Pricing Variance Swaps with Cash Dividends

Transcription

1 Prcng Varance Swaps wth Cash Dvdends Tmothy Klassen Abstract We derve a smple formula for the prce of a varance swap when the underlyng has cash dvdends. 1 Introducton The last years have seen renewed nterest n the modelng of dvdends for equty dervatves. Ths s partally due to the general ncrease n sophstcaton and maturty of the market, partally due to the ncreased dvdend yelds n the wake of the burstng of the dotcom bubble. More recently, the deflaton of the credt/housng bubble led to an ncreased awareness of dvdend rsk. There are two man questons relevant for dvdend modelng: One s the proper mx of stcky (cash-lke) and proportonal (yeld-lke) dvdends one should assume, the other s what exactly a sutable process for the stock prce would be that ncorporates the dvdend assumpton. Note that wthout answerng both of these questons one can not even begn to talk about nferrng some mpled volatlty from a market prce for a vanlla opton. Note also that ths queston s just as relevant for ndces as for sngle stocks just because the dvdends of an ndex come n frequent small amounts does not mean that they can be treated as yeld-lke. As far as vanlla opton prces are concerned, one could argue that dvdend (and volatlty) modelng s effectvely just an nterpolaton/extrapolaton algorthm between lqud and less lqud parts of the market. To some extent ths s true, but even then, mproper dvdend modelng can lead to problems when extrapolatng from the lsted to the OTC market (whch mght also mean a swtch from Amercan to European style exercse). Furthermore, the value of greeks depend on the dvdend modelng assumpton, even when dfferent models are perfectly calbrated to the same prces. Perhaps most mportantly, exotc structures can have very sgnfcant senstvty to the dvdend model (see eg [1]). Ths s even true for mld exotcs lke dvdend or varance swaps. The purpose of ths note s to dscuss ths ssue for varance swaps. The standard model used n the market follows Derman et al [2] and reles on the now well-known opton strp formula that (essentally) statcally replcates the varance swap n terms of vanllas. Its dervaton, whch wll be revewed as part of the generalzaton below, assumes that all dvdends are yeld-lke. In practce, even though the opton strp formula s the startng pont, one fnds that varance swaps always seem to trade below the opton strp prces. Varous recpes are used by dfferent dealers to cut off the opton strp to match the market. Practtoners have been debatng for some tme whether ths s purely a supply-demand ssue, or whether there are other, more fundamental, contrbutng factors. Ths motvated us to explore the effect of cash dvdends on varance swap prcng. We present a smple formula for the far value of a varance swap n terms of market prces of vanllas and 1 Electronc copy avalable at:

2 dvdends. The only other publshed research n ths drecton we are aware of s [3], where a rough approxmate formula s gven for the case of one cash dvdend and assumng rates, skews and volatlty term-structure are weak. We consder completely general dvdends, rates and volatlty skews and derve an exact formula to frst order n the (effectve) dvdend yeld. We start by layng out our notaton and assumptons. 2 Dvdends, Forwards, and SDEs Relatve to today, t =, let us denote by t > the ex-dvdend dates on whch the underler S t drops by some amount D. For a gven tme horzon T we wll ndex the dvdends as = 1,..., N T. The dvdend can be a mx of a cash amount, d, and a proportonal amount δ S t, vz. Here t denotes an nfntesmal nstant before tme t (at t = t the dvdend jump D = d + δ S t has already occurred). We also allow an arbtrary contnuous dvdend yeld va the drft µ t of the underler, whch one can thnk of as µ t = r t q t, where r t s the dscount rate and q t the dvdend yeld (any fundng spread can be absorbed n q t ). Concernng the frst queston of the ntroducton, the smplest reasonable assumpton about the mx of cash and proportonal dvdends s to assume a smooth (or stepwse) transton from pure cash dvdends, d >, δ =, to purely proportonal dvdends d =, δ >. Ths smply expresses the ntuton that dvdends tend to be stcky n the short term, but n the long run t s more reasonable to assume that they go up or down wth the stock prce. Ths assumpton seems sensble and s smple enough to allow for an analytc expresson for forward prces F t = E[S t ]. Namely, notng that between dvdend dates the forward grows wth rate µ t, and at a dvdend date drops by a factor 1 δ and a cash amount d we can wrte: t F t = exp( µ t dt) F t 1 (1 δ ) d t 1 (1) Ths recurson has the soluton F T = f p (T ) S :t T d (2) f p (t ) where we ntroduced the proportonal forward factor f p (t) := exp( t µ t dt) (1 δ ) (3) :t t whch s the proper dscount (growth) factor to use for the cash-dvdends (stock). Note that t s also the delta of the forward. The tme scale(s) of the cash-to-proportonal dvdend transton can be estmated ether from hstorcal data about stock prces and dvdends, or by observng how forwards of dfferent maturtes move when the spot S moves. The next queston s how exactly the stock evolves. There seem to be three man models n use among practtoners. We can present them n a somewhat unfed manner as follows. Wrte the stock prce as S t = (F t t )X t + t, (4) 2 Electronc copy avalable at:

