Using a synthetic strangle to Manage stock market risk Текст научной статьи по специальности «Экономика и бизнес»

pISSN 2073-8005 eISSN 2311-9438

Securities and Financial Markets

Translated Articlef

USING A SYNTHETIC STRANGLE TO MANAGE THE STOCK MARKET RISK

Sergei N. YASHIN

National Research Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russian Federation jashinsn@yandex.ru Corresponding author

Egor V. KOSHELEV

National Research Lobachevsky State University of Nizhny Novgorod,

Nizhny Novgorod, Russian Federation

ekoshelev@yandex.ru

Vladlen V. SOKOLOV

National Research Lobachevsky State University of Nizhny Novgorod,

Nizhny Novgorod, Russian Federation

sokolov2w@gmail.com

Article history:

Received 15 February 2017 Received in revised form 7 March 2017 Accepted 13 April 2017 Translated 10 July 2017 Available online 15 September 2017

JEL classification: C01, G01, G32

Keywords: synthetic option, strangle, option, equity risk

Abstract

Importance Options are a representative example of derivative securities that are applied to mitigate investors' risks. Considering contradictions and discrepancies arising from changes in the options price due to the fluctuating price for primary securities, rather than the flat exercise price for the option, investors seek possible combinations of securities so to reduce equity risks. The article considers a synthetic options strategy - a synthetic strangle. Objectives The research models a synthetic strangle and applies it in practice using stocks of LUKOIL Oil Company. Methods The research involves methods of logic and statistical analysis.

Results We apply a symmetrical binomial lattice to determine the synthetic strangle price. Based on a binomial model, we build the model of LUKOIL's stock price movement and consider the case of synthetic strangle duplication by constructing a portfolio of stocks and bonds, which generates the same cash flows as options. Conclusions and Relevance It is feasible to use a synthetic strangle in a situation when market stock price movements are uncertain and investors protect their capital against unexpected fluctuations in the stock market by purchasing a synthetic option. We also found break-even points for the investor.

© Publishing house FINANCE and CREDIT, 2017

The editor-in-charge of this article was Irina M. Komarova Authorized translation by Irina M. Komarova

Analyzing potential investment, investors consider two principal aspects - the expected return on investment and the risk. Any investor strives to maximize the first component and reduce the second one, using various methods. In this article, we do not focus on the expected return, but rather attempt to examine one of the investment risk mitigation methods. This method involves derivative securities and the options contract, in particular. Theoretical principles and options pricing are studied in proceedings of foreign authors, such as Z. Bodie [1], L. Brandao, J. Dyer, W. Hahn [2], A.K. Dixit, R.S. Pyndyck [3], R. Merton [4], J. Cox, S. Ross, M. Rubinstein [5, 6], J. Hull [7, 8], R. Schwartz [9], L.J. MacMillan [10]; and Russian scholars, such as A.N. Burenin1, A.S. Shvedov [11]. Options strategies and their use are reviewed in studies by Sh. De Kovno, Ch. Tucci2, J.F. Marshall, V.K. Bansal [12], Sh. Natenberg [13]. C. Bastian-Pinto, L. Brandao, L. Ozorio [14], G. Guthrie [15], J. Cox, S. Ross, M. Rubinstein [5, 6] also contributed to researches into binomial trees and their application to options. The conventional approach necessitates the exercise price of an option to be estimated. The exercise price of an option means the price that is to be paid for the underlying asset when the option is exercised. The option also depends on the time and risk [11].

Synthetic instruments stand for instruments that, if combined, create the totality of cash flows that reproduces the totality of cash flows generated from real instruments.

Following the strangle strategy, the buyer purchases a put option and a call option relating to the same underlying asset but with the different exercise price. Under this strangle strategy, the buyer pays the seller the amount equal to the value of two options (call and put). If the investors chooses to purchase this strategy, they have a yield only in case of the upward or downward swing in the price. If the investor decides to

fFor the source article, please refer to: Яшин С.Н., Кошелев Е.В., Соколов В.В. Применение синтетического стрэнгла для управления фондовым риском. Финансы и кредит. 2017. Т. 23. Вып. 21. C. 12141231. URL: https://doi.org/10.24891/fc.23.21.1214

1 Burenin A.N. Forvardy, f'uchersy, optsiony, ekzoticheskie ipogodnye proizvodnye [Forwards, futures, options, exotic and weather derivatives]. Moscow, Vavilov Scientific and Technological Society Publ., 2005, 534 p.

