time, whilst allowing for some amount of volatility arbitrage form to be picked up in the process. = 1 (σ 2 − σ 2) 2∂2 , d t 2 t,r i Xt xxVt dt (1) which generally is non-zero unless σi = σt,r . For reasons that will become clearer below, the importance of this result is of 2. The Fundamental Theorem of Derivative Trading such magnitude that Andreasen dubs it the Fundamental The- orem of Derivative Trading. Indeed, a more abstract variation 2.1. Model set-up of it will be the central object of study in this paper. Consider a financial market comprised of a risk-free money To the best of our knowledge, quantitative studies into the account as well as n risky assets, each of which pays out a effect of hedging with an erroneous volatility first appeared continuous dividend yield. We assume all assets to be infinitely in a paper on the robustness of the Black–Scholes formula by divisible as to the amount which may be held, that trading El Karoui et al. (1998). They viewed the result as a largely takes place continuously in time, and that no trade is subject to negative one: unless volatility is bounded (which it is not financial friction. Formally, we imagine the information flow of in any stochastic volatility model) then there is no simple this world to be captured by the stochastic basis (, F , F, P), super-replication strategy. Easier derivations of the Fundamen- where represents all possible states of the economy, P is the tal Theorem are encountered in the papers by Gibson et al. physical probability measure and F ={Ft }t≥0 is a filtration (1999), Mahayni et al. (2001), and Rasmussen (2001). Indeed, which satisfies the usual conditions.† The price processes of there seems to be an added awareness of the result being a the risky assets, Xt = (X1t , X2t ,...,Xnt) , are assumed to positive one: a decent forecast of volatility gives rise to a small follow the real dynamics‡ hedge error. Jointly, what characterizes these papers is their ∼ ∼ = [μ ( , ) + σ ( , ) ], binary partition of volatility into a ‘wrong’and ‘right’category. dXt DXt r t Xt dt r t Xt dW t (2)

Somewhat subtler treatments can be found in the unpublished where DX is the n×n diagonal matrix diag(X1t , X2t ,...,Xnt), works by Carr (2002) and Henrard (2001) in which the partition and W t = (W1t , W2t ,...,Wnt) is an n-dimensional standard becomes tripartite: concretely, these papers explicitly differen- F μ :[, ∞) × Brownian motion adapted to . Furthermore, r 0 tiate between a true governing volatility of the underlying asset, Rn+m → Rn and σ :[0, ∞) × Rn+m → Rn×n are determin- an implied volatility characterizing market consensus and a istic functions, sufficiently well-behaved for the SDE to have a hedge volatility which captures the personal belief of the option hedger. The benefits of this three-part structure are clearly †Specifically, it satisfies right-continuity, ∩s≥t Fs = Ft ∀t ≥ 0 enunciated in Ahmad and Wilmott (2005) and Wilmott (2007) (if we move incrementally forward in time there will be no jump in F P in which the associated P&L paths under the various choices of information), and completeness, i.e. 0 contains all null sets. volatility are exhibited, alongside mathematical (non-analytic) ‡The nomenclature ‘real dynamics’ is ripe with unfortunate connotations of Platonic realism (ontological significance of expressions for the mean implied P&L and its variance. mathematical objects). Strictly speaking, this is not what we require, The Fundamental Theorem has in other words received ex- but rather the expressive adequacy of a model: i.e. it’s ability to tensive treatment in the academic literature, yet it has never adequately capture the financial events unfolding. The Fundamental Theorem of Derivative Trading 517 unique strong solution (in particular, we assume the regularity is a matrix which takes values in Rn×n. conditions {h} ∈[ , ] s ∼ s ∼ Proof Let t t 0 T be the value process of the hedge port- | μ ( , )| < ∞, | σ ( , )|2 < ∞, DXu r u Xu du DXu r u Xu du folio long one option valued according to the implied market t t conception, {V i } ∈[ , ], and short {h =∇ V h} ∈[ , ] units (3) t t 0 T t x t t 0 T of the underlying with value process {X } ∈[ , ], where ∇ hold a.s. ∀t ≤ s, where the first norm is to be understood in t t 0 T x is the gradient operator. We suppose the money account B is the Euclidian sense, whilst the latter should be construed in the ∼ chosen such that the net value of the position is zero: matrical sense).† Finally, we define Xt as the n+m dimensional ( ; χ ) χ = (χ ,χ , ...,χ ) h = i + −∇ h • = , vector Xt t where t 1t 2t mt has the t Vt Bt x Vt Xt 0 interpretation of an m-dimensional state variable, the exact where • is the dot product. Now consider the infinitesimal dynamical nature of which is not integral to what follows.‡ change to the value of this portfolio over the interval [t, t +dt], In what , we consider the scenario of what happens when we where t ∈[0, T ).Fromtheself-financing condition we have hedge an option on Xt , ignorant of the existence of the state that variable χ , as well as the form of μ (·, ·) and σ (·, ·). Specif- t r r h = i + −∇ h • ( + ◦ ), ically, we shall imagine that we are misguided to the extent d t dVt rt Bt dt x Vt dXt qt Xt dt = ( ( , ), ( , ),..., ( , )) that we would model the dynamics of Xt as a local volatility where qt q1 t X1t q2 t X2t qn t Xnt codifies model with diffusion matrix σ h(t, Xt ). Similar assumptions the continuous dividend yields and ◦ is the Hadamard product.§ pertain to the market, although here we label the ‘implied’ Jointly, the two previous equations entail that diffusion matrix σ (t, X ) to distinguish it from our personal i t h = V i −∇ V h •( X −(r ι−q )◦ X t)−r V i t, belief. Irrespective of which dynamical specification is being d t d t x t d t t t t d t t d (6) made, we maintain that regularity conditions analogous to (3) where ι = (1, 1,...,1) ∈ Rn. remain satisfied. Finally, a cautionary remark: throughout these Now consider the option valued under σ h(t, Xt ); from the pages we use r and i to emphasize that the volatility is real multi-dimensional It¯o formula¶ we have that and implied, respectively, whilst h refers to an arbitrary hedge ∼ ∼ h ={∂ h + 1 [σ ( , ) ∇2 h σ ( , )]} volatility. For a comprehensible reading, it is incumbent that dVt t Vt 2 tr r t Xt DXt xxVt DXt r t Xt dt +∇ h • , the reader keeps these definitions in mind. x Vt dXt (7)

