MPRA Munch Personal RePEc Archve Hedgng reeks for a of otons usng lnear and quadratc rogrammng Panka Snha and Archt Johar Faculty of Management Studes, Unversty of elh, elh 5. February 200 Onlne at htt://mra.ub.un-muenchen.de/20834/ MPRA Paer o. 20834, osted 22. February 200 :26 UTC
Hedgng reeks for a Portfolo of Otons usng Lnear and Quadratc Programmng Panka Snha Faculty of Management Studes, Unversty of elh, elh and Archt Johar Formerly, eartment of Comuter Engneerng, eta Subhas Insttute of technology Unversty of elh, elh Abstract The am of ths aer s to develo a hedgng methodology for makng a of otons delta, vega and gamma neutral by takng ostons n other avalable otons, and smultaneously mnmzng the remum to be ad for the hedgng. A quadratc rogrammng soluton for the roblem s formulated, and then t s aroxmated to a lnear rogrammng soluton. A rototye for the lnear rogrammng soluton has been develoed n MS Excel usng BA.
.0 Introducton The am of hedgers s to use oton markets to reduce the rsk of ther. The change n the value of oton s subect to oton senstvtes summarzed n delta, gamma, vega, theta and rho. These reeks are mortant ndcators used n the rsk management of contanng otons, futures and stocks. Hull[] and Rendleman [2] dscuss ways to set u an otmal hedged. Paahrstodoulou [3] uts forward a lnear rogrammng hedgng otons strategy takng nto account greeks so that one's own belef about the underlyng assets s not requred. Horasanlı [4] extends the above aer for mult-asset settng to deal wth a of otons and underlyng assets. However, n the real world a erfectly hedged mght not be ossble. The above mentoned references do not dscuss about mnmzng the remum to be ad whle hedgng a comrsng of stocks, futures and otons. In the subsequent subsectons, we develo a methodology to hedge an exstng by makng t delta, vega and gamma neutral usng ostons n the other avalable otons n the market, and at the same tme mnmzng the remum to be ad for constructon of the hedge. A quadratc rogrammng soluton (whch s P-hard to solve) s formulated and then aroxmated to a lnear rogrammng soluton. A rototye for the lnear rogrammng soluton has been develoed n MS Excel usng BA. 2. reeks Calculaton Consder a consstng of n tyes of otons. For the th tye of oton, we defne the followng notatons: : umber of lots of the th tye : elta of a lot of otons of the th tye : ega of a lot of otons of the th tye : amma of a lot of otons of the th tye The overall elta, ega and amma of the s calculated as = = = n = n = n = 2
2.2 Hedgng to construct reek eutral Portfolo To make the elta, ega and amma neutral we need to select some otons from the gven otons such that = 0 = 0 = 0 For the th tye of oton from the gven otons, we defne the followng notatons. : umber of lots of the th tye : elta of a lot of otons of the th tye : ega of a lot of otons of the th tye : amma of a lot of otons of the th tye P : Premum ad/receved to buy a lot of otons of the th tye ow the overall elta, ega and amma of the after selectng some lots of otons from the gven otons to make t neutral can be calculated as + = + = + = Also whle selectng some otons from the gven otons (other avalable otons n the market), we need to ensure that the cost of settng u the hedge,.e., the remum ad P s mnmzed. Transacton costs,.e., brokerage fees ad etc wll not be consdered n ths entre analyss, whch can be easly ncororated n the below analyss, f needed. Let us defne a bnary varable whch s defned as follows: 0, f =, f th th lot of oton s not selected lot of oton s selected 3
Mnmze = = = = P = = = Where, s an nteger. subect to If we have a constrant on the maxmum number of lot of otons whch we can choose (say Z ) out of the gven otons, we can set u another lnear constrant such that = <= z But the roblem n the above set u s a quadratcally constraned quadratc rogram (QCQP) whch s P-Hard to solve n the general case. The above roblem can be aroxmated as Lnear Integer rogram f we relax the last constrant of choosng at-most z otons from the gven otons. Mnmze = = = = P = = = subect to Where s an nteger. 4
However snce truly hedged s are dffcult to obtan n real world busness scenaro snce an nteger soluton for the above roblem may not exst, for such cases we can ut a varance lmt on the reeks. Mnmze = P (+ (+ (+ Where s an nteger. subect to delta vega gamma ) <= ) <= = = ) <= = <= <= <= ( ( vega delta ) ( ) gamma Snce nteger lnear rograms are n many ractcal stuatons P-Hard to solve, another aroxmaton of the roblem can be develoed by relaxng the nteger constrants on and nstead mosng varance lmts on t. Mnmze = = = = P = = = subect to Round( ) <= * For examle f =3.97 or = 4.02 and varance lmt on s %, we assume that 4 lots of the otons were bought. In the case when a artcular oton does not meet the varance crteron, model s re-run gnorng ths oton. We now ut forward the last set of two constrants on the number of otons whch can be chosen to hedge the, at least one of whch must be chosen.. = n = ) 5