2 De Kovno Sh., Tucci Ch. Strategii hedzhirovaniya [Hedge Strategies]. Moscow, INFRA-M Publ., 1996, 208 p.

sell this strategy, a yield is generated only if the market does not fluctuate, remaining flat.

We use a binomial model to construct a synthetic strangle [16]. We resort to the binomial model, rather than the Black-Scholes model [17], since the investor combines a stock in question and risk-free bond in the synthetic strangle. In the mean time, the Black-Scholes model requires to use the risk-free interest rate only, without regarding the risk-free bond.

To clarify our further reasoning, we introduce the following denotations:

S 0 is the stock price as of the current moment of time;

u is the rate of growth in the stock price estimated as to a semiannual increase;

d is the rate of growth in the stock price estimated as to a semiannual decrease;

S0 d is the projected stock price estimated as to a semiannual increase;

S0 u2 is the projected stock price estimated as to a semiannual decrease;

S0 u is the projected stock price estimated as to a yearly increase;

S0 ud is the projected stock price estimated as to a semiannual increase and a subsequent semiannual decrease;

S0 du is the projected stock price estimated as to a semiannual decrease and a subsequent semiannual increase;

S0 d2 is the projected stock price estimated as to a yearly decrease;

K1 is the strike price for a call option exercised in a year;

K2 is the strike price for a put option exercised in a year;

Cuu is the price for a synthetic call option provided that the projected price doubles within a year;

Please cite this article as: Yashin S.N., Koshelev E.V., Sokolov V.V. Using a Synthetic Strangle to Manage Stock Market Risk. Digest Finance,

Cud is the price for a synthetic call option in case of NB 0 is the number of bonds needed to create

a semiannual increase and a semiannual decrease in the projected stock price;

Pdu is the price for a synthetic put option provided that the projected price demonstrates a two-fold drop within a year;

Pdd is the price for a synthetic put option in case of a semiannual decrease and a semiannual increase in the projected stock price;

O is the price for a synthetic strangle;

Ou is the price for a synthetic strangle in the year end in case of a semiannual increase in the stock price;

Od is the price for a synthetic strangle in the year end in case of a semiannual decrease in the stock price;

Ouu is the price for a synthetic strangle in the year end in case of a two-fold increase in the projected price within a year;

Oud is the price for a synthetic strangle in the year end in case of a semiannual increase and a subsequent semiannual decrease in the projected stock price;

Odu is the price for a synthetic strangle in the year end in case of a semiannual decrease and a subsequent semiannual increase in the projected stock price;

Odd is the price for a synthetic strangle in the year end in case of a two-fold decrease in the projected price within a year;

t is the half year number;

B0 is the value of a risk-free bond as of the current moment of time;

rf is a semiannual risk-free interest rate;

B 1= B0 (1 + rf) is the price for a bond as at the t = 1, i.e. in a half year;

B2= B0 (1 +rf)2 is the price for a bond as at the t = 2, i.e. in a year;

NS 0 is the number of stocks needed to create an equivalent portfolio as at the t = 0;

B ,0

an equivalent portfolio as at the t = 0 time;

NuS 0 is the number of stocks needed in a half year (t = 1) once the u-situation occurs;

NuB 0 is the number of bonds needed in a half year (t = 1) once the u-situation occurs;

NdS 1 is the number of stocks needed in a half year (t = 1) once the cf-situation occurs;

NdB 1 is the number of bonds needed in a half year (t = 1) once the f-situation occurs.

To measure the option, we use the binomial model described in the paper by C. Bastian-Pinto, L. Brandao, L. Ozorio [14].

This model requires to construct a symmetric lattice with a mathematical expectation that is a kind of the price trend. Fig. 1 and 2 indicate binomial movement.

The above figures contain the following denotations:

*

xt stands for values in the combined lattice; xt is a deterministic expected value of movement; a is the mean square variance of the value; ^ is the growth rate of the value; p is the probability of the event; A t is a time interval.

