Available online at www.sciencedirect.com Applied Mathematics and Computation 199 (2008) 804–810 www.elsevier.com/locate/amc Hedging strategy for a portfolio of options and stocks with linear programming Mehmet Horasanlı Istanbul University, Faculty of Business Administration, Avcilar, 34320 Istanbul, Turkey Abstract This paper extends the model proposed by Papahristodoulou [C. Papahristodoulou, Option strategies with linear programming, European Journal of Operational Research 157 (2004) 246–256] to a multi-asset setting to deal with a portfolio of options and underlying assets. General linear programming model is given and it is applied to Novartis, Sanoﬁ and AstraZeneca’s call and put options. A portfolio of options and their underlying assets is constructed under a hedging strategy that considers all the Greek letters such as delta, gamma, theta, rho and vega. The impact of each Greek constraint on the portfolio’s return is investigated considering the shadow prices. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Black and Scholes formula; Hedging; The Greek letters; Linear programming 1. Introduction Financial engineering means bundling and unbundling of ﬁnancial instruments such as equities, futures, options, ﬁxed income and swaps in order to maximize proﬁt. It can be argued that it is more important to get the hedging correct than to be precise in pricing of a contract [7, p. 99]. Therefore ﬁnancial engineers use trading strategies such as butterﬂy, straddle, strangle, bull and bear spreads, etc. to minimize their risk. These strategies are commonly used dealing with portfolio of options but every strategy is based on invertors’ beliefs and can change over time. A linear programming formulation can help us to identify the hedging problem easily with an objective function and many constraints. Once the model is setup, trading strategies can be drawn from the optimal solution of the LP model. In the literature, portfolio of options and stocks is constructed and an optimal mix is carried out by Rendleman [5]. In addition to this, Papahristodoulou [4] developed a linear programming model to determine option strategies to achieve delta, gamma, theta, rho, vega neutrality. In this paper, the model proposed by Papahristodoulou [4] is extended to a multi-asset setting to deal with a portfolio of options and underlying assets. The organization of the paper is as follows. E-mail address: horasan@istanbul.edu.tr 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.10.042 M. Horasanlı / Applied Mathematics and Computation 199 (2008) 804–810 805 In Section 2 the Black and Scholes formula is introduced and the Greek letters are derived. In Section 3 linear programming model whose optimum solution gives the hedging strategy that considers all Greek letters is constructed. Also, the data used as an input to the model is introduced in Section 3. Section 4 results of the solution of the linear programming model. Section 5 concludes the paper and gives insight for future works. 2. Black and Scholes formula and the Greeks The Black and Scholes formula for a European call option on a non-dividend paying stock where N() is the cumulative standard normal distribution function is given below: CðS t ; tÞ ¼ S t N ðd 1 Þ KerðT tÞ N ðd 2 Þ; where ð1Þ þ r 12 r2 ðT tÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ d 1;2 ¼ : r T t By using this formula, the value of a call option can be calculated easily by using the input parameters such as the stock price (St), strike price (K), volatility (r), risk-free interest rate (r) and time to maturity (T t). Putcall parity enables us to calculate the put option within the same arguments. Deriving the formula given by (1) is out of subject but it is well known that Black and Scholes [2] used no arbitrage arguments and they construct a very special portfolio by setting up a position with the option and a speciﬁc amount of shares of stock so called hedging. The Greek letters or Greeks brieﬂy, plays a key role in hedging. Each Greek letter measures option’s sensitivity with respect to a diﬀerent variable. Thus, the main objective of hedging is to manage the Greeks accurately. The delta of an option is the sensitivity of the option to the underlying [1, p. 94]. To calculate the delta, partial derivative with respect to St should be taken, ln D¼ S t K oC ¼ N ðd 1 Þ: oS t ð2Þ Delta hedging, due to Black and Scholes assumptions, means to hold