Cliquet Options and Volatility Models

Transcription

1 Clquet Optons and olatlty Models Paul Wlmott 1 Introducton Clquet optons are at present the heght of fashon n the world of equty dervatves. These contracts, llustrated by the term sheet below, are appealng to the nvestor because of ther protecton aganst downsde rsk, yet wth sgnfcant upsde potental. Cappng the mamum, as n ths globally floored, locally capped example, ensures that the payoff s never too extreme and therefore that the value of the contract s not too outrageous. From the pont of vew of the sell sde, amng to mnmze market rsk by delta hedgng, ther man exposure s to volatlty rsk. However, the contract s very subtle n ts dependence on the assumed model for volatlty. In ths bref note, I wll show how the contract value depends on the treatment of volatlty. In partcular, I shall show results for constant volatlty and volatlty ranges. The subtle nature of the clquet opton Tradtonally one measures senstvty to volatlty va the vega. Ths s defned as the dervatve of the opton value wth respect to a (usually constant) volatlty. Ths number s then used to determne how accurate a prce mght be should volatlty change. As part of one s rsk management, perhaps one wll vega hedge to reduce such senstvty. Ths s entrely reasonable when the contract n queston s an exchange-traded vanlla contract and one s measurng senstvty to the market s (mpled) volatlty. However, when t comes to the rsk management of exotc optons the senstvty to a constant volatlty s at best rrelevant and at worst totally msleadng. By now ths s common knowledge and I don t need to 78 Wlmott magazne

2 TECHNICAL ARTICLE dwell on the detals. It suffces to say that whenever a contract has a Gamma whch changes sgn (as does any nterestng exotc) vega may be small at precsely those places where senstvty to actual volatlty s very large. Confused? As a rule of thumb f you ncrease volatlty when Gamma s postve you wll ncrease a contract s value. At ponts of nflecton n the opton value (where Gamma s zero) the opton value may hardly move. But ths s senstvty to a parameter that takes the same value everywhere. What f you ncrease volatlty when Gamma s postve and decrease t when Gamma s negatve? The net effect s an ncrease n opton value even at ponts of nflecton. Skews and smles can make matters even worse, unless you are fortunate enough that your skew/smle model forecasts actual volatlty behavor accurately. The classcal references to ths phenomenon are Avellaneda, Levy & Para s (1995) and Lyons (1995) but also see Wlmott (). And the relevance to clquet optons? To see ths you just need to plot the formula ( ( max, mn Cap, S )) S 1 S 1 aganst S to see the non convex nature of the opton prce; Gamma changes sgn. Now comes the subtle part. The pont at whch Gamma changes sgn depends on the relatve move n S from one fng to the next. The pont of nflecton s not near any partcular value of S. The concluson has to be that any determnstc, volatlty surface model ftted to vanlla prces s not gong to be able to model the rsk assocated wth changng volatlty. Ths s true even f you allow the local volatlty surface to move up and down and to rotate. For ths reason we are gong to focus on usng the uncertan volatlty model descrbed n the above-mentoned references. In ths model the actual volatlty s chosen to vary wth the varables n such a way as to gve the opton value ts worst (or best) possble value. The actual volatlty s assumed to le n the range σ to σ +. The worst opton value s when actual volatlty s hghest for negatve Gamma and lowest for postve Gamma: { σ + f Ɣ< σ(ɣ) = σ f Ɣ>. Now let us look at the prcng of the clquet opton. 3 Path dependency, constant volatlty We wll be workng n the classcal lognormal framework for the underlyng ds = µs dt+ σ SdX. Assumng for the moment that volatlty s constant, or at most a determnstc functon of stock prce S and tme t, we can approach the prcng from the two most common drectons, Monte Carlo smulaton and partal dfferental equatons. A bref glance at the term sheet shows that there are none of the nastes such as early exercse, convertblty or other decson processes that make Monte Carlo dffcult to mplement. 3.1 Monte Carlo Monte Carlo prcng requres a smulaton of the rsk-neutral random walk for S, the calculaton of the payoff for many, tens of thousands, say, of paths, and the present valung of the resultng average payoff. Ths can be speeded up by many of the now common technques. Calculaton of the greeks s slghtly more tme consumng but stll straghtforward. 3. PDE To derve a partal dfferental equaton whch one then solves va, for example, fnte-dfference methods, requres one to work out the amount of path dependency n the opton and to count the number of dmensons. Ths s not dffcult, see Wlmott (). In all non trval problems we always have the two gven dmensons, S and t. In order to be able to keep track, before expry, of the progress of the possble opton payoff we also need the followng two new state varables where and S and, S = the value of S at the prevous fng = S = the sum to date of the bt nsde the max functon ( ( = max, mn Cap, S )) j S j 1. S j 1 j=1 Here I am usng the ndex to denote the fng just pror to the current tme, t. Ths s all made clear n the fgure. Snce S and are only updated dscretely, at each fng date, the prcng problem for (S, t, S, ) becomes t + 1 σ S + rs S S r = where r s the rsk-free nterest rate. In other words, the vanlla Black Scholes equaton. The twst s that s a functon of four varables, and must further satsfy the jump condton across the fng date (S, t, S, ) = and the fnal condton ( S, t +, S, + max (, mn (S, T, S, ) = max(, E ). (E 1, S S ))) Here E 1 s the local cap and E the global floor. (More general payoff structures can readly be magned.) Beng a four-dmensonal problem t s a toss up as to whether a Monte Carlo or a fnte-dfference soluton s gong to be the faster. S ^ Wlmott magazne 79