The Bastian-Pinto-Brandao-Ozorio model differs from the Cox one because the latter does not ingrain a drift rate into the lattice design, being a part of the probabilistic transition to each node. The researches of the authors [14] refer to cases demonstrating that this method allows for a more precise measurement of a real option.

To illustrate the model design, we use ordinary stocks of LUKOIL Oil Company, with their trends indicated in Table 1. Sometimes shareholders happen to choose whether they should retain or sell their stocks. As the price for this stock shows, the price is going down, but it is unclear whether this trend stems

from long-standing processes fluctuations in the market.

or some temporary

We assume this is a temporary reduction in the price and the investor does not intend to sell stocks, but rather plans to hedge the risk and benefit from a greater decrease or increase in prices for stocks.

To make up a synthetic strangle, the following steps should be accomplished:

1) determining the exercise price of an option (the strike price);

2) constructing a binomial model reflecting the movement of the stock price;

3) determining the price for call and put options;

4) making a synthetic strangle.

First, we determine the strike price. Whereas prices for stocks are unpredictable, prices are averaged to build forecasting models. Having found the average prices using the formula (P1+P2+P3)/3 , we can show in Fig. 2 that the price-to-earnings ratio (P/E ratio) of the stock demonstrates a more approximate normal distribution, thus allowing to gauge a confidence interval of future movements of stock prices.

Referring to Table 1, we set up a confidence interval of the stock price.

The P/E ratio of a stock per year is expressed as

(1 + i )52 - 1=',

• where iyear=-O.162

Annual standard deviation

l^week-At=0.0215714-57=0.156 .

where SEM =

V n

The strike price for a ^ put option is lower than the minimum value, being equal to RUB 1,850.

After we assess the strike prices for call and put options, we measure the option value and construct a synthetic strangle. We apply the above binomial model to determine the option value.

To build a binomial model, it is necessary to assess a mathematical expectation of the P/E ratio of the stock and standard deviation. Thus, we restate figures in Table 1 into P/E logarithms and find the mathematical expectation of the P/E ratio. The mathematical expectation of the P/E ratio is 0.915496. We also estimate a weekly standard deviation and compute it for a semiannual period.

Fig. 4 presents the binomial model of the stock price movement under the following conditions:

Mh£df year = 0.915496;

o„

t year

=0.11;

u =e

r-VAt.

,0.021571-5.09902

= 1.11627;

d =—=0.89584. u

equals

The confidence interval is assessed with the following formula:

x — ( 1.96-SEM); x + (1.96-SEM),

The example leads us to

-0.162±L96-°456; -0.162±0.04224. V52

Hence, in a year, prices for stocks will range from 1,852.317 to 2,048.964.

The strike price for a ^ call option shall be higher than the maximum price for a stock, i.e. being an out-of-money call. In our example, we accept the price of RUB 2,050.

To introduce a risk-free bond into our portfolio, we choose a government bond with the P/E ratio that would match a risk-free interest rate. In Russia, we may opt for the refinancing rate of 8.25 percent. The P/E ratio of bond 003-29010-nK approximates target traits, being equal to 8.677 percent3. The current market price for this bond is RUB 105 as of November 1, 2015.

Hence, we can compute the full semiannual P/E ratio rf using the formula below [18]:

(1 + rf )2=1 + rf annual; rf =V 1.08677-1=0.04248261; B 1= B 0 (1 + rf ) = 109.461; B 2 = B0 (1 + rf) 2 = 114.111.

Possessing all necessary data, we can make a synthetic strangle. In twelve months, the holder of this option strategy will be able to buy at the ^ price or sell at the ^ price the underlying stock of the contract:

Ouu=max {max (S0 u2 — K ,0), max (K—S 0 u2'0)}; Oud = max {max (S 0 ud — K ,0), max (K—S 0 ud ,0)}; Odu=max {max (S 0 du — K ,0), max (K — S 0 du ,0)]; Odd = max {max (S 0 d2 — K ,0), max (K — S0 d2,0)].