where we have used the fact that Xt is governed by (2). Mean- h while, Vt satisfies the multi-dimensional Black–Scholes 2.2. Theorem and derivation equation for dividend paying underlyings, Theorem 1 The Fundamental Theorem of Derivative h h h rt V = ∂t V +∇x V • ((rt ι − q ) ◦ Xt ) = ( , ) ∈ C1,2([ , ]×Rn) t t t t Trading Let Vt V t Xt 0 T be the 1 2 h + tr[σ (t, Xt )DX ∇ V DX σ h(t, Xt )]. (8) price process of a European option with terminal pay-off VT = 2 h t xx t t g(X T ), the underlying of which follows the real dynamics (2). Combining this expression with the It¯o expansion we obtain, We assume the option trades in the market at the (not neces- =− h + h +∇ h • ( − ( ι − ) ◦ ) σ = σ ( , ) 0 dVt rt Vt dt x Vt dXt rt qt Xt dt sarily uniquely determined) implied volatility i i t Xt . ∼ ∼ = + 1 [ σ ( , )σ ( , ) Now suppose we at time t 0 acquire such an option for 2 tr DXt r t Xt r t Xt i the implied price V and set out to -hedge our position. Said 2 h 0 − σ h(t, Xt )σ (t, Xt ) DX ∇ V ]dt, (9) hedge is performed under the notion that the volatility function h t xx t ought, in fact, to be of the form σ h = σ h(t, Xt ), leading to the where we have used the fact that the trace is invariant under h cyclic permutations of its constituent matrices. Finally, defin- fair price process Vt . Then the present value of the profit-&- ∼ loss we incur from holding such a portfolio over the interval ing rh(t, Xt ) as in (5), and adding (9)to(6) we obtain T =[ , ] 0 T is h = i − h − ( i − h) d t dVt dVt rt Vt Vt dt T ∼ P&Lh = V h − V i + 1 + 1 [ ( , ) ∇2 h] T 0 0 2 2 tr DXt rh t Xt DXt xxVt dt 0 ∼ t r du − t r du i h − t r du 2 h = e 0 u d(e 0 u (V − V )) × e 0 u tr[D (t, X )D ∇ V ]dt, (4) t t Xt rh t Xt xx t ∼ 1 2 h = ( , ) ∇2 + tr[DX rh(t, Xt )DX ∇ V ]dt. (10) where ru r u Xu is the locally risk free rate, xx is the 2 t t xx t Hessian operator, and Whilst a perfect hedge would render this infinitesimal value- ∼ ∼ ∼ change in the portfolio zero, this is clearly not the case here. (t, X ) ≡ σ (t, X )σ (t, X )−σ (t, X )σ (t, X ), (5) rh t r t r t h t h t In fact, upon discounting (10) back to the present (t = 0) and × integrating up the infinitesimal components, we find that net †Specifically, if x ∈ Rn and A ∈ Rn d , the Euclidian norm / profit-&-loss incurred over the lifetime of the portfolio is is defined as |x|≡( n x2)1 2, whilst the matrical norm is i=1 i T n d 2 1/2 − t |A|≡( A ) . h = ( 0 ru du( i − h)) i=1 j=1 ij P< d e Vt Vt ‡Nonetheless, a common assumption in the stochastic volatility 0 literature is obviously to let χ be driven by a stochastic differential ¯ equation of the form dχt = m(χt )dt + v(χt )dWt + v¯(χt )dWt , §Otherwise known as the entry-wise product. As per definition, if A ¯ where Wt is second standard Brownian motion (independent of and B are matrices of equal dimensions, then (A ◦ B)ij = AijBij. the first), and m, v and v¯ are dimensionally consistent, regularity ¶See for instance Björk (2009), p. 65. conforming vectors and matrices. See for instance Björk, Theorem 13.1 and Proposition 16.7. 518 S. Ellersgaard et al. T ∼ − t r du 2 h which is manifestly deterministic.† However, we observe that + e 0 u 1 tr[D (t, X )D ∇ V ]dt 2 Xt rh t Xt xx t this relies crucially on us holding the portfolio until expiry of 0 T − t the option. Day-to-day fluctuations of the profit-&-loss still = h − i + 1 0 ru du V0 V0 2 e vary stochastically (erratically) as it is vividly demonstrated 0 ∼ by combining equation (9) (where h = i) with equation (6) × [ ( , ) ∇2 h] . tr DXt rh t Xt DXt xxVt dt (where h = r): T − t ∼ h ≡ 0 ru du h r 1 2 i where P< e d t , and the last line makes use = [ ( , ) ∇ ] 0 d t 2 tr DXt ri t Xt DXt xxVt dt of the fact that V i = V h = g(X ). This is the desired result. T T T +∇ i − r • (μr − ι x Vt Vt t rt ∼ Remark 1 + ) ◦ + σ ( , ) , A few observations on this proof are in order: qt Xt dt DXt r t Xt dW t first, the relative simplicity of (4) clearly boils down to the assumption that the market is perceived to be driven by a local cf. the explicit dependence of the Brownian increment. As volatility model. If this assumption is dropped, equation (8)no for the profitability of the -hedging strategy, this is a com- longer holds. Secondly, it should be clear that the value of the plex issue which ultimately must be studied on a case-by-case P&L changes sign if we are short on the derivative and long basis. However, for options with positive vega,‡ it suffices to on the underlying. Thirdly, the market price of the derivative require that the real volatility everywhere exceeds the implied volatility. enters only though the initial price V0. That is because we look at the profit-&-loss accrued over the entire lifetime of the portfolio. The case of marking-to-market requires further 2.3.2. Hedging with the implied volatility. Suppose instead analysis and/or assumption. We will elaborate on this in the we hedge the portfolio using the implied volatility matrix following subsection. σ i (t, Xt ) ∀t ∈[0, T ], then the associated present-valued profit- Remark 2 From a generalist’s perspective, theorem 1 suffers &-loss is of the form T from a number of glaring limitations: for instance, the govern- − t ∼ i = 1 0 ru du [ ( , ) ∇2 i ] . ing asset price dynamics only considers Brownian stochas- P< 2 e tr DXt ri t Xt DXt xxVt dt ticity, the hedge is assumed to be a workaday -hedge, and 0 the option type is vanilla European in the sense that the ter- As we find ourselves integrating over the stochastic process minal pay-off is determined by the instantaneous price of the Xt , this profit-&-loss is manifestly stochastic. Notice though i underlying assets. Fortunately, the Fundamental Theorem can that d t here does not depend explicitly on the Brownian O( ) readily be extended in various directions: e.g. it can be shown increment (the daily profit-and-loss is dt ) which gives rise to point that ‘bad models cause bleeding—not blow-ups’. As that if Vt = V (t, Xt , At ) is an Asian option written on the for the profitability of the strategy, again this is a complex issue: continuous average At of the underlying process Xt , then the ∼ ( , ) ◦∇2 i Fundamental Theorem remains form invariant. In section 3, however, insofar as ri t Xt xxVt is positive definite a.s. we consider one particularly topical dynamical modification for all t ∈[0, T ], then we are making a profit with probability viz. the incorporation of possible market crashes through jump one. To see this, recall that the trace can be written as§ ∼ ∼ diffusion. [ ( , ) ∇2 i ]= ( ( , ) ◦∇2 i ) , tr DXt ri t Xt DXt xxVt Xt ri t Xt xxVt Xt ∼ ( , ) ◦∇2 i In particular, if ri t Xt xxVt is positive definite at all 2.3. The implications for -hedging times, i.e. ∼ From a first inspection, the Fundamental Theorem quite clearly ∀ ∈[ , ]∀ ∈ Rn : ( ( , ) ◦∇2 i ) > , t 0 T Xt Xt ri t Xt xxVt Xt 0 demonstrates that reasonably successful hedging is possible i then P< > 0. A sufficient condition for this to be the case even under significant model uncertainty. Indeed, as Davis ∼ ( , ) ∇2 i (2010) puts it ‘without some robustness property of this kind, is that ri t Xt and xxVt individually are positive definite it is hard to imagine that the derivatives industry could exist at ∀t, as demonstrated by the Schur Product Theorem. all’. In this section, we dive further into the implications of what happens to our portfolio, by considering the case where we hedge with (a) the real volatility, and (b) the implied volatility. 2.3.3. Wilmott’s hedge experiment. The points imbued in It is a standing assumption in this subsection that whatever is the previous two paragraphs are forcefully demonstrated in used as hedge volatility is of the form that allows the use of the event that there is only one risky asset in existence, the theorem 1 (see the discussion in remark 1 above). The reader can think of the cases of constant volatility. †Obviously, this can only be the case if there is no underlying state variable. ‡A clear example of vega being manifestly positive would be 2.3.1. Hedging with the real volatility. Suppose we happen European calls and puts, which satisfy the assumptions needed ∂ to be bang on our estimate of the real volatility matrix in our to derive the Black–Scholes formula. Explicitly, ν ≡ V = ∼ √ ∂σ −δ(T −t) -hedge, i.e. let σ (t, X ) = σ (t, X ) a.s. ∀t ∈[0, T ], then St e φ(d1) T − t > 0whereφ is the standard normal pdf ∼ h t r t ( , ) = and d1 has the usual definition. rr t Xt 0 and the present valued profit-&-loss amounts §This follows from the general identity for matrices A and B of to corresponding dimensions: x (A ◦ B)y = tr[DxADyB ] where x r = r − i , P< V0 V0 and y are vectors. The Fundamental Theorem of Derivative Trading 519

3 3

2.5 2.5

2 2

1.5 1.5

1 1 Mark-to-market (P&L) Mark-to-market (P&L)

0.5 0.5

0 0 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 Time [years] Time [years]

