Derivatives
Introduction
Derivative is a security or a financial asset, which derives its value from some specified underlying asset. A derivative does not have any physical existence but emerges out of a contract between two parties. These underlying assets may be shares, debentures, tangible commodities, currencies currency, interest rates etc. A derivative instrument is a financial instrument:
(i) Whose value changes in response to change in underlying asset?
(ii) That requires no initial investment or little initial investment relative to other types of contracts that have a similar response to change in market conditions; and
(iii) that is settled at a future date.
Financial instruments include primary instruments like shares, debentures, Bills etc. as well as derivative instruments like futures, options, swaps etc.
Objective
The derivative provide as hedging of price risk of financial transactions over a certain period. The main function served by derivative securities within the financial system is risk management. It is a contract to be settled in future. Hedging is taking a derivative position opposite to your exposure. Hedge ratio is the ratio of one position relative to other where risk is neutralized. Hedging does not remove losses. The best that can be achieved using hedging is the removal of unwanted exposure, that is, unnecessary risk. It is also true that hedged position will make less profit than the un-hedged position.
Type of derivatives
(i) Forward
A forward contract is an agreement made today between a buyer and seller to exchange the commodity or instrument for cash at a predetermined future date at a price agreed upon today. The agreed upon price is called as "Forward Price". The forward rate remains fixed irrespective of future rate fluctuations. The major shortcoming of forward contract is that involve the counter party risk. Counter party risk arises when one of the two parties of the transaction choose to declare bankruptcy, the other suffers.
(ii) Future
Futures are exchange-traded contracts to sell or buy financial instruments or physical commodities for future delivery at an agreed price. The agreed upon price is called the future price. Future markets are exactly like forward markets in terms of basic economies. However, future contracts are standardized and trading is centralized, so that future market is highly liquid. There is no counter party risk due to Marking-to -Market. Marking-to-market concept implies the daily adjustment of margin account to reflect profit or losses. Every day there is a winner and a loser, depending on the direction of price movement.
In general, the future price is greater than the spot price. The relationship between future prices is determined by the cost-of-carry. However, there might be factors other than cost-of-carry, which may influence this relationship. The difference is knows as the basis, where:
Basis = Spot - Future Price
Some authors define Basis as Future - Spot.
Although the spot price and future prices generally move in line with each other, the basis is not constant. This is known as basis risk. Over the life of a future contract the basis changes. It equals to Zero on the maturity date as the future price at maturity becomes same as spot price. This process is known as "Convergence". Hedgers' faces uncertainty as to what basis will prevail when they close out their future contracts. It is to be noted here that in a normal market, Future price is calculated on the basis of three factors i.e. Spot rate, interest and dividend or other return on underlying.
If the future price is greater than the spot it is called 'contango'. Under normal market conditions futures contracts are priced above the spot price as it includes cost of carry. This is known as "contango market". In this case basis will be negative.
If the spot price is greater than the future price it is called as "backwardation market" or "inverted market". This happens, when there are factors other than cost of carry influence the future price. In this case basis will be positive.'
Pricing index futures given expected dividend amount
While pricing stock future, effect of dividend payment during the future contract period may be carried out as above. But in case of index future we need to compute the amount of dividend receivable per unit of index. The holding is that dividend is payable on stock which is the part of index, hence we need to convert total dividend receivable on stock into equivalent amount of dividend receivable on per unit of index. It is calculated as below:
Spot value of index × weight of stock in the index × DPS
----------------------------------------------------.--------------
Current Market price of stock
After calculating dividend receivable per unit of index value of index future can be calculated as below:
Value of Index Future Contract = (Spot Price - DPS/ert) × ert or (1 + r)t
However, if the dividend flow through out the year is generally uniform, the cost of financing i.e. r will be reduced by annualized dividend yield on index. In this case the future price of index will be calculated as below:
Spot value of index x e(r-y) t or (1+r-y) t
Where y = Expected annualized dividend yield on index.