3 RUSBONDS. URL: http://www.rusbonds.ru

Showcasing the assessment of the option price and a choice between a call option and put option, we examine the above equation and substitute the values with ones we calculated before:

Ouu=max {max (2, 449.969-2, 050.0),} max{(1, 850-2, 449.969.0)}.

The first maximum max(2, 449.969-2, 050.0) denotes the price for a call option. This price is made up if one of two scenarios is chosen. Following the first scenario, the investor exercises the option if the price exceeds K2. According to the second scenario, the investor does not exercise the option if the stock price does not exceed the value. In the first scenario, the price for the option equals the difference between the stock price upon the exercise of the option, and the strike price. In the second scenario, the price for the option is zero.

The second maximum max(1, 850-2, 449.969.0) reflects the put option pricing. The price for a put option is formed like it is done in the example, being equal to zero.

Finally, the third maximum helps choose which option - a put or a call - is appropriate for the given situation. The highest of two prices is chosen:

Ouu=max {399.969.0}.

The maximum is 399.969. Therefore, in the uu-situation, a call option should be used, with its price being 399.969.

Relying on the figures from the example, we conclude the remaining options will be as follows:

Cud = Pud = °'

C du Pdu = 0'

Pdd=272.086.

As at t = 1, call and put options relate to the European type, generating neither income nor expenses. It has the following meaning:

Ou=Od=O.

We compose and solve three systems of equations. Each system of equations reflects nodes of the model built.

The first node is marked with a dashed line in Fig. 4. Assume we are at t = 1 now, and stock prices equal

S 0 u . At the following moment t = 2, stock prices can have one of the two values, increasing up to S0u2 or decreasing down to S0ud . Based on these scenarios, cash flows generated by the call-put option will amount to the volume Cuu or the value Cud . To create a portfolio of stocks and bonds that, as at t = 2, would generate the same cash flows like call and put options, in particular, Cuu and Cud , we apply the bond price as at t = 1, which is equal to Bt and the reverse flow for the following period accounting for B1 (1 + rf).

We present the system of equations, which will help to quantify how many stocks and bonds are needed for the equality and an equivalent portfolio:

So u2nS,1 + B1 (1 + rf )-"b,l = Cuu> S0 ud • ns ,1 + B1 (l + rf ) • nB ,1 = Cud •

Inserting the figures from our example, we arrive at the following system of equality:

J2, 449.969• nsf1 +114.111 • nB1=399.969, |1, 966.174 • ns ^+114.111 • nB¿=0.

The solution is as follows:

|ns f1 = 0.826732397, |nB f1 = -14.24489965.

The following system of equations results from the conditions that the stock price is S0 d as at t = 1. Likewise in the previous example, the stock price can grow up to S0ud or drop down to S0 d2. Based on this condition, the equivalent portfolio is made up so to equate it to Pdd as at t = 2 provided the stock price is unchanged. However, if the stock price rises, it equals to Cdu=Cud= 0.

Thus we arrive at the following equation:

S0 ud • nS ,1 + B1 (1 + rf ) • nB ,1 = Cud >

f B,1

0"" "S ,1

So d2 • ns,1 + B 1( 1 +rf ) • «B,1 = Pdd •

Inserting the specific figures from our example, the system of equations is as follows:

[1, 966.174255• nSf1 + 114.111 • nB1 = 0, |1, 577.914019 • nS1 + 114.111 • nB 1 = 272.086.

It leads us to the following solutions:

nS ,1=-0.70078298, nB1 = 12.07474543.

When the mix of the equivalent portfolio is known, we select shares of securities as at t = 0. These shares are selected so that related income, as at t = 1, precisely match expenses at the moment:

S0 u'nS ,0+ B 0( 1 + rf )'nB ,0 = S 0 u'nS ,1+ B1^nв,1,

S0 d'ns ,0+ B0 (1 + r

B ,0"

+ rf )"nB,0= s0 d-ns ,1+ B1-nB ,1.

Expressions in the left-hand part of the upper formula describe flows, which would proceed from holding of the stock and bond as at t = 1, if the stock price rises (falls). The opposite part of the formulae features necessary income, which would allow financing equivalent portfolios dependent on situations as at t = 1. Sourcing interim results and figures from the example, we arrive at:

Î2 , 397.37-nS,0 +109.461-nB,0=422.7225, |1, 923.962-nS,0+ 109.461-nB,„ = -26.5661.

