8

Derivatives and Their Valuation

Risk is inherent in an investment decision. The securities that are traded in the stock exchange inherently presume risk since their valuations are based on the future expectations of investors. The uncertainty in terms of the future reflects in the risk that is borne by the instrument. There are four basic steps in risk management that an investor should be aware of:

1. Understand the nature of various risks.
2. Define a risk management policy and quantify, if possible, the maximum risk that he is willing to take.
3. Measure the risks if quantifiable and identify the risk if not quantifiable.
4. Institute risk-absorption tactics into the portfolio to control and monitor all risk.

UNDERSTANDING RISK

Risks in an investment setting can arise due to several factors. The major categories of risk that can be found in the stock markets are described below.

Price or market risk is the risk of loss due to change in the market price of instruments traded in the stock exchange. The price risk can increase further due to the market liquidity risk, which arises when large positions in individual instruments or exposures reach more than a certain percentage of the market, instrument or issue. Such a large position could be potentially illiquid and not be capable of being replaced or hedged out at the current market value and as a result may be assumed to carry extra risk.

Counterparty or credit risk is the risk of loss due to a default of the counterparty in honouring its commitment in a transaction (credit risk). If the counterparty is situated in another country, this also involves a country risk, which is the risk of the counterparty not honouring its commitment because of restrictions imposed by the government though the counterparty is capable and willing to settle.

Operating risk is the risk that the investor may be exposed to financial loss either through human error, misjudgment, negligence, and misbehaviour, or through uncertainty, misunderstanding, and confusion as to responsibility and authority. The different kinds of operating risks are legal, regulatory, errors and omission, frauds, custodial, systems risk, dealing risk, and settlement risk.

Legal risk is the risk that the investor will suffer a financial loss because contract provisions are unenforceable or inadequately documented, or the precise relationship with the counterparty is unclear.

Regulatory risk is the risk of carrying out a transaction that is not as per the prevailing rules and laws of the country’s market regulator.

Errors and omissions are also inherent in financial operations. Errors and omissions may arise due to price, amount, value, date, currency, and buy/sell side, or settlement instructions.

Investor himself might enter into fraudulent practices such as front running, circular trading, undisclosed personal trading, insider trading, and routing deals to select brokers.

Custodial risk is the loss of prime documents due to theft, fire, water, termite, and so on. This risk is enhanced when the documents are in a physical form and occur especially in transit.

Systems risk arises due to significant deficiencies in the design or operation of supporting systems. This could also be due to the inability of systems to develop quickly enough to meet the rapidly evolving user requirements. Sometimes, the establishment of many diverse, incompatible system configurations, which cannot be effectively linked by the automated transmission of data and which require considerable manual intervention, may also lead to a systems risk.

Dealing risk arises from unsettled transactions due for all dates in future. If the counterparty were to go bankrupt one day, all unsettled transactions would have to be redone in the market at the current rates. The loss would be the difference between the original contract rate and the current rate. Dealing risk is, therefore, limited to only the movement in the prices and is measured as a percentage of the total exposure.

Settlement risk is the risk of the counterparty defaulting on the day of the settlement. The risk in this case would be 100 per cent of the exposure if valuation has been done before receiving the amount from the counterparty. In addition, the transaction would have to be redone at the current market rates.

DEFINE RISK POLICY

The investor has to decide the basic risk policy to be formulated to manage the several types of risks prevalent in the capital market. This may vary from taking no risk (cover all types of risk) to taking high risks (face all types of risks). Most investors would fall somewhere in between the two extremes. A risk management policy could be based on either a cost centre approach or a profit centre approach. A cost centre approach looks at exposure management as insurance against adverse movements. Here, the investor is not looking for optimisation of return or realisation of expenditure; the aim is to meet the budgeted or targeted risk-return rates. In a profit centre approach, the investor takes deliberate risks to make profits out of price movements.

RISK MEASUREMENT

There are different measures of price or market risk, which are mainly based on historical market values. For example, value at risk (VaR), revaluation, modelling, simulation, stress testing, back testing, and so on. VaR estimates the largest amount that a portfolio is likely to lose over a specified period of time at a specified level of confidence. Simulation repeatedly values current holdings based on market conditions that existed over a specific historical period of time. Stress tests help to understand the impact of extreme price movement scenarios. The effectiveness of VaR is assessed by a technique known as back testing, which counts the number of days when the losses are bigger than the estimated VaR figure.

RISK CONTROL

Control of Price Risk

Position limits are established to control the level of price or market risk taken by the investor. The market risk can also be reduced through diversification of instruments in a portfolio. This is known as reduction of systematic risk in a given portfolio. Price risk is also controlled through the hedging process.

Credit or counterparty risk can be controlled through the setting up of credit limits for members, dealers and market makers.

Control of Operating Risk

The control of operating risk can be through the establishment of an effective and efficient internal control structure over the trading and settlement activities, as well as implementing a timely and accurate management information system (MIS). Many tools are available to control operating risks. These include the following:

• Comprehensive market systems and operations manuals
• Proper structure of the market and adequate personnel
• Separation of trading function from settlement, accounting, and risk control functions.
• Enforcement of authority and limits
• Written confirmation of all verbal dealings and voice recording
• Legally binding agreements with counterparties
• Contingency planning
• Internal audit
• Daily reconciliation of outstanding positions
• Ethical standards and codes of conduct and dealing discipline.

While these tools can be used on the whole to control the risk in the investment environment for an investor, specifically, investors would be able to control the most prominent risk in investment, that is, price risk through hedging.

Hedging is the process of risk reduction. A hedger is someone with an existing risk due to price movement who uses derivatives as a means of reducing that risk. Derivatives are derived instruments from the base securities such as equity and bonds. A derivative is thus another financial instrument that is based on the performance of separately traded commodities or financial instruments. These derived instruments involve future commitments and hence they tend to the possibility of benefiting from otherwise favourable price (or rate) movements. Examples of derivative instruments are forward contracts, futures, options, and swaps. These hedging instruments not only provide protection against adverse price movements, they also preserve the ability of the investor to gain from beneficial price or rate movements.

Use of a derivative instrument in conjunction with another instrument (derivative or otherwise) is regarded as financial engineering. It can be used to construct investments with requisite risk-reward profiles to match the requirements of investors more closely than any existing investment instrument. The potential of financial engineering using derivatives is more to risk managers, bankers and those involved in investments, corporate finance, and treasury, since it is an important component of their expertise.

Operators in the derivatives market are hedgers (operators who want to transfer a risk component of their portfolio), speculators (operators who intentionally take the risk from hedgers in pursuit of profit), and arbitrageurs (operators who operate in different markets simultaneously in pursuit of profit and eliminate mispricing).

The main function of derivatives is the provision of a means of hedging for reducing an existing risk. Hedging is the buying and selling of futures contracts to offset the risks of changing prices in the cash markets, or where the commodities actually get bought and sold. This risk-transfer mechanism has made futures contracts virtually indispensable to investors, companies, farmers, and financial institutions.

Hedgers in commodity markets are individuals or companies that own a cash commodity, or are planning to own a cash commodity (corn, soyabeans, wheat, treasury bonds, notes, bills, equity, and so on). Therefore, these hedgers are concerned that the cost of the commodity may change before they either buy it or sell it. To alleviate some of that concern, they seek price protection by hedging the commodity. Anyone who wants protection against unwanted price changes in the cash market uses the futures markets for hedging. They could be farmers, merchandisers, producers, exporters, bankers, bond dealers, insurance companies, mutual funds, pension fund managers, portfolio managers, thrifts, manufacturers, or investors.

Speculators in the futures markets fulfil several economic functions by facilitating the marketing of commodities and trading in financial instruments. Speculators do not create risk; they assume it in the hope of making a profit. Speculators buy and sell derivatives simply to make profit, not to reduce risk. They buy when they believe futures or options are underpriced, and sell when they view them as overpriced. In a market without these risk takers, it would be difficult, if not impossible, for hedgers to agree on a price because the sellers (or short hedgers) want the highest price, while the buyers (or long hedgers) want the lowest possible price.

Speculators facilitate hedging, provide liquidity, tend to ensure accurate pricing, and can help to maintain price stability. If hedgers are net sellers, there will be a tendency for futures prices to fall which may generate profit opportunities for speculators. Speculators will then buy the underpriced futures. The purchases by speculators will allow net sales on the part of the hedgers. In effect, speculators fill the gap between sale and purchase by hedgers. In so doing, they tend to maintain price stability since they will buy into a falling market, and sell into a rising one (in the event of hedgers being net buyers).

Speculative transactions add to market liquidity. Market liquidity implies that the number of buying and selling activities are relatively large and are on a continuous basis. In such a liquid market, hedgers can make trade easily and protect their prices. In the absence of speculators, hedgers may have difficulty in finding counterparties and they might not be able to trade at their desired prices.

Speculators also help to improve the informational efficiency of markets. A market is efficient in this aspect when prices fully reflect all available, relevant information. Speculators consider all relevant information when deciding upon the appropriate price of a derivative contract. If the actual prices differ from those judged appropriate, speculators try to gain from the situation by buying underpriced derivatives (so their prices tend to rise), while selling overpriced derivatives. Such activities are expected, in the long run, to bring the market prices in line with the theoretical prices.

Speculators also help ensure the stability of the market. The price of commodities, for example, changes along with the supply and demand. Surplus harvest usually means a lower price for commodities. Higher prices may result from such factors as adverse weather conditions during the growing season or an unexpected increase in export demand. Financial instruments fluctuate in price due to changes in interest rates and various economic and political factors. Speculators by assuming these risks bring stability of prices to the market.

Apart from hedgers and speculators, there is a third category of traders known as arbitrageurs. They also ensure that the markets are liquid and the pricing accurate and enhance price stability. Arbitrage involves making profits from the difference in prices prevailing in two markets (cash market and derivative market). For example, futures share prices should show a positive relationship to actual share prices that are the underlying instruments. If this relationship does not hold good, an arbitrage opportunity arises. The buying of instruments in the cash market and the simultaneously sale of futures in the derivative market or vice versa will continue till this information is known to all the market players. This arbitrage automatically adjusts the over pricing or underpricing in the cash market relative to the derivative market and vice versa.