Portfolio hedging using indexes Futures
With the help of future contract we can hedge risk of underlying security by entering into reverse future contract of underlying. For example, for long position in underlying one can short future of that underlying & vice-versa. Similarly, index future contracts could be used for hedging a portfolio of underlying. An investor holding portfolio investment always worried about decline in value of portfolio due to decline in stock prices. Index futures can be used to hedge portfolio from potential downside due to a market drop. Index futures can also be used for increase the exposure in the market, which will also increase return. In that case we will proceed as below:
(a) Calculate current value of portfolio.
(b) Calculate portfolio beta.
(c) Find out spot value of index i.e. NIFTY or SENSEX.
(d) Calculate desired portfolio beta. Desired portfolio beta is based on desired return from portfolio.
(e) Calculate number of future contracts required for hedging the portfolio & achieving desired beta of portfolio.
Current value of portfolio × (b2 - b1) 1
-------------------------------------------- × ------------------
Current value of index Market lot of index
Where b, = Existing beta of portfolio b2 =Desired portfolio beta
Negative quantity indicates opposite exposure in future as compare to portfolio/security's exposure while positive quantity indicates same exposure as portfolio/security's exposure. In case of NIFTY market lot is of 100 indexes & in case of SENSEX, market lot is of 50 indexes.
Same approach will be adopted for hedging of a- particular security with index futures.
In this situation, whatever change occurs in the index, change in the value of investor's portfolio/security will be in accordance to desired portfolio beta.
Hedge ratio depends upon investor's attitude towards risk. In other words quantity of index futures is decided after considering the fact that how safe an investor wants to play. In case of complete or perfect hedging desired beta will be 0 that means there will be no loss/gain whatever happens in the market. In case of hedging, value of index futures to be bought/sold (as the case may be) will be calculated as follows:
Value of Portfolio/security × (b2 - b1) = Value of Index futures required for hedging
Number of index futures = Value of index futures (calculated as bove)'
------------------------------------------------.
Spot value of index × Market lot of index
(iv) Option Valuation/Pricing
Introduction:
An option is a claim without any liability for holder. It is a claim contingent upon the occurrence of certain conditions. An option is a contract that gives the holder a right without any obligation, to buy or sell an underlying asset at an agreed price on or before a specified date. The price at which option can be exercised is called as exercise price or strike price. The buyer/holder of the option has to pay a price called as option premium to the seller/writer of the option, whether or not the buyer/ holder exercises his option.
Option Terminology
(a) Call Option: Right to buy an underlying asset at strike price.
(b) Put Option: Right to sell an underlying asset at strike price.
(c) In-the-money: When strike price is better than market price.
(d) Out-of-the-money: When Strike price is worse than market price.
(e) At-the-money or Near-the-money: When strike price is same as market price.
(f) European option: Option is allowed to be exercised only on the maturity date.
(g) American option: Option can be exercised any time till the maturity date.
(h) Capped-style-options:
A capped option is an option with an established profit cap. The cap price is equal to the option's strike price plus a cap interval for a call option or the strike' price minus a cap interval for a put option. A capped option is automatically exercised when the underlying security closes at or above (for a call) or at or below (for a put) the option's cap price.
Option Premium:
The total cost (the price) of an option is called as premium. It consists of two values i.e. intrinsic value of option & Time value of option.
The amount by which an option is In-The-Money is referred to as intrinsic value. For Call options, this is the difference between the underlying stock's price and strike price. For put options, it is the difference between the strike price and the underlying ~e price. In other words intrinsic value in options is the In-The-Money portion of the option premium.
The portion of the option premium that is attributable to the amount of time remaining until the expiration of the option contract is referred to as Time Value. Basically, time value is the value the option has in addition to its intrinsic value.
Valuation of Option:
(i) Call Option:
Value of a call option at expiration is the maximum of the (share price minus the exercise price or Zero). The call option holder's opportunity to make profits is unlimited beyond the exercise price. If the share price on or below the exercise price, his payoff will be Zero as he will not exercise call option.