3 n terms of a postve martngale X t wth expected value E[X t ] = 1. Here t s a dvdend-dependent shft, and ts precse form s what dstngushes the three models. Note that by constructon the above wll always lead to the correct forward, E[S t ] = F t. Partal Hybrd Model [4]: The dea here s to consder the stock as consstng of a rskless pece t, the dvdends up to maturty T, and the fluctuatng remander: t = t,t := f p (t) :t<t T d f p (t ) Note that T =, whereas s the full dscounted dvdend sum up to maturty, whch s why ths model s sometmes referred to as the spot-adjustment model. Full Hybrd Model [5]: In the partal hybrd model the shft t s actually maturty-dependent, whch means that (4) does not really descrbe one consstent process; nstead t descrbes a separate stock process for each maturty. A consstent model, as descrbed n [5], s obtaned by lettng the shft contan all dvdends from tme t to nfnty: t = t, = f p (t) :t >t d f p (t ) In ths model one can really thnk of the stock S t as a hybrd of a cash component, the dscounted value of all cash dvdends from t to nfnty (the cash dvdend stream therefore has to be cut off at some pont n the future), and the fluctuatng remander. Spot Model [1, 6]: Ths model s defned by the SDE ds t = µ t S t dt (d + δ S t ) δ(t t ) + σ t S t dw t (5) where δ(t) s Drac s delta functon. As ponted out n [6], an accurate approxmate soluton of the European vanlla prcng equaton for ths SDE can be wrtten n an ntutve manner n terms of a spot and strke adjustment to the standard Black-Scholes formula (vz, subtractng the near-term dvdends from the spot, and addng the far-term dvdends to the strke). Equvalently, n terms of eqn (4), ths approxmate soluton corresponds to t δk t, δk t = f p (t) :t t t t d f p (t ) where δk t s the aforementoned strke adjustment (aka the far-term dvdends ) for tme t. All three models satsfy the desred property that the stock drops precsely by the cash dvdend amount across a dvdend date, S t S t = t t = F t F t = d. Otherwse, each of these models has some theoretcal or practcal qurk, whch perhaps explans why none of them s unversally preferred. The partal hybrd model seems to be popular among, for example, optons market makers (or at least used to be n the past), but snce t does not correspond to one consstent stochastc process for all maturtes, t s dffcult to use n an ntegrated vanlla and exotc busness where such consstency s mportant. The other two models do not have ths problem. The full hybrd model s a consstent and sensble approach. It does requre one to mantan a fnte tme scale for the transton from cash 3

