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Derivatives - Futures
Introduction
Overview
Derivatives have made the international and financial headlines in the past for mostly with their association with spectacular losses or institutional collapses. But market players have traded derivatives successfully for centuries and the daily international turnover in derivatives trading runs into billions of dollars.
Are derivative instruments that can only be traded by experienced, specialist traders? Although it is true that complicated mathematical models are used for pricing some derivatives, the basic concepts and principles underpinning derivatives and their trading are quite easy to grasp and understand. Indeed, derivatives are used increasingly by market players ranging from governments, corporate treasurers, dealers and brokers and individual investors.
Indian scenario
While forward contracts and exchange traded in futures has grown by leaps and bound, Indian stock markets have been largely slow to these global changes. However, in the last few years, there has been substantial improvement in the functioning of the securities market. Requirements of adequate capitalization for market intermediaries, margining and establishment of clearing corporations have reduced market and credit risks. However, there were inadequate advanced risk management tools. And after the ICE (Information, Communication, Entertainment) meltdown the market regulator felt that in order to deepen and strengthen the cash market trading of derivatives like futures and options was imperative.
Why have derivatives?
Derivatives have become very important in the field finance. They are very important financial instruments for risk management as they allow risks to be separated and traded. Derivatives are used to shift risk and act as a form of insurance. This shift of risk means that each party involved in the contract should be able to identify all the risks involved before the contract is agreed. It is also important to remember that derivatives are derived from an underlying asset. This means that risks in trading derivatives may change depending on what happens to the underlying asset.
A derivative is a product whose value is derived from the value of an underlying asset, index or reference rate. The underlying asset can be equity, forex, commodity or any other asset. For example, if the settlement price of a derivative is based on the stock price of a stock for e.g. Infosys, which frequently changes on a daily basis, then the derivative risks are also changing on a daily basis. This means that derivative risks and positions must be monitored constantly.
The purpose of this Learning Centre is to introduce the basic concepts and principles of derivatives.
We will try and understand
- What are derivatives?
- Why have derivatives at all?
- How are derivatives traded and used?
In subsequent lessons we will try and understand how exactly will an underlying asset effect the movement of a derivative instrument and how is it traded and how one can profit from these instruments.
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Options
Stock markets by their very nature are fickle. While fortunes can be made in a jiffy more often than not the scenario is the reverse. Investing in stocks has two sides to it -a) Unlimited profit potential from any upside (remember Infosys, HFCL etc) or b) a downside which could make you a pauper.
Derivative products are structured precisely for this reason -- to curtail the risk exposure of an investor. Index futures and stock options are instruments that enable you to hedge your portfolio or open positions in the market. Option contracts allow you to run your profits while restricting your downside risk.
Apart from risk containment, options can be used for speculation and investors can create a wide range of potential profit scenarios.
We have seen in the Derivatives School how index futures can be used to protect oneself from volatility or market risk. Here we will try and understand some basic concepts of options.
What are options?
Some people remain puzzled by options. The truth is that most people have been using options for some time, because options are built into everything from mortgages to insurance.
An option is a contract, which gives the buyer the right, but not the obligation to buy or sell shares of the underlying security at a specific price on or before a specific date.
'Option', as the word suggests, is a choice given to the investor to either honour the contract; or if he chooses not to walk away from the contract.
To begin, there are two kinds of options: Call Options and Put Options.
A Call Option is an option to buy a stock at a specific price on or before a certain date. In this way, Call options are like security deposits. If, for example, you wanted to rent a certain property, and left a security deposit for it, the money would be used to insure that you could, in fact, rent that property at the price agreed upon when you returned. If you never returned, you would give up your security deposit, but you would have no other liability. Call options usually increase in value as the value of the underlying instrument rises.
When you buy a Call option, the price you pay for it, called the option premium, secures your right to buy that certain stock at a specified price called the strike price. If you decide not to use the option to buy the stock, and you are not obligated to, your only cost is the option premium.
Put Options are options to sell a stock at a specific price on or before a certain date. In this way, Put options are like insurance policies
If you buy a new car, and then buy auto insurance on the car, you pay a premium and are, hence, protected if the asset is damaged in an accident. If this happens, you can use your policy to regain the insured value of the car. In this way, the put option gains in value as the value of the underlying instrument decreases. If all goes well and the insurance is not needed, the insurance company keeps your premium in return for taking on the risk.
With a Put Option, you can "insure" a stock by fixing a selling price. If something happens which causes the stock price to fall, and thus, "damages" your asset, you can exercise your option and sell it at its "insured" price level. If the price of your stock goes up, and there is no "damage," then you do not need to use the insurance, and, once again, your only cost is the premium. This is the primary function of listed options, to allow investors ways to manage risk.
Technically, an option is a contract between two parties. The buyer receives a privilege for which he pays a premium. The seller accepts an obligation for which he receives a fee.