Value of Call Option at expiration = Maximum (Share Price - Exercise price, 0)
Break Even Price = Exercise Price + Option Premium + Transaction cost (for buyer of option)' The call option holder's gain is call seller's loss.
(ii) Put Option:
Value of a Put option at expiration is the maximum of the (Exercise price minus the Share Price or Zero). The Put option holder's opportunity to make profits below the exercise price is very high but not unlimited since share price cannot fall below Zero. If the share price on or beyond the exercise price, his payoff will be Zero as he will not exercise Put option.
Value of Put Option at expiration = Maximum (Exercise Price - Share Price, 0)
Break Even Price = Exercise Price - Option Premium - Transaction cost (for buyer of option) The Put option holder's gain is put option seller's loss.
Factors determining option value:
Value of an option at expiration may be calculated as above. But how the value of the option will be, calculated with time to expiration? This is also called as option premium or option price. Such valuation depends on the following factors:
(i) Movement in price of underlying asset:
Since the options, being derivative instruments, derive their value from the underlying assets. Hence any movement in the price of underlying assets affects the option premium. The value of a call option would increase as the share price increase. Similarly, value of a Put option will increase as the share price decrease.
(ii) Exercise price of the option
Strike price is an important factor for the valuation of options. In case of call option, option premium will decline with increase in strike price. Similarly, in case of put option, option premium will increase with the increase in the strike price.
(iii) Volatility of Underlying Asset:
Volatility is another important factor, which decides the value of option. Volatility is a measure of how much an underlying price varies. Greater the volatility, greater the risk and the greater the value of an option.
(iv) Interest Rates:
The holder of an option pays or receives (as the case may be) exercise price not when he buys the option, rather, later on, when he exercise his option. Thus the present value of the exercise price depends on the interest rate and the time until the expiration of the option.
The value of a call option will increase with the rising interest rate since the present value of the exercise. price will fall. The effect is reversed in the case of a put option. The buyer of a put option receives the exercise price and therefore, as the interest rate increases, the value of the put option will decline.
(v) Time to option expiration:
The present value of the exercise price will be less if time to expiration is longer and consequently the value of the call option will be higher. The effect is reversed in the case of a put option. Further the possibility of greater volatility exists when the time to expiration is longer and the greater volatility makes option more valuable for buyer of the option.
(vi) Dividend from underlying asset
If any dividend is paid during the life of the option, the share price will go down by an amount reflecting the payment of dividend, which will cause decline in the value of a call option and increase in the value of a put option. The reason behind this concept is that after payment of dividend price of underlying asset declines but no adjustment is made in the strike price for such payment unless dividend amounts to be a certain percentage of strike prices.
5.0 Combination of Put Option, Call Option and Share/Future
A share/future, a put and a call can be combined together to create several pay-off opportunities,
(a) Covered option writing
A strategy in which one sells call option while simultaneously owning an equivalent long position in the underlying security/future known as covered call.
A strategy in which one write put option and simultaneously short an equivalent position in the underlying security /future.
When underlying security/future does not cover an option. it is called uncovered option.
(b) Protective Call
A Protective call is an attempt to protect a short sale of stock by buying a call in an equivalent position. The call acts as protection for the short sale. Theoretically the risk exposure to a short sale is unlimited; buying a call limits the risk. An investor can protect himself from a rise in price using the protective call. Also known a s synthetic put.
(c) Protective Put.
A protective put is an attempt to protect a long position in stock/future by buying a put in an equivalent position with strike price equal to purchase price of stock/future. Buying a put hedge-the risk of decline in price. Also known as Synthetic Call.
(d) Straddle
Buying a put and a call option at same exercise price. When one invest in straddle; he will be benefited whether the price of the share fall or rises. However, option premium & transaction cost is the net loss to the holder of the straddle. Hence, straddle is beneficial only if there is greater volatility in the share prices. If share price remains near to exercise price, holder of straddle may find himself in loss.
