Volatility trading and. volatility derivatives

Transcription

1 Volaly radng and volaly dervaves

2 Impled volales The only unobservable parameer n he Black-Scholes formulas s he volaly value, σ. By npung an esmaed volaly value, we oban he opon prce. Conversely, gven he marke prce of an opon, we can back ou he correspondng Black-Scholes mpled volaly. Requre roo fndng algorhms for he deermnaon of mpled volales. Several mpled volaly values obaned smulaneously from dfferen opons (varyng srkes and maures on he same underlyng asse provde he marke vew abou he volaly of he sochasc movemen of he asse prce.

3 Black wroe I s rare ha he value of an opon comes ou exacly equal o he prce a whch rades on he exchange. here are hree reasons for a dfference beween he value and prce: we may have he correc value, and he opon prce may be ou of lne; we may have used he wrong npus o he Black-Scholes formula; or he Black- Scholes may be wrong. Normally, all hree reasons play a par n explanng a dfference beween value and prce. Exreme vew pon: The marke prces are correc (n he presence of suffcen lqudy and one should buld a model around he prces.

4 Dfferen volales for dfferen srke prces Sock opons hgher volales a lower srke and lower volales a hgher srkes In a fallng marke, everyone needs ou-of he-money pus for nsurance and wll pay a hgher prce for he lower srke opons. Equy fund managers are long bllons of dollars worh of sock and wrng ou-of-he-money call opons agans her holdngs as a way of generang exra ncome.

5 Commody opons hgher volales a hgher srke and lower volales a lower srkes Governmen nervenon no worry abou a large prce fall. Speculaors are emped o sell pus aggressvely. Rsk of shorages no upper lm on he prce. Demand for hgher srke prce opons.

6 Volaly smles Ineres rae opons a-he-money opon has a low volaly and eher sde he volaly s hgher Propensy o sell a-he-money opons and buy ou-ofhe-money opons. For example, n he buerfly sraegy, wo a-he-money opons are sold and one-ou-of he-money opon and one n-he-money opon are bough.

7 Dfferen volales across me Supply and demand When markes are very que, he mpled volales of he near monh opons are generally lower han hose of he far monh. When markes are very volale, he reverse s generally rue. In very volale markes, everyone wans or needs o load wh gamma. Near-daed opons provde he mos gamma and he resulan buyng pressure wll have he effec of pushng prces up. In que markes no one wans a porfolo long of near daed opons. Use of a wo-dmensonal mpled volaly marx.

8 Floang volales As he sock prce moves, he enre skewed profle also moves. Ths s because wha was ou-of-he-money opon now becomes n-he-money opon. Example If an nvesor s long a gven opon and beleves ha he marke wll prce a a lower volaly a a hgher sock prce hen he may adjus he dela downwards (snce he prce apprecaon s lower wh a lower volaly.

9 Termnal asse prce dsrbuon as mpled by marke daa In real markes, s common ha when he asse prce s hgh, volaly ends o decrease, makng less probable for hgh asse prce o be realzed. When he asse prce s low, volaly ends o ncrease, so s more probable ha he asse prce plummes furher down. probably S sold curve: dsrbuon as mpled by marke daa doed curve: heorecal lognormal dsrbuon

10 Exreme evens n sock prce movemens Probably dsrbuons of sock marke reurns have ypcally been esmaed from hsorcal me seres. Unforunaely, common hypoheses may no capure he probably of exreme evens, and he evens of neres are rare and may no be presen n he hsorcal record. Examples 1. On Ocober 19, 1987, he wo-monh S & P 500 fuures prce fell 9%. Under he lognormal hypohess of annualzed volaly of 0%, hs s a 7 sandard devaon even wh probably (vrually mpossble.. On Ocober 13, 1989, he S & P 500 ndex fell abou 6%, a 5 sandard devaon even. Under he mananed hypohess, hs should occur only once n 14,756 years.