We will dwelve further into the mechanics of call/put options in subsequent lessons.
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Types Of Options
Call option
An option is a contract between two parties giving the taker (buyer) the right, but not the obligation, to buy or sell a parcel of shares at a predetermined price possibly on, or before a predetermined date. To acquire this right the taker pays a premium to the writer (seller) of the contract.
There are two types of options:
Call options
Call options give the taker the right, but not the obligation, to buy the underlying shares at a predetermined price, on or before a predetermined date.
Illustration 1:
Raj purchases 1 Satyam Computer (SATCOM) AUG 150 Call --Premium 8
This contract allows Raj to buy 100 shares of SATCOM at Rs 150 per share at any time between the current date and the end of next August. For this privilege, Raj pays a fee of Rs 800 (Rs eight a share for 100 shares).
The buyer of a call has purchased the right to buy and for that he pays a premium.
Now let us see how one can profit from buying an option.
Sam purchases a December call option at Rs 40 for a premium of Rs 15. That is he has purchased the right to buy that share for Rs 40 in December. If the stock rises above Rs 55 (40+15) he will break even and he will start making a profit. Suppose the stock does not rise and instead falls he will choose not to exercise the option and forego the premium of Rs 15 and thus limiting his loss to Rs 15.
Let us take another example of a call option on the Nifty to understand the concept better.
Nifty is at 1310. The following are Nifty options traded at following quotes.
|
Option contract |
Strike price |
Call premium |
| Dec Nifty |
1325 |
Rs 6,000 |
| |
1345 |
Rs 2,000 |
| |
|
|
| Jan Nifty |
1325 |
Rs 4,500 |
| |
1345 |
Rs 5000 |
A trader is of the view that the index will go up to 1400 in Jan 2002 but does not want to take the risk of prices going down. Therefore, he buys 10 options of Jan contracts at 1345. He pays a premium for buying calls (the right to buy the contract) for 500*10= Rs 5,000/-.
In Jan 2002 the Nifty index goes up to 1365. He sells the options or exercises the option and takes the difference in spot index price which is (1365-1345) * 200 (market lot) = 4000 per contract. Total profit = 40,000/- (4,000*10).
He had paid Rs 5,000/- premium for buying the call option. So he earns by buying call option is Rs 35,000/- (40,000-5000).
If the index falls below 1345 the trader will not exercise his right and will opt to forego his premium of Rs 5,000. So, in the event the index falls further his loss is limited to the premium he paid upfront, but the profit potential is unlimited.
Call Options-Long & Short Positions
When you expect prices to rise, then you take a long position by buying calls. You arebullish.
When you expect prices to fall, then you take a short position by selling calls. You arebearish.
Put Options
A Put Option gives the holder of the right to sell a specific number of shares of an agreed security at a fixed price for a period of time.
eg: Sam purchases 1 INFTEC (Infosys Technologies) AUG 3500 Put --Premium 200
This contract allows Sam to sell 100 shares INFTEC at Rs 3500 per share at any time between the current date and the end of August. To have this privilege, Sam pays a premium of Rs 20,000 (Rs 200 a share for 100 shares).
The buyer of a put has purchased a right to sell. The owner of a put option has the right to sell.
Illustration 2: Raj is of the view that the a stock is overpriced and will fall in future, but he does not want to take the risk in the event of price rising so purchases a put option at Rs 70 on 'X'. By purchasing the put option Raj has the right to sell the stock at Rs 70 but he has to pay a fee of Rs 15 (premium).
So he will breakeven only after the stock falls below Rs 55 (70-15) and will start making profit if the stock falls below Rs 55.
Illustration 3:
An investor on Dec 15 is of the view that Wipro is overpriced and will fall in future but does not want to take the risk in the event the prices rise. So he purchases a Put option on Wipro.
Quotes are as under:
Spot Rs 1040
Jan Put at 1050 Rs 10
Jan Put at 1070 Rs 30
He purchases 1000 Wipro Put at strike price 1070 at Put price of Rs 30/-. He pays Rs 30,000/- as Put premium.
His position in following price position is discussed below.
- Jan Spot price of Wipro = 1020
- Jan Spot price of Wipro = 1080
In the first situation the investor is having the right to sell 1000 Wipro shares at Rs 1,070/- the price of which is Rs 1020/-. By exercising the option he earns Rs (1070-1020) = Rs 50 per Put, which totals Rs 50,000/-. His net income is Rs (50000-30000) = Rs 20,000.
In the second price situation, the price is more in the spot market, so the investor will not sell at a lower price by exercising the Put. He will have to allow the Put option to expire unexercised. He looses the premium paid Rs 30,000.
Put Options-Long & Short Positions
When you expect prices to fall, then you take a long position by buying Puts. You are bearish.
