Showing posts with label options-trading-for-dummies. Show all posts
Showing posts with label options-trading-for-dummies. Show all posts

Introduction

As important as knowing the chances of a LEAP put winding up in the money is knowing whether the option in question is over- or underpriced. When it comes to long-term options, market makers use one of several analytic formulas to calculate option prices. This is because the computational complexity associated with multiperiod binomial models increases dramatically with the number of steps involved, thus restricting their applicability to standard, short-term option pricing.

Depending on the sophistication of the market maker, option pricing is typically determined either by straightforward use of the original pricing formula developed by Fischer Black and Myron Scholes in 1973 or by the use of later extensions of this formula by Robert Merton, Giovanni Barone-Adesi, and Robert Whaley. In a nutshell, the Black-Scholes model determines option prices for European-style options, ignoring the effects of dividends. The Merton model adjusts the Black-Scholes formula to take dividends into account; the Barone-Adesi and Whaley model adjusts the Black-Scholes-Merton formula for the early exercise potential present in American-style options.

The Black-Scholes Formula for Puts

The Black-Scholes formula for the put premium for European-style options with a zero dividend rate is given by:

where:

P is the price of the underlying issue
S is the strike (exercise price) of the option
r is the risk-free interest rate in decimal form
t is the time in years till expiration
v is the annualized volatility in decimal form
and h is given by the formula:

When t approaches zero, h takes on one of two values:

1. if the stock price P equals or is greater than the strike price S
2. if P is less than S, the result of which shows that the value of the put goes to zero if P equals or is greater than S, or it approaches S - P if P is less than S.

The integrals are the standard normal distribution, and the same curve-fitting formula that can be used for their evaluation. Note that the right-hand integral is the probability that the put will wind up in the money.

The Black-Scholes Formula for Calls

For completeness, the corresponding Black-Scholes formula for the call premium is given by:

where P, S, r, t, and h are as defined earlier for the put formula. If the put price has already been calculated, one can use the shortcut formula (called the put-call parity formula) for the call, as follows:

The Effect of Dividends on Option Premiums

To keep things simple and conservative, all put premiums used in conducting the ten-year economic simulations were calculated using a zero dividend rate. The effect of dividends is to lower the price of calls and to increase the price of puts. If I were to assume that dividends were paid continuously through the year (rather than in the usual quarterly distribution), the Merton variation of the Black-Scholes pricing formula can be used. This formula takes the following form for puts:

where Î´ is the annual dividend rate, and h is modified as follows:

The right-hand integral is again the probability that the put will wind up in the money and is the one adopted for that purpose.

Numerical Example

The effect that dividends have on option premiums, consider the case where the strike price and stock price are both \$100, the volatility is 0.35, the risk-free interest rate is 6 percent, there are 24 months till expiration, and the annual dividend rate is 2 percent. Applying the basic Black-Scholes formula yields a put premium for the zero-dividend case of \$13.314. For the Merton variation, do the following:

Since S = P = \$100, the parameter h is calculated as

Therefore, -h is -0.409112, and is -0.409112 + 0.4949748, or 0.0858628. From tables of the normal distribution or through the use of the curve-fitting formulas, we have N(-0.409112) = 1 - N(0.409112) = 0.341238 and N(0.0858628) = 0.534223. When the time to expiration is 2 years, e-0.02t is 0.960790 and e-0.06t is 0.886920. So from Equation the European put with its 2 percent annual dividend rate is calculated as

Thus, the annual 2 percent dividend has increased the price of the put by about \$1.28 over the two-year period involved, thus showing that the effect of dividends on option pricing can indeed be significant.The pricing of American options is so complex that market makers often restrict their pricing formulas to that of Black-Scholes or one of its simpler variations. The pricing of American-style options is examined.

Implied Volatility

Now that you've seen how to calculate the LEAP put premium for European-style options using the Black-Scholes-Merton formulas, the question arises as to whether one can work the process backwards and determine volatility if the premium is supplied. The answer to this question is both yes and no. To begin, I first note that it is not possible to solve for the volatility directly. However, the volatility can be determined on a trial-and-error basis using a Newton-Raphson iterative search technique. A relatively short BASIC program for doing this follows. It is then illustrated by two numerical examples.