When hedging in the futures markets, both profits and losses are possible—just as in owning the actual, physical commodity, or financial instrument. Hedgers “buy contracts” (go long) when they expect prices to increase, hoping to make an offsetting sale at a higher price, thus making a profit. Hedgers also “sell contracts” (go short) when they expect prices to fall, hoping to later make an offsetting purchase at a lower price, again resulting in a profit.

The profit potential is proportional to the amount of risk that is assumed and the speculator’s skill in forecasting price movement. Potential gains and losses are as great for the selling (going short) speculator as for the buying (going long) speculator. Whether long or short, speculators can offset their positions and never have to make, or take, delivery of the actual commodity.

In contrast to futures, options allow investors and risk managers to define risk and limit it to the cost of the premium paid for the right to buy or sell a contract. At the same time, options can provide the buyer with an unlimited profit potential. Options have a known and limited risk. Buying options contracts is an attractive investment for many individuals seeking to profit from significant price movements-either upward or downward in an increasingly volatile and often uncertain investment environment.

Speculators also have an advantage using options to hedge their risk, in that the facility of ignoring an option avoids the realisation of potentially large losses, an outcome that cannot be avoided with the other derivatives. However, these benefits of options, relative to the other derivatives, have to be paid for in the form of an option premium. A corresponding payment is not required when using forwards, futures, or swaps.

HEDGING INSTRUMENTS

Derivative is a product/contract which does not have any value on its own, that is, it derives its value from some underlying product/security. However, the availability of risk management products attracts more investors to the cash or spot market. Arbitrage between cash and futures markets fetches additional business to the cash market, improvement in delivery-based business, lesser volatility, and improved price discovery.

Derivative takes any of the following form—forward contracts, futures contracts, options, and swaps.

FORWARD CONTRACTS

A forward contract is one to one bi-partite contract, to be performed in the future, at the terms decided today (for example, the forward currency market in India). Forward contracts offer tremendous flexibility to design the contract in terms of the price, quantity, quality (in case of commodities), delivery time, and place. Forward contracts suffer from poor liquidity and default risk.

FUTURES CONTRACTS

Futures contracts are organised/standardised contracts, traded on the regulated exchanges. These contracts, being standardised and traded on the exchanges are very liquid in nature. In the futures market, clearing corporation/house provides the settlement guarantee.

Every futures contract is a forward contract. They are entered into through an exchange, traded on exchange, and a clearing corporation/house provides the settlement guarantee for trades. They are of standard quantity and standard quality (in case of commodities). They also have standard delivery time and place.

FORWARD/FUTURES CONTRACTS

The distinction between forward and futures contract are given in Table 8.1.

 

Table 8.1 Features of a Forward and Futures Contract

Features	Forward Contract	Futures Contract
Operational Mechanism	Not traded on exchange	Traded on exchange
Contract Specifications	Differs from trade to trade.	Contracts are standardised contracts
Counterparty Risk	Exists	Exists, but assumed by Clearing Corporation/ house
Liquidation Profile	Poor liquidity as contracts are tailor made contracts	Very high liquidity as contracts are standardised contracts
Price Discovery	Poor as markets are fragmented.	Better as fragmented markets are brought to the common platform

OPTIONS

Options are instruments whereby the option seller gives the option buyer the right to buy or sell a specific asset at a specific price on or before a specific date.

The option seller is one who gives/writes the option. He has an obligation to perform, in case option buyer desires to exercise his option.

An option buyer is one who buys the option. He has the right to exercise the option but no obligation.

There are two types of options—“calls” and “puts.” A call gives the holder of the option the right, but not the obligation, to buy the underlying instrument. Conversely, a put option gives the holder the right, but not the obligation, to sell the underlying instrument.

Call Option

Purchasing a call gives the investor a specific locked-in price for a future time. At this point, the investor has the right, but not the obligation, to buy a contract on a commodity or a financial instrument that is expected to increase in value. Thus, if the investor feels that the price of gold will increase in the future, the investor can hedge the inherent risk by buying a gold call option.

Put Option

Purchasing a put gives the investor a specific locked-in price at which point the investor has the right, but not the obligation, to sell a contract on a commodity or a financial instrument that the investor expects to decrease in value. For example, if an investor expects a decline in the price of silver in the near future, he can buy a silver put option.

A “call” is a way to profit if prices go up. A “put” is a way to profit if prices go down. Certain terms used in option trading are given below.

Premium is the cost that is paid by the option buyer to the seller for protecting the specific position that is expected in the future.

Strike price is the specific price at which the option gives the investor a right to buy a particular commodity or financial instrument, in the case of a call, or to sell the commodity or financial instrument in the case of a put. The strike price is stated in the option.

Expiration date is the date on which the option expires.

Exercise date is the date on which the option holder/buyer exercises the option.

Options are also categorised on the basis of their contractual time.

An American option is one that can be exercised any time on or before the expiry date.

An European option is one that can be exercised only on the expiry date.

An Asian option is one that can be exercised at the best price prevalent during the option duration.

The first derivative product to be introduced in the Indian securities market is the “index futures”. In the world, first index futures were traded in the US on the Kansas City Board of Trade (KCBT) on Value Line Arithmetic Index (VLAI).

TRADING IN THE FUTURES CONTRACT

When a futures deal is agreed between a buyer and a seller, an exchange takes over the role of the counterparty to both the buyer and the seller. In a futures market, transactions take place between two traders. But once the deal is registered, the exchange assumes the role of a buyer and a seller to each party respectively. This implies that there is no need to investigate the creditworthiness of the person or entity that is actually a party in the transaction. This makes the exchange take all default risks. The exchange protects itself from counterparty default risk by means of the variation and maintenance margins.

The margin system is central to the futures markets. There are three types of margins namely, initial margin, maintenance margin, and variation margin (daily margin).

Marking to market (daily margin) prevents the accumulation of counterparty debt and the maintenance margin is a limit below which daily margin will not fall.

The margin system results in substantial reduction of counterparty risk. As a result, dealers can confidently trade with any other trader. Another implication of the margin system is that futures are highly geared investments. For example, an initial margin of 1 per cent of the underlying instrument means that the exposure acquired is 100 times the initial money outlay.

Daily Margins

A Daily margins (variation margin, mark to market margin) is payable and receivable on a daily basis. It reflects the notional profit or loss made from a futures contract during the course of a day. If the futures price moves to the holder’s advantage, the holder will receive a variation margin from the exchange. If the futures price moves adversely, a payment must be made by the holder to the exchange. This process of realising profits and losses on a daily basis is known as marking to market.

Daily margins are collected to cover the losses that have already taken place on open positions. For this purpose the price for daily settlement is the closing price of the derivative instrument. The price for the final settlement is the closing price of the underlying instrument.

For daily margins, two legs of spread positions (short expiration and long expiration) would be treated independently. Members pay daily margins to the exchange before the market opens for trading on the next day. Daily margins are mostly paid in cash.

Maintenance Margins

Despite a mark to market margin maintenance, it is possible that the margin may reach a very low level, or even might become negative. To avoid this, the maintenance margin has been introduced. Maintenance margin is a lower limit below which the daily margin account cannot go. In case if the daily margin falls below the maintenance margin, the investor receives a margin call and is required to deposit additional funds. If the investor fails to deposit the additional funds, the securities held by the investors on which he is maintaining the margins will be realised and the amount will be used to maintain the required margin.

Initial Margins

The initial margin is a sum of money to be provided by both the buyer and the seller of a futures contract when they make their transaction. This margin is a small percentage of the face value of the contract. The initial margin is subject to variation and will depend on the volatility of the price of the underlying instrument concerned. Hence, the initial margins may be as little as 0.1 per cent or as much as 10 per cent of the value of the instrument to which the futures contract relates. One function of the initial margin is the provision of market discipline. The payment of the initial margin may deter poorly capitalised speculators from entering the market. In other words, margins to cover potential losses for one day are initial margins. They are to be collected on the basis of value at risk on 99 per cent of the days.

Different initial margins prevalent in the market are: naked long and short positions and spread positions Naked positions

where,

st is today’s volatility estimates [(st)² = dc (st − 1)² + (1 − dc) * (rt²)]

st−1 is the volatility estimates on the previous trading day.

dc is decay factor which determines how rapidly volatility estimates change (and is taken as 0.94 by J. R. Verma Committee Report, 1988).

rt is the return on the trading day [log(It/It−1)]

Since the volatility estimate ‘st’ changes everyday, the initial margin on open position (Naked position) will change every day.

Spread Positions

Spread positions are when the investor enters into two positions one near month and another far month for the some instrument at different prices. A flat rate of 0.5 per cent per month of spread on the far month contract is the margin requirement for spread positions. A minimum margin of 1 per cent and maximum margin of 3 per cent on spread positions has also been fixed by the NSE.

Margins are kept in the form of liquid assets. Liquid assets of brokers are cash, fixed deposits, bank guarantee, government securities, and other approved securities. Of the liquid assets, 50 per cent must be cash or cash equivalents.

CONCEPT OF BASIS IN FUTURES MARKET

Basis is defined as the difference between cash and futures prices. Basis can be either positive or negative. Basis may change its sign several times during the life of the contract. Basis ought to be around zero at maturity of the futures contract, that is, at that point of time both cash and future prices converge at maturity, as depicted in Figure 8.1.

Financial Futures

A financial future is a commitment to buy or sell, on a specified future date, a standard quantity of the underlying instrument at a futures price determined in the present. In practice, many contracts have no facility for the exchange of the underlying instrument. Futures markets are mostly independent of the underlying cash market, though they operate parallel to that market. For instance, ACC futures are different instruments from the ACC shares themselves, but ACC futures prices move in ways that are related to the movements in ACC share prices. However, since the futures markets are independent of the markets in the underlying instruments, it is possible for futures prices to show changes from underlying markets.

The main function of futures market is to provide a means of hedging. A hedger seeks to reduce an existing risk. This risk reduction could be achieved by taking a futures position that would tend to record a profit in the event of a loss on the underlying position (alternatively a loss in the case of a profit on the underlying position).