When you expect prices to rise, then you take a short position by selling Puts. You are bullish.
| |
CALL OPTIONS |
PUT OPTIONS |
| If you expect a fall in price(Bearish) |
Short |
Long |
| If you expect a rise in price (Bullish) |
Long |
Short |
SUMMARY:
|
CALL OPTION BUYER |
CALL OPTION WRITER (Seller) |
- Pays premium
- Right to exercise and buy the shares
- Profits from rising prices
- Limited losses, Potentially unlimited gain
|
- Receives premium
- Obligation to sell shares if exercised
- Profits from falling prices or remaining neutral
- Potentially unlimited losses, limited gain
|
|
PUT OPTION BUYER |
PUT OPTION WRITER (Seller) |
- Pays premium
- Right to exercise and sell shares
- Profits from falling prices
- Limited losses, Potentially unlimited gain
|
- Receives premium
- Obligation to buy shares if exercised
- Profits from rising prices or remaining neutral
- Potentially unlimited losses, limited gain
|
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Concepts
Option styles
Settlement of options is based on the expiry date. However, there are three basic styles of options you will encounter which affect settlement. The styles have geographical names, which have nothing to do with the location where a contract is agreed! The styles are:
European: These options give the holder the right, but not the obligation, to buy or sell the underlying instrument only on the expiry date. This means that the option cannot be exercised early. Settlement is based on a particular strike price at expiration. Currently, in India only index options are European in nature.
eg: Sam purchases 1 NIFTY AUG 1110 Call --Premium 20. The exchange will settle the contract on the last Thursday of August. Since there are no shares for the underlying, the contract is cash settled.
American: These options give the holder the right, but not the obligation, to buy or sell the underlying instrument on or before the expiry date. This means that the option can be exercised early. Settlement is based on a particular strike price at expiration.
Options in stocks that have been recently launched in the Indian market are "American Options".
eg: Sam purchases 1 ACC SEP 145 Call --Premium 12
Here Sam can close the contract any time from the current date till the expiration date, which is the last Thursday of September.
American style options tend to be more expensive than European style because they offer greater flexibility to the buyer.
Option Class & Series
Generally, for each underlying, there are a number of options available: For this reason, we have the terms "class" and "series".
An option "class" refers to all options of the same type (call or put) and style (American or European) that also have the same underlying.
eg: All Nifty call options are referred to as one class.
An option series refers to all options that are identical: they are the same type, have the same underlying, the same expiration date and the same exercise price.
|
Calls |
Puts |
| . |
JUL |
AUG |
SEP |
JUL |
AUG |
SEP |
| Wipro |
|
| 1300 |
45 |
60 |
75 |
15 |
20 |
28 |
| 1400 |
35 |
45 |
65 |
25 |
28 |
35 |
| 1500 |
20 |
42 |
48 |
30 |
40 |
55 |
eg: Wipro JUL 1300 refers to one series and trades take place at different
premiums
All calls are of the same option type. Similarly, all puts are of the same option type. Options of the same type that are also in the same class are said to be of the same class. Options of the same class and with the same exercise price and the same expiration date are said to be of the same series.
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Important Terms
(Strike price, In-the-money, Out-of-the-Money, At-the-Money, Covered call and Covered Put)
Strike price: The Strike Price denotes the price at which the buyer of the option has a right to purchase or sell the underlying. Five different strike prices will be available at any point of time. The strike price interval will be of 20. If the index is currently at 1,410, the strike prices available will be 1,370, 1,390, 1,410, 1,430, 1,450. The strike price is also called Exercise Price. This price is fixed by the exchange for the entire duration of the option depending on the movement of the underlying stock or index in the cash market.
In-the-money: A Call Option is said to be "In-the-Money" if the strike price is less than the market price of the underlying stock. A Put Option is In-The-Money when the strike price is greater than the market price.
eg: Raj purchases 1 SATCOM AUG 190 Call --Premium 10
In the above example, the option is "in-the-money", till the market price of SATCOM is ruling above the strike price of Rs 190, which is the price at which Raj would like to buy 100 shares anytime before the end of August.
Similary, if Raj had purchased a Put at the same strike price, the option would have been "in-the- money", if the market price of SATCOM was lower than Rs 190 per share.
Out-of-the-Money: A Call Option is said to be "Out-of-the-Money" if the strike price is greater than the market price of the stock. A Put option is Out-Of-Money if the strike price is less than the market price.
eg: Sam purchases 1 INFTEC AUG 3500 Call --Premium 150
In the above example, the option is "out-of- the- money", if the market price of INFTEC is ruling below the strike price of Rs 3500, which is the price at which SAM would like to buy 100 shares anytime before the end of August.
Similary, if Sam had purchased a Put at the same strike price, the option would have been "out-of-the-money", if the market price of INFTEC was above Rs 3500 per share.
