**Using options as investments Tool for any asset class.**

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**Essentials of options**

** Option **is a type of contract between 2 persons where one person grants the other person the right to buy or to sell a specific asset at a specific price within a specific time period. The most often options are used in the trading of securities.

** Option buyer **is the person who has received the right, and thus has a decision to make. Option buyer must pay for this right.

** Option writer **is the person who has sold the right, and thus must respond to the buyer’s decision.

*Types of option contracts:*

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**call option.**It gives the buyer the right to buy (to call away) a specific number of shares of a specific company from the option writer at a specific purchase price at any time up to including a specific date.

**put option.**It gives the buyer the right to sell (to put away) a specific number of shares of a specific company to the option writer at a specific selling price at any time up to including a specific date.

*Option contract specifies four main items:*

* *

*The company*whose shares can be bought or sold;*The number of shares*that can be bought or sold;- The purchase or selling price for those shares, known as the
(or*exercise price*);*strike price* - The date when the right to buy or to sell expires, known as
*expiration date.*

* *

*Types of call and put options:*

* *

- European options
- American options

** European options **can be exercised only on their expiration dates.

** American options **can be exercised any time during their life (defined by the option contract).

*The major advantages of investing in options:*

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- Possibility of hedging: using options the investor can “lock in the box” his/ her return already earned on the investment;
- The option also limits exposure to risk, because an investor can lose only a set amount of money (the purchase price of option);
- Put and call options can be used profitably when the price of the underlying security goes up or down.

*The major disadvantages of investing in options:*

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- The holder enjoys never interest or dividend income nor any other ownership benefit;
- Because put and call options have limited lives, an investor have a limited time frame in which to capture desired price behavior;
- This investment vehicle is a bit complicated and many of its trading strategies are to complex for the non-professional investor.

Further in this article we focus only on some fundamental issues of investing in stock options including some most popular strategies.

**Options pricing**

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The value of put or call options is closely related with the market value/ price of the security that underlies the option. This relationship is easily observed just before the expiration date of the option. The relationship between the intrinsic value of option and price of underlying stock graphically is showed in Fig. 1. (a – for call option, b – for put option). These graphs demonstrate the intrinsic value of the call and put options. In the case of call option (a), if the underlying stock price at the end of expiration period is less than the exercise price, intrinsic value of call option will be 0, because the investor does not use the option to buy the underlying stock at exercise price as he/ she can buy it for more favorable price in the market. But if the underlying stock price at the end of expiration period is higher than the exercise price, intrinsic value of call option will be positive, because the investor will use call option to buy the underlying stock at exercise price as this price is more favorable (lower) than price in the market. However it is not necessarily for the option buyer to exercise this option.

Instead the option writer can simply to pay buyer the difference between the market price of underlying stock and exercise price. In the case of put option (b), if the underlying stock price at the end of expiration period is higher than the exercise price, intrinsic value of put option will be 0, because the investor does not use the option to sell the underlying stock at exercise price as he/ she can sell it for more favorable price in the market. But if the underlying stock price at the end of expiration period is lower than the exercise price, intrinsic value of put option will be positive, because the investor will use put option to sell the underlying stock at exercise price as this price is more favorable (higher) than price in the market. In both cases graphs a and b demonstrates not only the intrinsic value of call and put options at the end of expiration date, but at the moment when the option will be used.

**Fig. 1 Intrinsic value of option**

** **

Exploring the same understanding of the intrinsic value of the call/ put option as it was examined above*, intrinsic value of the call/put options **can be more precisely estimated using analytical approach:*

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**IV****c ****= max **{ **0, P****s ****– E **}**, (1.1)**

**IV****p ****= max **{ **0, E – P****s**}**, (1.2)**

** **

here: IVc – intrinsic value of the call option;

IVp – intrinsic value of the put option;

Ps – the market price of the underlying stock;

E – the exercise price of the option;

max – means to use the larger of the two values in brackets.

*When evaluating the call and put options *market professionals frequently use the terms „in the money“, „out of money“, „at the money“. In table 1 these terms together with their application in evaluation of call and put options are presented. These terms are much more than only exotic terms given to options – they characterize the investment behavior of options.

