Volatility and Hedging Errors

Transcription

1 Volailiy and Hedging Errors Jim Gaheral Sepember,

2 Background Derivaive porfolio bookrunners ofen complain ha hedging a marke-implied volailiies is sub-opimal relaive o hedging a heir bes guess of fuure volailiy bu hey are forced ino hedging a marke implied volailiies o minimise mark-o-marke P&L volailiy. All praciioners recognise ha he assumpions behind fixed or swimming dela choices are wrong in some sense. Neverheless, he magniude of he impac of his dela choice may be surprising. Given he P&L impac of hese choices, i would be nice o be able o avoid figuring ou how o dela hedge. Is here a way of avoiding he problem?

3 An Idealised Model To ge inuiion abou hedging opions a he wrong volailiy, we consider wo paricular sample pahs for he sock price, boh of which have realised volailiy = 0% a whipsaw pah where he sock price moves up and down by 1.5% every day a sine curve designed o mimic a rending marke

4 Whipsaw and Sine Curve Scenarios Spo Pahs wih Volailiy=0% Days

5 Whipsaw vs Sine Curve: Resuls P&L vs Hedge Vol. 60,000,000 Whipsaw 40,000,000 0,000,000 (0,000,000) - Hedge Volailiy 10% 15% 0% 5% 30% 35% 40% (40,000,000) (60,000,000) (80,000,000) Sine Wave P&L

6 Conclusions from his Experimen If you knew he realised volailiy in advance, you would definiely hedge a ha volailiy because he hedging error a ha volailiy would be zero. In pracice of course, you don know wha he realised volailiy will be. The performance of your hedge depends no only on wheher he realised volailiy is higher or lower han your esimae bu also on wheher he marke is range bound or rending.

7 Analysis of he P&L Graph If he marke is range bound, hedging a shor opion posiion a a lower vol. hurs because you are geing coninuously whipsawed. On he oher hand, if you hedge a very high vol., and marke is range bound, your gamma is very low and your hedging losses are minimised. If he marke is rending, you are hur if you hedge a a higher vol. because your hedge reacs oo slowly o he rend. If you hedge a low vol., he hedge raio ges higher faser as you go in he money minimising hedging losses.

8 Anoher Simple Hedging Experimen In order o sudy he effec of changing hedge volailiy, we consider he following simple porfolio: shor $1bn noional of 1 year ATM European calls long a one year volailiy swap o cancel he vega of he calls a incepion. This is (almos) equivalen o having sold a one year opion whose price is deermined ex-pos based on he acual volailiy realised over he hedging period. Any P&L generaed by his hedging sraegy is pure hedging error. Tha is, we eliminae any P&L due o volailiy movemens.

9 Hisorical Sample Pahs In order o preemp criicism ha our sample pahs are oo unrealisic, we ake real hisorical FTSE daa from wo disinc hisorical periods: one where he marke was locked in a rend and one where he marke was range bound. For he range bound scenario, we consider he period from April 1991 o April 199 For he rending scenario, we consider he period from Ocober 1996 o Ocober 1997 In boh scenarios, he realised volailiy was around 1%

10 FTSE 100 since Trend Range 0 1/1/85 5//86 10/10/87 /7/89 7/18/90 1/6/91 4/5/93 9/13/94 /1/96 6/1/97 11/9/98

11 Range Scenario FTSE from 4/1/91 o 3/31/ Realised Volailiy 1.17% /3/91 4//91 6/11/91 7/31/91 9/19/91 11/8/91 1/8/91 /16/9 4/6/9 5/6/9

12 Trend Scenario FTSE from 11/1/96 o 10/31/ Realised Volailiy 1.45% /3/96 10/1/96 1/1/96 1/0/97 3/11/97 4/30/97 6/19/97 8/8/97 9/7/97 11/16/97

13 5,000,000 0,000,000 15,000,000 10,000,000 5,000,000 - (5,000,000) (10,000,000) (15,000,000) (0,000,000) (5,000,000) P&L vs Hedge Volailiy Range Scenario 5% 7% 9% 11% 13% 15% 17% 19% 1% 3% 5% Trend Scenario

14 Discussion of P&L Sensiiviies The sensiiviy of he P&L o hedge volailiy did depend on he scenario jus as we would have expeced from he idealised experimen. In he range scenario, he lower he hedge volailiy, he lower he P&L consisen wih he whipsaw case. In he rend scenario, he lower he hedge volailiy, he higher he P&L consisen wih he sine curve case. In each scenario, he sensiiviy of he P&L o hedging a a volailiy which was wrong by 10 volailiy poins was around $0mm for a $1bn posiion.