(e) Strangle
A strangle is an option strategy which involves the purchase of a call and a put that are both Slightly out of the money and have the same expiration date, and are of the same underlying security. The investor would use this strategy if he felts that the underlying security is volatile and is going to make a large move.
(f) Strip & Strap
A combination of two puts and a call is a "STRIP", while two calls and a put is "STRAP". This type -of combinations is formed on the basis of future expectation of price trends as well as for hedging purpose. For example, if it is expected that in near future market will head southward, strip will be a better option.
(g) Spread
A spread involves the purchase of one option and sale of another on the same stock. It comprises either all calls or all puts and not a combination of the two.
Option spread having different exercise prices but the same expiration date are called vertical spread. On the other hand, if the exercise prices are the same and the expiration dates are different, such combinations are called horizontal spreads or calendar spreads or time spreads. Spreads comprising options with different expiration dates and exercise prices are called diagonal spreads.
(h) Butterfly spread
The butterfly spread is a neutral options strategy position most often involving three different strike price calls. One call is bought at the lowest strike price. Two calls are sold at the middle strike price. And one call is bought at the highest strike price. It can also be implemented using puts. The investor would use this option strategy if he feels that the underlying security is not too volatile and won't experience too much of a net rise or fall by expiration.
(i) Ratio call spread
It is established by purchasing a certain amount of calls at one strike price and simultaneously selling more calls (ratio not defined, may be twice or 1.5 or 1.25.... depending on expected price behavior in future) than purchased at higher strike price. This spread is usually done for credit.
6.0 Put-Call Parity:
Put - Call Parity theorem implies relationship between put prices, call prices and the underlying asset/future. It may be in two forms:
(a) Synthetic Long:
Buying a call and selling a put at same strike price with same maturity date. In both the cases, if option is exercised by the holder delivery of underlying asset is to be taken.
(b) Synthetic Short:
Selling call and buying put at same strike price with same maturity date. In both the cases, if option is exercised by the holder delivery is to be given.
With the help of Put-Call parity, one can constitute a risk free portfolio. In the equilibrium with zero arbitrage opportunity, there will be a precise relationship between the market value of the Put and the call option and the stock. For example, we buy underlying asset or enter into long position in future. To constitute a risk free portfolio life will go for synthetic short that means buying a put an? selling a call.
If current share price is Rs. 100 & we buy underlying asset at current share price. To constitute a risk free portfolio, we will go for synthetic short at same strike price i.e. Rs. 100/- with same maturity date say after 3 months. If The expected share price after 3 months are Rs. 120 or 80, Ignoring the premium paid and earned and transaction costs, value of portfolio will be same as strike price or cost of long position i.e. Rs. 100/-.
It is a risk free portfolio since the outcome will be the same whatever happens to the share price. The hedge ratio according to this theorem is 1: 1: 1 that means equal quantity of share, calls put. Present value of portfolio can be calculated using risk free rate as discount factor.
A similar arrangement can be made using synthetic long. One can short the underlying or future and opt for synthetic long position in option that means buying a call option and writing a put option at same strike price. It will be a risk W portfolio since the outcome will be the same whatever happens to the share price. Present value of portfolio can be calculated using risk free rate as discount factor.
7.0 Option Delta:
The appropriate hedge ratio of stock to options is known as the option delta. A hedge position can be established by during or selling the stock and by writing options. The idea is to establish a risk less hedge position. Calculation of option delta requires the investor's expectation as to where the share price would be on the expiration of the option period.
Option delta is also a theoretical rate of change of an option premium with respect to change in the price of underlying asset.
Option delta also provides a probability estimate that an option will expire in-the money.