At-the-Money: The option with strike price equal to that of the market price of the stock is considered as being "At-the-Money" or Near-the-Money.
eg: Raj purchases 1 ACC AUG 150 Call or Put--Premium 10
In the above case, if the market price of ACC is ruling at Rs 150, which is equal to the strike price, then the option is said to be "at-the-money".
If the index is currently at 1,410, the strike prices available will be 1,370, 1,390, 1,410, 1,430, 1,450. The strike prices for a call option that are greater than the underlying (Nifty or Sensex) are said to be out-of-the-money in this case 1430 and 1450 considering that the underlying is at 1410. Similarly in-the-money strike prices will be 1,370 and 1,390, which are lower than the underlying of 1,410.
At these prices one can take either a positive or negative view on the markets i.e. both call and put options will be available. Therefore, for a single series 10 options (5 calls and 5 puts) will be available and considering that there are three series a total number of 30 options will be available to take positions in.
Covered Call Option
Covered option helps the writer to minimize his loss. In a covered call option, the writer of the call option takes a corresponding long position in the stock in the cash market; this will cover his loss in his option position if there is a sharp increase in price of the stock. Further, he is able to bring down his average cost of acquisition in the cash market (which will be the cost of acquisition less the option premium collected).
eg: Raj believes that HLL has hit rock bottom at the level of Rs.182 and it will move in a narrow range. He can take a long position in HLL shares and at the same time write a call option with a strike price of 185 and collect a premium of Rs.5 per share. This will bring down the effective cost of HLL shares to 177 (182-5). If the price stays below 185 till expiry, the call option will not be exercised and the writer will keep the Rs.5 he collected as premium. If the price goes above 185 and the Option is exercised, the writer can deliver the shares acquired in the cash market.
Covered Put Option
Similarly, a writer of a Put Option can create a covered position by selling the underlying security (if it is already owned). The effective selling price will increase by the premium amount (if the option is not exercised at maturity). Here again, the investor is not in a position to take advantage of any sharp increase in the price of the asset as the underlying asset has already been sold. If there is a sharp decline in the price of the underlying asset, the option will be exercised and the investor will be left only with the premium amount. The loss in the option exercised will be equal to the gain in the short position of the asset.
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Pricing of options
Options are used as risk management tools and the valuation or pricing of the instruments is a careful balance of market factors.
There are four major factors affecting the Option premium:
- Price of Underlying
- Time to Expiry
- Exercise Price Time to Maturity
- Volatility of the Underlying
And two less important factors:
- Short-Term Interest Rates
- Dividends
Review of Options Pricing Factors
The Intrinsic Value of an Option
The intrinsic value of an option is defined as the amount by which an option is in-the-money, or the immediate exercise value of the option when the underlying position is marked-to-market.
For a call option: Intrinsic Value = Spot Price - Strike Price
For a put option: Intrinsic Value = Strike Price - Spot Price
The intrinsic value of an option must be positive or zero. It cannot be negative. For a call option, the strike price must be less than the price of the underlying asset for the call to have an intrinsic value greater than 0. For a put option, the strike price must be greater than the underlying asset price for it to have intrinsic value.
Price of underlying
The premium is affected by the price movements in the underlying
instrument. For Call options - the right to buy the underlying at a fixed strike
price - as the underlying price rises so does its premium. As the underlying price falls so does the cost of the option premium. For Put options - the right to sell the underlying at a fixed strike
price - as the underlying price rises, the premium falls; as the underlying price falls the premium cost rises.
The following chart summarises the above for Calls and Puts.
The Time Value of an Option
Generally, the longer the time remaining until an option's expiration, the higher its premium will be. This is because the longer an option's lifetime, greater is the possibility that the underlying share price might move so as to make the option in-the-money. All other factors affecting an option's price remaining the same, the time value portion of an option's premium will decrease (or decay) with the passage of time.
Note: This time decay increases rapidly in the last several weeks of an option's life. When an option expires in-the-money, it is generally worth only its intrinsic value.
Volatility
Volatility is the tendency of the underlying security's market price to fluctuate either up or down. It reflects a price change's magnitude; it does not imply a bias toward price movement in one direction or the other. Thus, it is a major factor in determining an option's premium. The higher the volatility of the underlying stock, the higher the premium because there is a greater possibility that the option will move in-the-money. Generally, as the volatility of an under-lying stock increases, the premiums of both calls and puts overlying that stock increase, and vice versa.
Higher volatility=Higher premium
Lower volatility = Lower premium
Interest rates
In general interest rates have the least influence on options and equate approximately to the cost of carry of a futures contract. If the size of the options contract is very large, then this factor may take on
some importance. All other factors being equal as interest rates rise, premium costs fall and vice versa. The relationship can be thought of as an opportunity cost. In order to buy an option, the buyer must either borrow funds or use funds on deposit. Either way the buyer incurs an interest rate cost. If interest rates are rising, then the opportunity cost of buying options increases and to compensate the buyer premium costs fall. Why should the buyer be compensated? Because the option writer receiving the premium can place the funds on deposit and receive more interest than was previously anticipated. The situation is reversed when interest rates fall - premiums rise. This time it is the writer who needs to be compensated.