15 Quesions? Suppose you sell an opion a a volailiy higher han 1% and hedge a some oher volailiy. If realised volailiy is 1%, do you make money? No necessarily. I is easy o find scenarios where you lose money. Suppose you sell an opion a some implied volailiy and hedge a he same volailiy. If realised volailiy is 1%, when do you make money? In he wo scenarios analysed, if he opion is sold and hedged a a volailiy greaer han he realised volailiy, he rade makes money. This conforms o raders inuiion. Laer, we will show ha even his is no always rue.

16 Sale/ Hedge Volailiy Combinaions Sale Vol. Volailiy P&L Hedge Vol. Hedge P&L Toal Range Scenario 13% +$3.30mm 10% -$3.85mm -$0.55mm Trend Scenario 13% +$.0mm 17% -$3.07mm -$0.87mm

17 P&L from Selling and Hedging a he Same Volailiy 80,000,000 60,000,000 Range Scenario 40,000,000 0,000,000 - Trend Scenario 5% 7% 9% 11% 13% 15% 17% 19% 1% 3% 5% (0,000,000) (40,000,000) (60,000,000)

18 Dela Sensiiviies Le s now see wha effec hedging a he wrong volailiy has on he dela. We look a he difference beween $-dela compued a 0% volailiy and $-dela compued a 1% volailiy as a funcion of ime. In he range scenario, he difference beween he delas persiss hroughou he hedging period because boh gamma and vega remain significan hroughou. On he oher hand, in he rend scenario, as gamma and vega decrease, he difference beween he delas also decreases.

19 150,000, ,000,000 50,000,000 - (50,000,000) (100,000,000) (150,000,000) Range Scenario $ Dela Difference (0% vol. - 1% vol.)

20 60,000,000 40,000,000 0,000,000 - (0,000,000) (40,000,000) (60,000,000) (80,000,000) (100,000,000) (10,000,000) Trend Scenario $ Dela Difference (0% vol. - 1% vol.)

21 Fixed and Swimming Dela Fixed (sicky srike) dela assumes ha he Black-Scholes implied volailiy for a paricular srike and expiraion is consan. Then C δ = BS FIXED Swimming (or floaing) dela assumes ha he a-hemoney Black-Scholes implied volailiy is consan. More precisely, we assume ha implied volailiy is a funcion of relaive srike K S only. Then δ SWIM = S C BS C BS σ BS C BS K C BS σ BS + = S σ S S S σ K BS BS

22 An Aside: The Volailiy Skew Volailiy SPX Volailiy 6-Apr % 80.00% 70.00% 60.00% 50.00% 40.00% 30.00% 4/15/99 5/0/99 6/17/99 9/16/99 1/16/99 3/16/00 6/15/00 1/14/ % 10.00% Srike

23 Volailiy vs x 90.00% 80.00% 70.00% 60.00% 50.00% 40.00% 30.00% 0.00% 10.00% 0.00% /15/99 5/0/99 6/17/99 9/16/99 1/16/99 3/16/00 6/15/00 1/14/00 x = ln ( K F ) T

24 Observaions on he Volailiy Skew Noe how beauiful he raw daa looks; here is a very welldefined paern of implied volailiies. K / F ) When implied volailiy is ploed agains, all of he skew curves have roughly he same shape. T ln(

25 How Big are he Dela Differences? We assume a skew of he form From he following wo graphs, we see ha he ypical difference in dela beween fixed and swimming assumpions is around $100mm. The error in hedge volailiy would need o be around 8 poins o give rise o a similar difference. In he range scenario, he difference beween he delas persiss hroughou he hedging period because boh gamma and vega remain significan hroughou. On he oher hand, in he rend scenario, as gamma and vega decrease, he difference beween he delas also decreases. σ 0. 3 BS = x T

26 140,000,000 10,000, ,000,000 80,000,000 60,000,000 40,000,000 0,000,000 - (0,000,000) Range Scenario Swimming Dela-Fixed Dela

27 140,000,000 10,000, ,000,000 80,000,000 60,000,000 40,000,000 0,000,000 - (0,000,000) Trend Scenario Swimming Dela-Fixed Dela

28 Summary of Empirical Resuls Dela hedging always gives rise o hedging errors because we canno predic realised volailiy. The resul of hedging a oo high or oo low a volailiy depends on he precise pah followed by he underlying price. The effec of hedging a he wrong volailiy is of he same order of magniude as he effec of hedging using swimming raher han fixed dela. Figuring ou which dela o use a leas as imporan han guessing fuure volailiy correcly and probably more imporan!

