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Options strategies with the risk adjustment
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European Journal of Operational Research 192 (2009) 975–980 www.elsevier.com/locate/ejor
O.R. Applications
Options strategies with the risk adjustment Pei-wang Gao
*
Department of Finance, Guangxi University of Finance and Economics, 100 Mingxiu Road West, Nanning 530003, PR China Received 15 November 2006; accepted 6 October 2007 Available online 14 October 2007
Abstract This paper proposes a general linear programming model with risk bounds on all the Greek letters for the portfolio and then performs a new post-optimality analysis for the model. In the analysis, the risks can be adjusted by the investor to suit the needs of the market change. The applications of the model and the method to Ericsson’s options show that they are of practical interests. 2007 Elsevier B.V. All rights reserved. Keywords: Options; Portfolio; Risk; Linear programming
1. Introduction Options trading strategies may been taken on some patterns drawn up in advance (see for instance [3,4,10,11]). However, the patterns are simple, and therefore, difficult to meet the needs of the complicated market situation. For this reason, people applied various optimization models to finding the optimal options portfolio. In this area, Merton [6,7] does the pioneering work by using stochastic dynamic programming theory for continuous investment. Ross [9] introduced two simple deterministic and nondeterministic models, respectively. Korn and Trautmann [5] presented an expected utility maximization framework for optimally controlling a portfolio of options, which did not take into account the risk constraints and therefore was totally used for speculative purpose. Wu and Sen [12] used a discrete stochastic programming approach to develop currency option hedging models. One of the features of the model is that the portfolio incorporated the delta and gamma hedging constraints in the replication of contingent claims. Generally, if an investor is risk averter, he will include more constraints on the Greeks. Based on the fact, Papahristodoulou [8] developed a linear programming formulation for options strategies, which aims to pursue the maximum return under the conditions of the risk-neutrality on all the Greek letters of the Black–Scholes formula. The advantage of the model is that it can provide a quick and relatively simple solution in computation. But, it is not necessary to keep the risk exposure on all the Greek letters covered. Hull [4] argues that a zero gamma and a zero vega (or kappa, as used in [8]) are less easy to achieve in the concrete operation. In practice, every investor has his specific personality and beliefs about the market risk. However, there is still a lack of research concerning the determination of optimal options strategies based on the risk preference. This paper proposes a general linear programming model with risk bounds on all the Greek letters. As the risk bounds are enlarged in the model, the optimum return increases. So, a risk-return trade-off can be done for an investor by setting different risk bounds in the model. On the face of it, the model has analogous utility criteria in risk-return to the preference-based models. Especially, when some bounds approach to infinity, it means that the associated letters or constraints will be neglected; and when some bounds are set equal to zero, it leads to the associated Greek letters risk-neutral. Thus, the model is a generalization of the Papahristodoulou’s model, or the Papahristodoulou’s model is a particular case *
Tel.: +86 731 3825059. E-mail address: [email protected]
0377-2217/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.10.010
976
P.-w. Gao / European Journal of Operational Research 192 (2009) 975–980
of our model. Next, a new post-optimality analysis of the model is performed, by which one would adjust the risks to suit the needs of the market change. If he is rather risky, he can appropriately relax the risks. If he is risk averter, he can tighten up the risks. Therefore, the options strategies generated by the model and the method are of practical interests. An example based on Ericsson’s call and put options real data is given to demonstrate the application of the model and the method. The paper is organized as follows. In Section 2, we establish a linear programming model for the options portfolio. Next in Section 3 a new post-optimality analysis for the model is performed. Section 4 gives an example based on Ericsson’s call and put options real data to illustrate the application of the model and the method. Finally, in Section 5, we make a brief conclusion. 2. A linear programming model for the options portfolio For simplicity, we consider a portfolio containing n European options dependent on a single non-dividend-paying stock. Therefore, the theoretical value, denoted by VT, of any individual option in the portfolio can be calculated by the wellknown Black–Scholes pricing formulas (see [11] for a thorough analysis), that is, for a European call option, V T ¼ SN ðd 1 Þ X erT N ðd 2 Þ; and for a European put option, V T ¼ X erT N ðd 2 Þ SN ðd 1 Þ; Rx S r2 ÞT 2 pffiffiffi 2 , ffi ez2 dz is the cumulative probability function for a standardized normal variable, and d 1 ¼ lnðX Þþðrþ where N ðxÞ ¼ 1 p1ffiffiffi r T 2p pffiffiffiffi d 2 ¼ d 1 r T ; S is the current stock price, X is the exercise price, T is the time to expiration, r represents the volatility of the stock price, and r is the risk-free interest rate. From the Black–Scholes formulas above, we can also derive all the Greek letters for an option component in the portfolio below: For a European call option, D ¼ N ðd 1 Þ;