A position in the futures markets can be taken easily than positions in the underlying spot markets. For example, a position in stock index futures can be established by trading an index future. While, the construction of a portfolio of stocks in the spot market would not be so very easy. Futures markets hence may be more efficient than the underlying spot markets. Since futures prices react to new information faster than the spot prices, the future markets can be said to perform the ‘price discovery’ function. This price discovery function is particularly important when the underlying spot market is poorly developed or illiquid, that is, instruments are not frequently traded.

Stock Index Futures

Stock index futures are built on any market index. The underlying instrument is the index point movement. Stock index and futures movements are matched by compensatory cash flows. Futures contracts are available on many stock indices. In India, the BSE Sensex and S&P Nifty are the popular indices on which there is futures trading.

Rather than working on individual stocks in most cases, since it is the portfolio risk that is protected, stock index futures are used to reduce stock market risk. The anticipation is that losses arising from movements in market portfolio prices are offset by gains from similar trends in futures prices. When the investor foresees a bearish market and wants to protect the portfolio position, the futures position can be entered to safeguard the portfolio value. In such a case the investor may take a short (sell) position in index futures contracts. By entering into a short position, the investor guarantees a notional selling price of a quantity of stock at a specific date in the future. When share prices fall below this price and stock index futures also fall, the notional buying price on that date would be less than the predetermined notional selling price. The investor can close out the position in futures by taking a long (buy) position in the same number of contracts. The excess of the selling price over the buying price is given to the investor in cash in the form of a variation margin. This gain on the futures contracts is received on a daily basis as the futures price moves as per investor expectations.

On the other hand, if the market had been bullish, the investor would have gained from the portfolio of equities (cash market), but lost on futures dealings. Here too, the investor can successfully reduce the portfolio risk by ensuring a constant portfolio value.

The use of futures to hedge the risk of a rise/fall in stock prices does not require any alteration of the original portfolio. The profit /loss differentials are compensated in the derivative market. Hence, investors prefer such methods to changing the portfolio composition to suit the market movements.

Interest Rate Futures

The interest rate risk has to be borne by both the borrowers and lenders of funds. Borrowers lose from a rise in the future interest rates, while lenders lose when interest rates tend to fall in the future. Since both these players would like to minimise the risk component of their activities, the derivative contracts provide an opportunity to fix the desired interest rate for the future. Short-term interest rate futures, which frequently take the form of three-month interest rate futures, are suitable for hedging such risks. Interest rate futures are thus commitments to borrow or deposit for a pre-determined period from the date of the contract. Through these contracts, both the borrowers and lenders can hypothetically determine the interest rates for future periods.

Government bonds being long term debt instruments are subject to wide interest rate fluctuations and hence government bond futures are often the most sought after instruments in derivative exchanges. Fluctuations in long-term interest rates lead to very high volatile prices of these long-term bonds in the market. Since bond price movements arise from and reflect changes in long-term interest rates, government bond futures are termed as long-term interest rate futures. Investors who wish to manage their bond portfolio risk as well as reinvestment risk opt for futures contracts to offset their notional losses. Interest rate futures have been introduced recently in India and are not buoyant at present.

Currency Forward

Forward foreign exchange contracts are agreements between two parties for the exchange of two currencies on a future date at an exchange rate agreed in the present (the forward exchange rate).

A forward purchase is an agreement to buy foreign currency on a specified future date at a rate of exchange determined in the present. Similarly, a forward sale is an agreement to sell foreign currency on a specified future date at a rate of exchange determined in the present. These contracts remove uncertainty for the importer as well as the exporter as to how much future payables or receivables will be exchanged for the domestic currency.

If the forward price of a currency exceeds the spot price, that currency is quoted at a premium in the forward market. On the other hand, if the forward price is less than the spot price, the currency is quoted at a discount. The premiums and discounts are computed in terms of per cent per annum, determined from the following formula:

The first part, (premium/spot rate), expresses the premium as a proportion of the spot rate. The second part, (365/number of days to maturity), annualises this rate. It presents the relative premium to spot rate ratio in terms of the entire year though the contract is for a specified period within this one-year duration. For example, if the derivative is for a three-month forwards, this adjustment factor would be 4 (12 months/3 months), while for one-month forwards, the multiplication factor would be by 12. The final premium/discount is stated in terms of a percentage by the multiplication factor 100. This converts the decimals into a percentage measure.

Currency Futures

A currency futures contract enables the exchange of a standard amount of a particular currency on a specific future date, at a fixed exchange rate. Besides serving the same purpose of fixing a future exchange rate as in the forward contract, currency futures have the additional advantage of tradability. A contract can be closed out (cancelled) by buying a sold future or selling a future contract that was bought earlier. Thus there is a cancellation of the earlier contract and only the price differential exchanges hands.

In the market, there are both buyers and sellers of a specific futures contract. In order to avoid the risk, if a hedger buys futures, it is obvious that someone else must sell that contract, that is, there must be a risk taker. A hedger’s contract to buy currency on a specified date in the future at a price agreed upon in the present is matched by another user’s intention to sell that currency at that date and price.

If, for instance, the risk that the rupee may fall in value could be transferred to the seller of dollar futures who is willing to bear the risk. If the rupee falls (and hence the dollar rises against the rupee), the seller has to fulfill the commitment to sell the dollar at a price that is lower than the spot rate available at the time when the currency actually changes hands. The seller of the dollar futures does not enter into this contract with an intention to lose but he could either be a hedger wanting to avoid a rise in the rupee relative to the dollar or a futures trader willing to take risk in the expectation of making a profit.

Example—Financial Futures

Assume an investor expects to receive a lump sum of Rs 500,000 on March 15. He is aware of this a month before the money is due. The investor has long wanted to invest in the shares of Infosys. The market price of Infosys is expected to rise in the next few weeks. Currently Infosys is quoting in the market at Rs 3938.50. The investor buys a three-month futures contract at Rs 3951.10. If the Infosys shares trade at Rs 3951.80 and the futures price is Rs 3972.70 on March 15, the profit or loss position for the investor will be as follows.

Cash (spot) Market	Futures Market
February 14 The investor intends to buy Infosys shares for Rs 500,000 on March 15. The current market price is Rs 3938.50.	The investor buys 127 futures contract at a price of Rs 3951.10.
On March 15 the market price of infosys had risen to Rs 3951.80, the investor would be able to invest in 127 shares with an additional cash payment of Rs 1689.10. However, the future prices became Rs 3972.70 and buying at Rs 3951.10 and selling at Rs 3972.70 could close out the futures contract. The futures profit will be Rs 2743.20. This profit could be used towards the purchase of the desired Infosys share.
March 15 At Rs 3938.50 he would have spent Rs. 5,00,189.5 in buying 127 shares. Now he is required to pay Rs. 501878.60	There is a profit of Rs 2743.20 on the futures contract. This profit amount would reduce the additional fund requirement to purchase 127 shares
Purchase of 127 shares using Rs. 500,000 and profits from Futures Market Rs. 1878.60	Profits of Rs. 1,878.60 used for purchase of shares and excess cash balance Rs. 864.60

Example—Stock Index Futures

The S&P Nifty Futures Index stands at 950 on February 14 and a fund manager expects to receive Rs 900,000 on March 15. The fund manager fears that share prices will rise by March 15, implying that less shares will be bought with the Rs 900,000. Since the risk is that share prices will rise, the requisite futures position has to be one that would profit from a rise in share prices. Stock index futures are bought in contract sizes of 200. If the futures index were 950, Rs 900,000 worth of shares would imply a futures contract of roughly 947.368. Since trading is permitted in lots of 200, the futures contract to be entered would be 1000. The futures contract value would be 950 x 1000 = Rs 950,000. Hedging a purchase of Rs 900,000 of shares (at February 14 prices) requires the purchase of 5 futures contracts.

The futures profit, when added to Rs 900,000, provides the Rs 950,000 required to buy the quantity of shares that Rs 900,000 would have bought on February 14.

Cash (spot) Market	Futures Market
February 14 Fund manager expects to invest Rs 900,000 on March 15, but fears that the stock index will rise above the current 950, thus reducing the number of shares that could be bought with Rs 900,000.	Fund manager buys 1000 futures index at a future index of 950.
Suppose the index rises by 50 points by March 15. The quantity of shares that could have been bought for Rs 900,000 on February 14 now costs Rs 950,000. This could be regarded as a loss of Rs 50,000. However, if the futures index also rises by 50 points, there will be a futures profit of Rs 50,000.
March 15 The index has risen to 1000. The fund manager needs an extra Rs 50,000 to be able to buy the shares.	The futures are traded at 1000. There is futures profit of 50 index points for 1000 futures index = Rs 50,000.

In the following example, the portfolio holder fears a fall in equity prices and deals in the futures market to protect against a fall in the value of portfolio.

Cash (spot) Market	Futures Market
April 5 Holds a balanced portfolio of equities valued at Rs 1,000,000, but fears a fall in its value. The current index is 1000.	Sells 1000 May future index contracts at a price of 1000 points each. Thus the commitment is a notional sale of Rs 1,000,000 of stock on the May delivery date at the level of equity prices implied by the futures price on April 5. (Rs 1,000,000= 1000 x Rs 1000).
May 10 The index has fallen to Rs 950. Correspondingly, the value of the portfolio has declined to Rs 950,000.	Closes out the futures position by buying 1000 May futures index contracts at a price of Rs 950. The notional buying price of each contract is Rs 950.
Loss on portfolio = Rs 50,000	Gain from futures trading = Rs 50,000 (50 x 1000)

An investor who expects a future flow of funds can similarly enter into a futures contract to protect against a rise in the prices.