29 Some Theory Consider a European call opion sruck a K expiring a ime T and denoe he value of his opion a ime according o he Black-Scholes formula by. In + paricular, C T S K. We assume ha he sock price S saisfies a SDE of he form σ S bg b g BS = T where may iself be sochasic., ds S Pah-by-pah, we have = µ + σ, S d dz BSbg C

30 bg bg zt BS BS BS T = z 0 0 C T C 0 = dc = z T 0 R S T C S ds S BS BS S C BS + σ d + d C S R U ST C BS S ds 1 S C BS + S d VW ( σ ɵ σ ) S U V W where he forward variance ɵ.. = bg σ σ BS n s So, if we dela hedge using he Black-Scholes (fixed) dela, he oucome of he hedging process is: bg bg zt 1 BS BS 0 S 0 C T C = ( σ ɵ σ ) S C S BS d

31 In he Black-Scholes limi, wih deerminisic volailiy, dela-hedging works pah-by-pah because σ S = ɵ σ S In realiy, we see ha he oucome depends boh on gamma and he difference beween realised and hedge volailiies. If gamma is high when volailiy is low and/or gamma is low when volailiy is high you will make money and vice versa. Now, we are in a posiion o provide a counerexample o rader inuiion: Consider he paricular pah shown in he following slide: The realised volailiy is 1.45% bu volailiy is close o zero when gamma is low and high when gamma is high. The higher he hedge volailiy, he lower he hedge P&L. In his case, if you price and hedge a shor opion posiion a a volailiy lower han 18%, you lose money.

32 A Cooked Scenario

33 Cooked Scenario: P&L from Selling and Hedging a he Same Volailiy 0,000,000 15,000,000 10,000,000 5,000,000-5% 7% 9% 11% 13% 15% 17% 19% 1% 3% 5% (5,000,000) (10,000,000) (15,000,000)

34 Conclusions Dela-hedging is so uncerain ha we mus dela-hedge as lile as possible and wha dela-hedging we do mus be opimised. To minimise he need o dela-hedge, we mus find a saic hedge ha minimises gamma pah-by-pah. For example, Avellaneda e al. have derived such saic hedges by penalising gamma pah-by-pah. The quesion of wha dela is opimal o use is sill open. Traders like fixed and swimming dela. Quans prefer marke-implied dela - he dela obained by assuming ha he local volailiy surface is fixed.

35 Anoher Digression: Local Volailiy We assume a process of he form: ɵ σ, S ds S = µ d + σɵ, S dz wih a deerminisic funcion of sock price and ime. Local volailiies can be compued from marke prices of opions using σˆt,k C ˆ σ T, K = C T C K Marke-implied dela assumes ha he local volailiy surface says fixed hrough ime.

36 We can exend he previous analysis o local volailiy. bg bg zt L L L T = z 0 0 C T C 0 = dc = z T 0 R S T C S ds C S L L S C L + σ, d + d S R U ST C L S ds 1 S C L + S S d VW ( σ ɵ, σ, ) S So, if we dela hedge using he marke-implied dela L,he oucome of he hedging process is: C S U V W bg bg zt 1 L L 0 S S 0 C T C = ( σ ɵ, σ, ) S C S L d

37 Define Then b g C L Γ L S S S z T 1 L L 0 = ( σ S ɵ, σ, S ) Γ L 0 bg bg b g C T C S d b g Γ L S If he claim being hedged is pah-dependen hen is also b g pah-dependen. Oherwise all he Γ L S can be deermined a incepion. Wriing he las equaion ou in full, for wo local volailiy ~ surfaces Σ and Σ ˆ we ge z L L, S, S L 0 0 ~ T C C ɵ 1 Σ Σ = d z ds ( ~ σ ɵ σ ) E Γ S S ϕ S S d di di b g c h 0 Then, he funcional derivaive δ C δσ L ɵ, S Γb g 1 = E L S S ϕc S Sh 0

38 C L δ In pracice, we can se = 0 by bucke hedging. δσ, S Noe in paricular ha European opions have all heir sensiiviy o local volailiy in one bucke - a srike and expiraion. Then by buying and selling European opions, we can cancel he riskneural expecaion of gamma over he life of he opion being hedged - a saic hedge. This is no he same as cancelling gamma pah-by-pah. If you do his, you sill need o choose a dela o hedge he remaining risk. In pracice, wheher fixed, swimming or markeimplied dela is chosen, he parameers used o compue hese are re-esimaed daily from a new implied volailiy surface. Dumas, Fleming and Whaley poin ou ha he local volailiy surface is very unsable over ime so again, i s no obvious which dela is opimal.

39 Ousanding Research Quesions Is here an opimal choice of dela which depends only on observable asse prices? How should we price Pah-dependen opions? Forward saring opions? Compound opions? Volailiy swaps?

40 Some References Avellaneda, M., and A. Parás. Managing he volailiy risk of porfolios of derivaive securiies: he Lagrangian Uncerain Volailiy Model. Applied Mahemaical Finance, 3, 1-5 (1996) Blacher, G. A new approach for undersanding he impac of volailiy on opion prices. RISK 98 Conference Handou. Derman, E. Regimes of volailiy. RISK April, (1999) Dumas, B., J. Fleming, and R.E. Whaley. Implied volailiy funcions: empirical ess. The Journal of Finance Vol. LIII, No. 6, December Gupa, A. On neural ground. RISK July, (1997)