one option contract and sell delta quantity of the underlying asset. The main idea is to create the option synthetically. Since the value of the delta quantity of the underlying asset and the option is always equal, the overall value of the transaction is always zero. Basically, to hedge the option – or a portfolio of options – the value of the delta should be zero. This argument is valid for all other Greek letters. A delta-neutral portfolio would capture the small changes in the price of the underlying. But the bigger changes cause deviations from delta-neutrality. Gamma-neutrality is used to avoid this problem. The gamma of the option is the second order partial derivative of the value of the option with respect to the underlying asset. o2 C N 0 ðd 1 Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : ¼ ð3Þ 2 oS Str T t It is clear that gamma measures the rate of change of an option’s delta just because it is calculated by taking the ﬁrst order partial derivative of delta. The gamma always takes positive values. The value of the underlying so the value of the option changes as time passes by, the sensitivity of the option with time should be considered. The theta of an option is the sensitivity of the option price with time. So, to calculate the theta of an option, partial derivative with respect to time to maturity (T t) is taken, C¼ H¼ oC rS t N 0 ðd 1 Þ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rKerðT tÞ N ðd 2 Þ: oðT tÞ 2 T t ð4Þ Additionally, rho of an option is the sensitivity of the option price to the interest rate used in formula (1). So, to calculate the rho of an option, partial derivative with respect to risk-free interest rate (r) is taken, q¼ oC ¼ KðT tÞerðT tÞ N ðd 2 Þ: or ð5Þ 806 M. Horasanlı / Applied Mathematics and Computation 199 (2008) 804–810 The value of volatility can be calculated in two ways. Return and standard deviation parameters can be calculated by using historical values of the underlying, this yields the volatility of the option. Another way to compute the volatility is to use market prices of the option. One can get the market price of an option and calculate the corresponding volatility value so called implied volatility. Generally these two values do not coincide but tend to move together in the long run. Thus, the value of the portfolio against the changes in volatility should be achieved. The vega of an option is the sensitivity of the option price to volatility. So, to calculate the vega of an option, partial derivative with respect to volatility (r) is taken, pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ oC ¼ S t T tN 0 ðd 1 Þ: m¼ ð6Þ or 3. LP formulation and data It can be seen from Table 1 that the theoretical and market prices of options contracts diﬀer. But due to the no arbitrage restrictions, price of every ﬁnancial instrument converges to its theoretical value in the long run. For European type options the equivalence of the prices occurs at the expiry date. For investors, the diﬀerence between the theoretical and the market prices can cause possible gains or losses. Therefore, the objective function consists of the multiplication of the diﬀerence between the theoretical and market price of options within various strike prices and the amount of each option in the portfolio. Objective function is given as follows: Z MAX ¼ MðkÞ 2 X 2 X N X X i¼1 j¼1 ci ðptheor;k;e pmarket;k;e Þxi;j;k;e : ð7Þ k¼1 e¼1 The ﬁrst indices i stand for the type of the transaction where 1 denotes buying and 2 denotes selling the corresponding option. The second indices j denotes the type of the option, call option for 1 and put option for 2 is used, respectively. The third indices k denotes the options written on diﬀerent underlying assets taken in the portfolio. It is clear that N is the number of diﬀerent option contracts in the option portfolio. As each option is traded regarding to various strike prices, the last indices e stands for the diﬀerent strike prices for each option. So, M(k) denotes the number of diﬀerent strike price alternatives. The parameter c alters from +1 to 1 due to the speciﬁc option is bought or sold. It is clear that it takes the value of +1 when the option is bought and 1 when the same option is issued. Parameter x stands for the amount of each call or put option bought or sold at a speciﬁc strike price in the portfolio. To hedge the portfolio against the possible risks caused from the changes in the