C¼
N 0 ðd 1 Þ pffiffiffiffi ; Sr T
SrN 0 ðd 1 Þ pffiffiffiffi rX erT N ðd 2 Þ; 2 T pffiffiffiffi q ¼ XT erT N ðd 2 Þ; j ¼ S T N 0 ðd 1 Þ;
H¼
and for a European put option, D ¼ N ðd 1 Þ 1;
C¼
N 0 ðd 1 Þ pffiffiffiffi ; Sr T
SrN 0 ðd 1 Þ pffiffiffiffi þ rX erT N ðd 2 Þ; 2 T pffiffiffiffi q ¼ XT erT N ðd 2 Þ; j ¼ S T N 0 ðd 1 Þ:
H¼
Let P denote the value of the portfolio, and VM denote current market prices for these options in the portfolio. If the portfolio consists of an amount wi of buying option i and an amount wn+i of writing option i (1 6 i 6 n), then we have P¼
n X i¼1
wi V Ti
n X
wnþi V Ti :
ð2:1Þ
i¼1
Note that there is generally the difference between the theoretical value and the market value of an option. So we can gain profits of the portfolio, denoted by P, below: P¼
n n X X ðV Ti V Mi Þwi ðV Ti V Mi Þwnþi : i¼1
ð2:2Þ
i¼1
Again by (2.1) we can derive the values of all the Greek letters of the portfolio, denoted by DP,CP,HP,qP and jP, respectively, as follows:
P.-w. Gao / European Journal of Operational Research 192 (2009) 975–980
DP ¼ CP ¼
n X i¼1 n X
jP ¼
wi Ci
i¼1 n X
HP ¼ qP ¼
wi Di
i¼1 n X i¼1 n X
n X i¼1 n X
wi Hi
wi qi wi ji
i¼1
977
wnþi Di ; wnþi Ci ;
i¼1 n X
wnþi Hi ;
i¼1 n X
wnþi qi ;
i¼1 n X
wnþi ji :
i¼1
where the quantities with the subscript i signify those associated with option i. Now suppose that DL(60) and DU(P0) are the maximum negative and positive risks on D acceptable for an investor. We call them the risk lower and upper bounds of D, respectively. Similarly, let CL(60) and CU(P0) be the risk lower and upper bounds of C, HL(60) and HU(P0) be the risk lower and upper bounds of H,qL(60) and qU(P0) be the risk lower and upper bounds of q and jL(60) and jU(P 0) be the risk lower and upper bounds of j. If delta-neutrality is required before optimizing, it can been achieved by setting the sum of DP plus the number, K of shares purchased minus the number, S of shares sold equal to zero. Therefore, taking (2.2) as the objective function, we obtain a linear programming model, labelled as (PLP), for the portfolio w1, w2, . . . , w2n, K, S, of the form: ðPLPÞ
max
P¼
n X
ðV Ti V Mi Þwi
i¼1
s:t:
n X ðV Ti V Mi Þwnþi ; i¼1
DP þ K S ¼ 0; CL 6 CP 6 CU ;
HL 6 HP 6 HU ;
qL 6 qP 6 qU ;
jL 6 jP 6 jU ;
Bi ðw1 ; w2 ; . . . ; w2n ; K; SÞ 6 bi ; i ¼ 1; . . . ; p; wj P 0ðj ¼ 1; . . . ; 2nÞ; K P 0; S P 0; and
integral;
where Bi(w1, w2, , w2n, K, S) (i = 1, . . . , p) are assumed linear functions of w1, w2, . . ., w2n, K, S. The constraints may originate from the capital budget constraint, the limit with the trade volume of all options, a scale constraint between calls and puts, a precautious constraint against the dramatic change on the stock price, and so on. Obviously, when both of the risk lower and upper bounds on some letter are set equal to zero, it would lead to the letter risk-neutral. When the risk lower bound on some letter is set approaching to 1 and simultaneously the upper bound approaching to 1, it means that the constraint relating to the letter is dropped. Particularly, letting CL = CU = 0, HL = HU = 0, qL = qU = 0 and jL = jU = 0 in the model PLP would lead to the Papahristodoulou’s formulation for option strategies. 3. Options strategies with the risk adjustment Suppose that from the subjective personality, an investor first gives the largest values of the risk bounds on the Greeks, denoted by •LL and •LU, where • represents the Greeks. Furthermore, assume that the variables were treated as continuous. Introducing the slack variables, w2nþ1 ¼ CU CP P 0;
w2nþ2 ¼ CP CL P 0;
w2nþ3 ¼ HU HP P 0;
w2nþ4 ¼ HP HL P 0;
w2nþ5 ¼ qU qP P 0;
w2nþ6 ¼ qP qL P 0;
w2nþ7 ¼ jU jP P 0; w2nþ8 ¼ jP jL P 0; w2nþ8þi ¼ bi Bi ðw1 ; w2 ; . . . ; w2n ; K; SÞ P 0; i ¼ 1; . . . ; m; and letting w = (w1, . . . , w2n, K, S, w2n+1, . . ., w2n+8+m)T, b = (0, CU, CL, HU, HL, qU, qL, jU, jL, b1, . . . , bm)T, and then applying the simplex algorithm (see for instance [1]) to the model PLP, we would obtain an optimal basic solution wB ¼ b ¼ B1 b, wN ¼ 0 with the optimum value P*, where B* and N* are the optimal basic matrix and the nonbasic matrix, respectively. In the case of the optimality, the reduced costs corresponding to the nonbasic variables,
978
P.-w. Gao / European Journal of Operational Research 192 (2009) 975–980
cTN ¼ cTB B1 N cTN P 0, where cB and cN is the basic components and the nonbasic components of c, respectively. Thus, (2.2) can be expressed as P ¼ P cTN wN : Due to the market change, an investor thinks that the risks at the optimal solution to PLP may be too big and therefore should be further adjusted. At the moment, the investor can decrease the expected profit P to an appropriate level to meet his requirements for the risks. According to the theory about linear programming (see for instance [2]), when the objective function P varies as a parameter, it leads to a series of parallel shifts of the objective function hyperplane. Let the expected profit P not lower than a given value P0 with P0 < P*. Then the half space described by P P P0 and the feasible region of PLP intersect to form a convex polyhedron, denoted by SP. In the meaning of economy, each point in SP is one options strategy. So one can find the options strategies corresponding to the requirements in SP. Now assume that the goal of an investor is to minimize the weighted average on all the Greeks in SP. Then a model used for the post-optimality analysis, labelled by (POA), is below: ðPOAÞ
