The portfolio optimization problem is a basic problem of financial analysis. In the study, an optimization model for constructing an option's portfolio with a certain payoff function has been proposed. The model is formulated as an integer linear programming problem and includes an objective payoff function and a system of constraints. In order to demonstrate the performance of the proposed model, we have constructed the portfolio on the European call and put options of Taiwan Futures Exchange. The optimum solution was obtained using the MATLAB software. Our approach is quite general and has the potential to design option's portfolios on financial markets.
Interest to the options market steadily grows. In general case, brokers are creating financial portfolios based on a combination
of standard European call and put options, cash, and the underlying assets itself, which are not associated with an investor’s goal.
Sometimes this goal is simply to insurance and hedge [
In the study [
In the study [
In the study [8] authors proved that option’s combinations are very popular on major option markets. They show the most
popularly traded combinations in order of contract volume: straddles, ratio spreads, vertical spreads, and strangles. If European
options were available with every single possible strike, any smooth payoff function could be created [
The portfolio optimization problem is a basic problem of financial analysis. In modern portfolio theory, developed by
H. Markowitz, investors attempt to construct the portfolios by taking some alternatives into account: a) the portfolios with
the lowest variance correspond to their preferred expected returns and vice versa, b) the portfolios with the highest expected
returns correspond to their preferred variance. The Markowitz optimization is usually carried out by using historical data. The
objective is to optimize the security’s weight so that the overall portfolio variance is minimum for a given portfolio return. In this
approach, options and structured products do not have a chance to be included into optimal mean-variance portfolios, but they
have a place in optimal behavioral portfolios [
Another possible alternative is to apply some criterions to the payoff function as well as to the initial cost of a portfolio: for
example, market risk measure [
In the paper [
In the study [5] authors have described a class of stock and option strategies, involving a long or short position in a stock,
combined with a long or short position in an option. It was found that only the standard deviation, skewness, and kurtosis of the
returns distribution of the underlying stock affected the optimal strategy, i.e. yield maximum returns. In the study [
Authors [
In the paper [14] there is a proposed model for constructing a multi-period hedged portfolio which includes an European-type options. The objective function of optimization problem is recorded as the difference between the expected value of the portfolio and the expected regret of an investor. The model takes into account, the features of long-term investment: the risk aversion level is added into the objective function as well as the options contract with the different time to maturity were used.
In the paper [12] the two-step problem of optimal investment using the probability as an optimality criterion was studied. Various cases of distribution of returns were investigated. It was found that the structure of the optimal investment portfolio is almost identical despite of one or another distribution.
In the paper [13] a model for the construction of an option’s hedged portfolio was proposed under a fuzzy objective function. The model does not explicitly takes transaction costs into account, but the entered membership functions are aimed at minimizing transaction lots. Thus, the authors have implicitly tried to reduce the potential transaction costs.
In the study [4] proposed a model based on using exotic options – lookback call options. Authors denoted that lookback calls have positive payoffs for both maximum and minimum asset’s prices, and thus have features similar to a portfolio on call and put options.
In the paper [15] a computer-based system of identifying the informed trader activities in European-style options and their underlying asset was proposed, then the mathematical procedure of informed trader activity monitoring was built.
Previous studies considered the set of option’s prices as a price function of an underlying asset: (1 + δ) × S t, where |δ| ≤ 0.1. In contrast, in this paper the optimization is performed over the ask- and bid-prices and their combinations. We do not use the historical empirical distribution of returns.
Options are popular in Europe, USA, Russia, India and Taiwan [
The purpose of this study is to construct an option’s strategy with the piecewise linear payoff function. With this in mind, the purpose for this study can be defined as the following problem statement: How does the personal investor’s goal can be realized with an option’s portfolio? In order to answer this, two research questions have been put forward.
• Q1: How can the method of establishment of the option’s strategy be described from a theoretical perspective? • Q2: How to validate a proposed method on option’s market?
The remainder of this paper is organized as follows. In Section 2 we present the main definitions and assumptions which allow us to formulate the model as an optimization problem, including the objective function and the constraints for option’s strategy. Section 3 then demonstrates the results of numerical experiments, including data description, and the integer solution of the optimization problem. Finally, Section 4 presents conclusions and future research.