Cash (spot) Market	Futures Market
December 10 Anticipates receipt of Rs 1 million on January 10. Current index is 2200. Fears a rise in the index.	Buys 400 January futures contracts at a price of Rs 2200. The notional commitment is to pay Rs 880,000 (400 x Rs 2200) on a future date.
10 January The new index is 2500.	Closes out by selling 400 January futures index contracts at a price of Rs 2500. The investor has a receipt of Rs 1,000,000 (400 x 2500) from the contracts.
Requires an additional Rs 0.13636 million in order to buy the quantity of stock that Rs 1 million would have bought on December 10. Profits used from derivities Rs. 120,000 Additional capital brought in Rs. 16,363.	Profit from futures of Rs 120,000. (400 x Rs 300)

HEDGE RATIOS

In the market, it is not very often that the cash and future prices move in the same ratio. The price behaviour of the futures contract tends to differ from that of the underlying instrument. Hedge ratios become useful since they indicate the extent of variation in the futures price relative to the variation in the spot price. If the security to be hedged shows relatively large variations, then it is appropriate to take more futures contracts than in the case of a more stable instrument. While hedging a portfolio of securities, the entire portfolio might not have the same composition as that of the index to which it is related. The index could vary more or less than the portfolio. Thus, in such circumstances, it is essential to compute the hedge ratio to know the extent of hedge that an investor should enter into.

Assuming a linear relationship between the return from the spot prices and the future prices, the hedge ratio is defined as the slope of the regression line:

where,

ΔS = Change in spot price

α = Constant

β = Slope of the regression line

ΔF = Change in future price

β can be measured as follows:

where,

σ_S = Standard deviation on spot price returns

σ_F = Standard deviation on future price returns

r = correlation between spot price returns and future price returns

 

The beta of 1 implies that share prices move in tune with future prices. A beta higher than 1 implies that spot market change is higher than the future market change. In such circumstances, the value to be hedged also has to be higher since the underlying instrument’s variation is larger. Then only the loss in spot market can be offset equally from a profit from the futures market.

A beta of less than one implies that changes in the futures market is more than the changes in the spot market. Here, the investor can hedge the value to a relatively lesser amount since spot markets do not change in proportions higher than the futures market.

Example Assume that the beta of Bajaj Auto shares with Bajaj Auto futures is 0.78 and the alpha is 1.4. Suppose the price change in the spot market is −3.2, the expected change in the futures market will be −4.1. If the expected spot price is 453.80, a 3.2 per cent reduction from the current price of Rs 468.80, the futures price of Rs 469.30 would reach Rs 450.06. If the current holding in Bajaj Auto is 1000 shares, the value to be hedged by the investor will be Rs 468,800 * .78 = Rs 365,664. The number of futures contracts will be 779.

As a hedger, the investor would sell 779 future contracts at Rs 469.30. When, after three months, the predicted price movements are exact, the loss in the spot market after one month becomes an estimated Rs 15,000. The profit from the futures contract will be Rs 15,372.76 which will amply compensate for the loss in the spot market.

HEDGE POSITIONS FOR A PORTFOLIO

The beta of individual securities can be substituted by the beta of a portfolio. The beta factor of a portfolio of shares is the weighted average of the beta factors of the shares that constitute the portfolio.

If the portfolio beta is 1.25, then the portfolio tends to change by 25 per cent more than the stock index. Hedging the investment portfolio would require the value of the stock index futures contracts used to exceed the portfolio value by 25 per cent. If the portfolio beta is .75, then the portfolio changes by 25 per cent less than the stock index. Here for hedging the investment portfolio, the required value in the futures market may be less by 25 per cent of the investment portfolio value.

The calculation of the appropriate number of futures contracts to hedge will involve ascertaining the market exposure of the share portfolio. The market exposure of the share portfolio is not the same as its market value. Market value needs to be adjusted by the share betas. A high-beta shares will tend to display disproportionately high responsiveness to overall market movements. Conversely, share with betas of less than one will tend to be less volatile than the market as a whole. (The stock market taken as a whole would have a beta of one; stock index portfolios such as the BSE 500 are often treated as having betas equal to one.) Table 8.2 below shows hypothetical share betas and the corresponding market exposures, which are calculated by multiplying the market values of the shares by the betas.

 

Table 8.2 Hypothetical Share Betas and the Corresponding Market Exposures

The market exposure of the portfolio is Rs 2,106,957. If Nifty index is 926, then one futures contract would amount to (926 x 200) Rs 185,200. The required number of future contracts to hedge the portfolio would be (Rs 2,106,957/Rs 185,200) 11.38. This would amount to approximately 11 contracts.

OPTIONS

An option is the right to buy or sell a specified amount of a financial instrument at a predetermined price for a future date. The option involves the writer and buyer of the option contract. The buyer of the option contract retains the right but not the obligation to buy or sell a contract as per the terms. The writer of an option is the seller and receives a premium for undertaking the risk or willing to sell the option contract.

Any option contract that gives the buyer the right to buy a financial instrument at a future date at a predetermined price is a call option. The predetermined price is referred to in the market as the strike price. The option contract that gives the buyer the right to sell is termed a put option. In relation to the expiry of the contract, that is, the enforcement date of the contract, an option contract can be categorised either as an American option or an European option. An American option lets the buyer enforce the right at any time before the expiry date of the contract. The European option gives the right to the buyer only on the expiry date of the contract. The maturity date of the contract is called as the expiry date.

CALL OPTIONS

Equity options along with index options are traded in organised markets such as the National Stock Exchange and Bombay Stock Exchange. At the time of buying the option there will be at least two exercise prices available for the investors. For example, when the price of ACC shares was Rs 144 on November 14, 2002, the call option exercise price available was Rs 130, Rs 135, Rs 140, Rs 145, Rs 150, and Rs 155. If the holder of a call option decides to hedge it against rising price, buying the specified number of shares at a strike price will achieve this. For entering into such an option contract, the buyer of the option has to give the seller of the option a price called the option premium. For example, the various strike prices and the premiums for Reliance’s call options are given in Table 8.3. The closing price of Reliance Industries as on November 14, 2002, is Rs 262.80.

 

Table 8.3 Strike Prices and Premiums for Reliance’s Call Options

Exercise (Strike) Price (S)	Premium (p)
	28/11/2002	26/12/2002
240	23.70	30.00
250	15.25	20.10
260	7.70	13.05
270	3.65	8.85
280	1.75	5.95
300	0.35	2.65

 

Premiums are payable to the writer of the option at the time the option is bought. It is profitable to exercise a call option if the market price of the stock turns out to be higher than the strike price. If the market price is lower than the strike price, the call option holder will not exercise the right of the option since it will be a loss. In the event of the market price being lower than the strike price, the option holder is not obliged to exercise the right of the option, and presumably will not, since exercising would realise a loss. If the buyer wants to exercise the option on November 28, when the market price is Rs 264.20, at the strike price of Rs 240 for the November end contract, the net profit to the buyer will be Rs 262.80 − Rs 240 − Rs 23.70. The profit is Rs 0.50 per contract.

Example An investor buys a Rs 135 call option on ACC shares at a premium of Rs 10.20 per share when the share price is Rs 145.80. Since each option contract on NSE relates to 1,500 ACC shares, the cash outflow is Rs 15,300.

Subsequently, the share price rises to Rs 148.30. The investor can then exercise the right to buy at Rs 135. There is a gross profit of Rs 13.30 per share (Rs 19,950 per option contract). This gross profit of Rs 13.30 is the intrinsic value of the option. The net profit must take account of the Rs 15,300 premium paid for the option. The net profit is thus Rs 13.30 − Rs 10.20 = Rs 3.1 (Rs 4,650 per option contract). The investor has ensured that the effective price to be paid for the shares will not exceed Rs 145.20, that is, the strike price plus the premium paid for the option.

The Profit/Loss Profile at Expiry

Since an option buyer is not obliged to exercise an option, when the market price is not favourable to the strike price, the option buyer does not exercise the contract. However, the option premium that was paid at the time of entering into the contract will be a loss to the option buyer. Thus, the premium paid becomes the maximum loss that can be incurred by the option buyer. Theoretically, there is no upper limit to the share price and hence the buyer’s profit potential becomes unlimited. The following figure shows the profit/loss profile of a call option at expiry.

For example, the January Rs 200 L&T call option premium is Re 0.50. If the buyer holds the option to the expiry date and the share price turns out to be Rs 200.50 or less, the buyer will not exercise the option. The buyer does not get any profit from buying L&T shares from the writer since the shares can be bought from the market at that price or lower. Hence, below Rs 200, the option buyer loses Re 0.50 per share. This loss is constant since the buyer has locked the futures position at the buy rate of Rs 200.10. When the market price goes beyond the level of Rs 200.50, the buyer of the call option gets a profit that increases with the relative increase in the market price.

At a share price of Rs 200.50, the profit exactly offsets the premium paid. Hence, Rs 200.50 is the break-even price at which net profit is zero. This is a simple illustration and assumes that there is no spread in the buy-sell prices and that there is no transaction cost in the deal. In reality, there is a transaction cost and there can also be spread in the market.

At expiry, when an option results in a gross profit, it is said to be ‘in-the-money’. In a call option, the gross profit (intrinsic value) for the share will exist when the strike price is less than the market price. An option contract for which there is no gross profit or situations when the option will not be exercised, is said to be ‘out of-the-money’. In a call option, the ‘out of-the-money’ situation arises when the strike price is greater than the market price. Option contracts are ‘at-the-money’ when the market price is equal to the strike price. For example, if at the time of expiry for January Rs. 200 strike price L&T call option the market price is Rs 160, the option is ‘out of-the-money’. If the market price is Rs 210, the option is ‘in-the-money’. When the market price is Rs 200, the share is said to be ‘at-the-money’.

The Profit/Loss Profile Prior to Expiry

When the option contract expires, its price (premium) is expected to be equal to its intrinsic value (gross profit) if not for the time value of money. Usually, hence, the premium before expiry will include the intrinsic value and the time value. The excess of the price of the option over the intrinsic value is known as the time value. For example, ITC call options with a strike price of Rs 600 are traded at a premium of Rs 18.40 per share. The time to expiry is one month. The current market price is Rs 610.40. The intrinsic value is Rs 10.40 (excess of market price over strike price). The time value of this contract is Rs 8.00, hence the premium is Rs 18.40 (10.40 + 8.00). When an option is exercised on the expiry date, only the intrinsic value is realised. Before the expiry date, the seller of an option obtains a price/(premium) that incorporates the intrinsic and time value (see Figure 8.3).