4 X
max
kj ðlj1 w2nþ2j1 þ lj2 w2nþ2j Þ;
j¼1
s:t: wB þ B1 N wN ¼ b ; cTN wN 6 P P 0 ; wB P 0; wN P 0; where kj(j = 1, 2, 3, 4) are the weights associated with gamma, theta, rho and kappa, respectively, such that 0 6 kj 6 1 and P4 CU CL j¼1 wj ¼ 1. In addition, CU CL and CU CL are assigned to l11 and l12 as the weights on w2n+1 and w2n+2, respectively. SimU
L
U
L
U
L
q q j j ilarly, set l21 ¼ HUHHL ; l22 ¼ HH U HL ; l31 ¼ qU qL ; l32 ¼ qU qL ; l41 ¼ jU jL ; l42 ¼ jU jL . Note that the convex set SP is
bounded. Therefore, there certainly exist the solutions to the model POA. Given various different sets of values of the parameters kj(j = 1, . . . , 4), one can obtain options strategies with different risks combinations on the Greeks in SP by solving the model POA. Particularly, setting all kj(j = 1, . . . , 4) equal to 14 produces the options strategy with the ‘‘risk-average’’ meaning. If only one of the parameters such as k1 is set equal to 1 and 0 otherwise, the options strategy is generated with the smallest risk on Gamma. In general, the larger the value of kj for some j 2 {1, . . . , 4}, the stronger the adjustment to the risk on the associated Greek. In the post-optimality analysis above-mentioned, one can also add new constraints to the model POA to obtain the more abundant options strategies for the one’s selection. The options strategies generated by the post-optimality analysis may involve buying and selling simultaneously the same call or put option. Suppose that an options strategy contains an amount wi of buying some option and an amount wj of writing the same option. It is thought reasonable that when wi P wj, the strategy takes buying an amount wi wj of the option, and when wi < wj, the strategy takes selling an amount wj wi of the option. 4. Applications of the model to Ericsson’s call and put options real data As a contrast to the Papahristodoulou’s model, our model and the post-optimality analysis will be applied to an example based on Ericsson’s call and put options real data due to Papahristodoulou [8]. On February 13th 2001, one hour before the Stockholm Stock Exchange was closed, Ericsson’s various call and put options prices were observed as follows (see Table 1). Meanwhile, the stock price was trading at SEK 96. According to the data above-mentioned, the Greeks and the theoretical prices of Ericsson’s options can easily be computed (shown on Table 2 in [8] and here omitted). Using the same notations as in [8], we obtain a linear programming model (PLP1) for deciding the Ericsson’s portfolio as follows: Table 1 Ericsson’s call and put options prices Call options (April) Types Price
95 10.25
Types Price
95 8
100 7.75
Call options (June) 105 6
110 4.50
115 3.10
120 2.50
110 7.5
105 14.25
110 17
115 21
120 25
110 20
Put options (April) 100 10.75
115 6.5
120 4.75
Put options (June) 115 22.25
120 26.75
P.-w. Gao / European Journal of Operational Research 192 (2009) 975–980
max
979
P ¼ ð11:8 10:25ÞKC95A þ þ ð7:15 4:75ÞKC120J ð11:8 10:25ÞSC95A ð7:15 4:75ÞSC120J þ ð10:1 8ÞKP 95A þ þ ð29:55 26:75ÞKP 120J ð10:1 8ÞSP 95A ð29:55 26:75ÞSP 120J ;
s:t:
0:5815KC95A þ þ 0:3556KC120J 0:5815SC95A 0:3556SC120J 0:4185KP 95A 0:6444KP 120J þ 0:4185SP 95A þ þ 0:6444SP 120J þ K S ¼ 0; CL 6 0:01407KC95A þ þ 0:01032KC120J 0:01407SC95A 0:01032SC120J þ 0:01407KP 95A þ þ 0:01032KP 120J 0:01407SP 95A 0:01032SP 120J 6 CU ; HL 6 31:741KC95A 21:181KC120J þ 31:741SC95A þ þ 21:181SC120J 27:969KP 95A 16:445KP 120J þ 27:969SP 95A þ þ 16:445SP 120J 6 HU ; qL 6 7:9607KC95A þ þ 9:0156KC120J 7:9607SC95A 9:0156SC120J 9:0936KP 95A 30:561KP 120J þ 9:0936SP 95A þ þ 30:561SP 120J 6 qU ; jL 6 15:944KC95A þ þ 20:674KC120J 15:944SC95A 20:674SC120J þ 15:944KP 95A þ þ 20:674KP 120J 15:944SP 95A 20:674SP 120J 6 jU ; 0:5815KC95A þ þ 0:3556KC120J þ 0:4185SP 95A þ þ 0:6444SP 120J 6 1000; KC95A; ; SP 120J P 0; and integral:
The simplex algorithm (see for instance [1]) and the post-optimality analysis were programmed with MATLAB V6.5 and conducted to solve the problem (PLP1) on a HKSEE S262C. First of all, letting all the Greeks risk-neutral, and then setting the risk bounds to the maximum or the minimum: CLU = 1.0000, CLL = 0, HLU = 0, HLL = 1000.0, qLU = 20.000, qLL = 2000.0, jLU = 3000.0, jLL = 0 to solve the model PLP1, respectively, we find that the profit generated increases from SEK2181.3 to SEK2537.7. The detailed results are summarized in Table 2. If one thinks that the risks brought by the increase of the profit are too big, he can use the post-optimality analysis proposed to adjust the risks. Suppose that P0 is set equal to 2500, 2450 and 2400, respectively. Then solving the model POA on the ‘‘risk-average’’ meaning would produce the associated options strategies, shown in Table 3. From Table 3, we see that as the expected profit is gradually decreased, the risks on all the Greeks get lowered. Surprisingly, when the profit is only brought from SEK2537.7 down to SEK2400, theta is adjusted from the distant lower bound (1000) to the risk-neutrality, and meanwhile, the risks on gamma and rho get largely improved too by the post-optimality analysis. Therefore, the post-optimality analysis is of practical interests for an investor.