Numerical Examples

As examples of the results obtainable using the computer program, consider the two cases investigated earlier:

Stock Price: \$100
Strike Price: \$100
Risk-Free Interest Rate (e.g., .06): .06
Annual Dividend Rate (e.g., .02): 0
Time to Expiration in Months: 24

No.                            VOLD                           VNEW                         F
1                               0.50000                         0.38384                         7.4605
2                               0.38384                         0.35707                         1.6934
3                               0.35707                         0.35144                         0.3544
4                               0.35144                         0.35029                         0.0723
5                               0.35029                         0.35005                         0.0147
6                               0.35005                         0.35001                         0.0030

Implied Volatility: 0.35003
Stock Price: \$100
Strike Price: \$100
Risk-Free Interest Rate (e.g., .06): .06
Annual Dividend Rate (e.g., .02): .02
Time to Expiration in Months: 24

No                            .VOLD                          VNEW                              F
1                               0.50000                        0.38456                              7.3947
2                               0.38456                        0.35755                              1.7191
3                               0.35755                        0.35163                              0.3758
4                               0.35163                        0.35036                              0.0808
5                               0.35036                        0.35008                              0.0173
6                               0.35008                        0.35003                              0.0037

Implied Volatility: 0.35005

At first blush, this numerical method for determining volatilities from premiums looks like a great shortcut to the laborious process of calculating them from historical price information. All one has to do is to obtain the premium quoted, say, for a LEAP put, and then use the computer program to determine the volatility. The trouble with this is that it is not ordinarily possible to determine what pricing model the market maker used in the first place to arrive at the option premium quoted. The market maker may have used Black-Scholes, or the Merton variation, or a multiperiod binomial model, or even a more sophisticated model for the pricing of American-style options.

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## INTRODUCTION TO OPTIONS

An option is a contract written by a seller that conveys to the buyer the right — but not the obligation — to buy (in the case of a call option) or to sell (in the case of a put option) a particular asset, at a particular price (Strike price / Exercise price) in future. In return for granting the option, the seller collects a payment (the premium) from the buyer. Exchange- traded options form an important class of options which have standardized contract features and trade on public exchanges, facilitating trading among large number of investors. They provide settlement guarantee by the Clearing Corporation thereby reducing counter party risk. Options can be used for hedging, taking a view on the future direction of the market, for arbitrage or for implementing strategies which can help in generating income for investors under various market conditions

### OPTION TERMINOLOGY

#### • Index options:

These options have the index as the underlying. In India, they have a European style settlement. Eg. Nifty options, Mini Nifty options etc.

#### • Stock options:

Stock options are options on individual stocks. A stock option contract gives the holder the right to buy or sell the underlying shares at the specified price. They have an American style settlement.

#### • Buyer of an option:

The buyer of an option is the one who by paying the option premium buys the right but not the obligation to exercise his option on the seller/writer.

#### • Writer / seller of an option:

The writer / seller of a call/put option is the one who receives the option premium and is thereby obliged to sell/buy the asset if the buyer exercises on him.

#### • Call option:

A call option gives the holder the right but not the obligation to buy an asset by a certain date for a certain price.

#### • Put option:

A put option gives the holder the right but not the obligation to sell an asset by a certain date for a certain price.

Option price is the price which the option buyer pays to the option seller. It is also referred to as the option premium.

#### • Expiration date:

The date specified in the options contract is known as the expiration date, the exercise date, the strike date or the maturity.

#### • Strike price:

The price specified in the options contract is known as the strike price or the exercise price.

#### • American options:

American options are options that can be exercised at any time upto the expiration date.

#### • European options:

European options are options that can be exercised only on the expiration date itself.

#### • In-the-money option:

An in-the-money (ITM) option is an option that would lead to a positive cash flow to the holder if it were exercised immediately. A call option on the index is said to be in-the-money when the current index stands at a level higher than the strike
price (i.e. spot price > strike price). If the index is much higher than the strike price, the call is said to be deep ITM. In the case of a put, the put is ITM if the index is below the strike price.

#### • At-the-money option:

An at-the-money (ATM) option is an option that would lead to zero cashflow if it were exercised immediately. An option on the index is at-the-money when the
current index equals the strike price (i.e. spot price = strike price).

