The Long and Short of Stati Hedging with Fri tionsJohannes Siven∗ Rolf Poulsen†July 21, 2008Abstra tWe investigate stati hedging of barrier options with plain vanilla in the presen e ofbid/ask spreads. Spreads for options are mu h larger than for the underlying, andthis levels the playing �eld in relation to traditional dynami hedging. However,hedging with options is still the best way to redu e the risk of large losses.Keywords: Stati hedging, barrier option, bid/ask-spread.1 Introdu tionThe literature on stati hedging of barrier options is large,1 but surprisingly quiet onanything that has to do with market fri tions.2 In this paper we remedy that by ex-tending the risk-minimizing stati hedges from Siven & Poulsen (2008) to in orporatebid/ask-spreads. The method relies on simulation and numeri al optimization3 and isvery �exible with regard to the type of ontra t to be hedged, the hoi e of hedge in-struments, the model for the underlying, and the hoi e of risk-measure. To in orporate

∗Centre for Mathemati al S ien es, Lund University, Box 118, SE-22100 Lund, Sweden. Email:jvs�maths.lth.se†Centre for Finan e, University of Gothenburg, Box 640, SE-40530 Gothenburg, Sweden. E-mail:Rolf.Poulsen�e onomi s.gu.se1Early papers are Carr, Ellis & Gupta (1998) and Derman, Ergener & Kani (1995); a survey isPoulsen (2006). See also Carr & Lee (2008) for a re ent extension of the lassi al put- all-symmetryapproa h.2The �worst ase� approa h in Wilmott (2004, Chapter 58) is a notable ex eption.3The idea of hoosing a stati hedge in an optimal way, however, goes ba k at least to Avellaneda &Paras (1996). 1

bid/ask-spreads in the modelling, long and short positions in the hedge instruments haveto be treated separately, but the onvexity of the optimization problem is retained sothe numeri al solution is still straightforward.For dynami delta hedging, it is well-known that fri tions must be onsidered as anintegral part of the hedging problem. It is too expensive to use the no fri tions strategy,smile politely and pay the osts, see e.g. Whalley & Wilmott (1997). We demonstratethat when hedging barrier options with plain vanilla this is �even more true�.At a �rst glan e, bid/ask-spreads may seem less important when hedging stati allywith vanilla puts and alls rather than dynami ally with the underlying. This is naivethough, sin e the bid/ask-spreads are mu h larger for options than for the underlying� typi ally two orders of magnitude. We ompare dynami and stati hedging of anup-and-out all option under realisti assumptions on the bid/ask-spreads in simulationexperiments, and �nd that whether or not to stati ally hedge depends non-trivially bothon the size of the spreads for the di�erent ontra ts, and on the hoi e of risk-measure.The stati hedges do a better job of eliminating large losses, even for large vanilla optionspreads.The paper is stru tured as follows. In Se tion 2 we review the risk-minimizing ap-proa h to stati hedging from Siven & Poulsen (2008) and show how to extend it toin orporate �exible modelling of bid/ask-spreads. Se tion 3 des ribes how the bid/ask-spreads typi ally look in the urren y market where barrier options are most heavilytraded. Se tions 4 and 5 ontain simulation experiments and Se tion 6 on ludes.2 Stati hedging with bid/ask-spreadsWe now review the risk-minimizing stati hedges from Siven & Poulsen (2008) and showhow to extend them to take bid/ask-spreads into a ount.Stati hedging. Let (St)t∈[0,T ] be a pro ess modelling the mid-pri e of a tradedasset; the underlying. Let P denote the time-τ payo� of a ontingent laim, an exoti ,illiquid, or path dependent option on the underlying, where τ is a stopping time boundedby T . Furthermore, let H = (H1, . . . , HM) denote the time-τ payo� of M hedging in-struments: typi ally the market value of plain vanilla options, but more generally the ash �ows from any assets that an be exer ised or liquidated at time τ . A stati hedgestrategy is some c = (c1, . . . , cM) ∈ RM , where ck represents the position in the kth2

hedging instrument (ck > 0 is a long position), so the time-τ value of the orrespondinghedge portfolio is H · c =∑M