underlying asset price, delta-neutrality – a position with a delta of zero – should be achieved. The delta of a portfolio of options can be calculated as the weighted average of deltas of individual options in the portfolio denoted as d [3, p. 309]. Delta-neutrality constraint is formulated as follows: MðkÞ 2 X 2 X N X X i¼1 j¼1 k¼1 e¼1 ci d j;k;e xi;j;k;e þ 2 X N X i¼1 ci si;k ¼ 0: ð8Þ k¼1 To ensure the delta-neutrality, underlying assets are also added into the portfolio at a speciﬁc amount. The parameter s denotes the number of shares purchased or sold. c takes the value of +1 when the underlying is purchased and 1 when the same underlying is sold. The gamma-neutrality constraint is formulated as follows: MðkÞ 2 X 2 X N X X i¼1 j¼1 ci gj;k;e xi;j;k;e ¼ 0: ð9Þ k¼1 e¼1 The same arguments hold for gamma-neutrality except adding the underlying calculating the gamma of the portfolio. Since gamma-neutrality constraint appears together with delta-neutrality, share position is not included [4, p. 250]. Theta-, rho- and vega-neutrality is achieved by the same understanding, equaling the theta, rho and vega of the portfolio to zero. Thus, theta-, rho- and vega-neutrality constraints are formulated as follows. t stands for the theta of each underlying in the portfolio. The indices j indicates whether the option is call or put and e M. Horasanlı / Applied Mathematics and Computation 199 (2008) 804–810 807 Table 1 The Greeks and the market vs. theoretical prices of Novartis (NVS), Sanoﬁ (SNY) and AstraZeneca (AZN) Strike price Pmarket Ptheoretical Call options NVS 45 50 55 60 65 70 10.50 6.30 2.75 0.80 0.15 0.05 10.6945 6.1779 2.7743 0.9264 0.2299 0.0435 Put options NVS 45 50 55 60 65 70 0.15 0.75 2.31 5.50 9.40 15.20 Call options SNY 30 37.5 40 42.5 45 50 60 Delta Gamma Theta Rho Vega 0.9797 0.8582 0.5755 0.2694 0.0876 0.0206 0.0083 0.0378 0.0660 0.0556 0.0268 0.0084 2.8798 4.5895 5.6666 4.2034 1.9111 0.5776 12.2916 11.6765 8.2199 3.9534 1.3068 0.3104 1.4395 6.5854 11.4916 9.6869 4.6702 1.4575 0.0461 0.4518 1.9707 5.0452 9.2711 14.0071 0.0203 0.1418 0.4245 0.7306 0.9124 0.9794 0.0083 0.0378 0.0660 0.0556 0.0268 0.0084 0.4476 1.8871 2.6940 0.9605 1.6021 3.2057 0.3313 2.3489 7.2081 12.8772 16.9263 19.3252 1.4395 6.5854 11.4916 9.6869 4.6702 1.4575 13.10 5.20 4.52 2.90 1.70 0.45 0.05 13.2866 6.1409 4.1301 2.5313 1.4050 0.3220 0.0061 0.9993 0.9047 0.7779 0.6004 0.4101 0.1325 0.0040 0.0005 0.0333 0.0586 0.0760 0.0765 0.0422 0.0023 1.6418 0.0333 4.2578 4.7179 4.3551 2.2058 0.1152 8.4063 9.2879 8.3137 6.6038 4.6038 1.5248 0.0473 0.0566 3.8731 6.8054 8.8284 8.8863 4.8992 0.2720 Put options SNY 30 37.5 40 42.5 45 50 60 0.30 0.90 1.60 2.65 5.70 11.30 13.10 0.0010 0.2389 0.6892 1.5517 2.8866 6.7260 16.2549 0.0007 0.0953 0.2221 0.3996 0.5899 0.8675 0.9960 0.0005 0.0333 0.0586 0.0760 0.0765 0.0422 0.0023 0.0203 1.2744 2.0959 2.4208 1.9230 0.4966 3.1276 0.0090 1.2312 2.9067 5.3179 8.0191 12.5007 16.7833 0.0566 3.8731 6.8054 8.8284 8.8863 4.8992 0.2720 Call options AZN 30 35 40 45 50 55 60 65 70 19.20 16.80 10.00 4.30 2.20 0.60 0.25 0.05 0.05 18.4658 13.5545 8.7962 4.7312 2.0168 0.6765 0.1822 0.0406 0.0078 0.9999 0.9947 0.9397 0.7480 0.4513 0.2012 0.0681 0.0183 0.0041 0.0001 0.0024 0.0187 0.0499 0.0619 0.0439 0.0205 0.0070 0.0019 1.6257 2.0488 3.3365 5.2939 5.5240 3.6472 1.6448 0.5493 0.1452 8.4138 9.7418 10.3452 8.8822 5.5982 2.5589 0.8792 0.2383 0.0534 0.0105 0.3920 3.0656 8.1763 10.1455 7.1987 3.3674 1.1489 0.3078 Put options AZN 30 35 40 45 50 55 60 65 70 0.10 0.15 0.55 1.55 4.00 6.50 13.30 18.20 20.90 0.0002 0.0113 0.1753 1.0328 3.2407 6.8228 11.2509 16.0318 20.9213 0.0001 0.0053 0.0603 0.2520 0.5487 0.7988 0.9319 0.9817 0.9959 0.0001 0.0024 0.0187 0.0499 0.0619 0.0439 0.0205 0.0070 0.0019 0.0043 0.1571 1.1746 2.8617 2.8216 0.6746 1.5980 2.9638 3.6382 0.0015 0.0761 0.8752 3.7407 8.4273 12.8692 15.9513 17.9948 19.5823 0.0105 0.3920 3.0656 8.1763 10.1455 7.1987 3.3674 1.1489 0.3078 denotes the corresponding strike price, theta is calculated. Rho (r) and vega (v) parameters can be interpreted similarly. 