Table 2 Optimal portfolio with Ericsson’s options based on all the Greeks Constraints
Bounds
Risks
Optimal portfolio
Maximum profit
Gamma
CL = 0 CU = 0 HL = 0 HU = 0 qL = 0 qU = 0 jL = 0 jU = 0
0.0
Buy 6.87 call 115A Buy 153.55 call 120J Buy 1584.2 put 115J Sell 7.0 put 105A Sell 1691.8 put 110J Sell 47.25 shares
2181.3
Buy 277.9 call 120J Sell 121.2 call 120A Buy 1607.6 put 115J Sell 9.8 put 105A Sell 1613.3 put 110J
2537.7
Theta Rho Kappa Gamma Theta Rho Kappa
CL = 0.0 CU = 1.0 HL = 1000 HU = 0.0 qL = 2000 qU = 20.0 jL = 0.0 jU = 3000
0.0 0.0 0.0 0.8 1000 2000 3000
980
P.-w. Gao / European Journal of Operational Research 192 (2009) 975–980
Table 3 Strategies with Ericsson’s options by the post-optimality analysis P0
Risks
Options strategies
Profit
2500
Gamma = 0.5 Theta = 312.6 Rho = 1624 Kappa = 3000
Buy 325.1 call 120J Sell 189.0 call 120A Buy 1576.8 put 115J Sell 1592.9 put 110J
2500
2450
Gamma = 0.4 Theta = 0.0 Rho = 632.5 Kappa = 3000
Buy 379.3 call 120J Sell 125.9 call 95A Sell 71.8 call 120A Buy 1508.4 put 115J Sell 1558.2 put 110J
2450
2400
Gamma = 0.3 Theta = 0.0 Rho = 20.0 Kappa = 2728.8
Buy 382.2 call 120J Sell 173.6 call 95A Buy 1481.0 put 115J Sell 1556.4 put 110J Sell 8.4 shares
2400
5. Conclusion The analysis presented here extends the existing literature on options strategies. With the model and the method proposed above, one can take the options strategies in terms of one’s subjective personality, and meanwhile, adjust the risks to suit the needs of the market change. However, the absence of the transaction costs and margins required with options trade would significantly influence the reliability of the optimal solution. In addition, some coefficients in the model vary as time passes due to the volatility of the market. It deserves to be mentioned that the model is easily extended to dynamic, nondeterministic programming problems for options strategies with the transaction costs and margins. Thus, further research will be done thereafter. Acknowledgment This research was partially supported by the Scientific Fund of GUFE Grant No. 2006JYB001. References [1] G.B. Dantzig, Linear Programming and Extensions, Princeton University Press, 1963. [2] Pei-wang Gao, An efficient bound-and-stopped algorithm for integer linear programs on the objective function hyperplane, Applied Mathematics and Computation 185 (2007) 301–311. [3] D. Guppy, Introduction, in: W.F. Eng (Ed.), Options Trading Strategies that Work, Wrightbooks Pty Ltd., 1999. [4] John C. Hull, Fundamentals of Futures and Options Markets, Tsinghua University Press, 2001. [5] R. Korn, S. Trautmann, Optimal control of option portfolios and applications, OR Spektrum 21 (1999) 123–146. [6] R. Merton, Lifetime portfolio selection under uncertainty: The continuous case, Review of Economics and Statistics 51 (1969) 247–257. [7] R. Merton, Optimal consumption and portfolio rules in a continuous-time model, Journal of Economic Theory 3 (1971) 373–413. [8] Christos Papahristodoulou, Options strategies with linear programming, European Journal of Operational Research 157 (2004) 246–256. [9] Sheldon M. Ross, An Elementary Introduction to Mathematical Finance: Options and Other Topics, China Machine Press, 2004. [10] T.J. Watsham, Futures and Options in Risk Management, Beijing University Press, 2003. [11] Paul Wilmott, Sam Howison, Jeff Dewynne, The Mathematics of Financial Derivatives: A Student Introduction, Cambridge University Press, 1995. [12] J. Wu, S. Sen, A stochastic programming model for currency option hedging, Annals of Operations Research 100 (2000) 227–249.