Before the expiry date, when the market price is less than the strike price, the entire premium will be the time value of money. Time value is at its highest when the option is ‘at-the-money’. Time value declines as the option moves either ‘in’ or ‘out-of-the money’ and will approach zero as the market price of the share diverges substantially from the exercise (strike) price.

The following example of HLL call options can be given in support of the above points. HLL’s current market price is Rs 159.45. The nearest futures contracts, all maturing after 15 days in the market are as follows.

When the market price is close the strike price, the time value is highest (Rs 3.70). For the options with a strike price less than the market price, the intrinsic value is higher while the time value is very low. For strike prices that are above the market price Rs 170 and Rs 180 option contracts, the intrinsic value is nil while the premium paid is less than the premium paid for the options contract at Rs 160.

Market prices below the exercise price leads to a premium that consists of only time value. However, market value above the exercise price consists of both time and intrinsic value. This leads to the increase in the premium paid as market prices rise above the strike price. However, the premium component is mostly the intrinsic value and not the time value. This is because, the buyer of the call option is likely to exercise the calls immediately. The change in the option premium relative to the change in the market price is called as the gradient and is also known as the option delta. The delta approaches zero as the option becomes deeply ‘out-of-the-money’ (extremely low market price) and approaches one when it is deep ‘in-the-money’ (extremely high market price). The delta is approximately 0.5 when the option is ‘at-the-money’ (equal to the market price).

PUT OPTIONS

A put option gives the buyer of the contract the right, but not the obligation, to sell shares at a specified price prior to or on the expiry date of the option. A put option writer is obliged to buy the shares at a specified price on the expiry date when the option holder enforces the right to sell. Since the put option gives the buyer of the contract a right to sell, the option holder will sell the contract at expiry only when the market price is lower than the strike price. Since the strike price in this case is higher than the market price, the put holder can make a profit by buying the shares at market price and selling it to the writer of the put contract at a (higher) strike price. This situation gives an intrinsic value to the contract, since the price differential is the gross profit for the put option holder.

If, on the other hand, at the time of expiry, the put option strike price is lower than the market price, the holder will not sell the option contract to the writer, since the profit from the market is higher than the profit from the option writer. In this case, the option does not have any intrinsic value. The option buyer loses to the extent of premium paid at the time of entering into the contract.

The price of the put option is the premium that is paid for the contract. The premium for the put option consists of the intrinsic value and the time value. The put options also have at least two prices. The following table gives the different put options of Infosys that are exercisable after 15 days.

Exercise (Strike) Price (S)	Premium (p)
	28/11/2002	26/12/2002
3700	16.00	61.60
3900	70.25	138.00

The table shows the premiums of Infosys put options at the close of trading on November 14,2002. The price of Infosys shares was Rs 3,963.95. The dates indicate the expiry date of the put options.

Assume an investor buys a Rs 170 put option on BPCL shares at a premium of Rs 1.30 per share when the share price is Rs 184.75. Subsequently, the share price falls to Rs 160. The investor can exercise the right to sell at Rs 170. The gross profit from the option is Rs 10 per share. This gross profit of Rs 10 is the intrinsic value of the put option. The net profit from the option will be the intrinsic value less the premium paid for the put option. The net profit is thus (Rs 10 − Rs 1.30) Rs 8.70. The investor ensures that the sale price of BPCL share will not go below Rs 168.70.

The Profit/Loss Profile at Expiry

The premium is paid by the buyer of the put option at the time the option is purchased from the writer. As with the call option, the buyer of put option is also not obliged to exercise the right to sell the option contract when the market price is not favourable. When the contract is not enforced, the premium is the maximum loss the buyer of the option will incur. As in the example of November BPCL put contract, the Rs 170 put option faces a maximum loss of Rs 1.30.

The maximum profit is limited for a put option holder since the price cannot be less than zero. Theoretically, if it is assumed that the market price of the stock falls to zero, then the gross profit to the put option holder will be the strike price minus zero, that is, the strike price. This can never happen in the market. The net gain for the option holder is the strike price minus the premium paid. In the BPCL put option, if the market price of BPCL at the time of expiry is Rs 0, then the buyer receives (Rs 170 − Rs 1.30) Rs 168.70 per share. However, in the real market, a put option holder has to pay a minimum amount greater than zero in the market; hence, the maximum net profit will be below Rs 168.70.

Figure 8.4 explains this situation. When the market price is higher than the strike price of Rs 170, the put option holder incurs a consistent loss to the extent of the premium paid on the contract, that is, Rs 1.30. The net-profit or loss situation is nil to the put option holder when the market price is Rs 168.70 (Rs 170 − Rs 1.30). When the market price is below this break-even price, the put option holder makes a net profit from the trade.

The Profit/Loss Profile Prior to Expiry

At expiry, the put option has only intrinsic value. However, when put options are traded before expiry, the price of the put option includes the intrinsic value and the time value.

Figure 8.5 includes the profit/loss position of the put holder before the expiry date (dot and dash line). Since intrinsic value is the gross profit to be made from exercising the put option, it will be zero at or above the strike price of Rs 170. This is because the put option will not be exercised at or above the strike price of Rs 170. When the market value is below Rs 170, the intrinsic value of the put option will be equal to the difference between the stock price and the strike price. Prior to expiry, the price of an option will tend to differ from its intrinsic value because of the time value.

As the expiry date nears, the prior-to-expiry profits will tend to move closer to the at-expiry profits. This reflects the tendency for the time value of an option to decline with the passage of time, that is, as the expiry date nears.

When the share price exceeds the strike price, the put option is said to be out-of-the-money since the contract will not be exercised. A better price can be obtained by selling the shares in the market than by exercising the option. The out of-the-money put contract will reflect only the time value of the option. The time value declines as the option moves further deep out-of-the-money (extreme rise in market price). This reflects the decreasing likelihood of the market price declining at a steeper rate to cause the exercise of the option to become profitable.

Similarly, the put option will be in-the-money when the market price is lower than the strike price prior to expiry. The put option holder is likely to receive profits from exercising the contract at this market price. Here also, time value declines as the put option reaches a deep in-the-money (market price is near zero) situation. This is because there is a risk of losing the intrinsic value as time progresses.

The price of an in-the-money put option contains the intrinsic value of that option. The buyer of an in-the-money option bears this risk, whereas the buyer of an at-the-money put option does not. The risk borne increases as the option becomes deeper in-the-money. This risk is reflected in the time value. The buyer of an at-the-money option pays a higher price for time value than the buyer of an in-the-money option, with the price paid for time value declining as the option becomes deeper in-the-money.

The slope of the prior-to-expiry profit line is known as the delta and represents the ratio between the change in the price of the put option and the change in the market price of the share. In the case of put options, deltas are negative. The delta increases in absolute value as the option moves deeper in-the-money (falling market prices). This means that the delta approaches −1 when a put option becomes very deep in-the-money.

The delta decreases in absolute value as the option moves further out of-the-money. The prior-to-expiry profile approaches the horizontal line of the at-expiry profile since time value diminishes as the share price moves away from the strike price of the option. The delta tends towards zero, as the option becomes very deep out-of-the-money. The following table gives the put option contracts on a single day for Digital Global expiring on 26/06/2003.

The main factor influencing the time value is the relationship between the share price and the strike price of the option. Time value is at its highest when the share price is equal to the strike price. As the share price and strike price deviate largely in either direction, time value declines.

Another important factor influencing the time value of the option contract is the expected volatility of the share price. When the volatility of the underlying instrument is large, the possibility of making a substantial gross profit or loss from the options contract is high. Hence, the greater the expected volatility of the price of an underlying instrument, the greater will be the time value of an option on that share.

WRITING OPTIONS

The position of the buyer of the call option or put option had been discussed earlier. For every buyer of an option there has to be a seller. The seller of an option is also known as the writer of the option. While the buyer of an option is said to have a long option position, the writer of the option is said to have a short position. The profit/loss profile at expiry of a short (written) option is the exact mirror image of the long (bought) option. Hence, the profit of the buyer is the loss for the seller, and vice versa. These are shown through Figure 8.6.

The buyer of a call option has a loss potential limited to the premium paid, but unlimited profit potential. On the other hand, the writer of a call has a maximum profit equal to the premium received, but unlimited loss potential. The buyer of a put option has a maximum loss equal to the premium, which in other words is the maximum profit of the writer. The maximum profit of the put buyer (maximum loss of the put writer) occurs at a share price of zero.

For exchange traded options, writing an identical option, can close out a long option position. Similarly, buying an identical option can close out a written option.

Example The shares of Dr Reddy’s Laboratories trade at Rs 950 in the spot market. Put options with a strike price of Rs 975 are priced at Rs 26.4.

1. What is the intrinsic value of the options?

   The intrinsic value is Rs 975 − Rs 950 = Rs 25

2. What is the time value of the options?

   The time value is the premium minus the intrinsic value, Rs 26.4 − Rs 25 = Rs 1.4

3. What might cause the time value to increase with no change in intrinsic value?

   Since intrinsic value is not expected to change, the implication is the spot price does not change. Hence, the volatility in Dr Reddy’s shares could result in an increase in time value.

4. If the share price fell to Rs 900 on expiry date, what would be the profit/loss for the holder and writer of the options?

   The intrinsic value is Rs 75. The net profit to the buyer of the option contract is Rs 75 − Rs 26.4 = Rs 48.60. The loss to the option writer is the same, Rs 48.60.

5. What is the maximum loss for the writer (maximum profit for the buyer) of the options?

   The maximum loss for the writer would occur in the event of the share price falling to zero. It would be the strike price minus the premium received, Rs 975 − Rs 26.4 = Rs 948.6. The buyer would receive a net profit of Rs. 948.6 per share from the put option contract.

EXOTIC OPTIONS

Exotic option refers to different derivations of the basic option characteristics. Some of the exotic options are lookback options, Asian options, barrier options, option on an option, and so on.

Lookback call options give the right to buy at the lowest price traded during the life of the option. Lookback put options give the right to sell at the highest price traded by the underlying instrument.