k=1 ckHk. We let P (0) and H(0) = (H1(0), . . . , HM(0))denote the time-0 pri es of the target option and the hedging instruments.Risk-minimization. Generally, it is not possible to build a perfe t stati hedge ofthe payo� P , so it is natural to view hedging as an approximation problem: hoose astrategy c that makes the hedge error small in some sense. The quantitye−rτ (P − H · c)is the dis ounted P&L for an investor hedging a short position in the target optionwith the strategy c. We now take a onvex fun tion u (e.g. u(x) = x2) and de�ne arisk-minimizing hedge as a solution to4

minc∈D

E[u(e−rτ(P − H · c))], (1)where the set D of admissible hedging strategies is typi ally de�ned by linear onstraints:limits on the hedge weights and a budget restri tion, (time-0 ost of hedging portfolio)= H(0) · c ≤ onst. The sto hasti programming problem (1) an be solved with the so alled sample average approximation method: simulate N samples (P (n), H(n), τ (n)) ofthe triplet (P, H, τ), approximate the goal fun tion g(c) = E[u(e−rτ(P − H · c))] with

g(c) =1

N

N∑

n=1

u(e−rτ (n)

(P (n) − H(n) · c)), (2)and minimize this deterministi and onvex fun tion over D with some numeri al method.Simulating (P, H, τ) amounts to simulating paths of the underlying asset (this requiresassumptions about the real-world dynami s) and evaluating the payo� from the targetoptions and the hedge instruments (this requires assumptions about the pri ing mea-sure). The optimization problem is onvex and thus readily solved, most o�-the-shelfsoftware pa kages have e� ient solution algorithms. Not only is the sample averageapproximation method intuitively appealing, it an be shown, see Shapiro (2008) andthe referen es therein, that under mild onditions the minimizer of (2) onverges to asolution of (1) as N → ∞.4The hedge is not perfe t, so the expe tation should be taken w.r.t. the real-world probabilitymeasure. 3

In orporating bid/ask-spreads. The risk-minimizing approa h to stati hedging an easily be modi�ed to in orporate �exible modelling of bid/ask-spreads. Assumethat the M liquid ontra ts that are used as hedging instruments have time-τ mid-pri es Hm1 , . . . , Hm

M (supers ript m for �mid�). Denote the spread for the kth instrumentby sk and write H l,sk = Hm

k ∓ sk

2for the time-τ bid and ask pri es, where the supers ripts

l and s stand for �long� and �short� (a hedger long the kth instrument fa es the marketpri e H lk when liquidating, and vi e versa).Treating long and short positions separately, we onsider the ve tor

H = (H l1, . . . , H

lM , Hs

1 , . . . , HsM),and orrespondingly denote the (observable) ve tor of initial bid and ask pri es by H(0).A stati hedge portfolio is now a ve tor

c = (cl1, . . . , c

lM , cs

1, . . . , csM),where cl

k ≥ 0 and csk ≤ 0 respe tively orrespond to long and short positions in the kthhedging instrument. To ompute the risk-minimizing hedge, we just solve the minimiza-tion problem (1) with the extra onstraints that cl

k ≥ 0 and csk ≤ 0 for all k.In the modelling, the spreads sk may in prin iple depend on anything (the time

τ , the whole mid-pri e history, . . .) as long as they an be simulated together withP, Hm

1 , . . . , HmM and τ . However, in the experimental setting below, further simulations(not reported) indi ate that re�ned modelling of the time-τ spreads is of minor impor-tan e, it is the time-0 spreads that matter.If the kth hedging instrument is not liquidated but expires and is settled in ash, wehave sk = 0.53 Bid/ask-spreads in pra ti eBid/ask-spreads in the spot. For the underlying, bid/ask-spreads are approximatelyproportional to the spot level. So, if the time-t mid-pri e is St, we model the bid/ask-pri es as

St(1 − fu) and St(1 + fu),5One ould imagine a bene�t from letting options simply expire rather than liquidating; hedgea ura y would deteriorate, but osts would be saved. There is no su h gain in the setting of Se tions4 and 5 a ording to our experiments (not reported).4

where fu is the fri tion in the underlying. For major ex hange rates (su h as e/$ or¥/$) the bid/ask-spreads are quite tight; reasonable values for fu are between 0.0001and 0.0002, i.e. 1-2 basis points.Bid/ask-spreads for vanilla options. Wystup (2007, Se tion 3.2) des ribes howmarket makers typi ally determine spreads for vanilla options:1. Let CBS(T, St, K, σ) denote the time-t Bla k-S holes (or Garman-Kohlhagen) pri eof an expiry-T strike-K all-option, and let σ∗ denote the implied volatility for theat-the-money option. The vanilla spread is given by CBS(T, St, St, σ∗(1 + fv)) −