European Journal of Operational Research 192 (2009) 975–980 www.elsevier.com/locate/ejor
O.R. Applications
Options strategies with the risk adjustment Pei-wang Gao
*
Department of Finance, Guangxi University of Finance and Economics, 100 Mingxiu Road West, Nanning 530003, PR China Received 15 November 2006; accepted 6 October 2007 Available online 14 October 2007
Abstract This paper proposes a general linear programming model with risk bounds on all the Greek letters for the portfolio and then performs a new post-optimality analysis for the model. In the analysis, the risks can be adjusted by the investor to suit the needs of the market change. The applications of the model and the method to Ericsson’s options show that they are of practical interests. 2007 Elsevier B.V. All rights reserved. Keywords: Options; Portfolio; Risk; Linear programming
1. Introduction Options trading strategies may been taken on some patterns drawn up in advance (see for instance [3,4,10,11]). However, the patterns are simple, and therefore, difficult to meet the needs of the complicated market situation. For this reason, people applied various optimization models to finding the optimal options portfolio. In this area, Merton [6,7] does the pioneering work by using stochastic dynamic programming theory for continuous investment. Ross [9] introduced two simple deterministic and nondeterministic models, respectively. Korn and Trautmann [5] presented an expected utility maximization framework for optimally controlling a portfolio of options, which did not take into account the risk constraints and therefore was totally used for speculative purpose. Wu and Sen [12] used a discrete stochastic programming approach to develop currency option hedging models. One of the features of the model is that the portfolio incorporated the delta and gamma hedging constraints in the replication of contingent claims. Generally, if an investor is risk averter, he will include more constraints on the Greeks. Based on the fact, Papahristodoulou [8] developed a linear programming formulation for options strategies, which aims to pursue the maximum return under the conditions of the risk-neutrality on all the Greek letters of the Black–Scholes formula. The advantage of the model is that it can provide a quick and relatively simple solution in computation. But, it is not necessary to keep the risk exposure on all the Greek letters covered. Hull [4] argues that a zero gamma and a zero vega (or kappa, as used in [8]) are less easy to achieve in the concrete operation. In practice, every investor has his specific personality and beliefs about the market risk. However, there is still a lack of research concerning the determination of optimal options strategies based on the risk preference. This paper proposes a general linear programming model with risk bounds on all the Greek letters. As the risk bounds are enlarged in the model, the optimum return increases. So, a risk-return trade-off can be done for an investor by setting different risk bounds in the model. On the face of it, the model has analogous utility criteria in risk-return to the preference-based models. Especially, when some bounds approach to infinity, it means that the associated letters or constraints will be neglected; and when some bounds are set equal to zero, it leads to the associated Greek letters risk-neutral. Thus, the model is a generalization of the Papahristodoulou’s model, or the Papahristodoulou’s model is a particular case *
Tel.: +86 731 3825059. E-mail address: [email protected]
0377-2217/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.10.010
976
P.-w. Gao / European Journal of Operational Research 192 (2009) 975–980
of our model. Next, a new post-optimality analysis of the model is performed, by which one would adjust the risks to suit the needs of the market change. If he is rather risky, he can appropriately relax the risks. If he is risk averter, he can tighten up the risks. Therefore, the options strategies generated by the model and the method are of practical interests. An example based on Ericsson’s call and put options real data is given to demonstrate the application of the model and the method. The paper is organized as follows. In Section 2, we establish a linear programming model for the options portfolio. Next in Section 3 a new post-optimality analysis for the model is performed. Section 4 gives an example based on Ericsson’s call and put options real data to illustrate the application of the model and the method. Finally, in Section 5, we make a brief conclusion. 2. A linear programming model for the options portfolio For simplicity, we consider a portfolio containing n European options dependent on a single non-dividend-paying stock. Therefore, the theoretical value, denoted by VT, of any individual option in the portfolio can be calculated by the wellknown Black–Scholes pricing formulas (see [11] for a thorough analysis), that is, for a European call option, V T ¼ SN ðd 1 Þ X erT N ðd 2 Þ; and for a European put option, V T ¼ X erT N ðd 2 Þ SN ðd 1 Þ; Rx S r2 ÞT 2 pffiffiffi 2 , ffi ez2 dz is the cumulative probability function for a standardized normal variable, and d 1 ¼ lnðX Þþðrþ where N ðxÞ ¼ 1 p1ffiffiffi r T 2p pffiffiffiffi d 2 ¼ d 1 r T ; S is the current stock price, X is the exercise price, T is the time to expiration, r represents the volatility of the stock price, and r is the risk-free interest rate. From the Black–Scholes formulas above, we can also derive all the Greek letters for an option component in the portfolio below: For a European call option, D ¼ N ðd 1 Þ;