11 The marke behavor of hgher probably of large declne n sock ndex s beer known o praconers afer Oc., 87 marke crash. Impled volaly The marke prce of ou-of-he-money call (pus has become cheaper (more expensve han he Black- Scholes heorecal prce afer he 1987 crash because of he hckenng (hnnng of he lef-end (rgh-end al of he ermnal asse prce dsrbuon. 1.0 A ypcal paern of pos-crash smle. The mpled volaly drops agans X/S. X/S

12 Theorecal and mpled volales Theorecal volaly When valung an opon, a rader s heorecal volaly wll be a crcal npu n a prcng model. The sraegy of radng on heorecal volales nvolves holdng he opon unl expry common sraegy of opon users. Marke mpled volaly Volaly exrapolaed from, or mpled by, an opon prce. Tradng on mpled volaly nvolves mplemenng and reversng posons over shor me perods.

13 Prelmnary seps I s always necessary o provde prces of European opons of srkes and expraons ha may no appear n he marke. These prces are suppled by means of nerpolaon (whn daa range or exrapolaon (ousde daa range. A smooh curve s ploed hrough he daa pons (shown as crosses. The esmaed mpled volaly a a gven srke can be read off from he doed pon on he curve. Impled volaly X/S

14 Tme dependen volaly Gven he marke prces of European call opons wh dfferen maures (all have he srke prces of 105, curren asse prce s and shor-erm neres rae over he perod s fla a 5.6%. maury 1-monh 3-monh 7-monh Value Impled volaly 1.% 30.5% 19.4% Exend he assumpon of consan volaly o allow for me dependen deermnsc volaly σ(.

15 The Black-Scholes formulas reman vald for me dependen volaly excep ha s used o replace σ. T d T τ τ σ ( 1 How o oban σ( gven he mpled volaly measured a me of a European opon exprng a me. Now = mp d τ τ σ σ ( 1, (

16 so ha. (, ( ( = d mp σ τ σ τ Dfferenae wh respec o, we oban., (, ( (, ( ( mp mp mp + = σ σ σ σ

17 Praccally, we do no have a connuous dfferenable mpled volaly funcon, bu raher mpled volales are avalable a dscree nsans. Suppose we assume σ( o be pecewse consan over ( 1,, hen, ( mp σ,, ( ( (, ( (, ( ( mp mp d < < = = σ τ τ σ σ σ.,, ( (, ( ( ( mp mp < < = σ σ σ

18 Local volaly surface σ(s, The me dependence n mpled volaly can be urned no a volaly of he underlyng ha s me dependen. Queson: Can we deduce σ(s, from σ ( * mp, X, T?

19 Theorecal soluon Suppose a dsrbuon of European call prces of all srkes and maures are avalable, as denoed by V(X, T, we hen can oban he dependence of he volaly on srke and expry, based on he curren asse prce S * and curren me *. The rsk neural random walk for S s assumed o be ds S = rd + σ ( S, dz.

20 Impled volaly ree An mpled volaly ree s a bnomal ree ha prces a gven se of npu opons correcly. The mpled volaly rees are used: 1. To compue hedge parameers ha make sense for he gven opon marke.. To prce non-sandard and exoc opons. The mpled volaly ree model uses all of he mpled volales of opons on he underlyng - deduces he bes flexble bnomal ree (or rnomal ree based on all he mpled volales.

21 Volaly radng Tradng based on akng a vew on marke volaly dfferen from ha conaned n he curren se of marke prces. Ths s dfferen from poson radng where he rades are based on he expecaon of where prces are gong. Example A ceran sock s radng a $100. Two one-year calls wh srkes of $100 and $110 prced a $5.98 and $5.04, respecvely. These prces mply volales of 15% and %, respecvely. Sraegy Long he cheap $100 srke opon and shor of he expensve $110 srke opon.

22 Tradng volales Shor erm players Sensve o he marke prces of he opons. Ths s more of a speculave radng sraegy, applcable only o lqud opons markes, where he cos of radng posons s small relave o spreads capured n mpled volaly moves. Long erm players If a rader s heorecal value s hgher han he mpled volaly, he would buy opons snce he beleves hey are undervalued.

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24 Marke daa: Sock prce = $99, call prce = $5.46, dela = 0.5 porfolo A: 50 shares of sock; π A =0 = $4,550 porfolo B: 100 call opons; π B prof =0 = $546 sold lne: opon porfolo doed lne: sock porfolo sock prce 600 Boh porfolos are dela equvalen. Snce he opon prce curve s concave upward, he call opon porfolo always ouperforms he dela equvalen sock porfolo.