How do we measure the impact of change in each of these pricing determinants on option premium we shall learn in the next module.
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Greeks
The options premium is determined by the three factors mentioned earlier - intrinsic value, time value and volatility. But there are more sophisticated tools used to measure the potential variations of options premiums. They are as follows:
Delta
Delta is the measure of an option's sensitivity to changes in the price of the underlying asset. Therefore, its is the degree to which an option price will move given a change in the underlying stock or index price, all else being equal.
Change in option premium
Delta = --------------------------------
Change in underlying price
For example, an option with a delta of 0.5 will move Rs 5 for every change of Rs 10 in the underlying stock or index.
Illustration:
A trader is considering buying a Call option on a futures contract, which has a price of Rs 19. The premium for the Call option with a strike price of Rs 19 is 0.80. The delta for this option is +0.5. This means that if the price of the underlying futures contract rises to Rs 20 - a rise of Re 1 - then the premium will increase by 0.5 x 1.00 = 0.50. The new option premium will be 0.80 + 0.50 = Rs 1.30.
Far out-of-the-money calls will have a delta very close to zero, as the change in underlying price is not likely to make them valuable or cheap. An at-the-money call would have a delta of 0.5 and a deeply in-the-money call would have a delta close to 1.
While Call deltas are positive, Put deltas are negative, reflecting the fact that the put option price and the underlying stock price are inversely related. This is because if you buy a put your view is bearish and expect the stock price to go down. However, if the stock price moves up it is contrary to your view therefore, the value of the option decreases. The put delta equals the call delta minus 1.
It may be noted that if delta of your position is positive, you desire the underlying asset to rise in price. On the contrary, if delta is negative, you want the underlying asset's price to fall.
Uses: The knowledge of delta is of vital importance for option traders because this parameter is heavily used in margining and risk management strategies. The delta is often called the hedge ratio. e.g. if you have a portfolio of 'n' shares of a stock then 'n' divided by the delta gives you the number of calls you would need to be short (i.e. need to write) to create a riskless hedge - i.e. a portfolio which would be worth the same whether the stock price rose by a very small amount or fell by a very small amount.
In such a "delta neutral" portfolio any gain in the value of the shares held due to a rise in the share price would be exactly offset by a loss on the value of the calls written, and vice versa.
Note that as the delta changes with the stock price and time to expiration the number of shares would need to be continually adjusted to maintain the hedge. How quickly the delta changes with the stock price is given by gamma, which we shall learn subsequently.
Gamma
This is the rate at which the delta value of an option increases or decreases as a result of a move in the price of the underlying instrument.
Change in an option delta
Gamma =-------------------------------------
Change in underlying price
For example, if a Call option has a delta of 0.50 and a gamma of 0.05, then a rise of ±1 in the underlying means the delta will move to 0.55 for a price rise and 0.45 for a price fall. Gamma is rather like the rate of change in the speed of a car - its acceleration - in moving from a standstill, up to its cruising speed, and braking back to a standstill. Gamma is greatest for an ATM (at-the-money) option (cruising) and falls to zero as an option moves deeply ITM (in-the-money ) and OTM (out-of-the-money) (standstill).
If you are hedging a portfolio using the delta-hedge technique described under "Delta", then you will want to keep gamma as small as possible as the smaller it is the less often you will have to adjust the hedge to maintain a delta neutral position. If gamma is too large a small change in stock price could wreck your hedge. Adjusting gamma, however, can be tricky and is generally done using options -- unlike delta, it can't be done by buying or selling the underlying asset as the gamma of the underlying asset is, by definition, always zero so more or less of it won't affect the gamma of the total portfolio.
Theta
It is a measure of an option's sensitivity to time decay. Theta is the change in option price given a one-day decrease in time to expiration. It is a measure of time decay (or time shrunk). Theta is generally used to gain an idea of how time decay is affecting your portfolio.
Change in an option premium
Theta = --------------------------------------
Change in time to expiry
Theta is usually negative for an option as with a decrease in time, the option value decreases. This is due to the fact that the uncertainty element in the price decreases.
Assume an option has a premium of 3 and a theta of 0.06. After one day it will decline to 2.94, the second day to 2.88 and so on. Naturally other factors, such as changes in value of the underlying stock will alter the premium. Theta is only concerned with the time value. Unfortunately, we cannot predict with accuracy the change's in stock market's value, but we can measure exactly the time remaining until expiration.
Vega
This is a measure of the sensitivity of an option price to changes in market volatility. It is the change of an option premium for a given change - typically 1% - in the underlying volatility.