It gives the following solutions:

nS ,0 = 0.949052, nB0=-16.9239.

If we know the values of nS 0 and nB0 , we can be certain of measures to be undertaken at t = 0 and later on at t = 1 so to make up an equivalent portfolio in selling and purchasing stocks and bonds, with the price of the portfolio being the same as the purchase price of the call-put option:

ns,0-S +nB,0-B0=0.949052-2, 345.9— — 16.9239-106= 449.3722.

Obtaining all the necessary figures from the above equations, we put them into Table 2 and analyze how call and put options are duplicated. The first column indicates how many assets should be purchased or sold at a certain moment of time. Right from the onset t = 0, 16.9239 bonds and 0.949052 stocks are sold and purchased respectively. Net expenses account for RUB 449.3722.

We assume a situation when the stock price increases in the following period. At the moment, the stock price increases up to RUB 2,397.37, thus generating RUB

2,275.225 in income, while bonds entail RUB 1,852.50242 in expenditures. Therefore, the balance of income amounts to RUB 422.722573. However, to set an appropriate equality of balance, 0.82673 stocks shall be purchased and 14.245 bonds shall be concurrently sold. Thus, the purchase of stocks costs RUB 1,981.98353, while uncovered bonds are sold generating RUB 1,559.26096 in income. As at t = 1, income and expenses are observed to line up. As at t = 2, the expenses of RUB 1,625.499744 are incurred to sell the bonds, notwithstanding how the stock prices change. When the stock price grows, the stocks are sold generating RUB 2,025.46874 in revenue. As per the second scenario, the stock price drops, with the revenue being RUB 1,625.499744 only. So, in the uu-situation - stock price growth -the difference between income and expenses is RUB 399.969. In the ud-situation, when the stock price falls down to RUB 1,966.174, income from sale of stocks will match expenses for bonds, with the balance being zero. It is noticeable that the resultant balance of 399.969 and 0 precisely match cash flows from the above options. This equality is evidence of the correctness of the synthetic strangle. Now we should earmark it for buying or selling so to derive profit in a year.

We observe what happens, if the stock price decreases down to RUB 1,923.967, as at t = 1. Income from stocks are lower than that in the second case, but expenses for bonds remain equal RUB 1,852.5. To line up the difference between income and expenses (26.56), it is reasonable to sell stocks, derive income of RUB 1,348.28 and purchase bonds. This transaction costs RUB 1,321.71. In the following period, if the stock price increases up to RUB 1,966.174 (the du-situation), income from bonds equals expenses for stocks, with the balance being zero. This value completely matches the cash flow from the Cdu option. In the dd-situation, the balance of income and expenses should be Pdd. As seen in Table 2, the values match.

If the stock price remains within the range of ^ and ^ -RUB 1,850 through RUB 2,050 - at the final moment of the option exercise, call and put options will keep their in-the-money position both. Such options will not be exercised since they do not generate income, with the investor suffering losses in the amount of initial investment made in the option. In our case, it amounts to RUB -499.372. However, if the market undergoes certain changes that have a positive effect on the stock price and it reaches the projected amount

of RUB 2,449.969, the investor will be able to exercise the call and put options and earn the income amounting to RUB 2,449.969 - RUB 2,050 = = RUB 399.969. Considering the initial investment of RUB 499.372, the investor has a loss of RUB 99.403. Referring to the reverse situation, when stock prices drop down to RUB 1,577.914, the investor will suffer the loss of RUB 272.086 - 499.372 = = -227.286 anyway even if the investor exercises the call and put options and derives income from the difference of stock prices and the K2 price in the amount of RUB 272.086, since the initial investment exceeds the income.

We determine what prices are break-even points for the investor. For call and put options, they amount to RUB 2,050 + 499.372 = 2,549.372 and

RUB 1,850 - 499.372 = 1,350.628 respectively. The investor makes profit only when the prices go beyond these limits.

Those investors, who expect the stock prices to be unstable and go beyond the boundaries of prices as set before at the final moment of time, will opt for a long strangle strategy, i.e. buying call and put options.