Figure 1. Left: Delta hedging a portfolio assuming that σh = σr . The parameter specifications are: r = 0.05, μ = 0.1, σi = 0.2, σr = 0.3, S0 = 100, K = 100, q = 0andT = 0.25. The portfolio is rebalanced 5000 times during the lifetime of the option. Observe that whilst the P&L fluctuates randomly along the path of St due to the presence of dWt , the accumulated P&L at the maturity of the option is the deterministic = rT( r − i ) r = . i = . = . quantity T e V0 V0 . From the Black–Scholes formula, it follows that V0 6 583 and V0 4 615 so T =1 1 993. The fact that our 10 paths only approximately hit this terminal value is attributable to the discretization of the hedging which should be done in continuous time. Right: Delta hedging a portfolio assuming that σh = σi . The parameter specifications are as before. Evidently, the accumulated P&L i stays highly path dependent for the entire duration of the option. However, the curves per se are smooth, which highlights that d t does not depend explicitly on the Brownian increment. derivative is a European call option and all volatilities are seller; if one wins the other loses. That, however, is not true. assumed constant. Based on Wilmott and Ahmad, figure 1 Imagine a stock dynamics which is described by geometric clearly illustrates the behaviour of the profit-&-loss paths inso- Brownian motion with 15% volatility. A buyer is strongly far as we hedge with (a) the real volatility, and (b) the implied bullish and makes a directional bet by buying an at-the-money i volatility. Again, the main insights are as follows: hedging Vt call-option, which he is willing to pay 20% implied volatility with the real volatility causes the P&L of the portfolio to on. The buyer does not hedge his position, the market rallies, fluctuate erratically over time, only to land at a deterministic and he makes a nice profit. Meanwhile, the seller is right in her i value at maturity. On the other hand, hedging Vt with the 15%-volatility forecast, -hedges and makes a nice (arbitrage) implied volatility yields smoother (albeit still stochastic) P&L profit too. Needless to say, option writing can also imply losses curves. Nonetheless, here, there is no way of telling what the for both the buyer and the seller as we saw it during the financial P&L actually amounts to at maturity. crisis.‡ Rather perturbingly, both strategies blatantly suggest the relative ease with which we can make volatility arbitrage. Specifically, assuming that the historical volatility is a reasonable proxy for the real volatility, σhist ≈ σr , and that σhist >σi 2.4. Applications (σ <σ), it would suffice to go long (short) on the hedge hist i Due to the presence of the real volatility, the exact nature of portfolio for P(P< ≥ 0) = 1 and P(P< > 0)>0. which transcends our epistemic domain, one might reasonably The Wilmott experiment reflects the concerting approach ponder whether the Fundamental Theorem conveys any prac- (Hull and White 2016) of traders: Hedge with market implied tical points besides those of the preceding subsection. Using parameters/quantities. For the experiment’s strong smoothness two poignant (even if somewhat eccentric) examples, we will property of implied volatility hedging P&L to hold it is impor- argue that the gravity of the Fundamental Theorem propagates tant that implied volatility is assumed constant, or at least that it well into risk management and volatility surface calibration. does not diffuse randomly as it would in, e.g. the Heston model. Zero rates and dividends will be assumed throughout. Were the latter the case, then anything might happen. However, since implied volatility is intricately linked to (conditional Example 1 Let Vt (T, K ) be the price process of a European expectation of) time-integrated real volatility (squared),† it is strike K maturity T call or put option, written on an underlying likely to be quantitatively smoother than real volatility, so there which obeys geometric Brownian motion, dXt = Xt [μr dt + is hope that the conclusion of the idealized experiment carries σr dWt ], where μr ,σr are constants. Suppose we -hedge a over to empirical analysis. In section 4, we will see that this is long position on Vt at the implied volatility, σh = σi , then the indeed the case. Fundamental Theorem implies that Remark 3 T It is tempting to think of an option trade as a zero- i = 1 (σ 2 − σ 2) 2 i , P< 2 r i Xt t dt sum game (up to a risk-premium) between the buyer and the 0