Option contracts were originally developed and put into circulation in order to reduce financial risks (hedging, insurance),
associated with the underlying assets. In addition to existing standard option strategies [3] actively developing trading models,
oriented to the objectives of a particular trader (speculative trading) [5], construction of synthetic positions [
An option is a contract that will give an option holder a right to buy (or sell) the underlying asset. An options premium is the amount of money that investors pay for a call or put option.
A call option is a contract that will give its holder a right, but not the obligation, to purchase at a specified time, in the future, certain identified underlying assets at a previously agreed price. A put option is a security that will give its holder a right, but not the obligation, to sell at a specified time, in the future, certain identified underlying assets at a previously agreed price.
The strike price (exercise price) is defined as the price at which the holder of an options can buy (in the case of a call option) or sell (in the case of a put option) the underlying security when the option is exercised.
An American option may be exercised at the discretion of the option buyer at any time before the option expires. In contrast, a European option can only be exercised on the day the contract expires.
A covered option involves the purchase of an underlying asset (equity, bond or currency) and the writing a call option on that same asset. Short selling is the sale of a security that is not owned by a trader, or that a trader has borrowed.
A zero-cost option strategy is an option trading strategy in which one could take a free options position for hedging or speculating in equity, forex and commodity markets.
There are main types of option’s portfolio in real-world applications in terms of the time to expiration: American-, Europeantype options [16], and their combination.
The various option combinations represent strategies designed to exploit expected changes of the options values: the price of the underlying asset, its volatility, the time to expiration, the risk-free interest rate, the cost to enter [3, 8]. There is a large number of possible option combinations. When there are only two possible strike prices and two times to expiration we can design 36 combinations on one call and one put which may be either bought or sold. This number of combinations will increase significantly when any options values (strike prices, times to expiration, underlying assets) will be expanded insignificantly.
In this study we propose the strategy which involves European call and put options on the same underlying asset with the same maturity date T , but different strikes in a series. Let Kc = {kci ∈ Z>0, i ∈ I} and Kp = {kip ∈ Z>0, i ∈ I} be the call and put strikes, Kc, Kp are the increasing sequence of positive integers:
kci < kci+1, kip < kip+1, ∀i = 1, 2, . . . , n − 1, K = {Kc ∪ Kp} is the set of unique strikes, Z>0 = {x ∈ Z : x > 0} denotes the set of positive integers. Let the number of call and put options be
Xc = {xic ∈ Z : L ≤ xic ≤ U, L < 0, U > 0, i ∈ I},
Xp = {xip ∈ Z : L ≤ xip ≤ U, L < 0, U > 0, i ∈ I}, with xic, xip > 0 for buying, xic, xip < 0 for selling, if xic or xip equal to 0 it means that the contract does not include in the portfolio, L and U represents the lower and upper bounds of the integer search space, respectively, I = {1, 2, . . . , n} is the set of indices, and S t is a price of the underlying asset at calendar time, 0 ≤ t ≤ T , Sˆ T is an expected (forecasting) price of the underlying asset at the end of the strategy term, T (single period). Prices S t, Sˆ T ∈ R>0, where R>0 = {x ∈ R : x > 0} denotes the set of positive real numbers. We assume that the initial capital W is given and that no funds are added to or extracted from the portfolio, 0 ≤ t ≤ T .
In order to determine the number of call and put options X = {Xc, Xp} for the implementation of the individual investor goal we propose the following assumptions that have impact on the payoff V (T, X) and an initial cost C(t, X) of portfolio: • (i) the strategy should have protection on the downside and upside of strike prices, • (ii) the strategy should effectively limit the upside earnings and downside risk with a maximal loss, L, • (iii) the strategy should have the certain initial cost to enter C(t, X) at time t = 0.
To establish the strategy we propose to use a combination long and short positions in put and call contracts based on the same underlying asset with different strike prices. We consider that one can take a static position (buy-and-hold), and the portfolio can include xic, xip units of European call and put options, i ∈ I. Its value at time t is given by the formula
V (T, X) = n X xic(S t − kci)+ + xip(kip − S t)+, i=1
X+ = max(X, 0).