Asian options are exercised comparing the average market price of the underlying security with the strike price during the option period instead of the market price as on the expiry date.

Barrier options have limits as barriers apart from the strike price. Barrier options could be knock-in or knock out options or double barrier options. Knock-in options are exercised when a particular market price of the underlying security is reached. Knock-out options cease to exist when a particular price is reached. Double barrier options have both the upper and lower limits.

In an option on an option, an investor entering into a contract may take out an option to buy an option on the date on which the first option contract is due to be bought/sold. An option on an option is likely to be cheaper than the second option. The intrinsic value of an option on an option will be less than the intrinsic value of the second option because the delta of the second option would be less than one. Also, the volatility of such an option price will be less than the volatility of the underlying instrument. The option on an option may have a shorter period to expiry than the second option, hence the time value of the option on the option will be relatively low than the second option.

WARRANTS

Warrants are long-term call options issued by a company. Warrants usually have expiry dates that are five years or more in the future. Ordinary call options on the other hand, are exercisable within a month or two months. The maximum expiry date of an ordinary option instrument could be nine months. The company might issue warrants along with their debt instruments. These warrants are exercisable by the holder after the exercise date till the maturity of the instrument. When the warrants are exercised, the company issues shares at the predetermined price. The issuing company receives the money from the sale of the warrants. Thus warrants increase the number of shares for a company.

Some warrants are issued naked, that is, without the backup of corporate debt instruments. Since warrants do not pay any dividend or coupon interest, they only provide an issuing company with a source of finance that does not have servicing costs.

A bank can create covered warrants by writing third party warrants on the company’s securities without having any involvement in the company. Another type of third party warrant involves the company that is raising the finance, issuing warrants on the share of another company.

Videocon International’s warrants were traded in the Bombay Stock Exchange debenture section at Rs 3.50, Rs 2.85, and Rs 3.70 on January 25,2003. The closing price of Videocon International shares in the spot market was Rs 28.30. The bond instruments traded at Rs 187 on this date.

Similar to call options, warrants are said to be in-the-money when the market price is greater than the exercise price. Figure 8.7 shows the in-the-money, out-of-the-money, and at-the-money positions of a warrant along with the maximum and minimum values of a warrant.

Since the warrant is a right to buy, no holder of a warrant will be willing to pay more than the market price to exercise the warrant with the company. Hence, the theoretical maximum price of a warrant is the market price of the underlying instrument. The theoretical minimum value of the warrant is its intrinsic value. This is the amount by which the stock price exceeds the exercise price.

CONVERTIBLE BONDS

A convertible bond can be viewed as a corporate bond with an attached warrant (call option). Such a bond gives the bondholder the right to convert the debt instrument into shares at specified rates on the expiry date. Similarly, convertible preference shares are preference shares with the right to convert to ordinary shares. Some convertible instruments could provide the right to convert to other loan instruments rather than shares.

The number of shares for which a convertible bond can be exchanged is called the conversion rate. For example, a convertible bond may allow the conversion of Rs100 par value of debt into 5 shares. Multiplication of the conversion rate by the spot share price provides the conversion value. A share price of Rs 26 in this case would imply a conversion value of Rs 130.

The market value of a convertible instrument has to be higher than the conversion/investment values. The excess of the market value over the conversion/investment value is referred to as the premium. Figure 8.8 shows the relationship among the conversion, investment, and market values of a convertible instrument. It is assumed that the conversion rate is 5 and the investment value is Rs 90 per Rs 100 par value. It is further assumed that the conversion rate is Rs. 24.50.

The investment value is the market value of debt instrument, while the excess of the market value over the investment value corresponds to the option premium. In this example, the option has a strike price of Rs24.50. The excess of the conversion value over the investment value gives value the intrinsic value of the convertible warrant (26–24.50) = Rs. 1.50.

Convertibles are hybrid instruments in that they give a choice between bonds and shares. The percentage rate of dividend or coupon yield would be less than that of a simple bond. A dividend yield on the share that exceeds the coupon yield on the convertible instrument would induce conversion of the convertible into the share.

Conversion premium of the convertible instrument is the excess of the conversion price over the share price, expressed as a percentage of the share price.

            Conversion price = Market value of convertible/Number of shares on conversion

    Conversion premium (%) = {(Conversion price − Share price)/Share price} × 100

Mostly, due to the exercise of a right, the conversion premium would be positive. However, when the dividend on the share exceeds the coupon on the convertible (the dividend could increase due to better company performance), the premium need not be positive. If conversion dates are at distant intervals, (4 to 5 years), the yield could be lower on the convertible than on the share making it less valuable than the shares into which it might be converted. So also with a combination of these two factors, that is, a share dividend greater than the coupon of the convertible, together with a long time before the conversion date, could result in a negative conversion premium.

Example A convertible bond has a maturity of 10 years and pays an annual coupon of Rs 10. It has a conversion rate of 10 and the current share price is Rs 11. Conversion can take place on the sixth, seventh, and eighth years. The yield curve of the convertible is 12 per cent per annum.

1. Calculate the investment value of the convertible. Investment value

   Investment value
   = Rs 10/1.12 + Rs 10/(1.12)^2 + Rs 10/(1.12)^3 + Rs 10/(1.12)^4 + Rs 10/(1.12)^5 + Rs 10/(1.12)^6 + Rs 10/(1.12)^7 + Rs 10/(1.12)^8 + Rs 10/(1.12)^9 + Rs 10/(1.12)^10 + Rs 100/(1.12)^10
   = Rs 56.5 + Rs 32.2 = Rs 88.70

2. If at the end of the eighth year, one-and two-year interest rates were 5 per cent per annum and 5.5 per cent per annum respectively, what would be the price of the convertible?

   Price of the convertible at the end of the eight year
   Rs 10/(1.05) + Rs 110/(1.055)^2 = Rs 109.30

SWAPS

Swap contract is a spot purchase and simultaneous futures sale or a spot sale with a simultaneous buy from the future market. It is the agreed exchanges of future cash flows with spot cash flows.

The types of swaps prevalent in the market are interest rate swaps, currency swaps, and equity swaps. Interest rate swaps enable exchange of fixed to floating rates of return. Currency swaps enable exchange of currency between international markets.

Equity Swaps

An equity swap involves an agreement to exchange the returns on a stock index portfolio for a return of fixed interest payments. Investors desirous of exchanging a country’s interest return for a share market return of another country would usually enter into such equity swaps.

Assume investor A has a balanced portfolio of Indian shares but is bearish about the Indian stock market and is bullish on US interest rates. As an alternative to selling the portfolio, the investor could enter into an equity swap with another national interested in investing in Indian stock market. The swap illustrated in Figure 8.9 illustrates the swap of equity returns from India with the dollar interest return in USA. Investor A exchanges the return on BSE 100 portfolio for LIBOR + 1 per cent return on the US dollar deposit abroad. Similarly, investor B exchanges the return on US dollar deposit for a return on BSE 100 portfolio when his expectation of the emerging economy share market is bullish.

Investor B mostly is an American fund manager who wants to invest in the Indian stock market but does not have the expertise to evaluate Indian stocks. By entering into the equity swap the American fund manager simulates a balanced investment in Indian shares without getting involved in the analysis of individual Indian shares.

VALUATION OF INSTRUMENTS

Pricing Futures

Cost and Carry Model of Futures Pricing

The cost of carry forces the futures pricing to be different from that of the spot price. Cost of carry implies inclusion of three types of costs prevalent in a futures contract. The three types of costs are the financing cost, storage cost, and insurance cost. Hence, the computation of futures price based on this model will be:

    Futures price (Fair price) = Spot price + Cost of carry-Inflows

            FPtT = CPt + CPt * (RtT − DtT) * (T−t)/365

where,

FPtT—Fair price of the asset at time t for time T

CPt—Cash price of the asset

RtT—Interest rate accomnodating the storage and insurance cost at time t for the period up to T

DtT—Inflows in terms of dividend or interest between t and T

       t—Time at which futures are priced

       T—Contract expiry time

The assumptions behind the application of the above formula are no seasonal demand and supply exist in the underlying asset; storability of the underlying asset is not a problem; the underlying asset can be sold short and there are no taxes.

If the futures price is greater than the fair price, the trading strategy for the investor would be to buy in the cash market and simultaneously sell in the futures market. If the futures price, on the other hand, is less than the fair price, the strategy of the investor would be to sell in the cash market and simultaneously buy in the futures market. This arbitrage between cash and futures markets is expected to remain till prices in the cash and futures markets reach the point of equilibrium.

Example The spot price of Arvind Mills is Rs. 45.25. The futures on Arvind Mills trades at Rs. 45.50. The interest cost including the transaction and cost is estimated at 4% p.a. and the dividend inflow is expected to be 3% p.a. on the underlying instrument. Assuming the duration of the futures contract to be 20 days, determine the theoretical futures price. Is there an arbitrage possibility?

Since the future price is greater than the fair price, the investor could gain by buying in the spot market and selling in futures market.

STOCK INDEX FUTURES PRICES

The pricing of futures is based on arbitrage opportunities. When the underlying instrument can be held during the life of the futures contract, the arbitrage is referred to as a cash-and-carry arbitrage.

For such arbitrage to apply in pricing stock index futures, the arbitrager has to buy the portfolio of shares on which the index is based and hold it until the maturity date of the futures contract. Sometimes, the fund for holding the portfolio need not be owned but can be borrowed. This purchase of share with borrowed money involves a financial cost. If the cost of this futures position differs from the price of traded futures, an arbitrage opportunity may be available. If the actual futures price exceeds that of the financial cost, then a profit might arise from borrowing money and using it to buy the stock index portfolio and simultaneously selling futures. The proceeds from selling the futures would exceed the sum of money to be repaid resulting in the arbitrage profit.

If on the other hand, stock index futures are trading at a price below that of the financial cost, the arbitrager would buy futures and sell the stock index. Thus, in this case, the cash-and-carry arbitrage would involve buying futures while short selling index and depositing the proceeds in risk-free securities. The interest received from deposits when the futures contract matures is used to repay the financial cost. In this case the proceeds from depositing money less dividend obligations would exceed the amount required to buy the index futures, and this excess is the arbitrage profit.