†Think of the Black–Scholes model with time-dependent volatility or see Romano and Touzi’s Proposition 4.1 (Romano and Touzi 1997). ‡This observation is inspired by Antoine Savine. 520 S. Ellersgaard et al. where where 1{Xt >K } is the indicator function. The important point φ(di ) + i ≡ √ 1 , here is that (Xt − K ) may be reinterpreted as the terminal t Xt σi T − t pay-off of a strike K maturity t call option (obviously, the is the option’s gamma, φ : R → R+ is the standard normal specification σh = σi = 0 is paramount here). Substituting pdf and (11) into the infinitesimal form of the Fundamental Theorem, ∼ i ≡ √1 ( / ) + 1 σ 2( − ) . i 1 2 2 2 2 i d ln Xt K i T t d = (σ (t, X t ) − σ )X ∂ C (T, K )dt, 1 σi T −t 2 t 2 r i t xx t ∀ i > σ 2 > Since t t 0 the strategy is profitable if and only if r we find that σ 2 ∼ i . Furthermore, by maximizing the integrand with respect to + 1 2 2 i d((Xt −K ) +Bt −1{X >K } Xt ) = σ (t, X t )X δ(Xt −K )dt, Xt , we find that the P< is maximal when t 2 r t (12) ∗ 1 σ 2( − ) = 2 i T t , σ = Xt Ke where we once again have made use of i 0, alongside the fact that ∂ 1{ > } is the Dirac delta-function δ(X − K ). Specifically, upon evaluating the integral explicitly it can be x Xt K t Taking the risk neutral expectation of (12), conditional on F , shown that† 0 the left-hand side reduces to i = T K (σ 2 − σ 2). Q Q + Q max P< r i E [LHS]=E [d(Xt − K ) ]+E [dBt − 1{X >K }dXt ] X π σ t t 2 i Q + Q = dE [(X − K ) ]−E [1{ > }dX ] From a risk management point of view, the important point is t Xt K t = r ( , ) − EQ[EQ[ |F ]] that we can compute a confidence interval for the real volatility dC0 t K 1{Xt >K }dXt t based on historical observations. Hence, we can compute a = r ( , ) − EQ[ EQ[ |F ]] dC0 t K 1{Xt >K } dXt t confidence interval for the maximal profit-&-loss we might = dCr (t, K ), (13) face upon holding the hedge portfolio till expiry. 0 = Example 2 Let V = C (T, K ) be the price process of a where the second line uses dBt 0, whilst the third line uses t t EQ[( − )+] European strike K maturity T call option written on an under- the law of iterated expectations and the fact that Xt K lying price process X.Asin(2), we assume the fundamental is the time zero price of a strike K maturity t call option. ∼ ∼ Finally, the fourth line follows from the Ft -measurability of dynamics to be of the form dXt = Xt [μr (t, X t )dt + σr (t, X t ∼ 1{X >K }, whilst the fifth line exploits the martingale property ) ] ( + ) Q t dWt , where X t is defined as the 1 m -dimensional vector E [dXt ]=0. (Xt ; χ ) and χ is a state variable. Also, we suppose the inte- t ∼ As for the right-hand side, define the joint density grability condition E[ T σ 2(t, X )X 2dt] < ∞, and that there 0 r t t Q (σ 2, ) σ 2 Q fσ 2, x d dx exists an equivalent martingale measure, , which renders Xt r Xt a martingale (recall the risk-free rate is assumed zero)‡: ≡ Q({σ 2 ≤ σ 2 ≤ σ 2 + σ 2}∩{ ≤ ≤ + }), r d x Xt x dx ∼ = σ ( , ) Q. dXt r t X t Xt dWt then Now consider the admittedly somewhat contrived scenario EQ[ ]= 1 σ 2 2δ( − ) Q (σ 2, ) σ 2 RHS x x K fσ 2, x d dx dt of a -hedged portfolio, long one unit of the call, for which 2 2 r Xt R+ σh and σi are both zero.§ The associated value process is = 1 σ 2 2δ( − ) Q x x K fσ 2 i = i ( , ) + − ∂ h( , ) · 2 2 r t Ct T K Bt x Ct T K Xt R+ + Q = ( − ) + − { > } , 2 2 Xt K Bt 1 Xt K Xt (11) × (σ |Xt = x) f (x)dσ dx dt Xt = 1 2δ( − ) Q ( ) x x K fX x †We observe that a similar result can be found in Derman (2008). 2 R t ‡Obviously, such an existence claim is not altogether innocuous. + Indeed, the measure change is here further complicated by the fact × σ 2 Q (σ 2| = ) σ 2 fσ 2 Xt x d dx dt that we have not made formal specifications for the dynamical form R+ r of the state variable χ . However, insofar as we adopt the standard t Q dynamical assumption dχ = m(χ )dt + v(χ )dW + v¯(χ )dW¯ , ≡ 1 x2δ(x − K ) f (x) t t t t t t 2 Xt our existence claim is tantamount to positing the existence of a market R+ θ ∈ Rm ( ) = ( ) ( ) ∼ price of risk vector which renders L T L X T Lχ T a × EQ[σ 2( , )| = ] true martingale, where r t X t Xt x dx dt ∼ 1 2 Q Q 2 ∼ ∼ = K f (K )E [σ (t, X t )|Xt = K ]dt. (14) T μ ( , ) T μ2( , ) 2 Xt r ( ) ≡ − r t Xt − 1 r t Xt , L X T exp ∼ dWt ∼ dt Q Q 2 2 ∂ E [( − ) ]=−E [ ] 0 σ (t, X ) 0 σ (t, X ) Recalling that K Xt K 1{Xt >K } 1{Xt >K } , r t r t Q Q and −∂K E [1{X >K }]=E [δ(Xt − K )] we arrive at the t and Breeden–Litzenberger formula T T Lχ (T ) ≡ exp − θ dW¯ − 1 |θ |2dt . t t 2 t Q 2 r 0 0 f (K ) = ∂ C (t, K ). (15) Xt KK 0 §To be precise, the contrived part is the assumption that the call trades Combining equations (13), (14) and (15), we thus have that at zero volatility; less so that we hedge it at zero volatility. The latter r ∼ corresponds to a so-called stop-loss strategy,seeCarr and Jarrow dC0 1 2 r 2 Q 2 (t, K ) = ∂ C (t, K )K E [σ (t, X t )|Xt = K ], (1990). dt 2 KK 0 r The Fundamental Theorem of Derivative Trading 521 which using the change of notation† t = T amounts to the hedge error is further complicated under the model error frame- celebrated Dupire–Gyöngy–Derman–Kani formula work of the Fundamental Theorem. Stepping stones towards ∼ ∂ r ( , ) answering this question are found in Andreasen (2003) and Q 2 T C0 T K E [σ (T, X T )|XT = K ]= , (16) Davis (2010) both of whom consider a mono-dimensional r 1 K 2∂2 Cr (T, K ) 2 KK 0 implied-hedge scenario with perfect information about the jump –seeDerman and Kani (1998), Dupire (1994), Gyöngy (1986). diffusion component. Our contribution is to generalize their Conceptually, the important point here is that the right-hand results to a multi-dimensional framework with arbitrary (mis)- side, modulo some amount of interpolation,‡ is empirically specifications for the volatility and jump distribution. For an measurable, whence (16) provides a way of calibrating the overview of multi-dimensional jump-diffusion theory, we refer volatility surface to observed call option prices in the market. the reader to the appendix. Specifically, the formula enables us to square a local diffusion Suppose the real dynamics of the underlying price process model with the infamous skew/smile effect of implied volatil- obeys ities across different values for the strike and time to maturity, ∼ ∼ = [μ ( , ) + σ ( , ) ]+ , whilst delicately sidestepping the deeper issue as to why this dXt DXt r t Xt dt r t Xt dW t DXt− dY t (17) phenomenon prevails. where {Y t }t≥0 is an n-dimensional vector of independent com- Remark 4 In Wittgensteinian terms, we must ‘throw away pound Poisson processes. Specifically, the jth component is given by the ladder’to arrive at this final conclusion,Wittgenstein (1922), j prop. 6.54. Hitherto, we have assumed that the real parameters Nt j = j , (r) are fundamentally unobservable, whilst the implied param- Yt Zk eters (i) are those we are exposed to in the market. Yet, no such k=1 distinction exists in the works of Dupire et al., whence the r { j } λ { j } where Nt t≥0 is an intensity- j Poisson process, and Zk k≥1 superscript in (16) really ought to be dropped. is a sequence of relative jump-sizes, assumed to be i.i.d. square- Remark 5 The above derivation is arguably unconventional integrable random variables with cumulative distribution function (cdf) ν j : R →[0, 1]. For shorthand, we shall refer to the and neither rigorous nor the quickest way to demonstrate (16). vectors λ = (λ1,λ2,...,λn) and ν = (ν1,ν2,...,νn) as In fact, the entire point of setting σi = 0 is essentially to extract + the intensity and cdf of Y t . the It¯o-(Tanaka) formula applied to (Xt − K ) , from which Derman et al.’s derivation takes its starting point. We keep the Oblivious to the true nature of (17), we imagine that pric- derivation here, as it provides a curious glimpse into how two ing and hedging should be performed (with obvious notation) φ, λQ, νQ, σ ( , ), Q Q philosophically quite distinct theorems can be interconnected.§ under the tuple h h h t Xt , where is the risk neutral measure T Q = φ • d φ,λQ,νQ exp s dW s 3. The Gospel of the Jump 0 T n − 1 |φ |2 − (λQ − λ ) Following remark 2, it is worthwhile exploring how the Funda- 2 s ds h, j h, j T mental Theorem can be adapted to new terrain. For instance, it 0 j=1 is well known that Brownian motion in itself does not j Q Q n Nt λ ν ( j ) h, j d h, j Zk adequately capture the sporadic discontinuities that emerge in × dPh, (18) stock price processes. Hence, it is opportune to scrutinize the λ ν ( j ) j=1 k=1 h, j d h, j Zk effect of a jump diffusion process, which in turn will give {φ } rise to another valuable lesson on the profitability of imperfect such that t t≥0 is a bounded adapted n-dimensional process λQ, νQ hedging. (the so-called Girsanov kernel), and h h , respectively, rep- Q Already, it is a well-known fact that exact hedges generally resent the jump intensity and jump-size distribution under . ( ) do not exist in a jump economy where the true dynamics of Specifically, the price of an option with terminal pay-off g X T is determined as the underlying is perfectly disseminated (see Merton (1976) Q − T or the expositions by Privault (2013) and Shreve (2008)). It h = E [ t ru du ( )|F X ], Vt e g X T t is thus of some theoretical interest to see how this preexisting with the underlying supposedly driven by = [ ι + σ ( , ) Q] †We do this to emphasize that t is the maturity of the option (not its dXt DXt rt dt h t Xt dW t value at time t). Q + D [ Y − λ ◦ E Q [Z ]], ‡Exactly how to do this extrapolation has turned out to be sufficiently Xt− d t h ν 1 non-trivial to spurn numerous papers and successive quant-of-the- with Z = (Z 1, Z 2,...,Z n), and Q has been specified such year awards a-decade-and-a-half later, see Andreasen and Huge 1 1 1 1 (2011) (pure local volatility), Guyon and Henry-Labordère (2012) that (decorated stochastic volatility models). Q μ (t, Xt ) + λ ◦ EνQ [Z1]+σ h(t, Xt )φ = rt ι, (19) §Our motivation for establishing this curious connection stems from h h t Andreasen (2003) who declares the Dupire–Gyöngy–Derman–Kani is satisfied almost everywhere.¶ formula to be a corollary of the Fundamental Theorem (without proof). Whether this take on the situation ultimately stems from Q Dupire himself is perhaps dubious, but it is interesting to note that ¶It should be clear the is not uniquely determined. In fact, for (19)to λ = λQ = a variant of the Fundamental Theorem has appeared in his work at admit only one solution, we would require that either (i) h h 0 least since early 2003 (Dupire 2003). (there are no jumps), in which case we recover the standard Girsanov 522 S. Ellersgaard et al.