.e. the total number of otons n whch ostons can be taken should be less than the number of otons n the whch needs to be hedged. 2. ) Max( ) Max ( Where vares from to n and vares from to. By uttng ths constrant, we are ensurng that we do ncrease the remum ad and the corresondng rsk sgnfcantly. A rototye for the above model was develoed n MS Excel usng BA and Solver as an add-n lbrary. Illustraton: Consder the examle gven n Paahrstodoulou[3], the followng two sets of Ercsson otons were avalable as of 3 th Feb, 200. The stock rce was tradng at SEK 96 at the Stockholm Stock Exchange. The frst set of otons corresonds to Arl otons (days to exre were 66) and the second set corresonds to June otons.( days to exre were 22). The rsk free rate of nterest was 6%. Imlct volatlty was estmated as 57% for Arl otons and 55% for June otons. The three and sx month volatltes were 68% and 65% resectvely. We wsh to establsh a for a trader who wanted to set a artcular otons tradng strategy usng Arl otons and hedge the so formed usng June otons. Oton Tye o. of Otons Strke Prce Premum Pad/Receved er lot elta amma ega Call 0 95 0.25 0.585 0.0407 5.944 Call 32 00 7.75 0.58 0.0436 6.279 Call 0 05 6 0.4442 0.0423 6.26 Call 0 0 4.5 0.386 0.0373 5.563 Call 28 5 3. 0.3246 0.0295 4.685 Call 0 20 2.5 0.2736 0.099 3.586 Put 0 95 8-0.485 0.0407 5.944 Put 0 00 0.75-0.4887 0.0436 6.279 Put -25 05 4.25-0.5558 0.0423 6.26 Put -25 0 7-0.684 0.0373 5.563 Put 0 5 2-0.6754 0.0295 4.685 Put 0 20 25-0.7264 0.099 3.586 Table : Arl 200 otons of Ercsson 6
Oton Tye Strke Prce Premum Pad/Receved er lot elta amma ega Call 0 7.5 0.4448 0.0095 2.929 Call 5 6.5 0.3986 0.007 2.422 Call 20 4.75 0.3556 0.0032 20.674 Put 0 20-0.5552 0.0095 2.929 Put 5 22.25-0.604 0.007 2.422 Put 20 26.75-0.6444 0.0032 20.674 Table 2 : June 200 otons of Ercsson Fgure : Man Interface of the smulator If we take the constrant that sum of otons n whch ostons can be taken to hedge the should be less than or equal to the number of otons n the whch needs to be hedged (32 + 28 + 25 + 25 = 0 n ths examle), we get the followng results. Fgure 2 : Otons hedgng smulator tool results dectng reeks of the before and after hedgng, the remum ad whle dong so and the ostons n varous otons taken. 7
The oututs of oton hedgng smulator are summarzed n fgure 2. It s evdent from the fgure that the delta, gamma and vega of orgnal can be brought down sgnfcantly by selectng 57 short ostons n Call 0, short oston n Call 5, 7 long ostons each n Put0 and Put5, and 8 long ostons n 20. A remum of $765.75 has to be ad for ths hedgng strategy whch s mnmum, the delta, gamma and vega of resultng new are, -2.922, 0.0408, and -22.393. ow takng the second constrant that the maxmum number of new ostons n otons of one tye should be less than or equal to the number of otons of a artcular tye whch s maxmum n the exstng (32 n ths examle), we get the followng results: Fgure 3 : Otons hedgng smulator tool results dectng reeks of the before and after hedgng, the remum ad whle dong so and the ostons n varous otons taken wth new restrcton. The oututs of oton hedgng smulator for the above restrcton are summarzed n fgure 3. It s agan evdent from the outut that the delta, gamma and vega of orgnal have been brought down sgnfcantly by selectng 32 short ostons each n Call 0 and Call 5, long ostons n Call 20, 32 long ostons each n Put 0 and Put20, and 6 short ostons n Put5. A mnmum remum of $744.25 has to be ad for ths hedgng strategy, the delta, gamma and vega of resultng new are, -2.9794, 0.05328, and 0.609. Clearly n ths examle, the second constrant gves better results than the frst one. However, ths s not necessary and ths may vary from to. Hence, usng the above mentoned methodology of hedgng, the rsk of the can be hedged by reducng ts delta gamma and vega and at the same tme we can mnmze the remum to be ad for the creaton of the hedged. 8
References [] Hull, J. C. (2009). Otons, Futures, and Other ervatves, Prentce Hall. [2] Rendleman, R. J. (995). An LP aroach to oton selecton, Advances n Futures and Otons Research, 8, 3 52. [3] Paahrstodoulou, C. (2004). Oton strateges wth lnear rogrammng, Euroean Journal of Oeratonal Research, 57, 246 256. [4] Horasanlı, M. (2008). Hedgng strategy for a of otons and stocks wth lnear rogrammng, Aled Mathematcs and Comutaton, 99, 804 80. 9