Barone-Adesi, Giovanni

Elliott, Robert

RERO DOC Digital Library

Comput Econ (2007) 29:151–158DOI 10.1007/s10614-006-9069-8Cutting the hedgeGiovanni Barone-Adesi · Robert J. ElliottReceived: 5 October 2006/Accepted: 12 October 2006 /Published online: 30 January 2007© Springer Science+Business Media B.V. 20071 IntroductionDo you enjoy chores such as mowing the lawn or, as it is called in Canada,shovelling the snow? Below we discuss a simpler method of trimming thehedge, suggested by Barone-Adesi, Engle and Mancini (2004). Assuming theoption price is homogeneous our calculation ismodel independent and providesdelta hedge ratios immediately from market data. For strike prices whichare close enough our approximation will work for moderate departures fromhomogeneity. Indeed, independently of themodel the continuity of option priceas a function of asset price and strike price ensures that for c close to 1C(cS0, cK)equals approximately cC(S0,K). In fact we do not need to impose homogeneityeverywhere, but only as a local approximation. For common smiles foundempirically in SP index options our approximation holds well if the ratio ofthe increment in strike price over the asset price is less than 0.05, (in practicethis ratio is around 10/1000 = 0.01).The assumption of homogeneity is strong.When the logarithm of S(t) followsa general Levy process, under the risk-neutral measure, then C(S,K) is homo-geneous (of degree 1), though there is also a volatility smile/skew.However, ourmethod applies, and provides a very easy calculation of the delta from observeddata. Of course, for general Levy processes the delta dC/dS does not providea complete hedge, as the market is incomplete, (unless S(t) is just a BrownianG. Barone-Adesi (B)Institute of Finance, University of Lugano, via Buffi 13, 6900 Lugano, Switzerlande-mail: Giovanni.Barone-Adesi@lu.unisi.chR. J. ElliottHaskayne School of Business, University of Calgary, Calgary, Alberta, Canada T2N 1N4e-mail: relliott@ucalgary.ca152 Comput Econ (2007) 29:151–158motion or a single Poisson process with a drift). However, delta dC/dS is auseful quantity. Euler’s formulaC = SdC/dS + KdC/dKis a consequence of the homogeniety of C(S,K) and this has been used todetermine delta. However, our method is different.Hedge ratios are important in asset management. They are approximated asthe ratio of the change in an option price divided by the change in the under-lying. However, only one underlying price is observed today, together with itsassociated option prices. We determine option prices for different values of theunderlying by considering prices for different strikes and using the homogeneityof the option price.2 NotationSuppose S0 is the price today of the underlying asset and ST is the (unknown)price at time T, the expiration time of the option.Consider a strike price K and suppose the continuously compounded riskfree rate is r. With E denoting the expectation under the risk neutral measurethe price today of a European call isC(S0,K) = E[e−rT(ST − K)∗|S0].Our discussion also applies to puts. We assume that for c > 0,C(cS0, cK) =cC(S0,K). This is certainly the case in the Black–Scholes case.The usual hedge requires the evaluation of the delta∂C∂S(S0,K).For example, in the Black–Scholes framework (Elliott & Kopp, 1999),∂C∂S(S0,K) = (d1)where (x) = 1√2π∫ x−∞ ey2/2 dy is the standard normal distribution andd1 =n (S0K ) + (r + σ22 )Tσ√T.Here σ is the volatility of S which has to be estimated.The Black–Scholes model assumes σ is constant. However, observed marketprices imply that σ should possibly vary with K. This gives rise to the volatility‘smile’ when the implied value of σ is graphed against K.Comput Econ (2007) 29:151–158 153The delta is often approximated asCS= C(S0 + S,K) − C(S0,K)(S0 + S) − S0 .For example, this approximation is used in binomial tree models.3 HomogeneityToday there is only one price S0 given in the market. However, the marketoften quotes prices for different strikes K. We shall use the homogeneity of theoption price to deduce implied option prices for different values of S0.Suppose the option pricesC(S0,K1),C(S0,K) andC(S0,K2) are quoted todayin