Remark 6 We emphasize that (18) is a risk neutral measure transformation of the hedge dynamics with the associated measure Ph. This is to be contrasted with example 2 in section 2.4 in which Q is the risk neutral measure of the real dynamics. value of option

Theorem 2 The Fundamental Theorem of Derivative Trad- value of 1,2 n Δ -hedge ing with Jumps Let Vt = V (t, Xt ) ∈ C ([0, T ]×R ) be the price process of a European option with terminal pay- off VT = g(X T ), the underlying of which follows the real Value dynamics (17). We assume the option trades in the market at the (not necessarily uniquely determined) implied volatility σ i = σ i (t, Xt ). Now suppose we at time t = 0 acquire i Δ X Δ X such an option for the implied price V0 and set out to - t t hedge our position. Said hedge is performed under the notion Q Q X φ, λ , ν , σ ( , ), Q t h h h t Xt , leading to the fair price process h Underlying Vt . Then the present value of the profit-&-loss we incur from holding such a portfolio over the interval T =[0, T ] is Figure 2. Suppose we -hedge a long position in an option with a T convex pricing function. Insofar as a jump in the underlying occurs, − t h = h − i + 1 0 ru du Xt → Xt ± Xt , it follows that the value of the option will exceed P< V0 V0 2 e 0 the value of the -position. Hence, our net P&L benefits from such ∼ 2 h an occurrence. Obviously, the converse will be true if we hold a short × tr[DX rh(t, Xt )DX ∇ V ]dt, t t xx t position in the option. T n − t + 0 ru du h( , ) e j Vt t Xt− 0 j=1 h = ∂ h +∇ h • (( ι − ) ◦ ) rt Vt t Vt x Vt rt qt Xt h j − X , − Z j ∂ V N 1 2 j t N x j t d t + [σ ( , ) ∇ σ ( , )] t 2 tr h t x DXt xxVt DXt h t X Q Q h n − λ E [ V (t, x)]| = h, j j t x Xt− + λQ E [ ( , ◦ (ι + ˆ j )) h, j νQ V t x e j Z1 − EQ[ i ]∂ h , j=1 X j,t− Z1 x j Vt dt (20) − ( , ) − j ∂ ( , )] . V t x x j Z1 x j V t x x=Xt− where h( , ) ≡ h( , ◦ (ι + ˆ j )) − h( , ), j Vt t Xt− V t Xt− e j Z j V t Xt− Combining these three expressions as above yields the Nt desired result. represents the change in value of the option when the underlying jumps in the jth component, and eˆ is a unit vector in Rn j Remark 7 The last four lines in (20), which we denote by s.t. [eˆ j ]k = δ j,k. P&LJ , represent the present-valued profit-&-loss brought about Sketch Proof The proof runs in parallel with that of theorem by our inability to hedge the jump risk completely. A more 1. Specifically, the analogue of expression (6)is compact way of writing this result is attained by considering the ( × j ) h i h cont. associated Poisson random measure J j dt dz with intensity d = dV −∇x V • (dX − (rt ι − q ) ◦ Xt dt) j j t t t t t measure E[J j (dt × dz )]=λ j dtdν j (z ) for j = 1, 2,...,n. −∇ h • − i , x Vt dY t rt Vt dt Specifically, upon defining the pseudo-compensated random cont. measure where dXt is the continuous part of (17) i.e. ∼ ∼ ˜ ( × j ) ≡ ( × j ) − λQ νQ ( j ), cont. = [μ ( , ) + σ ( , ) ]. Jh, j dt dz J j dt dz h, j dt d h, j z (21) dXt DXt r t Xt dt r t Xt dW t for j = 1, 2,...,n, we see that the jump contribution to the Furthermore, in analogy with (7) and (8) we have the It¯o formula profit-and-loss may be written as ∼ ∼ T n h ={∂ h + 1 [σ ( , ) ∇2 h σ ( , )]} − t dVt t Vt tr r t Xt DXt xxVt DXt r t Xt dt = 0 ru du h( , ) 2 P&LJ e j Vt t Xt− +∇ V h • dXcont., 0 R = x t t j 1 n j h ˜ − z X , −∂ V J , (dt × dz ), (22) + [ h( , ◦ (ι + ˆ j )) − h( , )] j , j t x j t h j j V t Xt− e j Z j V t Xt− dNt Nt j=1 where ‘pseudo’ is used to emphasize that (21) is a real-world and the partial integro-differential equation for pricing pur- Poisson random measure compensated by a mis-specified poses intensity measure term: it is neither a martingale measure under P nor under the listed Q. Only if we remove parameter (λQ, νQ) = uncertainty on the jump parameters in the sense h h φ = σ −1( ι − μ ) σ = Q Q theorem with t h r h , or (ii) when h 0(thereare (λ , ν ) do we recover the Q martingale measure property. Q Q Q only jumps) and ν = ν = δ1 (the simple Poisson process case) in In particular, for the Merton specification (λ , ν ) = (λ, ν) λQ = ι − μ P which case h rt h. the compensated random measures become martingale mea- The Fundamental Theorem of Derivative Trading 523 sures, which in turn implies E[P&LJ ]=0 insofar as the as rt = μ + εt , where μ is the mean return, and εt has the integrand is square integrable. interpretation of a hetereoskedastic error. In particular, εt is Conceptually, the takeaway message from the formulation construed to be the product between a white noise process, (22) is that if V is convex in all of its components (a property zt ∼ N(0, 1) and a daily standard deviation, σt , which obeys it will inherit from the pay-off function under mild conditions) the relation then ∀ j : V >∂x V X j whence the integrand in P&LJ j |ε − | 2 ε − is positive. Thus, our hedge portfolio actually benefits from σ 2 = α + α σ 2 + α t 1 − + α t 1 , log t 0 1 log t−1 2 σ π 3 σ jumps in either direction of any of the underlying price process. t−1 t−1 Conversely, if we had shorted the option, the hedge profit (23) α ,α ,α α would obviously take a hit in the event of a jump (in Talebian where 0 1 2 and 3 are constants. The resulting volatility terms, holding a hedge portfolio with a short option position process is illustrated by the light grey curve in figure 3. corresponds to ‘picking pennies in front of a steam roller’).† A few remarks on the estimated volatility processes are in A vivid illustration of this point is provided in figure 2 for an order. First, we clearly see that volatility can change dramati- option written on a single underlying. cally during the lifetime of a portfolio. We also see that implied volatility typically is higher than actual volatility. This oft- reported result can be explained theoretically by the stochastic 4. Insights from empirics: on arbitrage and erraticism volatility having a market price of risk attached, see for instance Henderson et al. (2005). Finally, there is a clear negative cor- Inspired by Wilmott’s theoretical hedge experiment, we now relation between stock returns and volatility during the finan- look into the empirical performance of -hedging strategies cial turmoil which followed the Lehman default in September based on (I) forecasted implied volatilities and (II) forecasted 2008. All in all, reality unsurprisingly turns out to be a bit more actual (i.e. historical) volatilities. Specifically, we are inter- complicated than the set-up in Wilmott’s experiment. Still and ested in the properties of the accumulated P&L, insofar as we all, do its main messages carry over? To test this, we perform -hedge, till expiry, a three-month call-option on the S&P500 a hedge experiment with the following design: index, initially purchased at-the-money. We investigate a to- • For any given portfolio, we compute the daily implied tality of 36 such portfolios over disjoint intervals between July {σ imp}63 volatilities t t=1 and the daily actual volatilities 2004 and July 2013. This involves market data on both the {σ act}63 t t=1 as outlined above. We assume there are 63 underlying index and on options. Daily data on the S&P500 trading days over a three months period (labelled by index are readily and freely available. For option data, we t = 1, 2,...,63) and let St , rt and qt denote the time combine a 2004–2009 data-set from a major commercial bank‡ t value of the index, interest rate and dividend yield. with more recent prices from OptionMetrics obtained via the • For each of the two hedging strategies x ∈{σ imp,σact} Wharton Financial Database. σ act <σimp we do the following: If 1 1 we short the call Whilst ATM call option prices straightforwardly are (γ =−1); otherwise, we go along the call (γ =+1) obtained from the data-set, the (forecasted) implied and actual in accordance with the remark made in the section volatilities require a bit of manipulation. In case of the former, on Wilmott’s hedge experiment. Then, we set up the we define the daily implied volatility, over the lifetime of = − γBS( ) + delta neutral portfolio 1 B1 1 x1 S1 the portfolio, as the ATM implied volatility of corresponding imp γ CBS(σ ) s.t. = 0, where BS(x ) is the well- tenor obtained at the portfolio purchasing date (the resulting 1 1 1 1 1 known Black–Scholes delta. volatility process is illustrated by the black curve in figure 3). • For t = 2, 3,...,63 we do the following: compute the In case of the latter, we require a suitable volatility model time t value of the portfolio set up the previous day: fitted to historical data in order to predict the ‘actual’ volatility ˜ r − t BS q − t BS imp t = Bt−1e t 1 −γ (xt )St e t 1 +γ C (σ ). process. Specifically, we define the daily actual volatility, over t ˜ t t The quantity dP&L = − − defines the profit- the lifetime of the portfolio, as the conditional expectation t t t 1 &-loss accrued over the interval [t − 1, t]. Next, we of a volatility model which has been fitted to market data rebalance the portfolio such that it, once again, is delta- from the previous portfolio period. In this context, we observe neutral, = B −γBS(x )S +γ CBS(σ imp), where that models with lognormal volatility dynamics generally have t t t t t t t B is chosen in accordance with the self-financing con- more empirical support than, say, Heston’s model.§ The Expo- t dition: ˜ = . nential GeneralAutoregressive Conditional Heteroskedasticity t t • Finally, at the option expiry, we compute the termi- model (EGARCH(1,1)) has proven particularly felicitous in the nal P&L, as well as its lifetime quadratic variation, context of S&P 500 forecasting¶—a result we assume applies 63 |dP&L |2/63. universally for each of the 36 portfolios investigated. Thus, we t=1 t hold it to be the case that daily log returns, rt , can be modelled The 36 hedge error (or P&L) paths and the distributions of the quadratic variation of the two methods are shown in †Taleb (2010), p.19, ‘Most traders were just “picking pennies in front figure 4. Table 1 reports descriptive statistics and statistical of a steam roller,” exposing themselves to the high impact rare event tests of various hypotheses. yet sleeping like babies, unaware of it.’ First, we note (top panels figure 4) that even though ‡The bank shall remain nameless, but the data can be implied volatility typically is above actual volatility, this far downloaded from http://www.math.ku.dk/~rolf/Svend/http://www. math.ku.dk/~rolf/Svend/. from creates arbitrage. Hedge errors for the two methods read- §See Gatheral et al. (2014) and their references. ily become negative. A primary explanation for this is the ¶See Awartani and Corradi (2005). randomness of volatility. Our -hedged strategy only makes us 524 S. Ellersgaard et al.