C¼
N 0 ðd 1 Þ pffiffiffiffi ; Sr T
SrN 0 ðd 1 Þ pffiffiffiffi rX erT N ðd 2 Þ; 2 T pffiffiffiffi q ¼ XT erT N ðd 2 Þ; j ¼ S T N 0 ðd 1 Þ;
H¼
and for a European put option, D ¼ N ðd 1 Þ 1;
C¼
N 0 ðd 1 Þ pffiffiffiffi ; Sr T
SrN 0 ðd 1 Þ pffiffiffiffi þ rX erT N ðd 2 Þ; 2 T pffiffiffiffi q ¼ XT erT N ðd 2 Þ; j ¼ S T N 0 ðd 1 Þ:
H¼
Let P denote the value of the portfolio, and VM denote current market prices for these options in the portfolio. If the portfolio consists of an amount wi of buying option i and an amount wn+i of writing option i (1 6 i 6 n), then we have P¼
n X i¼1
wi V Ti
n X
wnþi V Ti :
ð2:1Þ
i¼1
Note that there is generally the difference between the theoretical value and the market value of an option. So we can gain profits of the portfolio, denoted by P, below: P¼
n n X X ðV Ti V Mi Þwi ðV Ti V Mi Þwnþi : i¼1
ð2:2Þ
i¼1
Again by (2.1) we can derive the values of all the Greek letters of the portfolio, denoted by DP,CP,HP,qP and jP, respectively, as follows:
P.-w. Gao / European Journal of Operational Research 192 (2009) 975–980
DP ¼ CP ¼
n X i¼1 n X
jP ¼
wi Ci
i¼1 n X
HP ¼ qP ¼
wi Di
i¼1 n X i¼1 n X
n X i¼1 n X
wi Hi
wi qi wi ji
i¼1
977
wnþi Di ; wnþi Ci ;
i¼1 n X
wnþi Hi ;
i¼1 n X
wnþi qi ;
i¼1 n X
wnþi ji :
i¼1
where the quantities with the subscript i signify those associated with option i. Now suppose that DL(60) and DU(P0) are the maximum negative and positive risks on D acceptable for an investor. We call them the risk lower and upper bounds of D, respectively. Similarly, let CL(60) and CU(P0) be the risk lower and upper bounds of C, HL(60) and HU(P0) be the risk lower and upper bounds of H,qL(60) and qU(P0) be the risk lower and upper bounds of q and jL(60) and jU(P 0) be the risk lower and upper bounds of j. If delta-neutrality is required before optimizing, it can been achieved by setting the sum of DP plus the number, K of shares purchased minus the number, S of shares sold equal to zero. Therefore, taking (2.2) as the objective function, we obtain a linear programming model, labelled as (PLP), for the portfolio w1, w2, . . . , w2n, K, S, of the form: ðPLPÞ
max
P¼
n X
ðV Ti V Mi Þwi
i¼1
s:t:
n X ðV Ti V Mi Þwnþi ; i¼1
DP þ K S ¼ 0; CL 6 CP 6 CU ;
HL 6 HP 6 HU ;
qL 6 qP 6 qU ;
jL 6 jP 6 jU ;
Bi ðw1 ; w2 ; . . . ; w2n ; K; SÞ 6 bi ; i ¼ 1; . . . ; p; wj P 0ðj ¼ 1; . . . ; 2nÞ; K P 0; S P 0; and
integral;
where Bi(w1, w2, , w2n, K, S) (i = 1, . . . , p) are assumed linear functions of w1, w2, . . ., w2n, K, S. The constraints may originate from the capital budget constraint, the limit with the trade volume of all options, a scale constraint between calls and puts, a precautious constraint against the dramatic change on the stock price, and so on. Obviously, when both of the risk lower and upper bounds on some letter are set equal to zero, it would lead to the letter risk-neutral. When the risk lower bound on some letter is set approaching to 1 and simultaneously the upper bound approaching to 1, it means that the constraint relating to the letter is dropped. Particularly, letting CL = CU = 0, HL = HU = 0, qL = qU = 0 and jL = jU = 0 in the model PLP would lead to the Papahristodoulou’s formulation for option strategies. 3. Options strategies with the risk adjustment Suppose that from the subjective personality, an investor first gives the largest values of the risk bounds on the Greeks, denoted by •LL and •LU, where • represents the Greeks. Furthermore, assume that the variables were treated as continuous. Introducing the slack variables, w2nþ1 ¼ CU CP P 0;
w2nþ2 ¼ CP CL P 0;
w2nþ3 ¼ HU HP P 0;
w2nþ4 ¼ HP HL P 0;
w2nþ5 ¼ qU qP P 0;
w2nþ6 ¼ qP qL P 0;
w2nþ7 ¼ jU jP P 0; w2nþ8 ¼ jP jL P 0; w2nþ8þi ¼ bi Bi ðw1 ; w2 ; . . . ; w2n ; K; SÞ P 0; i ¼ 1; . . . ; m; and letting w = (w1, . . . , w2n, K, S, w2n+1, . . ., w2n+8+m)T, b = (0, CU, CL, HU, HL, qU, qL, jU, jL, b1, . . . , bm)T, and then applying the simplex algorithm (see for instance [1]) to the model PLP, we would obtain an optimal basic solution wB ¼ b ¼ B1 b, wN ¼ 0 with the optimum value P*, where B* and N* are the optimal basic matrix and the nonbasic matrix, respectively. In the case of the optimality, the reduced costs corresponding to the nonbasic variables,
978
P.-w. Gao / European Journal of Operational Research 192 (2009) 975–980
cTN ¼ cTB B1 N cTN P 0, where cB and cN is the basic components and the nonbasic components of c, respectively. Thus, (2.2) can be expressed as P ¼ P cTN wN : Due to the market change, an investor thinks that the risks at the optimal solution to PLP may be too big and therefore should be further adjusted. At the moment, the investor can decrease the expected profit P to an appropriate level to meet his requirements for the risks. According to the theory about linear programming (see for instance [2]), when the objective function P varies as a parameter, it leads to a series of parallel shifts of the objective function hyperplane. Let the expected profit P not lower than a given value P0 with P0 < P*. Then the half space described by P P P0 and the feasible region of PLP intersect to form a convex polyhedron, denoted by SP. In the meaning of economy, each point in SP is one options strategy. So one can find the options strategies corresponding to the requirements in SP. Now assume that the goal of an investor is to minimize the weighted average on all the Greeks in SP. Then a model used for the post-optimality analysis, labelled by (POA), is below: ðPOAÞ