Table 1

Prices for LUKOIL Oil Company's stocks, RUB

Date Price

January 5, 2015 2,461

January 12, 2015 2,736

January 19, 2015 2,849

January 26, 2015 2,789.9

February 2, 2015 3,120.1

February 9, 2015 3,135

February 16, 2015 2,990

February 23, 2015 2,981.2

March 2, 2015 2,800

March 9, 2015 2,659

March 16, 2015 2,710

March 23, 2015 2,577

March 30, 2015 2,824.8

April 6, 2015 2,634.9

April 13, 2015 2,576

April 20, 2015 2,645

April 27, 2015 2,645.6

May 4, 2015 2,691.6

May 11, 2015 2,595

May 18, 2015 2,537

May 25, 2015 2,461.3

June 1, 2015 2,570

June 8, 2015 2,506.9

Please cite this article as: Yashin S.N., Koshelev E.V., Sokolov V.V. Using a Synthetic Strangle to Manage Stock Market Risk. Digest Finance,

292 2017, vol. 22, iss. 3, pp. 286-297.

https://doi.Org/10.24891/df.22.3.286

Other investors, who, by contrast, believe stock prices to be more or less stable, make profit by opening short strangle positions, i.e. selling strangles.

If the highest level of profit is RUB 449.3722 and the loss has no limits upon the sale of the strangle, the purchase of the strangle reverses the situation for the investor, when the highest loss equals the initial investment of RUB 449.3722, while the profit is unlimited.

Fig. 5 provides the payment schedule for short strangles relating to stocks with the K1 u K2 strike prices, maturing in year. They are depicted with a dashed line.

Hence, the proposed synthetic strangle model demonstrates the way the investor can significantly reduce the risk associated with the underlying asset. Forecasting future prices for assets, investors diverge in two directions. Some of them open long strangles trying to avoid an unnecessary risk and benefit from big swings in prices, while the others start selling the positions believing that prices will not change dramatically and generate some income, albeit limited.

June 15, 2015 2,474.4

June 22, 2015 2,401.7

June 29, 2015 2,425.3

July 7, 2015 2,465

July 13, 2015 2,465.4

July 20, 2015 2,468.9

July 27, 2015 2,537.6

August 3, 2015 2,520

August 10, 2015 2,561.1

August 17, 2015 2,412

August 24, 2015 2,520

August 31, 2015 2,458

September 7, 2015 2,487.6

September 14, 2015 2,403.8

September 21, 2015 2,240

September 28, 2015 2,140

October 5, 2015 2,400.1

October 12, 2015 2,303.8

October 19, 2015 2,336.7

October 26, 2015 2,320

November 2, 2015 2,511.8

November 9, 2015 2,485

November 16, 2015 2,540

November 23, 2015 2,539.4

November 30, 2015 2,489.9

December 7, 2015 2,450

December 14, 2015 2,327.1

December 21, 2015 2,310.2

Source: Moscow Stock Exchange data

Table 2

Duplication of call and put options

Payments as of the given moment

assets t = 0 t = 1 t = 2

u d uu ud du dd

«S.0 =0,95 -2,226.38 2,275.22 1,825.94 0 0 0 0

nB,o = —16,9: 1,777.01 -1,852.5 -1,852.5 0 0 0 0

n"s ,0=0,83 0 -1,981.98 0 2, 025.47 1,625.5 0 0

«B-1=—14,25 0 1,559.26 0 -1,625.5 -1,625.5 0 0

nS ,1 = -0,7 0 0 1,348.28 0 0 -1,377.86 -1,105.78

nB1 = 12,07 0 0 -1,321.71 0 0 1,377.86 1,377.86

Source: Authoring

Figure 1

Steps of the symmetrical lattice construction

Source: [14] Figure 2

Symmetrical lattice node

Source: [14]

Figure 3

Normal distribution of weekly P/E ratio of LUKOIL's stock

Figure 4

A price movement model of LUKOIL's stock

Source: Authoring

Figure 5

Payment schedule for a synthetic strangle

600 + Source: Authoring

Acknowledgments

The article was supported by the Russian Foundation for Humanities, grant No. 15-02-00102. References

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