3 45 4 S However, the structure of the payoff, and the assumpton of lognormalty, mean that a smlarty reducton s possble, takng the problem down to only three dmensons and thus comfortably wthn the doman of usefulness of fnte-dfference methods. The smlarty varable s ξ = S S. The opton value s now a functon of ξ, t and. The governng equaton for (ξ, t, ) (loose notaton, but the most clear) s The jump condton becomes t + 1 σ ξ ξ + r ξ r =. ξ (ξ, t, ) = ( 1, t +, + max (, mn (E 1,ξ 1)) ) and the fnal condton s (ξ, T, ) = max(, E ). All of the results that I present are based on the fnte-dfference soluton of the partal dfferental equaton. The reason for ths s that I want to focus on the volatlty dependence, n partcular I need to be able to mplement the uncertan volatlty model descrbed above and ths s not so smple to do n the Monte Carlo framework. (The reason beng that volatlty depends on Gamma n ths model and Gamma s not calculated n the standard Monte Carlo mplementaton.) S t t 4 Results The followng results are based on the clquet opton descrbed n the term sheet. In partcular, t s a fve-year contract wth annual fngs, a global floor of 16% and local caps of 8%. The nterest rate s 3% and there are no dvdends on the underlyng. To understand the followng you must remember that the clquet value s a functon of three ndependent varables, ξ, and t. I wll be showng plots of value aganst varous varables at certan tmes before expry. These wll assume a constant volatlty. Then we wll look at the effect of varyng volatlty on the prces. 4.1 Constant volatlty In the followng fve plots volatlty s everywhere 5%. Ths plot shows the clquet value aganst and ξ at 4.5 years before expry. The contract has thus been n estence for sx months. At ths stage there have been no fngs yet and the state varable only takes the value. The non convex contract value can be clearly seen. At 3.5 years before expry, and therefore 1.5 years nto the contract s lfe, the value s shown n the next fgure, above. The state varable now ranges from zero to 8%. One year later, see the next fgure above, the contract s exactly half way through ts lfe. The state varable les n the range zero to 16%. For small values of the opton value s very close to beng the present value of the 16% floor. Ths represents the small probablty of gettng a payoff n excess of the floor at expry. After beng n estence for 3.5 years, and havng only 1.5 years left to run, the clquet value s as above. Now ranges from zero to 4%. When s zero there s no chance of the global floor beng years before expraton years before expraton Wlmott magazne