Change in an option premium
Vega = -----------------------------------------
Change in volatility
If for example, XYZ stock has a volatility factor of 30% and the current premium is 3, a vega of .08 would indicate that the premium would increase to 3.08 if the volatility factor increased by 1% to 31%. As the stock becomes more volatile the changes in premium will increase in the same proportion. Vega measures the sensitivity of the premium to these changes in volatility.
What practical use is the vega to a trader? If a trader maintains a delta neutral position, then it is possible to trade options purely in terms of volatility - the trader is not exposed to changes in underlying prices.
Rho
The change in option price given a one percentage point change in the risk-free interest rate. Rho measures the change in an option's price per unit increase -typically 1% - in the cost of funding the underlying.
Change in an option premium
Rho = ---------------------------------------------------
Change in cost of funding underlying
Example:
Assume the value of Rho is 14.10. If the risk free interest rates go up by 1% the price of the option will move by Rs 0.14109. To put this in another way: if the risk-free interest rate changes by a small amount, then the option value should change by 14.10 times that amount. For example, if the risk-free interest rate increased by 0.01 (from 10% to 11%), the option value would change by 14.10*0.01 = 0.14. For a put option the relationship is inverse. If the interest rate goes up the option value decreases and therefore, Rho for a put option is negative. In general Rho tends to be small except for long-dated options.
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Options Pricing Models
There are various option pricing models which traders use to arrive at the right value of the option. Some of the most popular models have been enumerated below.
The Binomial Pricing Model
The binomial model is an options pricing model which was developed by William Sharpe in 1978. Today, one finds a large variety of pricing models which differ according to their hypotheses or the underlying instruments upon which they are based (stock options, currency options, options on interest rates).
The Black & Scholes Model
The Black & Scholes model was published in 1973 by Fisher Black and Myron Scholes. It is one of the most popular options pricing models. It is noted for its relative simplicity and its fast mode of calculation: unlike the binomial model, it does not rely on calculation by iteration.
The intention of this section is to introduce you to the basic premises upon which this pricing model rests. A complete coverage of this topic is material for an advanced course
The Black-Scholes model is used to calculate a theoretical call price (ignoring dividends paid during the life of the option) using the five key determinants of an option's price: stock price, strike price, volatility, time to expiration, and short-term (risk free) interest rate.
The original formula for calculating the theoretical option price (OP) is as follows:
Where:
The variables are:
S = stock price
X = strike price
t = time remaining until expiration, expressed as a percent of a year
r = current continuously compounded risk-free interest rate
v = annual volatility of stock price (the standard deviation of the short-term returns over one year).
ln = natural logarithm
N(x) = standard normal cumulative distribution function
e = the exponential function
Lognormal distribution: The model is based on a lognormal distribution of stock prices, as opposed to a normal, or bell-shaped, distribution. The lognormal distribution allows for a stock price distribution of between zero and infinity (ie no negative prices) and has an upward bias (representing the fact that a stock price can only drop 100 per cent but can rise by more than 100 per cent).
Risk-neutral valuation: The expected rate of return of the stock (ie the expected rate of growth of the underlying asset which equals the risk free rate plus a risk premium) is not one of the variables in the Black-Scholes model (or any other model for option valuation). The important implication is that the price of an option is completely independent of the expected growth of the underlying asset. Thus, while any two investors may strongly disagree on the rate of return they expect on a stock they will, given agreement to the assumptions of volatility and the risk free rate, always agree on the fair price of the option on that underlying asset.
The key concept underlying the valuation of all derivatives -- the fact that price of an option is independent of the risk preferences of investors -- is called risk-neutral valuation. It means that all derivatives can be valued by assuming that the return from their underlying assets is the risk free rate.
Limitation: Dividends are ignored in the basic Black-Scholes formula, but there are a number of widely used adaptations to the original formula, which I use in my models, which enable it to handle both discrete and continuous dividends accurately.
However, despite these adaptations the Black-Scholes model has one major limitation: it cannot be used to accurately price options with an American-style exercise as it only calculates the option price at one point in time -- at expiration. It does not consider the steps along the way where there could be the possibility of early exercise of an American option.
As all exchange traded equity options have American-style exercise (ie they can be exercised at any time as opposed to European options which can only be exercised at expiration) this is a significant limitation.
The exception to this is an American call on a non-dividend paying asset. In this case the call is always worth the same as its European equivalent as there is never any advantage in exercising early.
Advantage: The main advantage of the Black-Scholes model is speed -- it lets you calculate a very large number of option prices in a very short time. Since, high accuracy is not critical for American option pricing (eg when animating a chart to show the effects of time decay) using Black-Scholes is a good option. But, the option of using the binomial model is also advisable for the relatively few pricing and profitability numbers where accuracy may be important and speed is irrelevant. You can experiment with the Black-Scholes model using on-line options pricing calculator.