the market and K1 < K < K2.In fact suppose K1 = αK and K2 = βK. Now the option price is assumedhomogeneous in S0 and K. That is, for c > 0C(cS0, cK) = cC(S0,K).Therefore, for the observed pricesC(S0,K1) and C(S0,K2)we haveC(S0,αK) = αC(α−1S0,K)C(S0,βK) = βC(β−1S0,K).Writeα−1S0 = S0 + 1Sβ−1S0 = S0 + 2SsoC(S0 + 1S,K) = α−1C(S0,αK)C(S0 + 2S,K) = β−1C(S0,βK)where the quantities on the right are available from market prices. To compen-sate for any ‘smile’ effect we calculate ratios C/S for strikes, K1, K2 aboveand below K.154 Comput Econ (2007) 29:151–158Write−K =C(S0 + 1S,K) − C(S0,K)(S0 + 1S) − S0= α−1C(S0,K1) − C(S0,K)α−1S0 − S0+K =C(S0 + 2S,K) − C(S0,K)(S0 + 2S) − S0= β−1C(S0,K2) − C(S0,K)β−1S0 − S0 .Our estimate for delta is thenK := −K + +K2.By considering the upper and lower estimates, −K and +K, and then averag-ing the ‘smile’ effect should be reduced.4 ExampleSuppose the price today of the underlying is S0 = 180.Further, suppose the prices of 6month options with strikes 210, 220 and 230are quoted as:C(180, 210) = 6.75C(180, 220) = 4.75C(180, 230) = 3.29.Using the homogeneity of the option price we deduceC(188.57, 220) = 220210C(180, 210) = 7.07andC(172.17, 220) = 220230C(180, 230) = 3.15.Comput Econ (2007) 29:151–158 155Therefore,−220 =C(172.17, 220) − C(180, 220)172.17 − 180= 4.75 − 3.157.83= 0.204and+220 =C(188.57, 220) − C(180, 220)188.57 − 180= 0.271.Consequently an approximate value for the delta is220 = −220 + +2202= 0.238.The above option prices are given by Black–Scholes with T = 0.5, r= 5% andσ = 30%. The Black–Scholes delta is then 0.23 so our approximation is reason-ably accurate.We now provide two sets of data (Appendix Tables 1, 2) in which our methodis used to calculate the hedging ratios. In the first we compute theBlack–Scholesprices associated with a volatility smile. The volatility, sigma, takes values from0.15 down to 0.11 and then increases again to 0.14. These values are associ-ated with strikes X which take values from 900 to 1100 based on a constantunderlying price of 1000. With an expiration of 3months, 0.25 of a year, theBlack–Scholes call prices are calculated as well as the N(d1) from the Black–Scholes model. The approximate hedging ratio H of this paper is also foundand the error between H and N(d1) given.The two graphs (Figs. 1, 2) represent the volatility smile and the error inthe hedge ratio. We note that the error between the Black–Scholes delta N(d1)and our approximate delta is negative when the volatility is decreasing andpositive when the volatility is increasing. For the given values the hedge ratioreaches a maximum at the strike X = 1030. The final column of values givesthe butterfly spread value and we see that when the strike is 1030 the butterflyspread has a negative price, which represents an arbitrage opportunity.Therefore our hedge ratio, based on option prices, appears to deteriorate whenthese prices allow for arbitrage opportunities. However, the Black–Scholesmodel is itself an approximation and the hedge ratios given by N(d1) areapproximations.The second set of figures provides Black–Scholes hedge ratios N(d1) andour corresponding approximate values H when the volatility is constant, that iswhen there is no smile. In this case the errors are much smaller and constant.156 Comput Econ (2007) 29:151–158Fig. 1 Volatility Smile 0.0000 0.150010.0000 0.140020.0000 0.135030.0000 0.130040.0000 0.125050.0000 0.120060.0000 0.115070.0000 0.112080.0000 0.110090.0000 0.1100100.0000 0.1120110.0000 0.1150120.0000 0.1150130.0000 0.1200140.0000 0.1220150.0000 0.1250160.0000 0.1270170.0000 0.1300180.0000 0.1320190.0000 0.1350Fig. 2 Delta Error 10 -0.0463571220 -0.03766243630 -0.04414547940 -0.05143379950 -0.05945467660 -0.05246033970 -0.03728681380 -0.01516508690 0.020018894100 0.048986563110 0.029705333120 0.046086335130 0.066630216140 0.042604593150 0.040903934160 0.035701882170 0.033844371180 0.028626893190 0.041151685Acknowledgements The authors are grateful to Peter Carr, Emanuel Derman, Farshid Jamshi-dian and John van derHoek for their helpful comments. The financial support of FinRisk, a networkof the Swiss National Fund, is gratefully