Ac = {aic ∈ R>0, i ∈ I}, Bc = {bic ∈ R>0, i ∈ I}, Ap = {aip ∈ R>0, i ∈ I}, Bp = {bip ∈ R>0, i ∈ I},
bic < aic, bip < aip, i ∈ I and aic > aic+1, aip < aip+1, bic > bic+1, bip < bip+1,
C(t, X) = n X xic · gc(xic) + xip · gp(xip), i=1 gc(xic) = aic ∈ Ac, if xic > 0,
bic, ∈ Bc, if xic ≤ 0, the first term is the value of the call option payoff and the second is the value of the put option payoff, and
Let the best ask- and bid-prices for buying and selling of call and put options at time t = 0 be which we will name the input constants: i = 1, 2, . . . , n − 1. The initial cost portfolio at time t = 0 can be expressed as where the functions gc(xic) and gp(xip) are defined as (1) (2) (3) i ∈ I. Thus taking into account Eq. (1) and Eq. (2) the overall profit and loss at time T will be the final payoff minus the initial cost. So the objective function can be expressed as which is a linear function of the decision variable X = {Xc, Xp}. The objective functional F(X) maps the entire stochastic process (cash flow) to a single real number
F(X) : Zn 7→ R, where Zn 7→ Z× · n· · ×Z.
Using the conditional functions gc(·) and gp(·) in the objective function Eq. (5) leads us to solve a sequence of optimization problems. There are four input parameter values for each call and put options: (Ac or Bc) and (Ap or Bp). In this case, the number of permutations based on the selection between alternative prices (ask or bid) and possible contracts (call or put) equal to N = 2n × 2n = 22n. In the numerical examples section of this article (Section 3) we will use the ask- and bid-prices quoted on the Taiwan Futures Exchange (TAIFEX).
Let C denote the set of all 2 × n-tuples of elements of given ordered sets of ask- and bid-prices Ac, Bc, Ap and Bp. The set C can be expressed as
C = {(x1, x2, . . . , xn, y1, y2, . . . , yn) : xi = aic or bic, yi = aip or bip, i ∈ I}.
Thus C = {c1, c2, . . . , cN } is the set of ordered permutations without replacement of two n-elements sets Ac, Bc and two n-elements sets Ap, Bp.
The calculation of the portfolio’s terminal payoff under each price combination in the vector notation takes the form: max{Xc⊤((Sˆ T − Kc)+ − Gc(Xc)) + X⊤p((Kp − Sˆ T )+ − Gp(Xp))} = max{FC(X)},
X X where C denotes the set of ordered permutations of model input constants Eq. (6), and Gc(Xc), Gp(Xp) are the vector notation of conditional functions defined in Eq. (3) and Eq. (4). The objective function Eq. (7) maximizes the option’s payoff over the holding period [0, T ].
Each objective function FC(X) from the set Eq. (7) is the piecewise linear function. Taking into account the assumptions mentioned in Section 2 we should determine the slope of the objective function Eq. (7) in the unique strike intervals. We separately investigate the intervals 0 ≤ S T ≤ k1, k2 ≤ S T ≤ k3, . . ., km ≤ S T < +∞, here k1 = min(Kc, Kp) is the smallest strike and km = max(Kc, Kp) is the largest strike.
The horizontal slope of the function (7) in the first closed interval [0, k1] and the left-closed interval [km, +∞) are specified respectively by: Positive and negative slopes of the function (7) in the interior intervals [kq, kq+1] are provided by: n X xc
i i=1 n X xp
i i=1 = = 0, if S T ∈ [0, k1], 0, if S T ∈ [km, +∞).
X xic − i:kci≤kq
X j:k pj≥kq+1 x pj is ≥ 0, if kq ≤ k, ≤ 0, if kq > k, gp(xip) = aip ∈ Ap, if xip > 0,
bip, ∈ Bp, if xip ≤ 0, here k ∈ K is an inflection point of the function (7).
The next balance constraint defines the bound of the downside risk with a maximal loss, L, over the holding period 0 ≤ t ≤ T : The objective function value (7) at the terminal time T must be positive:
V(T, X) = −L.