CBS(T, St, St, σ∗(1 − fv)), where fv is the vanilla fri tion.2. This spread is used for all the vanilla options with the same expiry-date, i.e. anexpiry-T vanilla option with time-t mid-pri e Ct has bid and ask pri es Ct(1 −vanilla spread/2) and Ct(1 + vanilla spread/2).For vanilla options on major ex hange rates, realisti values for the vanilla fri tion fvare between 0.01 and 0.02 (i.e. 1-2 per entage points) in the interbank market, twi ethat for institutional and orporate lients and even higher for retail lients.4 Hedging of barrier options: base aseWe now perform simulation experiments to ompare stati and dynami hedging withbid/ask-spreads that are realisti for a major ex hange rate market. As target optionwe hoose a daily monitored kno k-out all option with 40 days to expiry, strike at-the-money and barrier 4% out-of-the-money.6 That is, at time τ it pays

P =

{0 τ < T,

(ST − K)+1ST <B τ = T,where τ is either the time of the �rst barrier rossing or expiry, whi hever omes �rst: τ =

min{min{t ∈ {0, dt, 2dt, . . . , T}; St ≥ B}, T}, with dt = 1/252. As hedge instrumentswe use vanilla options that are liquidated if the barrier is rossed and otherwise heldto expiry. To qualify in a stati hedge portfolio, a vanilla all must have the samestrike as the barrier option, or a strike at or beyond the barrier, otherwise it will give6This was the typi al urren y barrier option Danske Bank had on their books during 2004 and2005. 5

a time-T ontribution when the barrier option is not kno ked out. We use vanilla allswith strikes 100, 104, 106, 108 and the same expiry-date as the barrier option � Siven &Poulsen (2008) demonstrate that very little is to gain from in luding vanilla alls withother expiry-dates than T .As data-generating model we use the standard Bla k-S holes model. As far as therisk-minimizing stati hedging is on erned, we ould just as easily handle other models;7we make this hoi e to justify the use of the dynami bandwidth hedging strategy fromWhalley & Wilmott (1997). The strategy is a ni e and simple way to take the fri tionin the underlying into a ount:- Start out by holding ∆BS(0) units of the underlying, and do nothing as long as theposition in the underlying stays within the band that de�nes the no-trade region:∆BS(t) ± f 1/3

u

(3Γ2

BS(t)St

2γ

)1/3

, (3)where ∆BS and ΓBS are the usual no fri tions Bla k-S holes delta and gamma forthe barrier option, and the parameter γ e�e tively ontrols the trade-o� betweenaversion to osts and aversion to risk.- If the position in the underlying goes outside the no-trade region, buy or sell theminimal ne essary number of units needed to bring the position ba k to the edgeof the no-trade region.We hoose volatility σ = 0.1 and ost of arry r = 0 under both the real world and therisk neutral measure.8 These hoi es are not really important � the results from thesimulations look very mu h the same for other values of σ and r. With these parameters,the ben hmark (fair) value of the barrier option is P (0) = E[e−rτP ] = 0.35.Risk-minimizing hedging with bid/ask-spreads. We onsider three risk mea-sures: quadrati (u(x) = x2), positive part (u(x) = x+) and ES5%, the Expe ted 5%Shortfall.97Siven & Poulsen (2008) demonstrates that risk-minimizing stati hedging works �ne in the Batesmodel (sto hasti volatility and jumps) as well as in an in�nite intensity Levy model.8A detailed analysis of this distin tion is beyond the s ope of this paper � and the e�e ts minor forthe ases we onsider � so our real world measures �just happen� to be the pri ing martingale measure.9The Expe ted a% Shortfall is the expe ted loss in urred in the a% worst ases, see Tas he (2002).This risk-measure is not immediately on the form E[u(·)], but Ro kafeller & Uryasev (2000, Theorem1) show how minimization of ESa% an be formulated as a linear programming problem (at the ost ofextra variables and onstraints). 6