1.75 Implied vol E-GARCH vol 1600 S&P500 Index 1.5 1400

1.25 1200

1000 1

800

Volatility 0.75

600 S&P 500 Index

0.5 400

0.25 200

0 0

Jul04Oct04Jan05Apr05Jul05Oct05Jan06Apr06Jul06Oct06Jan07Apr07Jul07Oct07Jan08Apr08Jul08Oct08Jan09Apr09Jul09Oct09Jan10Apr10Jul10Oct10Jan11Apr11Jul11Oct11Jan12Apr12Jul12Oct12Jan13Apr13

Figure 3. The top grey curve is the S&P500 Index plotted from July 2004 to July 2013 [units on right hand axis]. The tic-dates on the time { }36 axis have deliberately been chosen to match the purchasing dates ti i=1 of the 36 delta-hedged portfolios under investigation (each of which is of three months’duration). The light grey curve is the actual (stochastic) volatility estimated from a lognormal volatility model. Specifically, every time segment between purchasing dates [ti , ti+1) reflects a mean Monte Carlo simulated forecast based upon an EGARCH(1,1) fitted to market data from the previous time segment [ti−1, ti ). Finally, the black curve is the three-month ATM implied volatility. Specifically, every time segment between purchasing dates [ti , ti+1) is a static forecast based upon ATM implied volatility data from the purchasing date ti . Both volatility curves have their units on the left hand axis.

Hedging with Daily E-GARCH Volatilities Hedging with Daily Implied Volatilities 30 30

20 20

10 10

0 0 Mark-to-market (P&L) Mark-to-market (P&L) -10 -10

-20 -20

0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 Time [years] Time [years]

Figure 4. Panels (a) (actual) and (b) (implied) show the path-for-path hedge error behaviour for the 36 non-overlapping three-month hedges. Dotted paths correspond to cases where we initially take a long position in the option. a profit if realized volatility ends up ‘on the right side’of initial benchmark for the collateral that would need to be posted on a implied volatility. And that we don’t know for sure until after hedged short call option position. From column three in table the hedging period is over; we have to base our decisions on C1 the average option price is $49.2. Comparing this to the forecasts; initial forecasts even, for the fundamental theorem means (∼7.7; remember this is over a three-month horizon) and to apply. Notice though that the averages for both hedge errors standard deviations (∼15.5; ditto) of the hedge errors in table 1 are significantly positive. This shows that there is a risk pre- shows that the gains are also significant in economic√ terms. Put ( · 7.7 − . )/( · 15.5 ) mium that can be picked up, most often by selling options and differently, the crude calculation 4 49.2 0 02 4 49.2 -hedging them. Because the hedge is not perfect, this com- gives annualized Sharpe-ratios around 1. pensation is anticipated. The question is, is it financially signif- If we look just at the terminal hedge errors, then the differ- icant? In theory, the hedged portfolio has an initial cost of zero, ence in riskiness (as measured by standard deviation) between so it is not obvious how to define a rate of return, but the initial hedging with actual and hedging with implied volatility is in option price would seem a reasonable (possibly conservative) no way statistically significant (the p-value for equality of The Fundamental Theorem of Derivative Trading 525

Table 1. Summary statistics and hypothesis tests for different hedge strategies.

Quantity Mean (m) Std. Dev. (sd) Notes (hypotheses tests)

Hedge error, 7.7 17.3 The mean hedge error is statistically greater than zero actual volatility (p-value = 1%) when we hedge with the actual vol. forecast. Hedge error, 7.7 15.6 The mean hedge error is statistically greater than zero implied volatility (p-value = 1%) when we hedge with the implied vol. forecast. We cannot reject the hypothesis that the standard deviations sdact = sdimp are equal (p-value = 55%). Quadratic var., 1.2 2.1 actual volatility Quadratic var., 0.81 2.0 The mean quadratic variation of the hedge error when implied volatility we hedge with the actual vol. forecast is statistically less than the quadratic variation when we hedge with the implied vol. forecast (p-value = 1.4%).

variances is 55%). Also, the correlation between the terminal is in the positive; nonetheless, qua a significant dispersion the hedge error from the two approaches is 0.97. However, if we profit readily turns negative: the statistical arbitrage accord- consider the quadratic variations as the measure of riskiness, ingly relies on us being willing to take some significant hits then the picture changes. The average quadratic variation of the along the way. Indeed, this is without even factoring in the non- implied hedge error (0.81) is only two-thirds of the average negligible role of transaction costs. On the other hand, there quadratic variation of the actual hedge error (1.2) (a paired is strong evidence that hedging at the implied volatility does t-test for equality yields a p-value of 1.4%). yield smoother P&L paths. All in all this shows that volatility arbitrage is difficult, but One final remark: this paper can be seen as an exhaustive the following insight from Wilmott’s experiment stands: if you exposition of which volatility to use when delta hedging. To are in the business of hedging, then the use of implied volatility keep the length manageable and the presentation self-contained should make you sleep better at night. we have ignored an aspect that is of both theoretical and practical importance. It can be posed thus: Which delta should I use? In models where the underlying and the volatility are 5. Conclusion correlated a strong case can be made for using the so-called risk-minimizing delta, which is in broad terms is the usual delta In the world of finance, no issue is more pressing than that of plus the (underlying, volatility)-correlation times the volatility hedging our risks, yet remarkably little attention has been paid of volatility times the Vega of the target option, see for in- to the risk brought about by the possibility that our models stance Poulsen et al. (2009), Andreasen (2013), or Hull and might be wrong. To remedy this deplorable situation, we have White (2016). We leave the connection of this theory to the in this paper derived a meta-theorem that quantifies the P&L of Fundamental Theorem of Derivative Trading, theoretically as a -hedged portfolio with an erroneous volatility specification. well empirically, to future research. Meta- to the extent that one of the constituent parameters (the real volatility) is transcendental; yet, also a theorem with some very concrete ‘real world’ corollaries. For instance, a specific Acknowledgements case was investigated in which the implied volatility gives rise to smooth (i.e. O(dt)) P&L-paths, whilst any other hedge O( ) All authors gratefully acknowledge support from the Danish volatility yields erratic (i.e. dWt ) P&Lpaths. In a somewhat Strategic Research Council, Program Committee for Strate- quirkier context, the Dupire–Gyöngy–Derman–Kani formula gic Growth Technologies, via the research center HIPERFIT: for volatility surface calibration was shown to be a corollary. Functional High Performance Computing for Financial Infor- Whilst the theorem proved in section 2 is more general than mation Technology (hiperfit.dk) under contract number [10- the versions typically found in the literature, it does not go 092299]. far enough. Extensive empirical support has been added to the case of discontinuities in the stock price process: thus, in the Gospel of the Jump we extended the Fundamental Theorem Disclosure statement to include compound Poisson processes, which came with the revelation that jumps unambiguously hurt you when you try to No potential conflict of interest was reported by the authors. hedge short put and call option positions. One of the most conspicuous implications of the Funda- mental Theorem is undoubtedly the apparent ease with which Funding arbitrage can be made: e.g. in the constant parameter framework of Wilmott’s experiment, a free lunch is guaranteed in- This work was supported by the Danish Strategic Research Coun- {σ ,σ } cil, Program Committee for Strategic Growth Technologies, via the sofar as we can establish max r i (in case of the former, research center HIPERFIT: Functional High Performance Computing we go long on the option—otherwise, we short it). Studying for Financial Information Technology (hiperfit.dk) under contract this strategy empirically, we find that the mean P&L indeed number [10-092299]. 526 S. Ellersgaard et al.