acknowledged.AppendixThis is an example of an artificial smile, calibrated at values typical of thestandard and poor index, to highlight the differences of our approximate hedg-ing from Black–Scholes’ DeltasTable 1 Hedging the smileSigma Time Stock N(d1) X Call Approx. Error Butterflyprice H. ratio (convexity)0.15 0.2500 1000.00 0.9322 900.0000 105.67290.14 0.2500 1000.00 0.9245 910.0000 95.8443 0.9708 -0.0464 0.42642727Comput Econ (2007) 29:151–158 157Table 1 continuedSigma Time Stock N(d1) X Call Approx. Error Butterflyprice H. ratio (convexity)0.135 0.2500 1000.00 0.9073 920.0000 86.4423 0.9450 −0.0377 0.139700980.13 0.2500 1000.00 0.8862 930.0000 77.1799 0.9304 −0.0441 0.176718920.125 0.2500 1000.00 0.8602 940.0000 68.0942 0.9116 −0.0514 0.223498630.12 0.2500 1000.00 0.8283 950.0000 59.2321 0.8877 −0.0595 0.28215820.115 0.2500 1000.00 0.7893 960.0000 50.6521 0.8417 −0.0525 0.67902330.112 0.2500 1000.00 0.7385 970.0000 42.7511 0.7758 −0.0373 0.686778210.11 0.2500 1000.00 0.6783 980.0000 35.5369 0.6935 −0.0152 1.000953040.11 0.2500 1000.00 0.6097 990.0000 29.3236 0.5896 0.0200 1.107120130.112 0.2500 1000.00 0.5378 1000.0000 24.2175 0.4888 0.0490 0.919784950.115 0.2500 1000.00 0.4685 1010.0000 20.0311 0.4388 0.0297 0.080679370.115 0.2500 1000.00 0.4011 1020.0000 15.9255 0.3550 0.0461 1.561908980.12 0.2500 1000.00 0.3445 1030.0000 13.3817 0.2779 0.0666 −0.048688990.122 0.2500 1000.00 0.2908 1040.0000 10.7893 0.2482 0.0426 0.619007270.125 0.2500 1000.00 0.2453 1050.0000 8.8158 0.2044 0.0409 0.221812320.127 0.2500 1000.00 0.2042 1060.0000 7.0642 0.1685 0.0357 0.457909770.13 0.2500 1000.00 0.1709 1070.0000 5.7705 0.1370 0.0338 0.133960080.132 0.2500 1000.00 0.1409 1080.0000 4.6107 0.1123 0.0286 0.325639520.135 0.2500 1000.00 0.1175 1090.0000 3.7766 0.0764 0.0412 0.335983340.14 0.2500 1000.00 0.1015 1100.0000 3.2785The error pattern reflects the smile pattern. It is negative for declining volatility, positive otherwise.It reaches a maximum at strike 1030. At that price there is a violation of convexity that leads to anarbitrage opportunity (the butterfly spread has a negative price). The Black–Scholes hedge ratioscalibrated at the implied volatilities are in any case approximations, the true model being unknownTable 2 Hedging the smileSigma Time Stock N(d1) X Call Approx. Error ButterflyPrice H. ratio (convexity)0.2 0.2500 1000.00 0.8731 900.0000 109.97600.2 0.2500 1000.00 0.8486 910.0000 101.6298 0.8482 0.0004 0.283147340.2 0.2500 1000.00 0.8216 920.0000 93.5669 0.8212 0.0004 0.30816190.2 0.2500 1000.00 0.7920 930.0000 85.8120 0.7916 0.0003 0.331174540.2 0.2500 1000.00 0.7600 940.0000 78.3884 0.7597 0.0003 0.351574470.2 0.2500 1000.00 0.7259 950.0000 71.3163 0.7256 0.0002 0.36882850.2 0.2500 1000.00 0.6900 960.0000 64.6130 0.6898 0.0002 0.382504450.2 0.2500 1000.00 0.6525 970.0000 58.2923 0.6524 0.0001 0.392288530.2 0.2500 1000.00 0.6139 980.0000 52.3638 0.6139 0.0001 0.397995710.2 0.2500 1000.00 0.5746 990.0000 46.8333 0.5746 0.0000 0.399572870.2 0.2500 1000.00 0.5349 1000.0000 41.7024 0.5349 −0.0001 0.397095020.2 0.2500 1000.00 0.4952 1010.0000 36.9686 0.4953 −0.0001 0.390755280.2 0.2500 1000.00 0.4560 1020.0000 32.6256 0.4562 −0.0002 0.38084970.2 0.2500 1000.00 0.4176 1030.0000 28.6634 0.4178 −0.0002 0.367758440.2 0.2500 1000.00 0.3803 1040.0000 25.0690 0.3806 −0.0003 0.351924540.2 0.2500 1000.00 0.3444 1050.0000 21.8264 0.3448 −0.0003 0.33383190.2 0.2500 1000.00 0.3102 1060.0000 18.9178 0.3106 −0.0004 0.313983650.2 0.2500 1000.00 0.2779 1070.0000 16.3231 0.2783 −0.0004 0.292882190.2 0.2500 1000.00 0.2476 1080.0000 14.0212 0.2480 −0.0004 0.271011540.2 0.2500 1000.00 0.2194 1090.0000 11.9904 0.2198 −0.0004 0.248822820.2 0.2500 1000.00 0.1934 1100.0000 10.2085158 Comput Econ (2007) 29:151–158ReferencesBarone-Adesi, G., Engle, R., & Mancini, L. (2004). GARCH options in incomplete markets,Manuscript, Lugano.Elliott, R. J., & Kopp, P. E. (1999). Mathematics of financial markets. Springer-Verlag, (Secondedition 2004).

Cutting the hedge

http://doc.rero.ch/record/317884/files/10614_2006_Article_9069.pdf

Cutting the hedge

Abstract

Similar works

Full text

Available Versions

RERO DOC Digital Library