Parameter Valuer 0σ 0.1S0 100K 100B 104T 40/252dt 1/252N 10,000fu 0, 0.0001, 0.0002fv 0, 0.005,. . . ,0.04Table 1: Default parameters for the simulation experiments.Table 2 reports the risk-minimizing hedges orresponding to the di�erent risk-measuresand the budget restri tion H(0) · c ≤ P (0), for di�erent values of the vanilla fri tion fv:we learly see how the hedge weights de rease with higher fri tion. This is not surpris-ing: the time-0 transa tion ost from a portfolio is h× (vanilla spread)/2, where h is thesum of absolute hedge weights:h =

M∑

k=1

|clk| + |cs

k|.A � on eivable � 20% mark-up on the barrier option orresponds to h ≈ 5 for fv =

0.01 and only h ≈ 2 for fv = 0.02. So stati hedging may be very expensive in thepresen e of realisti transa tion osts: Table 2 shows that all the zero-fri tion risk-minimizing portfolios have h ≥ 15.To ompare dynami and stati hedging in the presen e of fri tions, we evaluatethe performan e of the bandwidth hedging strategy from Whalley & Wilmott (1997).Varying the risk-aversion parameter γ gives a urve in a ost/risk-spa e, see Figure 1.We �x γ = 1 in what follows.Figure 2 shows the performan e of the dynami hedges for fu = 0.0001 and 0.0002together with the performan e of the risk-minimizing stati hedges for di�erent budgetrestri tions and vanilla fri tion fv = 0, 0.005, . . . , 0.04. Look at the performan e of thehedges orresponding to the budget restri tion (time-0 ost of hedge) ≤ P (0). Stati hedging is superior to dynami hedging as long as fv ≤ 0.015 for the quadrati risk-7

Risk- Vanilla fri tionmeasure Strike 0% 1% 2% 3% 4%100 0.9 0.7 0.6 0.5 0.5104 -6.2 -4.0 -2.7 -2.0 -1.7p

E[(·)2] 106 10.0 4.2 1.8 0.6 0108 -5.6 0 0 0 0100 1.0 0.6 0.5 0.5 0.4104 -6.8 -2.7 -1.5 -1.3 -1.1E[(·)+] 106 10.0 2.2 0 0 0108 -3.8 0 0 0 0100 0.9 0.7 0.7 0.6 0.6104 -5.7 -4.1 -2.3 -2.2 -2.1ES5% 106 7.6 3.7 0 0 0108 -2.8 0 0 0 0Table 2: Risk-minimizing portfolios for the budget restri tion H(0) · c ≤ P (0).

1 1.1 1.2 1.3 1.4

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

E[total transaction costs]/P(0)

E[(

⋅)2 ]0.5 /P

(0)

1 1.1 1.2 1.3 1.40.2

0.3

0.4

0.5

0.6

0.7

E[total transaction costs]/P(0)

E[(

⋅)+]/P

(0)

1 1.1 1.2 1.3 1.43.5

4

4.5

5

5.5

6

E[total transaction costs]/P(0)

ES

5%/P

(0)

Figure 1: Risk (not ounting transa tion osts) vs. expe ted total transa tion ost forthe dynami hedges, for the values 0.01, 0.1, 1, 10, 100 of the risk-aversion parameter γ.The urves orrespond to fu = 0.0001 (solid) and 0.0002 (dashed), the stars orrespondto γ = 1.measure, but inferior for any fv ≥ 0.005 in the ase of the positive part risk-measure.For the risk-measure ES5% though, the stati hedges are better than the dynami hedgeseven for very large values of fv. The reason is that ES5% looks only at the tail: the risk-minimizing hedges manage to avoid very large losses, while the dynami hedge does not.We on lude that whether or not it is better to hedge stati ally or dynami ally dependsboth on the relative sizes of fu and fv and on the hoi e of risk-measure.Figure 2 also illustrates that the performan e of the risk-minimizing hedges improveswhen the budget restri tion is relaxed, in all ases ex ept one: when fv = 0, the budgetrestri tion is not binding for the hedge orresponding to the quadrati risk-measure. Thisis �nan ially weird, and re�e ts the fa t that the quadrati risk-measure punishes gains8

1 1.2 1.4 1.6 1.8

0.8

1

1.2

1.4

1.6

budget/P(0)

E[(

⋅)2 ]0.5 /P

(0)

1 1.2 1.4 1.6 1.80

0.1

0.2

0.3

0.4

0.5

0.6

budget/P(0)

E[(

⋅)+]/P

(0)

1 1.2 1.4 1.6 1.80

1

2

3

4

budget/P(0)

ES

5%/P

(0)