V(T, X) > 0, S T = Sˆ T . (4) (5) (6) (7) (8) (9) (10) (11) (12)
The model has the liquidity constraints, we assume that an investor can buy at least U and sell at least L contracts at each strike price kci, kip ∈ K:
L ≤ xic, xip ≤ U, L < 0, U > 0, i ∈ I. (13)
The inflection point k introduced in Eq. (10), the maximal loss L, and the expected (forecasting) price Sˆ T Eq. (12) should be specified by an investor. Thus to address research question Q1 we formulated the integer linear programming of option’s portfolio selection problem (7), subject to the portfolio constraints (8)-(13).
We assume that an investor can use the money received from the sale of some contracts to buy other contracts in the portfolio, then the initial cost of portfolio C(t, X) Eq. (2) can be either a positive number or zero, or even negative number. In the Section 3 we will represent the series of numerical experiments for these three cases. Thus, the system of constraints (8)-(13) can be (optionally) extended with the constraint on the initial cost of portfolio:
C(t, X) T 0.
Another series of numerical experiments will be conducted to define the sensitivity of the solution on the liquidity constraints Eq. (13).
To address research question Q2 we apply the optimization problem (7)–(13) and construct the option’s strategy with the certain payoff function on the derivatives of Taiwan Futures Exchange. All options in the Taiwan’s market are European-style. The expiration periods of TXO options have spot month, the next two months, and the next two quarterly calendar months2. We will be designing option’s portfolio from the daily closing ask- and bid-prices for TXO options3. We select TXO options; they are liquid assets and comprise above 60% of trading volume of TAIFEX. Here we have taken a single date, May 16, 2016, selected at random, to illustrate the portfolio design in practice.
The strikes and the ask- and bid-prices of options are required inputs to the portfolio optimization model. The available TXO prices are denominated in New Taiwan Dollars (NTD). The TAIFEX index closed at 8, 067.60 on May 16, 2016 (Fig. 1), and the May options contracts expired 9 days later, on May 25, 2016. The option price equals to S 0 = 8, 067.60 NTD at May 16, 2016. Next, we will consider two cases for the expected price, Sˆ T . Suppose that the price will significant move up to 1) S T = 8,300.00 NTD, 2) Sˆ T = 8,400.00 NTD at May 25, 2016. ˆ
The investor then wants to monetize his position, C(t, X) = 100 NTD at the time of purchase, t = 0, and to limit the maximum loss by L = −100 NTD, if the price of underlying asset will come out of the certain range from 8,000 to 8,400 NTD, respectively. The margin requirement for short positions and transaction costs are not accounted. To establish the proposed strategy we use 12 strike prices: sequential n = 6 strike prices are corresponding to call and sequential n = 6 strike prices are corresponding to put
Kc = (8050, 8150, 8250, 8350, 8400, 8500)
Kp = (7850, 7950, 8050, 8150, 8250, 8350) at the same expiration date May 25, 2016. The central strike of the option is K = 8050, is marked with bold. The number of combinations of ask- and bid-price for the 12 options equal to 2n × 2n = 26 × 26 = 4096, thus the cardinality of set of feasible 2http://www.taifex.com.tw/eng/eng4/Calendar.asp 3http://www.taifex.com.tw/eng/eng2/TXO.asp Strike Price 7850 7950 8050 8150 8250 8350 8400 8500 max{FC(X)}
X Total number of contracts 7 -8 4 -3 -9 9 400 64 portfolios |C| = 4096. The cardinality of the set |K| = |Kp ∪ Kc| is 8, therefore, we have 7 pairs of sequential strike prices [kq, kq+1], and that these pairs produce the system of 7 inequalities from Eq. (10), and we should add two equalities on the first closed interval and the left-closed interval from Eq. (8). In our example, we assumed that one can buy or sell at least L = −10, U = 10 contracts at each strike price. This assumption does not limit our approach because the total volume (TtlVol, Fig. 1) is bigger for all strike prices. Then we calculate the price for call and put in accordance with the specific strike (Fig. 1). Next, we maximized the objective function FC(X), proposed in Eq. (7) with the system of constraints (8)–(13). The optimal portfolio was obtained in approximately two minutes on a personal computer, using the MATLAB software.
