Hedging strategy for a portfolio of options and stocks Mehmet Horasanlı

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: [email protected]
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.
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