4 TECHNICAL ARTICLE.5 years before expraton years before expraton.5 years before expraton exceeded and so the contract value there s exactly the present value of 16%. Sx months before expry the opton value s as shown above. ranges from zero to 3% and for any values below 8% the contract s agan only worth the present value of 16%. 4. Uncertan volatlty The above shows the evoluton of the opton value for constant volatlty. There s dffuson n the ξ drecton and a jump condton to be appled at every fng. The amount of the dffuson s constant. (Or rather, s constant on a logarthmc scale.) To prce the contract when volatlty s uncertan we must use a volatlty that depends on (the sgn of) Gamma. Some results are shown below. In the next fgure s plotted the contract value aganst ξ at fve years before expry wth =. Fve calculatons have been performed. 1. The frst lne to examne s the fuscha-colored lne n the fgure, the mddle lne. Ths corresponds to a constant volatlty of 5%. Ths s the base case wth whch we compare other prces.. The second lne to examne s the blue one, close to the mddle lne. Ths s the clquet value wth a constant volatlty of just %. 3. The thrd case has a constant volatlty of 3%. Ths s the orange lne n the fgure. 4. The fourth lne s green, representng the clquet value when the volatlty s allowed to range between and 3%, takng a value locally that mamzes the clquet value overall 5. The ffth and fnal curve s the purple one for whch volatlty has agan been allowed to range from to 3% but now such that t gves the opton ts lowest possble value..4 Uncertan versus constant volatlty σ = % σ = 5% σ = 3% % < σ < 3% (Best) % < σ < 3% (Worst) ^ Wlmott magazne 81

5 The frst observaton to make s how close the constant volatlty curves are,.e. curves 1 3. As stated above, a good rule of thumb s that hgh volatlty and postve Gamma gve a hgh opton value. Because Gamma changes sgn n ths contract a result of ths s that there s a ξ value at whch the contract value does not appear to be senstve to the volatlty. In ths case the value s around.95, close to the pont of nflecton. Now ask yourself the followng queston. Do I beleve that volatlty s a constant, and ths constant s somewhere between % and 3%? Or do I beleve that volatlty s hghly uncertan, but s most lkely to stay wthn the range % to 3%? If you beleve the former, then the calculaton we have just done, n curves 1 3, s relevant. If, on the other hand, you thnk that the latter s more lkely (and who wouldn t?) then you must dscard the calculatons n curves 1 3 and consder the whole spectrum of possble opton values by lookng at the best and worst cases, curves 4 and 5. Such calculatons show that the real senstvty to volatlty s much, much larger than a nave vega calculaton would suggest. The next table shows how the clquet value (fve years before expry at = and ξ = 1) vares wth the allowed range for volatlty. The table s to be read as follows. When volatlty takes one value only, read along the dagonal, the orange cells, to see the contract values. For example, when the volatlty s % the contract value s And when the volatlty s 7% the contract value s.176. Now consder a range of possble volatltes. Suppose you beleve volatlty wll not stray from the range % to 7%. The worst case s s n the yellow cells, n ths case The best case s to be found n the clear cells,.183. So, when volatlty ranges from to 7% the correct range for the contract value s.1647 to.183. When volatlty s a constant, but a constant between % and 7%, the contract value range s =.13 or.75% relatve (to md prce) range. When volatlty s allowed to varyng over the 7% range we fnd that the contract value tself has a value range of =.183 or 1.5% relatve (to md prce) range. The true senstvty to volatlty s 14 tmes greater than that estmated by vega. 5 Code sample: Clquet wth uncertan volatlty, n smlarty varables Below s some sual Basc code that can be used for prcng these clquet optons n the uncertan volatlty framework. The range for volatlty s olmn to olmax, the dvdend yeld s Dv, rsk-free nterest rate IntRate, the local cap s Strke and the global floor Strke1. Expry s Expry. The numercal parameter s NumAssetSteps, the number of steps n the S and drectons. Ths program clearly leaves much to be desred, for example n the dscretzaton, the treatment of the jump condton etc. But t does have the beneft of transparency. A: The tmestep s set so that the explct fnte dfference method s stable. If the tmestep s any smaller than ths the method wll not converge. B: Here the payoff s set up, the dependent varable as a functon of the ndependent varables. C: The tmesteppng engne. Delta and Gamma are dscretzed versons of the frst- and second-order dervatves wth respect to S. Ths part of the code also treats the uncertan volatlty. See how the volatlty depends on the sgn of Gamma. D: The boundary condtons, for ξ = and large ξ. E: Updatng the next step back n the grd. F: Here the code tests for a fng date. G: Across fng dates the updatng rule s appled. Ths s really the only pont n the code that knows we are prcng a clquet opton. mn max REFERENCES Avellaneda, M, Levy, A & Para s, A (1995), Prcng and hedgng dervatve securtes n markets wth uncertan volatltes. Appled Mathematcal Fnance Avellaneda, M & Para s, A (1996), Managng the volatlty rsk of dervatve securtes: the Lagrangan volatlty model. Appled Mathematcal Fnance Lyons, TJ (1995), Uncertan volatlty and the rsk-free synthess of dervatves. Appled Mathematcal Fnance Wlmott, P (), Paul Wlmott on uanttatve Fnance. John Wley & Sons Ltd 8 Wlmott magazne