As a result, the optimum solution in the case Sˆ T = 8400, C = 100 is {cazll {cazll X = (4, −8, 10, −8, 5, 7, 0, 0, −3, 8, −5, 0) | } | }
{puzt X = (7, −8, 4, −3, −9, 9, 0, 0, −7, 9, 3, −5) | } | } {puzt the first six elements correspond to call options, the second six – to put options, the total number of contracts are 58 out of which 34 are for buying and 24 are for selling, with objective function value equal to 700 NTD (bold in Table 1). The optimum solution in the case Sˆ T = 8300, C = 100 is – the total number of contracts are 64 out of which 32 are for buying and 32 are for selling, with objective function value equal to 400 NTD (bold in Table 1). From the Table 1 one can see that the maximum values of the payoff function are achieved at C = 100 NTD, and the minimum values at C = −100 NTD.
At the end of the strategy term, May 25, 2016, the price of underlying index increased to S T = 8,396.20 NTD. The forecast come true, the amount of loss was limited by the maximal loss L = −100.
The initial cost C(t, X) Eq. (2) can be either a positive number or zero, or even negative number. We calculated the alternative portfolios with the different initial costs C(t, X) = {−100, 0, 100} with fixed liquidity constraints |L| = U = 10 Eq. (13), and then the values of liquidity constraints were varied |L| = U = {10, 50, 100} with the fixed initial cost C(t, X) = 100, results are represent in Tables 1, 2. Table 2 shows the strong dependence: with the increase in the number of liquidity constraints, i.e. |L| = U, the maximum value of the payoff function grows too. Fig. 2 show the payoff functions from the proposed option’s portfolio, taking into account, the different values of: a), b) the initial cost C(t, X), Eq. (2) and c), d) liquidity constraints L, U, Eq. (13), respectively. Our strike prices are the x-axis and the payoff functions are the y-axis.
The expansion of the boundaries on the liquidity constraints |L| = U ∈ {10, 50, 100} makes the options portfolio more attractive from the point of view of the terminal payoff amount. On the other hand, there is the difficulty of forming a strategy in view of the buy/sell of a sufficiently large number of underlying assets, a transaction which is difficult to implement for a short time.
In this study, the description of the method of construction of the option’s strategy with the piecewise linear payoff function is carried out. We take into account the next set of assumptions of the proposed option’s strategy: strategy should effectively limit the upside earnings and downside risks; strategy should have an initial cost to enter; strategy should have protection on the downside and upside of the underlying asset price.
To address research question Q1 we have formulated the mathematical model as an integer linear programming problem which includes the system of constraints in the form of equalities and inequalities. The optimum solution was obtained using the MATLAB software.
To address research question Q2 we have demonstrated the possibility of the proposed model on the European-style TXO options of Taiwan Futures Exchange.
In the study we do not use the historical empirical distribution of returns. Our approach is statical, quite general and has the potential to design option’s portfolios on financial markets.
In option’s strategies, in addition to the forecast of the price of the underlying asset, various parameters can be taken into
account: exercise price, volatility, the time to expiration, the risk-free interest rate, option premium, transaction costs. In this case,
even an insignificant change of the parameter’s values can lead to a significant change in the number of possible combinations
of option’s strategies. In our numerical experiments the total number of contracts for different cases varieties from 28 to 64
(Table 1) and from 58 to 488 (Table 2). In papers [
The further research of our study can be continued in the following directions. At first, it is a portfolio optimization under transaction costs (exchange commissions, brokerage fees) and the margin requirement for short positions which are essential in the options market. At second, it is using options with different time to the expiration. At third, it is necessary to extend the system of constraints and add the budget constraint. Such extensions allows us to make the proposed approach more realistic and flexible.
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V. Kurochkin (National Research University Higher School of Economics, Russia) and Mr. Ashu Prakash (Indian Institute of Technology, Kanpur,
India) for valuable comments and suggestions that improved the work and resulted in a better presentation of the material.
grateful to Dr. Sergey V. Kurochkin (National Research University Higher School of Economics, Russia) and Mr. Ashu Prakash
(Indian Institute of Technology, Kanpur, India) for valuable comments and suggestions that improved the work and resulted in a
better presentation of the material.
[
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