4 X
max
kj ðlj1 w2nþ2j1 þ lj2 w2nþ2j Þ;
j¼1
s:t: wB þ B1 N wN ¼ b ; cTN wN 6 P P 0 ; wB P 0; wN P 0; where kj(j = 1, 2, 3, 4) are the weights associated with gamma, theta, rho and kappa, respectively, such that 0 6 kj 6 1 and P4 CU CL j¼1 wj ¼ 1. In addition, CU CL and CU CL are assigned to l11 and l12 as the weights on w2n+1 and w2n+2, respectively. SimU
L
U
L
U
L
q q j j ilarly, set l21 ¼ HUHHL ; l22 ¼ HH U HL ; l31 ¼ qU qL ; l32 ¼ qU qL ; l41 ¼ jU jL ; l42 ¼ jU jL . Note that the convex set SP is
bounded. Therefore, there certainly exist the solutions to the model POA. Given various different sets of values of the parameters kj(j = 1, . . . , 4), one can obtain options strategies with different risks combinations on the Greeks in SP by solving the model POA. Particularly, setting all kj(j = 1, . . . , 4) equal to 14 produces the options strategy with the ‘‘risk-average’’ meaning. If only one of the parameters such as k1 is set equal to 1 and 0 otherwise, the options strategy is generated with the smallest risk on Gamma. In general, the larger the value of kj for some j 2 {1, . . . , 4}, the stronger the adjustment to the risk on the associated Greek. In the post-optimality analysis above-mentioned, one can also add new constraints to the model POA to obtain the more abundant options strategies for the one’s selection. The options strategies generated by the post-optimality analysis may involve buying and selling simultaneously the same call or put option. Suppose that an options strategy contains an amount wi of buying some option and an amount wj of writing the same option. It is thought reasonable that when wi P wj, the strategy takes buying an amount wi wj of the option, and when wi < wj, the strategy takes selling an amount wj wi of the option. 4. Applications of the model to Ericsson’s call and put options real data As a contrast to the Papahristodoulou’s model, our model and the post-optimality analysis will be applied to an example based on Ericsson’s call and put options real data due to Papahristodoulou [8]. On February 13th 2001, one hour before the Stockholm Stock Exchange was closed, Ericsson’s various call and put options prices were observed as follows (see Table 1). Meanwhile, the stock price was trading at SEK 96. According to the data above-mentioned, the Greeks and the theoretical prices of Ericsson’s options can easily be computed (shown on Table 2 in [8] and here omitted). Using the same notations as in [8], we obtain a linear programming model (PLP1) for deciding the Ericsson’s portfolio as follows: Table 1 Ericsson’s call and put options prices Call options (April) Types Price
95 10.25
Types Price
95 8
100 7.75
Call options (June) 105 6
110 4.50
115 3.10
120 2.50
110 7.5
105 14.25
110 17
115 21
120 25
110 20
Put options (April) 100 10.75
115 6.5
120 4.75
Put options (June) 115 22.25
120 26.75
P.-w. Gao / European Journal of Operational Research 192 (2009) 975–980
max
979
P ¼ ð11:8 10:25ÞKC95A þ þ ð7:15 4:75ÞKC120J ð11:8 10:25ÞSC95A ð7:15 4:75ÞSC120J þ ð10:1 8ÞKP 95A þ þ ð29:55 26:75ÞKP 120J ð10:1 8ÞSP 95A ð29:55 26:75ÞSP 120J ;
s:t:
0:5815KC95A þ þ 0:3556KC120J 0:5815SC95A 0:3556SC120J 0:4185KP 95A 0:6444KP 120J þ 0:4185SP 95A þ þ 0:6444SP 120J þ K S ¼ 0; CL 6 0:01407KC95A þ þ 0:01032KC120J 0:01407SC95A 0:01032SC120J þ 0:01407KP 95A þ þ 0:01032KP 120J 0:01407SP 95A 0:01032SP 120J 6 CU ; HL 6 31:741KC95A 21:181KC120J þ 31:741SC95A þ þ 21:181SC120J 27:969KP 95A 16:445KP 120J þ 27:969SP 95A þ þ 16:445SP 120J 6 HU ; qL 6 7:9607KC95A þ þ 9:0156KC120J 7:9607SC95A 9:0156SC120J 9:0936KP 95A 30:561KP 120J þ 9:0936SP 95A þ þ 30:561SP 120J 6 qU ; jL 6 15:944KC95A þ þ 20:674KC120J 15:944SC95A 20:674SC120J þ 15:944KP 95A þ þ 20:674KP 120J 15:944SP 95A 20:674SP 120J 6 jU ; 0:5815KC95A þ þ 0:3556KC120J þ 0:4185SP 95A þ þ 0:6444SP 120J 6 1000; KC95A; ; SP 120J P 0; and integral:
The simplex algorithm (see for instance [1]) and the post-optimality analysis were programmed with MATLAB V6.5 and conducted to solve the problem (PLP1) on a HKSEE S262C. First of all, letting all the Greeks risk-neutral, and then setting the risk bounds to the maximum or the minimum: CLU = 1.0000, CLL = 0, HLU = 0, HLL = 1000.0, qLU = 20.000, qLL = 2000.0, jLU = 3000.0, jLL = 0 to solve the model PLP1, respectively, we find that the profit generated increases from SEK2181.3 to SEK2537.7. The detailed results are summarized in Table 2. If one thinks that the risks brought by the increase of the profit are too big, he can use the post-optimality analysis proposed to adjust the risks. Suppose that P0 is set equal to 2500, 2450 and 2400, respectively. Then solving the model POA on the ‘‘risk-average’’ meaning would produce the associated options strategies, shown in Table 3. From Table 3, we see that as the expected profit is gradually decreased, the risks on all the Greeks get lowered. Surprisingly, when the profit is only brought from SEK2537.7 down to SEK2400, theta is adjusted from the distant lower bound (1000) to the risk-neutrality, and meanwhile, the risks on gamma and rho get largely improved too by the post-optimality analysis. Therefore, the post-optimality analysis is of practical interests for an investor.