Option Delta = (Value of option at Hs)-(Value of option at Ls)
--------------------------------------------------------
Hs- Ls
Where = Hs = Higher share price & Ls = Lower share price (both expected at expiration)
For example, if option delta for call is 1/2 it means buying one underlying/future and selling two calls. In this case overall position is perfectly hedged in the sense of providing the same value at the end of the period, regardless of the stock price outcome.
For example, If Strike price is Rs. 200/- and expected share prices are Rs. 300 & 100. Option delta 'for call will be:
100 - 0 100 1
----------- = ----- = -----
300 - 100 200 2
Here, option delta is 1/2, that means buying 1 underlying/future and writing 2 calls. In this situation, value of portfolio for both the prices i.e. Rs. 300 & Rs. 100 will be same i.e. Rs. 100/-.
Other interpretations of option delta are:
If there is Rs. 1/- increase in the price of underlying asset, option premium will increase by R·s. 0 50/- & vice- versa.
There is 50% probability that option will be expiring in the money.
Simultaneously, option delta may be used for short sale in underlying or future with writing a put. Buying underlying /future and buying put can also form alternative strategy. Simultaneously, selling underlying / future and buying a call.
7.1 When option contracts are more than 1:
When a trader has position in different types of option contracts e.g. Long call or Short call or Short Put or. Long Put and wants to establish a risk less hedge position. In such cases we can ascertain the net equivalent exposure in the underlying/future for all such option contracts with the value of Net Option Delta. To ascertain Net Option Delta we have to calculate value of option delta for each type of contract.
Long calls & Short puts have positive delta and Short Calls & long puts have negative delta. The rationale behind assigning negative sign to long put & short call is that in both the cases delivery has to be given, if holder exercises his option.
Numbers of contracts of each type are multiplied by the option delta and net result of such calculation may be obtained. The net result is called as Net Option Delta.
Interpretation:
With the help of net option delta & put call parity theorem or option delta theorem, one can constitute risk free portfolio.
For example, if net option delta is 0.5 (positive), it means that trader's total exposure is equivalent to 0.5 long positions in the underlying/future contracts. To create a risk free portfolio, he may take help of put-call parity concept or option delta concept. He may go for synthetic short in the same proportion, which will create a risk free portfolio. Similarly, option delta may be used to constitute risk free portfolio.
8.0 Models for Option Valuation:
An option buyer has to pay some premium to acquire right without any obligation, which is expressed in terms of the option price. There are various models to derive this price. If the secondary market prices deviate from this, it would imply the presence of arbitrage opportunities. However it is to be noted that pricing models give an approximate idea about the true option price. The price observed in the market is outcome of demand-supply principle and may differ from the price calculated with the help of these models.
8.1 Black & Scholes model for option valuation:
In 1973, the Black & Scholes developed this model. Prof. Robert Metron had further contributed in this model. Prof. Metron and Prof. Scholes have been awarded Nobel Prize in Economics in 1997 for having developed a pioneering formula for the valuation of options. After introduction of this model, the quest for a mathematical model for valuation of options ended. This model has been accepted as the basic option valuation model all over the world. This model basically deals with valuation of call options.
Assumptions:
(a) The rates of return on a share are log normally distributed.
(b) The risk free rate is known and is constant during the life of the option.
(c) The market is efficient and there are no transaction costs and taxes.
(d) There is no dividend to be paid on the share during the life of the option.
(e) The call option is the European option.
(f) Share prices moves randomly in continuous time.
Co = S × N (d1) - E × N (d2)
------
ert
Where =
Co = the value of a call option
S = the current market value of the share
E = The Strike price
e = the exponential constant,
r = rate of interest (p.a.)
t = time to expiration (in years)
N (d1) = the cumulative normal probability distribution function
d1 = Ln(S/E) + [rf + sd2/2] t
------------------------------
sd *
/t
d2 = d1 - sd*t
Where
(a) Ln = the natural logarithm.
(b) SD = Standard deviation i.e. volatility. It should be annualized. If daily SD is given it need to be annualized. Annualized SD will be T Daily SD × Number of trading days per year. On an average it can be assume that there are 250 trading days in a year.