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Q h < ≤ for the defined measure . To this end, define Y t as the vector Y t measurable functions. Suppose s t T then i with the upper limit of the summation, Nt , replaced by some fixed E [ ( ) ( − )]=E [ ( ) ( − )ξ ] i λQ,νQ f Y s g Y t Y s λ,ν f Ys g Yt Y s t number h ∈ N0 for all i. Then the RHS can be written as = Eλ,ν[ f (Ys)ξs]Eλ,ν[g(Y t −Y s)ξt /ξs] = Eλ,ν[ f (Ys)ξs]Eλ,ν[g(Y t −Y s)ξt ] ⎡ ⎤ = E Q Q [ f (Y )]·E Q Q [g(Y −Y )]. i λ ,ν s λ ,ν t s dy N Q Q d Q t λ ν ( i ) − y (λ −λ )T ⎢ i d i Zk ⎥ e i=1 i i Eλ,ν ⎣ f (Y ) ⎦ t λ ν ( i ) i=1 k=1 i d i Zk ⎛ ⎞ ∞ ∞ dy − dy (λQ−λ ) A.2. Measure changes in jump-diffusion dynamics = i= i i T ··· P ⎝ { i = i }⎠ e 1 NT h 1= dy = To appreciate the gravity of this result, consider the jump-diffusion ⎡ h 0 h =0 i 1 ⎤ dynamics of the n-dimensional stock price process d N i Q Q d ⎢ y t λ dν (Zi ) y ⎥ i i k i i = [μ( , ) + σ( , ) ]+ θ( , −) × Eλ,ν ⎣ f (Y t ) {N = h }⎦ dXt DXt− t Xt dt t Xt dWt DXt t Xt dY t λ ν ( i ) T = = i d i Z = (A1) i 1 k 1 k i 1 = ( , ,..., ) μ :[ , ∞)×Rn → where DX − diag X1,t− X2,t− Xn,t− , 0 t × × ∞ ∞ dy Rn σ :[, ∞) × Rn → Rn dw θ :[, ∞) × Rn → Rn dy − dy (λQ−λ ) , 0 and 0 . = i= i i T ··· P i = i e 1 NT h For the purposes of no-arbitrage pricing we want to determine the Q h1=0 hdy =0 i=1 measure such that the discounted process ⎡ ⎤ t dy i Q Q ∗ − h λ ν ( i ) := 0 ru du , h i d i Zk Xt e Xt × Eλ,ν ⎣ f (Y ) ⎦ t λ ν ( i ) i=1 k=1 i d i Zk is a martingale. From Ito’s lemma, it can readily be deduced that ∗ ∗ ∞ ∞ dy −λ dX = D [(μ(t, Xt ) − rt ι + θ(t, Xt )λ ◦ Eν[Z ])dt dy Q i T (λ )hi t Xt 1 − (λ −λ )T e i T = e i=1 i i ··· + σ (t, X )dW ] ki ! t t d = ∗ h1=0 h y =0 i 1 + D θ(t, X −)(dY − λ ◦ Eν[Z ]dt), ⎡ ⎤ Xt− t t 1 d i Q Q y h λ dν (Zi ) θ( , )λ ◦ E [ ] × E ⎣ ( h) i i k ⎦ where we have added and subtracted t Xt ν Z1 dt,where λ,ν f Y t dy λ ν ( i ) Z := (Z1, Z2,...,Z ) . Using the measure transformation in the i=1 k=1 i d i Zk 1 1 1 1 Q theorem above this transforms to ∞ ∞ dy −λ T Q e i (λ T )hi ∗ ∗ Q i dX = D [(μ(t, X ) − r ι + θ(t, X )λ ◦ E Q [Z ] = ··· t Xt t t t ν 1 ki ! Q h1=0 hn=0 i=1 + σ ( , )φ ) + σ ( , ) ] ⎡ ⎤ t Xt t dt t Xt dWt dy hi Q i ∗ Q dν (Z ) + θ( , −) ˜ . × E ⎣ ( h) i k ⎦ DX − t Xt dY t λ,ν f Y t t ν (Zi ) i=1 k=1 d i k ∗ Q φ , λQ, νQ, Q Hence, Xt is a -martingale iff the tuple t is chosen Q ∞ ∞ dy −λ T Q such that e i (λ T )hi = ··· i ··· Q μ( , ) + θ( , )λ ◦ E Q [ ]+σ( , )φ = ι, ki ! Rh1 Rhdy t Xt t Xt ν Z1 t Xt t rt (A2) h1=0 hdy =0 i=1 almost everywhere. Needless to say, the infinite number of tuples d i Q y h dν (zi ) which satisfies the no-arbitrage condition (A2) ruins our chances of × ( h) i k ν ( i ) f yt d i z establishing unique prices for financial derivatives depending on the dν (zi ) k i=1 k=1 i k underlying jump diffusion dynamics. This is obvious by recalling that Q ∞ ∞ dy −λ Q the discounted price process of V also should be a martingale, whence: i T (λ )hi e i T = ··· ··· T i ! 1 dy − ru du k Rh Rh V (t, Xt ) = E ,λQ,νQ e t g(XT ) , (A3) h1=0 hdy =0 i=1 t dy hi where g(XT ) is the terminal pay-off. × ( h) νQ( i ) f yt d i zk i=1 k=1 Appendix 2. PIDE methods ∞ ∞ dy = ··· Q i = i E ( h) Theorem 4 NT h λQ,νQ f Y t The Partial Integro-Differential Pricing Equation (PIDE) 1 n= = Consider a jump diffusion dynamics of the form (A1), and let V = h =0 h 0 i 1⎛ ⎞ t V (t, Xt ) be a derivative the value of which is contingent upon it. Let ∞ ∞ dy Q Q φ , λ , ν , Q be a tuple such that the no-arbitrage condition (A2) = ··· Q ⎝ {N i = hi }⎠ t T is satisfied. Then 1= dy i=1 h 0 h ⎡=0 ⎤ Q dX = D [r ιdt + σ(t, X )dW ] dy t Xt t t t ⎣ i i ⎦ Q × E Q Q ( ) { = } + θ( , )[ − λ ◦ E Q [ ] ]], λ ,ν f Y t NT h DXt− t Xt− dY t ν Z1 dt i=1 and Vt = V (t, x) satisfies the PIDE r V = ∂ V + r x •∇ V + 1 tr[σ (t, x)D ∇2 V D σ(t, x)] t t t t t x t 2 x xx t x dy + E λQ{ ( , ◦ (ι + θ ( , ) i )) − ( , )} νQ i V t x :,i t x Z1 V t x = which by the law of total probability corresponds to the LHS, i 1 EλQ,νQ [ f (Y T )] as desired. To show independence of increments Q −(Dxθ(t, x)λ ◦ Z1) •∇x V (t, x) , under Q,letξs = dQ/dP(s), and let f and g be two bounded 528 S. Ellersgaard et al.