The formula for fair value premium is

where,

FP—Fair value premium

I—Stock index

r—Interest rate

y—Percentage yield on the index portfolio

d—Number of days in the futures position

 

For example, if index value (I) = 2400, treasury market rate (r) = 5% p.a., index yield (y) = 4% p.a. and d = 15 days.

Then,

The index futures value = stock index + fairvalue premium

                    = 2400 + .98

                    = 2400.98

Short cash and carry involves selling index and buying futures. In this case, the excess of interest over dividends is a net inflow and this gain would be the arbitrage profit.

Let us assume the current level of the Indian stock index is 5100 and that of the German index is 2300. The settlement date is 10 days from now in both cases. Assume interest rates in India and Germany are 4.68 per cent per annum and 6.75 per cent per annum respectively. The estimated rate of dividend yield on the Indian portfolio over the next 10 days is 4% p.a.

On the basis of this information it is possible to calculate fair prices for futures on the Indian stock index and the German stock index. Note that while the Indian index is based only on share prices the German index is a total return index and includes dividend return as reinvestment, hence German portfolio yield is 0% p.a.

In the case of the Indian index:

    Fair future price = spot index * {1 + [(r − d) * (n/365)]}

                                = 5100 * {1 + [(0.0468 − 0.04) * (10/365)]}

                                = 5100.95

In the case of the German index:

    Fair future price = spot index * {1 + [r * (n/365)]}

                                = 2300 * {1 + [0.0675 * (10/365)]}

                                = 2304.25

THE BLACK-SCHOLES OPTION PRICING MODEL

The basic Black-Scholes model relating to a call option of non-dividend paying shares can be expressed as follows:

where,

C—call option price

S—share price

N(d1) and N(d2)—cumulative normal distribution functions

K—strike price

e—exponential (which has the constant value of 2.7182818)

r—annualised risk-free interest rate

t—time to expiry (in years)

σ—annualised standard deviation of share returns (volatility) as a decimal.

The term (In) in the formula stands for the natural logarithmic value of (S/K).

The expression e^^−rt refers to continuous compounding of interest rates for time duration t. This determines the present value of a future sum of money discounted on a continuous basis.

The statistical distributions used by Black-Scholes Model is shown in Figure 8.10. N(d), the cumulative normal distribution function, is based on a standardised normal distribution. Cumulative probability states that a normally distributed variable (share price volatility) will be less than ‘d’ standard deviations above the mean.

The volatility of stock returns over the period to option expiry, σ * square root (t), corresponds to the value of 1 on the horizontal axis. The values of dl and d2 are in units of volatility and are points on the horizontal axis of the standardised normal distribution. N(d1 ) is the area under the distribution to the left of dl and N(d2) is the area to the left of d2. N(d1) is the probability of ‘d’ being ‘d1’ or less. N(d2) is the probability of ‘d’ being ‘d₂’ or less. N(d1) can be interpreted as the probability of the call option being in-the-money at expiry.

Example On November 14, Bajaj Auto shares have a market price of Rs 442.35. The option expiry date is November 28,2002. The interest rate is 5 per cent per annum and the estimated volatility of the share is 10 per cent per annum. Use the Black-Scholes (non-dividend payment) option pricing model to calculate fair prices for a) Rs 440, b) Rs 400, and c) Rs 450 strike price call options expiring in November.

The Black-Scholes equation for non-dividend poring shares is:

So

    ln (S/K) = 0.005327; rt = (.05 * (14/360)) = .001944; std. * sqrt(t) = 0.01972

            e ^–rt = 0.0998057

1. Strike price Rs 440

   d1 = [(0.005327 + 0.001944)/0.01972] + (0.01972/2) = 0.378574

   d2 = [(0.005327 + 0.001944)/0.01972] − (0.01972/2) = 0.358854

   C = [442.35 * N(0.378574)] − [440 * 0.998057 * N(0.358854)]

   C = ( 442.35 * 0.647498) – (400 * 0.998057 * 0.640148)

   C = 286.4207 − 281.1179 = Rs 5.3028

   The fair price of an at-the-money call option is Rs 5.3028 per share.

2. Strike price Rs 400

   d1 = [(0.100637 + 0.001944)/0.01972] + (0.01972/2) = 5.211683

   d2 = [(0.100637 + 0.001944)/0.01972] − (0.01972/2) = 5.191962

   C = [442.35 * N(5.211683)] − [440 * 0.998057 * N(5.191962) ]

   C = ( 442.35 * 1) − (400 * 0.998057 * 1)

   C = 442.35 − 399.2229 = Rs 43.12702

   The fair price of an in-the-money call option is Rs 43.12702 per share.

3. Strike price Rs 450

   d1 = [(−0.01715 + 0.001944)/0.01972] + (0.01972/2) = − 0.76101

   d2 = [(−0.01715 + 0.001944)/0.01972] − (0.01972/2) = − 0.78073

   C = [442.35 * N(−0.76101)] − [440 * 0.998057 * N(−0.78073)]

   C = (442.35 * 0.223326) − (450 * 0.998057 * 0.217481)

   C = 98.78836 − 97.67645 = Rs 1.111906

The fair price of an out-of-the-money call option is Rs 1.111906 per share.

VARIATIONS ON THE BASIC BLACK-SCHOLES MODEL

European-style Call Options on Dividend Paying Shares

The Black-Scholes model can be used to value call options on dividend paying shares. To the basic model, the present value of dividend to be received is adjusted. Here the formula is rewritten by replacing S by dividend adjusted S.

According to the dividend discount model, the current stock price is the present value of all future expected dividends. However, dividends accruing before the expiry date will be incorporated into the current share price, but not on the share price on the expiry date. European call options are exercised on the expiry date; hence the market price in the formula is the dividend-adjusted market price. In other words, the present value of expected dividends prior to expiry needs to be subtracted from the current share price in order to ascertain the market price to be used in the Black-Scholes model.

Example The shares of Gujarat Ambuja Cements are quoted in the market at Rs 159.95 on January 29,2003. The annual dividend yield on its shares is 6 per cent. If the volatility is 35 per cent, compute the call premium for Rs 170 strike price expiring on February 27,2003. The interest rate is 10 per cent per annum.

The Black-Scholes equation for dividend paying shares is

            C = S^@N(d1) − Ke^−rt N(d2)

            d1 = (ln(S^@/K) + rt/std. sqrt(t)) + 0.5 std.sqrt(t)

            d2 = (ln(S^@/K) + rt/std. Sqrt(t)) − 0.5 std. sqrt(t)

So

            S = 159.95; r = .1; t = (29/360); d = .06; std. = .35; sqrt(t) = .283823

            Se^ ^−dt = (159.95 * e^ ^−.0048) = (159.95 * 0.9952) = Rs 159.18 = S^@

            ln (S^@/K) = − 0.06576; rt = (.10* (29/360)) = .0081; std. * sqrt(t) = 0.09934

            d1 = [(− 0.06576 + 0.0081)/0.09934] + (0.09934/2) = − 0.53133

            d2 = [(− 0.06576 + 0.0081)/0.09934] − (0.09934/2) = − 0.63066

            C = [159.18 * N(− 0.53133)] − [170 * e^−.0081 * N(− 0.63066)]

            C = (159.18 * 0.297596) − (170 * 0.9919 * 0.26413)

                  = (159.18 * 0.297596) − (168.62* 0.26413)

            C = 47.37103 − 44.54184 = Rs 2.829187

The fair price of this out of-the-money call option is Rs 2.83 per share.

European-style Call Options on Stock Index

Stock index portfolios can be expected to yield continuous stream of dividends since the shares comprising the index would declare dividends at different point of time. Here too, adjustment in market price to the expected future dividend receipts has to be made to compute the option value. The distinction with respect to dividends from an index is that while computing d1 and d2, the expected annualised dividend yield is subtracted from the risk-free interest rate.

where,

C—Call option price

S—Spot stock index

e—Exponential (natural logarithm)

d—Expected rate of dividend yield (as a decimal)

t—Time to expiry

N(d1) and N(d2)—Cumulative normal distribution functions

K—Strike price

r—Risk-free interest rate (as a decimal)

σ—Volatility (annualised standard deviation of returns).

Example The Nifty index stands at 1046.20 on January 29,2003. Compute the call premium for 1060 Nifty options with an expiry date on February 27,2003. The interest rate is 10 per cent and the Nifty companies yield an annualised dividend of 7 per cent. The volatility of the index is 0.42.

The Black-Scholes equation for dividend paying shares is:

            C = S.^@N(d1) − Ke^ ^−rt N(d2)

            d1 = (ln(S/K) + (r − d)t/std. sqrt(t)) + 0.5 std.sqrt(t)

            d2 = (ln(S/K) + (r − d)t/std. Sqrt(t)) − 0.5 std. sqrt(t)

So,

            S = 1046.20; e^ ^−dt = (1046.20 * e^ ^−.0564) = (1046.20 * 0.99437) = Rs 1040.317

            ln (S/K) = − 0.01331; (r − d)t = (.03* (29/360)) = .00242; std. * sqrt(t) = 0.11921

            d1 = [(− 0.01331 + 0.00242)/0.11921] + (0.11921/2) = − 0.03005

            d2 = [(− 0.01331 + 0.00242)/0.11921] − (0.11921/2) = − 0.14926

            C = [1046.2* e^ ^−.0564 * N(− 0.03005)] − [1060* e^ ^−.0081 * N(− 0.14926)]

            C = (1040.317 * 0.488012) − (1060 * 0.9919 * 0.440674)

            C = (1040.317 * 0.488012) − (1051.41 * 0.440674)

            C = 507.6869 − 463.3668 = Rs 44.32017

The fair price of an out of-the-money call option index is Rs 44.32017 per share.

European-style Call Options on Futures

Futures themselves do not yield an income return such as dividends or interest but the cash relating to the futures contract can yield a return. If a futures position is bought, the money relating to the value of that futures contract can be deposited to yield a risk-free return. Hence, instead of a return in the form of dividend, it can be presumed that a futures contract will yield a risk-free rate of return.