(c) Present value is calculated on the basis of continuous compounding basis.
(d) N (d1) is called as hedge ratio or option delta (according to this model).
Value of N (d1) & N (d2) may be obtained from statistical table for normal distribution i.e. Z table.
If value of (d) is positive
(In case of one tail Z table starting with Z=0=0.00) = N (d) = 0.50 + Value of Z for d
(In case of one tail Z table starting with Z=0=0.50) = N (d) = 1.00 - Value of Z for d
(In case of Two tail Z table) = N = (d) = Value of Z for d
If value of (d) is negative,
(In case of one tail Z table starting with Z=O=O.OO) = N (d) = 0.50 - Value of Z for d
(In case of one tail Z table starting with Z=0=0.50) = N (d) = Value of Z for d
(In case of Two tail Z table) = N (d) = Value of Z for d
8.1.1 Value of put option under Black & Scholes Model:
Though this model was developed for valuation of call options. However, with the help of following formula we may calculate value of put option as well.
Value of put option
+ Current Share price - Value of call option = Exercise price -----------------
ert
BS model is very popular model for valuation of options. In spite of some of unrealistic assumptions and require complicated computation, it helps us in finding out the fair value of call option. It also helps us in constitution of risk free portfolio without affected by risk perception of the investor. This model does not require the investor's expectation as to where the share price would be on the expiration of the call period.
8.1.2 Effect of dividend payment during the option period:
In above model, it has been assumed that share on which option has been created does not involve dividend during the life of the option. This may not be in practice. The share price will go down by an amount reflecting the payment of dividend, which will cause decline in the value of a call option and increase in the value of a put option. Hence B-S model need slight modification. The share price (S) will be modified as below:
Modified S = [S (as above) - DPS ]
---------
ert Where
t = time to ex-dividend date (in years)
8.1.3 Ordinary Share as an option:
Ordinary share is also one sort of call option as it has limited liability. The feature of limited liability gives a right to shareholders to default on debt.
If the firm's value is more than the payment that is due, the shareholders will make payment since they shall be left with a positive value of their equity and keep the firm. However, in reverse case, they will default and let the debt holders keep the firm. Value of Equity = Value of Call option (BS model). We can put it in the following way:
(i) Company is treated as underlying asset.
(ii) Current value of underlying asset may be taken as "S".
(iii) The cost of share may be taken as option premium.
(iv) The debt contract gives shareholders a hidden right to default on debt without any liability. Hence, they have a call option. While the debt holders are the sellers of call option.
(v) The amount of debt to be repaid is the exercise price.
(vi) Maturity of debt is the time to expiration.
8.2 Risk Neutrality
This is an alternative way to look at the option valuation. We can assume that investors are risk neutral. Therefore, for their investment in share, they would simply expect a risk free rate of return. In other words, their expected return is equal to risk free rate of return.
Current Value of option:
P × Value of option at Hs + (1-P) × Value of option at Ls
--------------.-------------------------------------------.--------------
(1 + risk free rate of return) tor ert
Where
n = time to expiration in years
P = Probability of price increase. It may be calculated with the help of following equation:
Risk Free rate of return or expected return (Rf) = (P × % increase in price) + ((1-P) × (% decrease In price) Where
(i) % decrease will be shown in minus.
(ii) Rf = risk free rate or return as given in the problem. It will be adjusted for period of option.
(iii) For calculation of present value; continuous compounding may also be used by modifying the formula.
8.3 Binomial Model for Option Valuation:
The Binomial Model for option valuation is based on the assumption that in a given time period, price of underlying asset can move to one of the two possible prices. This model also assumes that investor can borrow or lend an amount at a risk free rate of interest.
This model is based on the concept of Replicating Portfolio, which refers to a portfolio, which generates the same payoff at expiration as a call/put option.