∇ ∇2 ι := where x is the gradient operator, xx is the Hessian operator, ( , ,..., ) ∈ Rn θ ( , ) = ∂ ( , ) +∇ ( , ) • ( ι) 1 1 1 , and :,i t x denotes the ith column of the t V t Xt dt x V t Xt DXt rt dt θ( , ) = θ = I Q matrix t x . Particularly, when n dy and is the identity −∇ ( , ) • ( θ( , )λ ◦ E Q [ ]) x V t Xt DXt− t Xt− ν Z1 dt matrix then + 1 tr[σ (t, X )D ∇2 V (t, X )D σ (t, X )]dt = ∂ + •∇ + 1 [σ ( , ) ∇2 σ( , )] 2 t Xt xx t Xt t rt Vt t Vt rt x x Vt 2 tr t x Dx xxVt Dx t x dy n Q Q i + λ E Q (V (t, x ◦ (ι + λ E Q [V (t, x ◦ (ι + eˆ Z )) i ν i ν i 1 i=1 i=1 i i + θ:, (t, x)Z )) − V (t, x)) t − ( , ) − ∂ ( , )], i 1 x=X − d V t x xi Z1 xi V t x t +∇ ( , ) • ( σ ( , ) Q) where eî is a unit vector in the ith direction. x V t Xt DXt t Xt dWt dy Proof Suppose a jump occurs in the ith component of the compound i i + (V (t, Xt− ◦ (ι + θ:, (t, Xt−)Z ))−V (t, Xt−))dN i = i + i i N i t Poisson process Y t : Yt Yt− Zt . From the governing dynamics i=1 t = ◦ (A1), this means that the stock price process jumps by Xt Xt− Q i (ι + θ ( , ) i ) − λ EνQ (V (t, x ◦ (ι + θ:,i (t, x)Z )) − V (t, x)) = dt . :,i t Xt− Zt . Defining the continuous SDE i 1 x Xt− cont. := [ ι + σ( , ) Q] dXt DXt rt dt t Xt dWt Q Under Q, the expectations of the diffusion term and the compen- − D θ(t, X −)λ ◦ E Q [Z ] t], Xt− t ν 1 d sated jump terms (the last three lines) vanish. Furthermore, since we find by Ito’s lemma, ∗ − t := ru du ( , ), Vt e 0 V t Xt dV (t, Xt ) ∗ is a Q martingale; dV should be driftless. These facts jointly imply = ∂ ( , ) +∇ ( , ) • cont. t t V t Xt dt x V t Xt dXt that + 1 [σ ( , ) ∇2 ( , ) σ( , )] tr t Xt DXt xxV t Xt DXt t Xt dt − ( , ) + ∂ ( , ) + ι •∇ ( , ) 2 rt V t Xt t V t Xt rt DXt x V t Xt dy Q −∇ ( , ) • ( θ( , −)λ ◦ E Q [ ]) + ( ( , + ) − ( , )) i x V t Xt DXt− t Xt ν Z1 V t Xt− Xt V t Xt− dNt i=1 + 1 [σ ( , ) ∇2 ( , ) σ( , )] 2 tr t Xt DXt xxV t Xt DXt t Xt = ∂t V (t, Xt )dt +∇x V (t, Xt ) • (D rt ι)dt Xt dy Q Q i + λ E Q (V (t, x ◦ (ι + θ:, (t, x)Z ))−V (t, x)) = , +∇ V (t, X ) • (D σ(t, X )dW ) i ν i 1 x=X − 0 x t Xt t t = t Q i 1 −∇ V (t, X ) • (D θ(t, X −)λ ◦ E Q [Z ])dt x t Xt− t ν 1 which essentially is what we wanted to show. + 1 [σ ( , ) ∇2 ( , ) σ( , )] 2 tr t Xt DXt xxV t Xt DXt t Xt dt dy i i + (V (t, Xt− ◦ (ι + θ:, (t, Xt−)Z )) − V (t, Xt−))dN i N i t i=1 t The Fundamental Theorem of Derivative Trading 529

Appendix 3. Data

Table C1. The first column lists the purchasing dates of the 36 contracts. Column two shows the ATM strikes at which the contracts are purchased and column three show the prices at which this happens. The fourth column gives the terminal P&L for each contract, when the hedge is performed with an ‘actual’ (EGARCH(1,1)) volatility forecast. Column five likewise but when the hedge is with the implied N | |2/ = volatilities. Finally, columns six and seven give the quadratic variation, defined as i=1 dP&Li N,whereN 63 is the number of trading days, for the entire actual and implied paths, respectively.

actual implied actual implied Contract ATM strike Option price P< P< Q.V. Q.V. 07-Jul-2004 1118.3 36.5852 12.2045 15.1591 0.5269 0.2615 05-Oct-2004 1134.5 33.0392 5.8372 5.0520 0.1683 0.1386 05-Jan-2005 1183.7 34.7050 11.4080 13.6705 0.1975 0.1759 06-Apr-2005 1184.1 34.9985 7.3072 9.0917 0.3162 0.1693 06-Jul-2005 1194.9 34.4864 11.9818 10.5282 0.2974 0.0894 04-Oct-2005 1214.5 37.8141 7.2779 7.4261 0.5384 0.1670 06-Jan-2006 1285.4 37.1621 12.6952 12.4934 0.1539 0.1406 07-Apr-2006 1295.5 38.2703 0.0765 0.5022 0.2827 0.2444 07-Jul-2006 1265.5 45.5356 15.3714 13.7452 0.3655 0.1974 05-Oct-2006 1353.2 42.7682 12.6179 12.6400 0.0904 0.0945 08-Jan-2007 1412.8 45.4682 −5.0096 2.1741 2.4476 1.0569 09-Apr-2007 1444.6 47.0689 19.4885 7.4564 0.7699 0.0865 09-Jul-2007 1531.8 55.8378 −11.4976 −7.5524 1.8603 1.1396 05-Oct-2007 1557.6 63.1625 1.6451 −1.2115 1.2330 0.4542 09-Jan-2008 1409.1 74.2874 9.6117 9.6158 1.1975 0.6555 09-Apr-2008 1354.5 66.2276 17.3617 19.0049 0.8019 0.6270 09-Jul-2008 1244.7 62.8179 −56.9636 −47.0345 8.0872 10.4193 07-Oct-2008 996.2 83.8510 55.3847 51.8900 9.7721 6.4129 09-Jan-2009 890.3 69.9489 14.1892 3.2637 3.0947 0.4083 10-Apr-2009 856.6 62.9702 30.2400 27.2551 0.5701 0.4336 09-Jul-2009 882.7 49.8464 −12.4499 −9.8467 0.1245 0.1039 07-Oct-2009 1057.6 49.0640 17.0496 18.0507 0.2944 0.2135 07-Jan-2010 1141.7 42.2410 16.4989 16.4106 0.2595 0.1990 09-Apr-2010 1194.4 36.6784 −10.3121 −9.5031 0.5463 0.5578 09-Jul-2010 1078.0 52.2001 15.6833 17.6455 3.0501 0.3326 07-Oct-2010 1158.1 50.6050 20.7394 19.8607 0.1926 0.2166 06-Jan-2011 1273.8 43.6970 9.4015 11.7384 0.3762 0.2400 07-Apr-2011 1333.5 44.7866 13.3942 13.8116 0.3490 0.3055 08-Jul-2011 1343.8 43.0900 −3.8722 3.8883 0.1692 0.3196 06-Oct-2011 1165.0 73.4417 14.3245 16.8015 0.8601 0.7112 06-Jan-2012 1277.8 53.9770 −17.4158 −21.6739 0.3472 0.1853 09-Apr-2012 1382.2 48.9735 −9.7517 −9.9641 0.4760 0.4018 09-Jul-2012 1352.5 47.5814 15.8417 15.4475 0.3181 0.3184 05-Oct-2012 1460.9 42.9608 11.2648 9.0422 3.0925 0.8156 09-Jan-2013 1461.0 43.7355 17.6935 14.7094 0.2747 0.1360 11-Apr-2013 1593.4 39.2535 9.4000 6.6261 1.0037 0.7546