4 to proportonal dvdends, otherwse the t can get very large, leadng to large mpled volatltes (snce they only apply to the pece S t t ). However, even wth reasonable dvdend assumptons can mpled volatltes (and deltas) be sgnfcantly dfferent from the more Black-Scholes-lke values of the other models, whch mght make some people uncomfortable. One mght also be concerned that n ths model mpled volatltes for a gven maturty depend on what one assumes about the cash-dvdend stream after maturty. Ths s unlke the spot model, but fully consstent wth the logc of the hybrd model. The spot model as defned above has the ssue that there s a non-zero probablty for the stock to go negatve at some dvdend date. In practce ths s not much of a problem [6], except for very large maturtes, f the cash dvdend stream s not sutably truncated. One could modfy the SDE to floor the stock at zero, whch one wll effectvely do anyhow f one uses a numercal method (fnte dfference or tree) to solve the prcng PDE rather than use the spot-strke-adjustment approxmaton. 1 In ths modfed verson, or n the spot-strke adjustment approxmaton, the spot model s perhaps the most wdely used approach. For the purpose of prcng varance swaps wth cash dvdend the spot model s the most convenent. Compared to the hybrd models, one can easly obtan analytcal results, as we now descrbe. 3 Prcng Varance Swaps The prce of a (newly ssued) varance swap s smply related to the expected value of the total varance up to some maturty T, defned by w T := ( ) ln 2 Sj+1 (6) S j j Here the sum s over all samplng dates (we reserve the ndex for sums over dvdend dates). In the usual contnuum lmt of frequent samplng (for a fxed physcal tme T ), we have w T σ 2 t dt + N T =1 ln 2 ( n terms of the (possbly stock-dependent and/or otherwse stochastc) local volatlty σ t of S t. The above holds up to terms down by factor of O( T ), n terms of the samplng tme nterval t, relatve to the ones shown. Note that the second term would be absent f the log-return n (6) were dvdend-corrected, e the numerator replaced by S j+1 +D j+1. In practce ths s n fact usually done for sngle stock varance swaps, but not for ndces. In any case, ths term s quadratc n the effectve dvdend yeld, and we wll only work to frst order n ths yeld, so we neglect t here (ts precse contrbuton s easy to work out along the lnes below; t s numercally small n realstc cases). From Ito s lemma for processes wth jumps we have d ln S t = ds(c) t S t 1 2 (ds (c) t ) 2 S 2 t + S t S t ) ( ) St δ(t t ) ln dt S t 1 Then, however, the expresson for the forward, eqn (2), does not hold exactly anymore for large maturtes. Wth sutably (realstcally) truncated cash dvdends t wll stll be qute accurate, though. (7) 4

5 where S (c) t s the contnuous part of the stock process. In the spot model, generalzed to allow σ t n (5) to be spot-dependent and/or stochastc, ths mples E[ln(S T /S )] = or, to the order we are workng, µ t dt 1 2 σ 2 t dt + N T =1 E[ln(1 D /S t )] 1 2 E[w T ] = 1 2 E[ σ 2 t dt] = µ t dt E[ln(S T /S )] + N T =1 E[ln(1 D /S t )] (8) The frst term on the rght-hand sde (rhs) can be calculated from eqn (1) as µ t dt = N T =1 ln F (t ) + d F (t 1 ) (1 δ ) + ln F T F tn (9) Notng that 1 D /S t = (1 δ )S t /(S t + d ), we see that the 1 δ factors n the frst and last term on the rhs of (8) wll cancel, and we are left to calculate E[ln(S T /S )] and a sum of terms of the form E[ln(S t /(S t + d ))]. To ths end recall a general result for the expected value of a (generalzed) functon of the stock prce, E[f(S T )] = f(s ) + (F T S )f (S ) + f (K) ˆV OTM (T, K) dk (1) Here S s a reference spot that can, n prncple be chosen to take any value. f, f denote the frst, respectvely, second dervatve of f, and ˆV OTM (T, K) denotes the un-dscounted value of the out-of-the-money vanlla opton (wth respect to S ) of maturty T and strke K. In other words, ˆV OTM (T, K) = { ˆP (T, K) f K < S Ĉ(T, K) f K > S n terms of un-dscounted call and put prces, Ĉ respectvely ˆP. The above s easly proved after expressng the expectaton value n terms of the mpled densty. It follows after a couple of partal ntegratons, that can be performed under mld assumptons about the decay of the densty n the wngs and the functon f (chefly that f s contnuous at S T = S, but the above formula can be generalzed to the case where even ths does not hold). The most convenent choce here s to take S to be the forward of the approprate maturty, so that the second term on the rhs of (1) vanshes. Puttng everythng together, usng that for f(s) = ln(s/(s +d)) we have f (S) = 1/S 2 +1/(S +d) 2 2d/S 3 as a legtmate approxmaton to the frst order n d/s we are workng, as well as notng the cancellaton of the terms nvolvng the forwards on the rhs of (8), we fnally get smply 1 2 E[w T ] = ˆV OTM (T, K) dk K 2 NT 2 =1 ˆV OTM (t, K) d K dk K 2 + O((d/S ) 2 ) (11) Ths generalzes the results of [2] where the second term s absent, whle stll expressng everythng n terms of observable vanllas prces and forwards (up to strke-maturty nterpolaton/extrapolaton ssues), and the cash-dvdend stream. In addton to the standard opton strp at maturty, t now 5