Value of Call Option
The following steps are involved to calculate value of call option according to Binomial. Model:
v Calculate call option delta.
v To calculate borrowing /lending amount
v To calculate value of call option
v To find out arbitrage opportunity if any, and to constitute a risk free portfolio.
The calculation of borrowing/lending amount is based on strategy of replicating portfolio. Replicating portfolio with call option may be form as follows:
v Buying call option (Option)
v Buying underlying on the basis of option delta & borrowing at risk free rate. (Replicating portfolio)
Or
v Writing a call option (option)
v Selling underlying on the basis of option delta & lending at risk free rate (Replicating portfolio)
Once a replicating portfolio has decided, lending or borrowing amount can be calculated. Holding is that replicating portfolio generates same payoff at expiration after considering borrowing/lending amount as call option. Alternatively borrowing or lending amount can be calculated as follows:
B = Amount borrowed or lent
1
-------------------- × (option delta × Ls - C2)
(1 + r)n or ert
C2 = Value of call option at lower share price
To calculate value of call option
Co = Option delta x Current Market Price of underlying - B
(iii) To find out arbitrage opportunity if any, and to constitute a risk free portfolio
(a) If market value of call is greater than the value calculated as above, one can write a call option and buy underlying on the basis of option delta with the help of borrowed money at risk free rate of interest & enjoy the benefit of arbitrage. In this situation, his payoff at maturity will remain same irrespective of price of underlying at expiration.
(b) If market value of call is less than the value calculated as above, one can buy a call option and set. underlying on the basis of option delta & lend at risk free rate of interest & enjoy the benefit of arbitrage. In this situation also, his cash flow position will remain same on maturity date irrespective of price of underlying.
Value of Put Option
The following steps are involved to calculate value of put option according to Binomial Model:
v Calculate Put option delta.
v To calculate borrowing/lending amount.
v To calculate value of put option
v To find out arbitrage opportunity if any, and to constitute a risk free portfolio
The calculation of borrowing/lending amount is based on strategy of replicating portfolio. Replicating portfolio with put option may be form as follows:
v Buying put option (Option)
v Selling underlying on the basis of option delta & lending at risk free rate. (Replicating portfolio)
Or
v Writing a put option (option)
v Buying underlying on the basis of option delta & borrowing at risk free rate (Replicating portfolio)
Once a replicating portfolio has decided, lending or borrowing amount can be calculated. Holding is that replicating portfolio generates same payoff at expiration after considering borrowing/lending amount as put option. Alternatively borrowing or lending amount can be calculated as follows:
B = Amount borrowed or lent
1
---------------- × (option delta × Ls + C2)
(1 + r) or ert
(ii) To calculate value of put option
Po = B - Option delta x Current Market Price of underlying
(iii) To find out arbitrage opportunity if any, and to constitute a risk free portfolio
(a) If market value (option premium) of put is greater than the value calculated as above, one can write a put option and sell underlying on the basis of option delta & lend at risk free rate of interest & keep benefit of arbitrage. In this situation, his payoff will remain same on maturity irrespective of price of underlying.
(ii) If market value of put is less than the value calculated as above, one can buy a put option and buy underlying on the basis of option delta with the help of borrowed money at risk free rate of interest & enjoy gain from arbitrage. In this situation also, his payoff will remain same on maturity date. Hence, it not only constitutes a risk free portfolio but also gives arbitrage benefits.
The BM model is simple in its approach, however it involves excessive subjective ness. It's assumption that there are only two possibilities for share price is impracticable and hypothetical.
8.4 General Model for option valuation
Current Share price - Value of Call option + Value of Put Option =
Exercise price
------------------
ert,
9.0 Portfolio Hedging using Index Option
We have discussed that with the help of various strategies, we can hedge risk for underlying security/future contracts. An investor holding portfolio investment always worried about decline in value of portfolio due to decline in stock prices. Index options can be used to protect ones portfolio from potential downside due to a market drop.
To protect the value of a portfolio from falling below a particular level one should buy the right number of put options with the right strike price. We can put it in the following way:
(a) Calculate current value of portfolio.