808 M. Horasanlı / Applied Mathematics and Computation 199 (2008) 804–810 MðkÞ 2 X N X X2 X i¼1 j¼1 MðkÞ 2 X N X X2 X i¼1 j¼1 ð10Þ ci rj;k;e xi;j;k;e ¼ 0; ð11Þ ci vj;k;e xi;j;k;e ¼ 0: ð12Þ k¼1 e¼1 MðkÞ 2 X N X X2 X i¼1 j¼1 ci tj;k;e xi;j;k;e ¼ 0; k¼1 e¼1 k¼1 e¼1 Adding the non-negativity constraint for each variable and solving the linear programming model it can be seen that unbounded solution is obtained. Nevertheless, limiting the amount of transactions by adding a scale constraint, unique solution is achieved. Scale constraint is limiting the sum of buying call options delta, selling put options delta and the amount of underlying asset bought [4, p. 251]. The scale constraint is formulated as follows: MðkÞ MðkÞ N X N X N X X X c1 d 1;k;e x1;1;k;e þ c2 d 2;k;e x2;2;k;e þ c1 s1;k 6 L: ð13Þ k¼1 e¼1 k¼1 e¼1 k¼1 The ﬁrst sum states the sum of deltas of the call options bought whereas the second sum states sum of deltas of the put options sold. The last term indicates the number of shares bought for each underlying. The parameter L, limits the size of the portfolio. Adding the scale constraint the problem is ready to solve and a unique solution is now achieved. To test the model given by Eqs. (7)–(13), option contracts of Novartis (NVS), Sanoﬁ (SNY) and AstraZeneca (AZN) maturing on January 2008 are observed on September 19th 2007, in New York stock exchange. Market prices (Pmarket) of option contracts for diﬀerent strike prices are observed and noted in Table 1. Table 2 Optimal portfolios of Novartis, Sanoﬁ and AstraZeneca’s options Model Shadow price Type and volume of transaction Maximum proﬁt Delta Gamma Theta Rho Vega Scale 168.264 1474.810 45.539 10.487 26.687 378.103 Sell Buy Sell Buy Buy Sell 1456.161 15,233.700 26.105 393.353 1679.195 390,651.900 Call Call Call Put Put Put NVS 70 SNY 60 SNY 37.5 SNY 37.5 AZN 35 AZN 30 37,810.260 Delta Gamma Rho Vega Scale 2.301 2730.308 14.097 18.766 451.873 Sell Sell Buy Buy Sell 1095.147 117.277 12,814.700 812.592 463,604.800 Call Put Call Put Put NVS 70 NVS 45 SNY 60 SNY 37.5 AZN 30 45,187.290 Delta Gamma Theta Rho Scale 555.530 1489.620 19.667 23.605 555.530 Buy Buy Buy Sell Buy 91.321 9443.005 1029.865 567,714 5.457 Put Call Put Put Shares NVS 45 SNY 60 SNY 37.5 AZN 30 NVS 55,552.970 Delta Gamma Vega Scale 178.646 2500.250 16.930 454.064 Sell Buy Buy Sell 1221.441 13,350.780 785.292 465,968.900 Call Call Put Put NVS 70 SNY 60 SNY 37.5 AZN 30 45,406.380 Delta Gamma Scale 263.922 640.619 621.303 Buy Buy Sell 8685.235 18,867.920 652,590.600 Call Put Put SNY 60 AZN 35 AZN 30 62,130.280 Delta Scale 10.975 1008.975 Sell Sell 25,000 1,000,000 Call Put SNY 60 AZN 30 100,897.500 M. Horasanlı / Applied Mathematics and Computation 199 (2008) 804–810 809 In addition to this, historical volatilities of each underlying asset are calculated for 2007. Historical volatilities were computed annually as 20.2465% for NVS, 22.2401% for SNY and 24.9689% for AZN. The number of days to expiry is 104 and the risk-free interest rate is 5.49%, given these parameters and historical volatilities, theoretical prices (Ptheoretical) of call and put options for each underlying is calculated and noted in Table 1. Using the same input parameters, the Greeks of each call and put options for various strike prices are also calculated by the given Eqs. (2)–(6) and noted in Table 1. Closing prices of underlying assets are observed as 54.95 for NVS, 42.82 for SNY and 48.00 for AZN. The model given by Eqs. (7)–(13) is solved by LINGO – optimization software enables to solve maximum– minimum problems within constraints – and results are noted in Table 2. 4. Results The problem is solved iteratively within diﬀerent constraints added. First of all, the full model – the model with all constraints – is solved and iteratively a constraint is dropped from the model until the simplest model – the model consists of the objective function and delta, scale, non-negativity constraints – achieved. The results are gathered in Table 2. The ﬁrst column of Table 2 shows the constraints added to the model for each state. The second column gives the shadow prices for each constraint. Shadow prices help us to identify whether the optimum value tends to rise or fall if the corresponding Greek risk is increased. The third column gives the optimum hedging strategy and the last column shows corresponding proﬁt. Observe that the formulation assumes one option is equivalent to one share. To obtain the equivalent amount of shares, these values must therefore be divided by 100 [4, p. 253]. For example for the full model, the optimum strategy is to sell call options equivalent to 1456.161 shares (14.56161 call options) of Novartis within strike price of 70, buy call options equivalent to 15,233.700 shares (15.2337 call options) of Sanoﬁ within strike price of 60, sell call options