Figure 2: Risk (in luding transa tion osts in urred after time-0) vs. budget for thedynami and risk-minimizing hedges, for di�erent risk-measures. Dynami hedges: fu =

0.0001 (star) and 0.0002 ( ir le). Stati hedges: ea h urve orresponds to a parti ularlevel of vanilla fri tion, fv = 0, 0.005, . . . , 0.04.0.02 0.025 0.03 0.035 0.040

0.2

0.4

0.6

0.8

1

1.2

x 10−3

fv

f u

0 0.01 0.02 0.03 0.040

0.2

0.4

0.6

0.8

1

1.2

x 10−3

fv

f u

0.04 0.05 0.06 0.07 0.080

0.2

0.4

0.6

0.8

1

1.2

x 10−3

fv

f u

Figure 3: Points (fv, fu) where the dynami and the stati hedges perform equally well,for the risk-measures orresponding to u(x) = x2 (left) and u(x) = x+ (middle) andES5% (right).and losses symmetri ally. Even worse, for high budgets, the quadrati risk-minimizinghedges perform better when fv = 0.005 than in the no-fri tions ase. This is a malign onsequen e of the symmetry in this risk-measure: the hedger is redu ing the risk bytaking long and short positions in the same instruments, thus throwing away money bypaying unne essary transa tion osts!Figure 3 shows the points (fv, fu) where stati and dynami hedging performs equallywell. For fv = 0.02, the stati hedges are ompetitive only if fu ≥ 0.0005 for thequadrati and fu ≥ 0.0008 for the positive part risk-measure � in pra ti e, the fri tionin the underlying is mu h smaller. The pi ture is di�erent for the risk-measure ES5%:the dynami hedge start to ompete only for very high levels of vanilla fri tion.9

−4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

4

Value of dynamic portfolio

P−

H⋅c

−4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

4

Value of dynamic portfolio

P−

H⋅c

−4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

4

Value of dynamic portfolio

P−

H⋅c

Figure 4: S atter plots of time-τ value of the dynami al hedge portfolio plotted againstthe residual P − H · c, for the risk-measures E[(·)2] (left), E[(·)+] (middle) and ES5%(right) � for a perfe t hedge, all points would be on the solid line. The stati hedgesall have time-0 pri e H(0) · c = P (0) and the fri tions are fv = 0.0005 and fu = 0.0001.5 Hedging of barrier options: extensionsFinally, we run a ouple of additional experiments.Stati + dynami hedging. A natural idea is to do stati hedging, and then hedgethe residual dynami ally.10 This is straightforward to implement; just view P − H · cas the time-τ payo� from a ontingent laim. However, dynami hedging of the residualdoes not improve performan e, it may even make things worse. Figure 4 shows s atter-plots of the time-τ value of the dynami hedging portfolio against the residual P −H · c.The explanation for the poor performan e of the dynami hedge is that this residual isquite nasty; even more so than mu h the original barrier option payo�. This is be auseH ·c ≈ P and we take the di�eren e. Con eptually, the failure of the dynami hedge tellsus something about the optimality of the stati hedge: if there was some stru ture leftthat was easy to delta hedge, it ould be taken are of by improving the stati portfolio.Beyond Bla k-S holes. As a simple empiri al sanity he k of the hedge results weex hange the Bla k-S holes model's normal return distribution. Spe i� ally, we swit hthe daily returns to independent draws from the un onditional empiri al distribution ofdaily ¥/$ returns over the period 1991�2008. This distribution is skewed (skewness 0.5)10This has been suggested for instan e in Avellaneda & Paras (1996).

10

0 0.01 0.02 0.03 0.041

1.2

1.4

1.6

1.8

fv

E[(

⋅)2 ]0.5 /P

(0)

0 0.01 0.02 0.03 0.040.1

0.2

0.3

0.4

0.5

0.6

0.7

fv

E[(

⋅)+]/P

(0)

0 0.01 0.02 0.03 0.041.5

2

2.5

3

3.5

4

4.5

5

fv

ES

5%/P

(0)