The variant of the Black- Scholes model applicable to futures options can be expressed as:

where,

F is futures price.

The variation from the usual Black-Scholes formula is that the term (rt) is not added to the natural log of (F/K). This is because the option on futures are assumed to earn an interest rate of ‘r’, and hence the term ‘r–r’ will be zero.

Valuing an European-style Put Option

The put option can be valued using the following formula:

Example On November 14,2002 Bajaj Auto shares have a market price of Rs 442.35. The option expiry date is November 28,2002. The interest rate is 5 per cent per annum and the estimated volatility of the share is 10 per cent per annum. Use the Black-Scholes (non-dividend payment) option pricing model to calculate fair prices for a) Rs 440 b) Rs 430 c) Rs 450 strike price put options expiring in November.

The Black-Scholes equation for non-dividend poring shares is:

            C = Ke^ −rt [1 − N(d2)] − S.[1 − N(d1)]

            d1 = (ln(S/K) + rt /std. sqrt(t)) + 0.5 std.sqrt(t)

            d2 = (ln(S/K) + rt /std. Sqrt(t)) − 0.5 std. sqrt(t)

1. Strike price Rs 440

   ln (S/K) = 0.005327; rt = (.05 * (14/360)) = .001944; std. * sqrt(t) = 0.01972

     d1 = [(0.005327 + 0.001944)/0.01972] + (0.01972/2) = 0.378574

     d2 = [(0.005327 + 0.001944)/0.01972] − (0.01972/2) = 0.358854

     C = [440 * e^−.001944 * {1 − N(0.358854)}] − [442.35 * {1− N(0.378574)}]

     C = (440 * 0.9978 * 0.359852) ( 442.35 * 0.35250)

     C = (399.11 * 0.359852) − (442.35 * 0.35250)

     C = 158.0274 − 155.9293 = Rs 2.09811

   The fair price of an at-the-money put option is Rs 2.09811 per share.

2. Strike price Rs 430

   ln (S/K) = 0.028316; rt = (.05 * (14/360)) = .001944; std. * sqrt(t) = 0.01972

     d1 = [(0. 0.028316 + 0.001944)/0.01972] + (0.01972/2) = 1.544356

     d2 = [(0. 0.028316 + 0.001944)/0.01972] − (0.01972/2) = 1.524635

     C = [430 * e^−.001944 * {1 − N(1.524635)} ] − [442.35 * {1 − N(1.544356)}]

   C = (430 * 0.9978 * 0.06368) ( 442.35 * 0.061251)

   C = (429.05 * 0.06368) − (442.35 * 0.061251)

   C = 27.32709 − 27.09443 = Rs 0.232655

   The fair price of an out of-the-money put option is Rs 0.232655 per share.

3. Strike price Rs 450

   ln (S/K) = −0.01715; rt = (.05 * (14/360)) = .001944; std. * sqrt(t) = 0.01972

     d1 = [(− 0.01715 + 0.001944)/0.01972] + (0.01972/2) = − 0.76101

     d2 = [(− 0.01715 + 0.001944)/0.01972] − (0.01972/2) = − 0.78073

     C = [450 * e^−.001944 * {1 − N(− 0.78073)}] − [442.35 * {1 − N(− 0.76101)}]

     C = (450 * 0.9978 * 0.7825) ( 442.35 * 0.7767)

     C = (449.01 * 0.7825) − (442.35 * 0.7767)

     C = 351.4494 − 343.5616 = Rs 7.887756

The fair price of an in-the-money put option is Rs 7.887756 per share.

COMPUTING VOLATILITY

Volatility can be computed by determining the daily variance for a sample-traded days and annualising the resultant figure. For example, the daily variance of returns for Castrol is .000606 (January 2000 to December 2001). Annualising this figure, that is, multiplying it by the approximate number of traded days (250), gives us the variance of 0.1515. The square root of this figure is the sigma, or the volatility, that is, 0.3764. Assuming a sample of 30 days (January – February 2001) is used, the historical volatility measure for Castrol will be 0.263309 or 26 per cent as shown in the following table. Hence, the historical volatility measure will depend on the sample data set used.

COMPUTING IMPLIED VOLATILITY

The volatility that is implied in the market using Back-Scholes model can be computed by taking the market premium to be the fair price of the call and deriving the volatility. Implied Volatility computation assumes that the market premium is correct. The actual volatility from past historical returns from prices is the historical volatility. On the other hand, the Black-Scholes model-determined volatility assuming the market premium as the best price for the option is the implied volatility.

BINOMIAL OPTION PRICING MODEL

The limitation of the Black-Scholes model is that it cannot be applied to an American-style option. Binomial models can incorporate the possibility of early exercise, and hence can be used for the valuation of American-style options. Another strength of binomial models, relative to the Black-Scholes model, is that they can consider changes in interest rates and volatility over time.

Example Assume that a share, currently priced at Rs 100, can either rise by a factor (1 + x) (x = 10%), which equals 1.1, or fall by a factor l/(1 + x) that is, 90.91 for each duration of the option. The possible share prices after 1 and 2 periods are shown in Figure 8.11.

Assume that a one-period call option has a strike price of K while the share price, S, may rise by (1+x) or fall by (1/(1+x) during the period. At the end of the period, the option is worth either

In other words, the price of the call option using binomial model can be stated as

where,

p (probability) can be worked out by equating the weighted end-of-period prices to the spot price. This is stated as: S = p * C_(1+x) + (1 – p) * C_(1/(1+x))

For the illustration,     100 = p * 110 + (1 – p) * 90.91

Solving for p,     p = 0.476 and (1 – p) = 0.524

Thus, the value of the spot price, is the discounted weighted average of the two probable option prices occurring at the end of time. Figure 8.12 illustrates the formula.

Substituting the values of p and (1 − p) in the call option formula, the price of the call option will be

Incorporating Time Value and Dividend Flows

When the spot price is expected to yield a rate of return, then the revised current price for computing the probability values will be S * (1+r), where S is the spot price and ‘r’ is the risk-free return for that time period.

Example Assuming the risk-free return is 3 per cent for the time duration, the revised probabilities are

100(1.03) = p * 110 + (1 − p) * 90.91

        p = 0.6333 and (1 − p) = 0.3677

Hence, the call price will be

Similarly, for making an adjustment for receipt of dividend, the current price will be adjusted for receipt of dividend apart from the adjustment for time value. Here the revised spot price will be S * (1+r−d), where S is the spot price, ‘r’ is the risk-free rate of return and ‘d’ is the dividend yield.

Example Assuming the risk free return to be 3 per cent and dividend paid as 2 per cent for the duration, the revised probabilities are:

100 * (1+.03−.02) = p * 110 + (1−p) * 90.91

        p = 0.5285 and (1−p) = 0.4715

Hence, the price of the call is

In the two-period model, similar probabilities are expected to occur at the second stage and hence the expected prices can be worked out. The generalised model for evaluating the two-period binomial model is given in Figure 8.13.

Similarly, C_{((1/(1 + x))} equals the discounted weighted average of the two possible end-of-second-period option prices when there has been a downward stock price movement in the first period:

The current call option price, C, equals the discounted weighted average of the two possible first period option prices:

SUMMARY

The introduction of derivative products in the Indian stock market has increased the use of carefully built risk management tools for investors. Derivative products are derived instruments that have a future contractual potential based on the underlying asset performance.

Derivative products could take the form of forward contracts, futures, and option contracts. Further, the option contract may be a call option, put option, or exotic option. Other options are the convertible bonds and warrants.

The basic valuation model used for ascertaining the worth of a derivative is the cost and carry forward model. Besides, for valuing options, Black-Scholes and binomial models are available.

CONCEPTS

• Synthetic forward/ futures
• Bond futures
• Bid offer spread
• Equivalent portfolio
• Equity swap
• Forward rate agreements
• Index options
• Fair future price

SHORT QUESTIONS

1. What is a swap agreement?
2. What are index options?
3. What is an equivalent portfolio?
4. What is portfolio hedge?
5. Distinguish between American and European options.

ESSAY QUESTIONS

1. Explain the Black-Scholes model and highlight its limitations.
2. Explain the binomial model of pricing options
3. Explain the possible opportunities for swap agreement.

PROBLEMS

1. An investor expects Rs 10,00,000 three months hence. The current Nifty index is 1036.40. The investor expects an increase in index by 50 points. Three-month Nifty futures are quoted at 1045. Can the investor make use of the futures contract? If so, how? What is the profit/loss to the investor when the index touches the investor’s expectation in three months?
2. An investor has Rs 50,00,000 worth of portfolio in the stock market. There is an expectation of a bearish market spell in three months. What risk management opportunities are available to the investor?
3. Tata Power is quoted in the spot market at Rs 104.95. Three-month interest rates are 5 per cent. The company is expected to pay a dividend of Rs 5 for the quarter. What should the futures price be if the investor wants to lock the position for one month?
4. The following stocks are held in the portfolio of an investor.
   

   If the Nifty index is 1055, determine the number of futures contracts required to protect the portfolio.

5. The Grasim call option for Rs 300 is Rs 8.50. At expiry date, the market price of the share is Rs 312. What is the profit/loss situation to the buyer and writer of the call option?
6. The VSNL put option (strike price, Rs 110), is priced at Rs 5. As on expiry date, the market price of the share is Rs 120. What is the profit/loss to the buyer and writer of the put option?
7. Cipla has a market price of Rs 890. The volatility on the share is .32; the risk-free interest rate is 5 per cent. What would be the price of the call with a strike price of Rs 880, if the expiry date is 20 days ahead? Assume there has not been any dividend announcement.
8. Sterlite has a market price of Rs 68. The volatility of the share is .21, the risk free rate is 5 per cent. What would be the price of a put with a strike price of Rs 60 if the expiry date is 27 days hence? Use Black-Scholes model (non-dividend paying company).
9. Telco is presently quoted at Rs 143. An investor expects a 10 per cent increase in the price of Telco one month hence. Use the binomial model to price the call.
10. MTNL is quoting in the market at Rs 102. An investor expects a 5 per cent decline in the price of the share one month hence. Use the binomial model to price the put.