(b) Calculate portfolio beta.
(c) Find out market value of index i.e. NIFTY or SENSEX.
(d) Calculate number of put options required to hedge portfolio:
Current value of portfolio x portfolio beta 1
---------------------------------------------- x -------------------
Current value of index Market lot of index
In case of NIFTY market lot is of 100 indexes & in case of SENSEX, market lot is of 50 indexes.
(e) Determine the strike price of put option on index. Strike price will depend upon, how safe investor wants to play.
For-example, if current value of NIFTY is 1600 & portfolio beta of investor is 1.25, it means that if index fall by 10%, value of portfolio will decline by 12.5%. If 12.50% fall in value of portfolio is affordable to investor, he can fix the strike price of put option on NIFTY, below 10% of current value of index i.e. 1440. In this situation, whatever decline occurs in the market, value of investors' portfolio will not decline by more than 12.50% in any case.
(v) Interest rate Derivatives.
Fluctuations in interest rate affect a firm's cash flows by affecting interest income on financial assets and interest expenses on liabilities. For undertaking fluctuations in interest rates causes corresponding fluctuations in operating earnings and rates of return on projects. Hence an effective assessment and management of interest rate exposure is essential. In interest rate derivatives, underlying is interest rate. There are number of derivative instruments which can be used for the purpose of hedging for interest rate fluctuations. For example, FRAs, Interest rate futures, Interest rate options, Swaps etc.
(a) Forward Rate Agreements
A FRA is a notionally an agreement between two parties in which one of them (the seller of the FRA), contracts to lend to the other (the buyer), a specified amount of funds, in a specific currency, for a specified period starting at a specified future date, at an interest rate fixed at the time of agreement In practice, actual lending or borrowing of the underlying principal does not take place but only the interest rate is locked in.
If the settlement rate (the rate with the which the contract rate is compared) on the settlement date (the day on which the settlement rate is determined' is above the contract rate, the seller compensates the buyer for the difference in interest on agreed upon principal amount for the duration of the period in the contract and vice versa.
The compensation is paid up front on the settlement day and therefore has too be suitably discounted since interest payment on short-term loans is at maturity of the loan.
(b) Interest Rate Options
A call option on interest rate gives the holder the right to borrow funds for a specified duration at a specified interest rate without an obligation to do so.
A put option on interest rate gives the holder the right to invest funds for a specified duration at a specified return, without an obligation to do so.
Interest rate options include Caps, Floors and Collars. On occasion, borrowers want to cap their short term, floating-rate borrowing costs. A cap is a call option, which is bought by borrower and lender becomes writer. If interest rates rise beyond some specified ceiling, the borrower exercises this call option and pays no more. It can be arranged directly with a lender or an investment bank for a price. If it is arranged through investment banker, the investment banker to the lender pays the 'excess over capped interest rate directly or via borrower.
A floor is a put option, which is written by borrower and lender, becomes holder. If interest rate falls below the floor, lender exercises this option and borrower pays the floor rate.
A Collar is a combination of a cap and a floor with variation only in middle range. The advantage of a collar is that the cost is much less than it is for a cap.
(c) Interest Rate Futures
It is one of the most successful financial innovations of 70s. The underlying asset is a debt instrument such a TB, Bond, time deposit and so on. It is used to hedge interest rate risk. For instance, a corporation planning to issue CP in the near future can use TB futures to protect itself against an increase in interest rate.
(d) Interest Rate Swaps Discussed earlier.
(vi) Credit derivatives
In the recent past, markets have developed for credit derivatives. The idea is to unbundling the default risk of a loan or security from its other attributes. The original lender no longer needs to bear this risk; it can be transferred to others for a price. The party who wishes to transfer is known as the protection buyer. The protection seller assumes the credit risk and receives a premium for providing this insurance. The premium is based on the probability and likely severity of default. For example; Forfaiting.