6 T (yrs) σ f (%) 2(σ f /σ f, 1)/T (%) n Table 1: Numercal results for far volatltes σ f for varous maturtes T, assumng a flat mpled volatlty of 2%, a $.2 dvdend pad every.1 years (wth the frst one at.5 years) and forwards of $1 at all dvdend dates. The thrd column should be q eff when the approxmaton σ f σ f, (1 1 2 q efft ) s accurate. The last column s the number of standard devatons (n terms of the ATM vol; chosen symmetrcally on the call and put sde) one has to use n the no-dvdend formula to match the cash-dvdend far volatlty. nvolves a strp for each cash-dvdend date. Note that the contnuous and dscrete proportonal dvdends do not explctly appear n the fnal result (to the order we are workng). Market quotes for varance swap prces are usually expressed n terms of a far volatlty σ f (aka far strke ), defned by E[w T ] = σf 2 T. In contrast to the conventon-ndependent total varance, t requres a choce of tme-conventon. The market conventon s busness tme ; we assume T has been chosen approprately. To gan some ntuton, note the followng. Each of the ntegrands n (11) s peaked around the forward of the relevant maturty (roughly, at least for small maturtes). The ntegrand s not perfectly symmetrc around ths peak, but at least approxmately so f the skew s weak. If there s just one cash dvdend d just before maturty, we have σ f σ f, (1 d/f T ), where σ f, denotes the far volatlty wth no cash dvdends. If there s just one dvdend at tme t before maturty, and rates, skews and term-structure are weak, then σ f σ f, (1 t d T F t ) (cf. [3]). As a fnal example, f there s a large number of cash-dvdends and they are roughly equally dstrbuted n tme, then under the same assumptons we have σ f σ f, (1 1 2 q efft ), n terms of the effectve dvdend yeld of the cash-dvdends, q eff. As seen n table 1, ths approxmaton s pretty accurate n the flat case for equally-spaced dvdends. For any real-lfe applcaton one should of course use the exact (to frst-order n q eff ) eqn (11). 2 Note that a larger skew wll ncrease the magntude of the correcton term, especally for larger maturtes. All the usual remarks about how to value varance swaps on the run, greeks, etc can be appled to (11). Fnally, a remark relevant for the market practce of cuttng off the opton strp. In table 1 we also show, for q eff = 2%, roughly the yeld of the SPX, the mpled number of standard devatons n one has to use n the no-dvdend formula to match (11). These numbers are not far from typcal market-mpled values for the SPX. They are qute nsenstve to the overall level of volatlty, but wth nonzero skew and rates they change somewhat, of course. A detaled study s beyond the scope of ths note (snce t requres a dscusson of how to extrapolate volatltes n the wngs, precse dvdend streams, how to transton from cash to proportonal dvdends, effect of stochastc nterest rates for very large maturtes, etc), and s left for future work. 2 Perhaps wth some bundlng of the cash dvdends for ndces (whle mantanng the forwards at the relevant maturtes), f one wants to avod dong too many one-dmensonal ntegrals. 6

7 4 Concluson We presented a smple formula, eqn (11), for the value of a varance swap when the underler has cash dvdends. It s exact to frst order n the cash-dvdend yeld. The effect of cash dvdends s szable and goes at least some way towards explanng (and perhaps obvatng) the market practce of mantanng a term-structure of mpled standard devatons at whch to cut off the opton strp n the standard zero-dvdend formula. Acknowledgement: I would lke to thank Hans Buehler and Jm Gatheral for dscussons. References [1] V. Frshlng, A Dscrete Queston, Rsk, Jan. 22, p [2] K. Demeterf, E. Derman, M. Kamal and J. Zou, A Gude to Varance Swaps, Rsk, Jun. 1999, p [3] Z. Duanmu, Constant Dollar Investment Strategy, talk at Global Dervatves, Pars, May 27. [4] J. Hull, Optons, Futures, and Other Dervatves, 6th edton, Pearson Prentce Hall, 26. [5] M. Overhaus et al, Equty Hybrd Dervatves, Wley, 27. [6] M. Bos and S. Vandermark, Fnessng Fxed Dvdends, Rsk, Sep. 22, p