**Table 1**

Intrinsic values of put and call options, estimated **using **formulas 1.1 and 1.2 reflect what an option should be worth. In fact, options very rarely trade at their intrinsic values. Instead, they almost always trade at the price that exceeds their intrinsic values. Thus, put and call options nearly always are traded at the premium prices. ** Option premium **is the quoted price the investor pays to buy put or call option.

Option premium is used to describe the market price of option.

** The time value (TV) **reflects the option’s potential appreciation and can be calculated as the difference between the option price (or premium, Pop) and intrinsic value (IVop):

**TV = P****op ****– IV****op ****(1.3)**

Thus, **the premium for an option **can be understood as the sum of its intrinsic value and its time value:

**P****op ****= IV****op ****+ IV****op ****(1.4)**

**Using options. Profit and loss on options.**

** **

Fig.1 shows the intrinsic values of call and put options at expiration.

However, for the investor even more important is the question, what should be his/ her profit (or loss) from using the option? In order to determine profit and loss from buying or writing these options, the premium involved must be taken into consideration. Fig.2, 3, 4 demonstrates the profits or losses for the investors who are engaged in some of the option strategies. Each strategy assumes that the underlying stock is selling for the same price at the time an option is initially bought or written.Outcomes are shown for each of 6 strategies. Because the profit obtained by a buyer of option is the writer’s loss and vice versa, each diagram in Fig. 2, 3 and 4 has a corresponding mirror image.

Fig. 2 shows the profits and losses associated with buying and writing a call respectively. Similarly, Fig.3 shows the profits and losses associated with buying and writing a put, respectively. If we look at the graphs in these figures we identify that the kinked lines representing profits and losses are simply graphs of the intrinsic value equations (1.1. 1.2), less the premium of the options.

** Thus, the profit or loss of using options is defined as **difference between the intrinsic value of the option and option premium:

**Profit (or loss) on call option = IVc – P****op ****= max **{**0, P****s ****– E**} **– P****op ****=**

**= max **{**– P****op****, P****s ****– E – P****cop**}**, (1.5)**

**Profit (or loss) on put option = IV****p ****– P****op ****= max **{**0, E – P****s**}**– P****op ****=**

**= max **{**– P****op****, E – P****s ****– P****pop**}**, (1.6)**

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here Pcop – premium on call option;

Ppop **– **premium on put option.

**Fig. 2. Profit/ loss on the call options**

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**Fig. 3. Profit/ loss on the put options**

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Fig. 4 illustrates a more complicated option strategy known as ** straddle. **This strategy involves buying (or writing) both a call and put options on the same stock, with the options having the same exercise price and expiration date. The graph in Fig.4 representing profit and loss from the strategy “Buy a put and a call” can be easily derived by adding the profits and the losses shown in Fig. 2 (Buy call) and 3 (Buy put); profit and loss from the strategy “Write a put and a call” can be derived by adding the profits and the losses shown in Fig. 2 (Write call) and 3 (Write put).

**Fig. 4. Profit/ loss on a straddles**

** **

For more precise valuation of options some fairly sophisticated options pricing models have been developed. The most famous of them is ** Black-Scholes model for estimating the fair value of the call options **published by Black and Scholes in 1973.

*5 main parameters used in Black-Scholes model:*

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- Continuously compounded risk free rate of return expressed on the annual basis;
- Current market price of the underlying stock;
- Risk of the underlying common stock, measured by the standard deviation of the continuously compounded annual rate of return on the stock;
- Exercise price of the option;
- Time remaining before expiration, expressed as a fraction of a year.

Many active options traders use the complex formulas of this model (see Annex 1) to identify and to trade over- and under valuated options.

**Annex 1**.

**Black-Scholes formula for estimating the fair value of the call options**

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**Vc = N(d****1****) * P****s ****– [E*R* T* N(d****2****)] / e, (1)**

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here Ps – current market price of the underlying stock;

E – exercise price of the option;

R – continuously compounded risk free rate of return expressed on the annual basis;

T – time remaining before expiration, expressed as a fraction of a year

N(d1) and N(d2) denote the probabilities that outcomes of less d1 and d2 respectively.

**N(d****1 ****) = ln (P****s****/E) + [(R + 0,5 **δ**)*T] / **δ **√ ****T; (2)**

**N(d****2 ****) = ln ( P****s****/E) + [(R – 0,5 **δ**)* T] / –**δ **√ ****T, (3)**

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here: δ – risk o the underlying common stock, measured by the standard deviation of the continuously compounded annual rate of return on the stock.