equivalent to 26.105 shares (0.26105 call options) of Sanoﬁ within strike price of 37.5, buy put options equivalent to 393.353 shares (3.93353 put options) of Sanoﬁ within strike price of 37.5, buy put options equivalent to 1679.195 shares (16.79195 put options) of AstraZeneca within strike price of 35 and sell put options equivalent to 390,651.9 shares (3,906.519 put options) of AstraZeneca within strike price of 30. This position yields to a proﬁt of 37,810.260. The shadow price for the delta-constraint is negative. Thus, if we increase the delta risk by one unit, the proﬁt decreases by 168.264 units. The lower shadow prices indicate less inﬂuence to the model whereas if we drop a constraint with a higher shadow price, the maximum proﬁt value increases dramatically. Alternatively, positive shadow prices means increasing the right hand side of the corresponding constraint yields an increase in the objective value. For example if we increase the right hand side of the scale constraint by one unit – taking L + 1 instead of L – the maximum proﬁt increases by 378.103 approximately. The shadow price for gamma-constraint is signiﬁcantly greater than the other ones. This can be seen the impact of excluding the gamma-constraint from the model. Accordingly, excluding gamma from the model, the proﬁt increases by 62%. Iteratively a constraint – a Greek – is dropped from the model. The second model consists of ﬁve constraints by excluding the theta. Dropping the theta, proﬁt increases by 7377.03. But if we choose to drop the vega constraint ﬁrst, proﬁt increases by 17,742.71. This is consistent with the shadow prices of each constraint. The model with delta, gamma, theta, rho and scale constraints is interesting because, 5.457 shares of Novartis should be hold in the portfolio. The simplest model consists of delta and scale constraints. Optimum trading strategy is to sell call options equivalent to 25,000 shares (250 call options) of Sanoﬁ within strike price of 60 and sell call options equivalent to 1,000,000 shares (10,000 call options) of AstraZeneca within strike price of 30. This strategy yields to a proﬁt of 100,897.5. 5. Conclusions In this paper, the model proposed by Papahristodoulou [4] is extended to a multi-asset setting to deal with a portfolio of options and underlying assets. General linear programming model is given and applied to Novartis, Sanoﬁ and AstraZeneca’s call and put options. 810 M. Horasanlı / Applied Mathematics and Computation 199 (2008) 804–810 Although linear programming models are easy to setup and solve with aid of computers, it has many disadvantages in dealing with options. Since returns on options are non-linear, forcing them to model with linear constraints is the most important handicap of the model. Also, the model uses constant historical volatility based on the past performance of the underlying but it is a well known subject that volatility changes over time. Another limitation of the model is that it ignores transaction costs. However transaction costs play a key role in dynamic hedging. Despite all the limitations of the model, linear programming provides many advantages to the investor. The main advantage is easy of use and it can deal with many constraints. By using duality analysis one can achieve alternative optimum solutions or shadow prices. Most importantly, shadow prices help investors to quantify the proﬁt change if a speciﬁc constraint is included or excluded to the model [6, p. 123–124]. Additionally, shadow prices identiﬁes whether the optimum value increases or decreases, if the corresponding Greek risk is increased. Finally, further research can be implemented on adding transaction costs to the model. References [1] [2] [3] [4] [5] [6] [7] M. Baxter, A. Rennie, Financial Calculus: An Introduction to Derivative Pricing, Cambridge University Press, UK, 2003. F. Black, M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy 81 (1973) 637–654. J.C. Hull, Options Futures and Other Derivatives, ﬁfth ed., Prentice-Hall, NJ, 2003. C. Papahristodoulou, Option strategies with linear programming, European Journal of Operational Research 157 (2004) 246–256. R.J. Rendleman, An LP approach to option portfolio selection, Advances in Futures and Options Research 8 (1995) 31–52. H. Taha, Operations Research, Prentice-Hall Inc., USA, 1997. P. Wilmott, Paul Wilmott on Quantitative Finance, vol. 1, John Wiley and Sons Inc., England, 2000.