25 Long volaly rade Whchever way he sock prce moves, he holder always make a prof. Ths s he essence of he long volaly rade. By rehedgng, one s forced o sell n rsng markes and buy n fallng marke rade n he oppose drecon of he marke rend. Where s he cach The opon loses me value hroughou he lfe of he opon. Long volaly sraegy Compeon beween he orgnal prce pad and he subsequen volaly experenced. If he prce pad s low and he volaly s hgh, he long volaly player wll wn overall.

26 Vega rsk Vega s defned as he change n opon prce caused by a change n volaly of 1%. Shorer daed opons are less sensve of volaly npus. Tha s, vega decreases wh me. Near-he-money opons are mos sensve and deep ou-of-he-money opons are less sensve.

27 Gamma radng and vega radng Tme decay prof: Gamma radng (asse prce poson gamma (mpled volaly Ne prof from realzed volaly poson gamma (asse prce [(realzed volaly (mpled volaly ] Vega radng Ne prof from changes n mpled volaly vega (curren mpled volaly orgnal mpled volaly

28 Maury and moneyness The ably of ndvdual dervave posons o realze profs from gamma and vega radng s crucally dependen on he average maury and degree of moneyness of he dervaves book. For a-he-money opons, long maury opons dsplay hgh vega and low gamma; shor maury opons dsplay low vega and hgh gamma. For ou-of-he-money opons, long maury opons dsplay lower vega and hgh gamma, and shor maury opons hgher vega and lower gamma.

29 Balance beween gamma-based and vega-based volaly radng 1. If a rader desres hgh gamma bu zero vega exposure, hen a suable poson would be a large quany of shor maury a-he-money opons hedged wh a small quany of long maury a-he-money opons.. If a rader desres hgh vega bu zero gamma exposure, hen a suable poson would be a large quany of long maury a-he-money opons hedged wh a small quany of shor maury a-he-money opons.

30 Long gamma holdng a sraddle A rader beleves ha he curren mpled volaly of a-he-money opons s lower han he expecs o be realzed. He may buy a sraddle: a combnaon of an a-he-money call and an a-he-money pu o acqure a dela neural, gamma poson.

31 If hs predcon s correc, he can prof n wo ways: 1. The res of he marke begn o agree wh hm, hen he opon prce wll mark up. He gans by unwndng hs opon poson.. The marke connues o prce opons a 15%,. He keeps he porfolo dela neural (dela calculaed based on marke volaly. Hs rehedgng prof wll exceed he me decay losses. Tradng msprced opons If opons are offered a an mpled volaly of 15% and a manager beleves ha he real volaly s gong o be hgher n he fuure, say, 5%. How o prof? He should se up a dela neural porfolo.

32 Log conrac The selemen prce s gven by log S T. The presen value of he Log conrac durng s lfe s almos equal o 1 ( ISD σ me o maury, where ISD s he volaly mpled n he prce of he Log Conrac and σ s he realzed volaly.

33 Advanages of he Log conrac There s a need for smple opon produc ha enable nvesors o rade vews on fuure volaly. For he Log conrac: I s easy o dela hedge. A rader who s long one Log conrac wll dela hedge by shorng $1 worh of he underlyng asse. The performance of a dela hedged Log conrac depends only on he oucome volaly and no on he hedger s forecas of volaly. The dynamc sraegy s a sable sraegy ha depends only n he level of he asse prce, no on he me o maury. When dela hedged, s a pure volaly play, unlke a dela hedged opon.

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36 Varance swap conrac The ermnal payoff of a varance swap conrac s noonal (v srke where v s he realzed annualzed varance of he logarhm of he daly reurn of he sock. = = ln 1 n S S n N v µ

37 Varance swap conrac (con d where n = number of radng days o maury N = number of radng days n one year (5 µ = realzed average of he logarhm of daly reurn of he sock = 1 n n1 S+ 1 ln = = 0 S 1 n ln S S n 0.

38 The payoff could be posve or negave. The objecve s o fnd he far prce of he srke, as ndcaed by he prces of varous nsrumens on he rade dae, such ha he nal value of he swap s zero. Observe ha v = N n 1 ( ln n ln n S+ 1 S+ 1 S S.