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**Portfolio protection with options. Hedging.**

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Hedging with options is especially attractive because they can give protection against loss or the stock protects the option against loss.

A ** hedger **is an individual who is unwilling to risk a serious loss in his or her investing position and takes the actions in order to avoid or lessen loss.

** Using options to reduce risk. **Suppose the investor currently holds the shares of the company X in his portfolio. The price of the share is 8,5 EURO. Looking to the future the investor is not sure in what direction the price of the share will change. If the price will grow, may be to 9 EURO, by holding these shares investor could receive the profit. But if the price will fall, may be to 8 EURO, the investor could suffer the loss.One way to avoid this downside risk is to sell the shares. The problem is that the investor may regret this action if the fall of the price of the share does not occur and investor has forgone the opportunity to earn a profit. An alternative approach is to retain the shares and buy a put option. This option will rise in value as the share price falls. If the share price increases the investor gains from his/ her underlying shareholding.

The hedging reduces the dispersion of possible outcomes to the investor. There is a floor below which losses cannot be increased, while on the upside the benefits form any increasing in the price of the share is reduced due to the option premium paid. But if the price of the share stands still 8,5 EURO, however, the investor may feel that the option premium he/ she paid to insure against an adverse movement at 0,85 EURO or 10 percent of the share price was excessive. If the investor will keep buying this type of “insurance”, though it can reduce investor’s portfolio returns during the longer time (for example, a year), substantially.

** Using options to reduce losses. **Suppose that the investor wants to buy the attractive shares of company B to his / her portfolio. The current market price of this share is 9 EURO. The investor is fairly sure that this share will rise in price, but is not so confident as to discount possibility of a fall in price. May be the price of the share fall to 8,5 EURO? How could the investor behave? He/ she can either exercise a direct purchase of these shares in the market at current price or to purchase a call options with underlying stock of company B. If the price of the share does fall significantly, the size of the loss is greater with the share purchase, because the option loss is limited to the option premium paid.

** Hedging portfolios of shares using index options. **Large investors usually manage varied portfolios of shares so, rather than hedging individual shareholdings with options they may hedge their portfolios through the options on the entire index of shares

** Index option **is based on stock index instead of an underlying stock. When index option is exercised, settlement is made by cash payment, not delivery of shares.

The most often index options are settled on the bases of such indexes as Standard&Poor’s 500 (USA); FTSE 100 (UK); DAX (Germany), CAC (France), NIFTY 50 , Sensex etc.

Suppose, the investor manage a well diversified portfolio of shares and currently is concerned that the market may fall over the next 3 months. One of possible investment strategies for the investor is to buy the put option on the stock index. If the market does fall, losses on the portfolio will be offset by gains on the value of the index put option. If the portfolio is unhedged, the investor suffers from the market fall substantially. But it is important to remember about the expenses of the insurance of portfolio: when the options premiums are high (during periods of market volatility caused by economic crises), hedging of the portfolio of stocks with index options over longer period could be expensive.

Using hedging strategies very important characteristic is the hedge ratio of the portfolio. *Hedge ratio **is a number of stocks to buy or sell with options such that the future portfolio value is risk-free. *** The hedged portfolio **consists of m purchased shares and n options written (issued) on these shares.

*Hedge ratio (HR) can be estimated using formula:*

*HR = m / n, ***( 1.5)**

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here: m – number of shares in the portfolio;

n – a number of options written on the shares in the portfolio.

** Riskless (perfect) hedge **is when for m and n are chosen such a values which allow in each moment given to compensate the decrease in prices of the stocks by increase in value of options. This meaning of hedge ratio is called as a

**But**

*perfect hedge ratio.**perfect hedge ratio could be achieved only under following assumptions:*

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- There are no transaction costs in the market;
- There are no taxes;
- The numbers of all traded securities is unlimited (including fractional

numbers);

- All the securities are available for trading permanently (24 hours) and at any moment.

It is obvious that today even in the high developed markets these assumptions can not be realized. Thus any hedged portfolio and its hedge ratio reflects only the particular level of the “insurance” of the investor against the market risk.

Lokesh Madan

thanks for explain option very nice way kindly continue with call and option