The Binomial Model
The binomial model breaks down the time to expiration into potentially a very large number of time intervals, or steps. A tree of stock prices is initially produced working forward from the present to expiration. At each step it is assumed that the stock price will move up or down by an amount calculated using volatility and time to expiration. This produces a binomial distribution, or recombining tree, of underlying stock prices. The tree represents all the possible paths that the stock price could take during the life of the option.
At the end of the tree -- ie at expiration of the option -- all the terminal option prices for each of the final possible stock prices are known as they simply equal their intrinsic values.
Next the option prices at each step of the tree are calculated working back from expiration to the present. The option prices at each step are used to derive the option prices at the next step of the tree using risk neutral valuation based on the probabilities of the stock prices moving up or down, the risk free rate and the time interval of each step. Any adjustments to stock prices (at an ex-dividend date) or option prices (as a result of early exercise of American options) are worked into the calculations at the required point in time. At the top of the tree you are left with one option price.
Advantage: The big advantage the binomial model has over the Black-Scholes model is that it can be used to accurately price American options. This is because, with the binomial model it's possible to check at every point in an option's life (ie at every step of the binomial tree) for the possibility of early exercise (eg where, due to eg a dividend, or a put being deeply in the money the option price at that point is less than the its intrinsic value).
Where an early exercise point is found it is assumed that the option holder would elect to exercise and the option price can be adjusted to equal the intrinsic value at that point. This then flows into the calculations higher up the tree and so on.
Limitation: As mentioned before the main disadvantage of the binomial model is its relatively slow speed. It's great for half a dozen calculations at a time but even with today's fastest PCs it's not a practical solution for the calculation of thousands of prices in a few seconds which is what's required for the production of the animated charts in my strategy evaluation model
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Trading Strategies
Bull Marker Strategies
Calls in a Bullish Strategy
An investor with a bullish market outlook should buy call options. If you expect the market price of the underlying asset to rise, then you would rather have the right to purchase at a specified price and sell later at a higher price than have the obligation to deliver later at a higher price.
The investor's profit potential buying a call option is unlimited. The investor's profit is the the market price less the exercise price less the premium. The greater the increase in price of the underlying, the greater the investor's profit.
The investor's potential loss is limited. Even if the market takes a drastic decline in price levels, the holder of a call is under no obligation to exercise the option. He may let the option expire worthless.
The investor breaks even when the market price equals the exercise price plus the premium.
An increase in volatility will increase the value of your call and increase your return. Because of the increased likelihood that the option will become in- the-money, an increase in the underlying volatility (before expiration), will increase the value of a long options position. As an option holder, your return will also increase.
A simple example will illustrate the above:
Suppose there is a call option with a strike price of Rs 2000 and the option premium is Rs 100. The option will be exercised only if the value of the underlying is greater than Rs 2000 (the strike price). If the buyer exercises the call at Rs 2200 then his gain will be Rs 200. However, this would not be his actual gain for that he will have to deduct the Rs 200 (premium) he has paid.
The profit can be derived as follows
Profit = Market price - Exercise price - Premium
Profit = Market price - Strike price - Premium.
2200 - 2000 - 100 = Rs 100
Puts in a Bullish Strategy
An investor with a bullish market outlook can also go short on a Put option. Basically, an investor anticipating a bull market could write Put options. If the market price increases and puts become out-of-the-money, investors with long put positions will let their options expire worthless.
By writing Puts, profit potential is limited. A Put writer profits when the price of the underlying asset increases and the option expires worthless. The maximum profit is limited to the premium received.
However, the potential loss is unlimited. Because a short put position holder has an obligation to purchase if exercised. He will be exposed to potentially large losses if the market moves against his position and declines.
The break-even point occurs when the market price equals the exercise price: minus the premium. At any price less than the exercise price minus the premium, the investor loses money on the transaction. At higher prices, his option is profitable.
An increase in volatility will increase the value of your put and decrease your return. As an option writer, the higher price you will be forced to pay in order to buy back the option at a later date , lower is the return.
Bullish Call Spread Strategies
A vertical call spread is the simultaneous purchase and sale of identical call options but with different exercise prices.
To "buy a call spread" is to purchase a call with a lower exercise price and to write a call with a higher exercise price. The trader pays a net premium for the position.
To "sell a call spread" is the opposite, here the trader buys a call with a higher exercise price and writes a call with a lower exercise price, receiving a net premium for the position.
An investor with a bullish market outlook should buy a call spread. The "Bull Call Spread" allows the investor to participate to a limited extent in a bull market, while at the same time limiting risk exposure.
To put on a bull spread, the trader needs to buy the lower strike call and sell the higher strike call. The combination of these two options will result in a bought spread. The cost of Putting on this position will be the difference between the premium paid for the low strike call and the premium received for the high strike call.
The investor's profit potential is limited. When both calls are in-the-money, both will be exercised and the maximum profit will be realised. The investor delivers on his short call and receives a higher price than he is paid for receiving delivery on his long call.