6 TECHNICAL ARTICLE Opton Explct Functon clquet(olmn, olmax, Dv, IntRate, Strke1, Strke, NumFxes, Fng, Expry, NumAssetSteps) ReDm (-NumAssetSteps To NumAssetSteps) Dm max, AssetStep, TStep, Step, Delta, Gamma, Theta, Tm, ol, qafter, frac, 1, As Double Dm, j, k, M, after, kafter, N, NumSoFar, NumSteps As Integer ReDm jtest(1 To NumFxes) As Integer max = Applcaton.Max(olMn, olmax) AssetStep = 1 / NumAssetSteps TStep =.95 * AssetStep ^ / max ^ / ^ ' Ths ensures stablty of the explct method M = Int(Expry / TStep) + 1 TStep = Expry / M Step = AssetStep NumSteps = Int(Strke / Step) * NumFxes A ReDm ( To NumSteps) ReDm Old(-NumAssetSteps To NumAssetSteps, To NumSteps) ' Frst dmenson centred on = 1 ReDm New(-NumAssetSteps To NumAssetSteps, To NumSteps) NumSoFar = 1 For j = 1 To NumFxes - 1 jtest(j) = Int(j * Fng / TStep) ' Used n testng whether fng date has been passed Next j For k = To NumSteps (k) = k * Step For = -NumAssetSteps To NumAssetSteps () = 1 + AssetStep * ' = corresponds to = 1 Old(, k) = Applcaton.Max(Strke1, (k) + _ Applcaton.Max(, Applcaton.Mn(Strke, () - 1))) ' Payoff Next Next k For j = 1 To M For k = To NumSteps For = -NumAssetSteps + 1 To NumAssetSteps - 1 Delta = (Old( + 1, k) - Old( - 1, k)) / / AssetStep ' Central dfference Gamma = (Old( + 1, k) - * Old(, k) + Old( - 1, k)) / AssetStep / AssetStep ol = olmax If Gamma > Then ol = olmn ' olatlty depends on Gamma n the uncertan volatlty model Theta = IntRate * Old(, k) -.5 * ol * ol * () * () * Gamma _ - (IntRate - Dv) * () * Delta ' The Black-Scholes equaton New(, k) = Old(, k) - TStep * Theta Next New(-NumAssetSteps, k) = Old(-NumAssetSteps, k) * (1 - IntRate * TStep) ' Boundary condton at = New(NumAssetSteps, k) = New(NumAssetSteps - 1, k) ' Boundary condton at = nfnty. Delta = For = -NumAssetSteps To NumAssetSteps Old(, k) = New(, k) Next Next k If jtest(numsofar) = j Then ' Test for a fng date For = -NumAssetSteps To NumAssetSteps For k = To NumSteps qafter = (k) + Applcaton.Max(, Applcaton.Mn(Strke, () - 1)) ' The updatng rule kafter = Int(qafter / Step) frac = (qafter - Step * kafter) / Step 1 = = If kafter < NumSteps Then 1 = New(, kafter) = New(, kafter + 1) End If Old(, k) = (1 - frac) * 1 + frac * ' The jump condton. Lnear nterpolaton Next k Next NumSoFar = NumSoFar + 1 End If E B F G C D Next j clquet = Old ' Output the whole array W End Functon Wlmott magazne 83