Figure 5: Vanilla fri tion vs. risk with an empiri al ¥/$ return distribution. The threepanels orrespond to di�erent risk measures. The horizontal lines orrespond to dynami hedging with underlying fri tions of 1 (dotted) and 2 (dashed) basis points, and thesolid lines to the risk-minimizing stati hedges, omputed with the budget ristri tionH(0) · c ≤ P (0).and onsiderably more heavy-tailed (ex ess kurtosis 4.2) than the normal distribution.11We then look at the performan e of the risk-minizing hedges based on this empiri aldistribution (remember, as soon as we an simulate, we an ompute optimal hedges)and ompare it to that of a bandwidth hedger who uses σ = 0.1, whi h is extremely loseto both the annualized standard deviation of the returns and the Bla k-S holes impliedat-the-money volatility. The results are shown in Figure 5. The risk is higher than inthe Bla k-S holes ase for both the dynami and the stati hedges, but omparable; lookat the left-most endpoints of the urves in Figure 2. As e onomists we are tempted saythat there is a Pareto improvement of stati hedges over dynami hedges: never worse,sometimes better. For the quadrati risk-measure, the stati hedges are ompetitive forlarger values of fv (up to 0.03, ompared to 0.015 with normal reurns), while results areas before for the positive part risk-measure and ES5%.6 Con lusionWe showed that with an optimization approa h to stati hedging it is easy to work withbid/ask-spreads. We then performed a simulation study to ompare dynami and stati hedging of a reverse barrier option, assuming bid/ask-spreads similar to what is typi ally11We altered the mean return ever-so-slightly to make the underlying a martingale (so te hni ally, wesimulate under a U.S.-martingale measure with zero arry), and used simulation to onsistently pri ethe plain vanilla options. 11

observed in major urren y markets. The results showed that the playing �eld betweendynami and stati hedging is now quite level: whether it is optimal to hedge stati allyor dynami ally depends on the relative sizes of the spreads in the spot and for the vanillaoptions, and on the hoi e of risk-measure. For the tail fo using risk-measure Expe tedShortfall, stati hedging with options is the way to go, but dynami al delta hedging istypi ally superior for the quadrati and the positive part risk-measures. The explanationis that while bid/ask-spreads are signi� antly larger (typi ally two orders of magnitude)for the vanilla options than for the spot, the stati hedges are good at removing the verylarge hedge errors. The on lusion that hedging of large risk �goes well� with fri tionsis also rea hed in Selmi & Bou haud (2003), although in a quite di�erent setting. Of ourse, there are other problems with dynami hedging of barrier options than the ostsasso iated with the bid/ask-spreads: Taleb (1996) dis usses in detail so- alled liquidityholes and other problems that o ur in pra ti e.Referen esAvellaneda, M. & Paras, A. (1996), `Managing the volatility risk of portfolios of deriva-tive se urities: the Lagrangian un ertain volatility model', Applied Mathemati alFinan e 3, 21�52.Carr, P., Ellis, K. & Gupta, V. (1998), `Stati Hedging of Exoti Options', Journal ofFinan e 53, 1165�1190.Carr, P. & Lee, R. (2008), `Put-Call Symmetry: Extensions and Appli ations', forth- oming in Mathemati al Finan e .Derman, E., Ergener, D. & Kani, I. (1995), `Stati Options Repli ation', Journal ofDerivatives 2, 78�95.Poulsen, R. (2006), `Barrier Options and Their Stati Hedges: Simple Derivations andExtensions', Quantitative Finan e 6, 327�335.Ro kafeller, R. T. & Uryasev, S. (2000), `Optimization of onditional value-at-risk',Journal of Risk 2, 21�41.Selmi, F. & Bou haud, J. P. (2003), `Alternative Large Risks Hedging Strategies forOptions', Wilmott Magazine 1(4), 64�67.12

Shapiro, A. (2008), `Sto hasti programming by Monte Carlo simulation methods',Mathemati al Programming, Series B 112, 183�220.Siven, J. & Poulsen, R. (2008), Auto-Stati for the People: Risk-minimizing Hedges ofBarrier Options. Submitted; google se ond author for latest version.Taleb, N. (1996), Dynami Hedging: Managing Vanilla and Exoti Options, John Wiley& Sons.Tas he, D. (2002), `Expe ted Shortfall and Beyond', Journal of Banking and Finan e26, 1519�1533.Whalley, A. E. & Wilmott, P. (1997), `An asymptoti analysis of an optimal hedgingmodel for option pri ing with transa tion osts', Mathemati al Finan e 7, 307�324.Wilmott, P. (2004), Paul Wilmott on Quantitative Finan e, John Wiley & Sons.Wystup, U. (2007), FX Options and Stru tured Produ ts, John Wiley & Sons.

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