The investors's potential loss is limited. At the most, the investor can lose is the net premium. He pays a higher premium for the lower exercise price call than he receives for writing the higher exercise price call.
The investor breaks even when the market price equals the lower exercise price plus the net premium. At the most, an investor can lose is the net premium paid. To recover the premium, the market price must be as great as the lower exercise price plus the net premium.
An example of a Bullish call spread:
Let's assume that the cash price of a scrip is Rs 100 and you buy a November call option with a strike price of Rs 90 and pay a premium of Rs 14. At the same time you sell another November call option on a scrip with a strike price of Rs 110 and receive a premium of Rs 4. Here you are buying a lower strike price option and selling a higher strike price option. This would result in a net outflow of Rs 10 at the time of establishing the spread.
Now let us look at the fundamental reason for this position. Since this is a bullish strategy, the first position established in the spread is the long lower strike price call option with unlimited profit potential. At the same time to reduce the cost of puchase of the long position a short position at a higher call strike price is established. While this not only reduces the outflow in terms of premium but his profit potential as well as risk is limited. Based on the above figures the maximum profit, maximum loss and breakeven point of this spread would be as follows:
Maximum profit = Higher strike price - Lower strike price - Net premium paid
= 110 - 90 - 10 = 10
Maximum Loss = Lower strike premium - Higher strike premium
= 14 - 4 = 10
Breakeven Price = Lower strike price + Net premium paid
= 90 + 10 = 100
Bullish Put Spread Strategies
A vertical Put spread is the simultaneous purchase and sale of identical Put options but with different exercise prices.
To "buy a put spread" is to purchase a Put with a higher exercise price and to write a Put with a lower exercise price. The trader pays a net premium for the position.
To "sell a put spread" is the opposite: the trader buys a Put with a lower exercise price and writes a put with a higher exercise price, receiving a net premium for the position.
An investor with a bullish market outlook should sell a Put spread. The "vertical bull put spread" allows the investor to participate to a limited extent in a bull market, while at the same time limiting risk exposure.
To put on a bull spread, a trader sells the higher strike put and buys the lower strike put.
The bull spread can be created by buying the lower strike and selling the higher strike of either calls or put. The difference between the premiums paid and received makes up one leg of the spread.
The investor's profit potential is limited. When the market price reaches or exceeds the higher exercise price, both options will be out-of-the-money and will expire worthless. The trader will realize his maximum profit, the net premium
The investor's potential loss is also limited. If the market falls, the options will be in-the-money. The puts will offset one another, but at different exercise prices.
The investor breaks-even when the market price equals the lower exercise price less the net premium. The investor achieves maximum profit i.e the premium received, when the market price moves up beyond the higher exercise price (both puts are then worthless).
An example of a bullish put spread.
Lets us assume that the cash price of the scrip is Rs 100. You now buy a November put option on a scrip with a strike price of Rs 90 at a premium of Rs 5 and sell a put option with a strike price of Rs 110 at a premium of Rs 15.
The first position is a short put at a higher strike price. This has resulted in some inflow in terms of premium. But here the trader is worried about risk and so caps his risk by buying another put option at the lower strike price. As such, a part of the premium received goes off and the ultimate position has limited risk and limited profit potential. Based on the above figures the maximum profit, maximum loss and breakeven point of this spread would be as follows:
Maximum profit = Net option premium income or net credit
= 15 - 5 = 10
Maximum loss = Higher strike price - Lower strike price - Net premium received
= 110 - 90 - 10 = 10
Breakeven Price = Higher Strike price - Net premium income
= 110 - 10 = 100
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Bear Market Strategies
Puts in a Bearish Strategy
When you purchase a put you are long and want the market to fall. A put option is a bearish position. It will increase in value if the market falls. An investor with a bearish market outlook shall buy put options. By purchasing put options, the trader has the right to choose whether to sell the underlying asset at the exercise price. In a falling market, this choice is preferable to being obligated to buy the underlying at a price higher.
An investor's profit potential is practically unlimited. The higher the fall in price of the underlying asset, higher the profits.
The investor's potential loss is limited. If the price of the underlying asset rises instead of falling as the investor has anticipated, he may let the option expire worthless. At the most, he may lose the premium for the option.
The trader's breakeven point is the exercise price minus the premium. To profit, the market price must be below the exercise price. Since the trader has paid a premium he must recover the premium he paid for the option.
An increase in volatility will increase the value of your put and increase your return. An increase in volatility will make it more likely that the price of the underlying instrument will move. This increases the value of the option.
Calls in a Bearish Strategy
Another option for a bearish investor is to go short on a call with the intent to purchase it back in the future. By selling a call, you have a net short position and needs to be bought back before expiration and cancel out your position.
For this an investor needs to write a call option. If the market price falls, long call holders will let their out-of-the-money options expire worthless, because they could purchase the underlying asset at the lower market price.