Table 2 Optimal portfolio with Ericsson’s options based on all the Greeks Constraints
Bounds
Risks
Optimal portfolio
Maximum profit
Gamma
CL = 0 CU = 0 HL = 0 HU = 0 qL = 0 qU = 0 jL = 0 jU = 0
0.0
Buy 6.87 call 115A Buy 153.55 call 120J Buy 1584.2 put 115J Sell 7.0 put 105A Sell 1691.8 put 110J Sell 47.25 shares
2181.3
Buy 277.9 call 120J Sell 121.2 call 120A Buy 1607.6 put 115J Sell 9.8 put 105A Sell 1613.3 put 110J
2537.7
Theta Rho Kappa Gamma Theta Rho Kappa
CL = 0.0 CU = 1.0 HL = 1000 HU = 0.0 qL = 2000 qU = 20.0 jL = 0.0 jU = 3000
0.0 0.0 0.0 0.8 1000 2000 3000
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P.-w. Gao / European Journal of Operational Research 192 (2009) 975–980
Table 3 Strategies with Ericsson’s options by the post-optimality analysis P0
Risks
Options strategies
Profit
2500
Gamma = 0.5 Theta = 312.6 Rho = 1624 Kappa = 3000
Buy 325.1 call 120J Sell 189.0 call 120A Buy 1576.8 put 115J Sell 1592.9 put 110J
2500
2450
Gamma = 0.4 Theta = 0.0 Rho = 632.5 Kappa = 3000
Buy 379.3 call 120J Sell 125.9 call 95A Sell 71.8 call 120A Buy 1508.4 put 115J Sell 1558.2 put 110J
2450
2400
Gamma = 0.3 Theta = 0.0 Rho = 20.0 Kappa = 2728.8
Buy 382.2 call 120J Sell 173.6 call 95A Buy 1481.0 put 115J Sell 1556.4 put 110J Sell 8.4 shares
2400
5. Conclusion The analysis presented here extends the existing literature on options strategies. With the model and the method proposed above, one can take the options strategies in terms of one’s subjective personality, and meanwhile, adjust the risks to suit the needs of the market change. However, the absence of the transaction costs and margins required with options trade would significantly influence the reliability of the optimal solution. In addition, some coefficients in the model vary as time passes due to the volatility of the market. It deserves to be mentioned that the model is easily extended to dynamic, nondeterministic programming problems for options strategies with the transaction costs and margins. Thus, further research will be done thereafter. Acknowledgment This research was partially supported by the Scientific Fund of GUFE Grant No. 2006JYB001. References [1] G.B. Dantzig, Linear Programming and Extensions, Princeton University Press, 1963. [2] Pei-wang Gao, An efficient bound-and-stopped algorithm for integer linear programs on the objective function hyperplane, Applied Mathematics and Computation 185 (2007) 301–311. [3] D. Guppy, Introduction, in: W.F. Eng (Ed.), Options Trading Strategies that Work, Wrightbooks Pty Ltd., 1999. [4] John C. Hull, Fundamentals of Futures and Options Markets, Tsinghua University Press, 2001. [5] R. Korn, S. Trautmann, Optimal control of option portfolios and applications, OR Spektrum 21 (1999) 123–146. [6] R. Merton, Lifetime portfolio selection under uncertainty: The continuous case, Review of Economics and Statistics 51 (1969) 247–257. [7] R. Merton, Optimal consumption and portfolio rules in a continuous-time model, Journal of Economic Theory 3 (1971) 373–413. [8] Christos Papahristodoulou, Options strategies with linear programming, European Journal of Operational Research 157 (2004) 246–256. [9] Sheldon M. Ross, An Elementary Introduction to Mathematical Finance: Options and Other Topics, China Machine Press, 2004. [10] T.J. Watsham, Futures and Options in Risk Management, Beijing University Press, 2003. [11] Paul Wilmott, Sam Howison, Jeff Dewynne, The Mathematics of Financial Derivatives: A Student Introduction, Cambridge University Press, 1995. [12] J. Wu, S. Sen, A stochastic programming model for currency option hedging, Annals of Operations Research 100 (2000) 227–249.