#### • Out-of-the-money option:

An out-of-the-money (OTM) option is an option that would lead to a negative cash flow if it were exercised immediately. A call option on the index is out-of-the-money when the current index stands at a level which is less than the strike
price (i.e. spot price < strike price). If the index is much lower than the strike price, the call is said to be deep OTM. In the case of a put, the put is OTM if the index is above the strike price.

#### • Intrinsic value of an option:

The option premium can be broken down into two components - intrinsic value and time value. The intrinsic value of a call is the amount the option is ITM, if it is ITM. If the call is OTM, its intrinsic value is zero. Putting it another way, the intrinsic value of a call is Max[0, (S t — K)] which means the intrinsicvalue of a call is the greater of 0 or (S t — K). Similarly, the intrinsic value of a put is Max[0, K — S t ],i.e. the greater of 0 or (K — S t ). K is the strike price and S t is the spot price.

#### • Time value of an option:

The time value of an option is the difference between its premium and its intrinsic value. Both calls and puts have time value. An option that is OTM or ATM has only time value. Usually, the maximum time value exists when the option is ATM. The longer the time to expiration, the greater is an option's time value, all else equal. bAt expiration, an option should have no time value.

### OPTIONS PAYOFFS

The optionality characteristic of options results in a non-linear payoff for options. In simple words, it means that the losses for the buyer of an option are limited, however the profits are potentially unlimited. For a writer (seller), the payoff is exactly the opposite. His profits are limited to the option premium, however his losses are potentially unlimited. These non-linear payoffs are fascinating as they lend themselves to be used to generate various payoffs by using combinations of options and the underlying. We look here at the six basic
payoffs (pay close attention to these pay-offs, since all the strategies in the book are derived out of these basic payoffs).

### Payoff profile of buyer of asset: Long asset

#### Payoff for investor who went Long ABC Ltd. at Rs. 2220

In this basic position, an investor buys the underlying asset, ABC Ltd. shares for instance, for Rs. 2220, and sells it at a future date at an unknown price, S t . Once it is purchased, the investor is said to be "long" the asset.  shows the payoff for a long position on ABC Ltd.The figure shows the profits/losses from a long position on ABC Ltd.. The investor bought ABC Ltd. at Rs. 2220. If the share price goes up, he profits. If the share price falls he loses.

### Payoff profile for seller of asset: Short asset

#### Payoff for investor who went Short ABC Ltd. at Rs. 2220

In this basic position, an investor shorts the underlying asset, ABC Ltd. shares for instance, for Rs. 2220, and buys it back at a future date at an unknown price, S t . Once it is sold, the investor is said to be "short" the asset.  shows the payoff for a short position on ABC Ltd.The figure shows the profits/losses from a short position on ABC Ltd.. The investor sold ABC Ltd. at Rs. 2220. If the share price falls, he profits. If the share price rises, he loses.

### Payoff profile for buyer of call options: Long call

A call option gives the buyer the right to buy the underlying asset at the strike price specified in the option. The profit/loss that the buyer makes on the option depends on the spot price of the underlying. If upon expiration, the spot price exceeds the strike price, he makes a profit. Higher the spot price, more is the profit he makes. If the spot price of the underlying is less than the strike price, he lets his option expire un-exercised. His loss in this case is the premium he paid for buying the option.  gives the payoff for the buyer of a three month call option (often referred to as long call) with a strike of 2250 bought at a premium of 86.60.

#### Payoff for buyer of call option

The figure shows the profits/losses for the buyer of a three-month Nifty 2250 call option. As can be seen, as the spot Nifty rises, the call option is in-the-money. If upon expiration, Nifty closes above the strike of 2250, the buyer would exercise his option and profit to the extent of the difference between the Nifty-close and the strike price. The profits possible on this option are potentially unlimited. However if Nifty falls below the strike of 2250, he lets the option expire. His losses are limited to the extent of the premium he paid for buying the option.

### Payoff profile for writer (seller) of call options: Short call

A call option gives the buyer the right to buy the underlying asset at the strike price specified in the option. For selling the option, the writer of the option charges a premium. The profit/loss that the buyer makes on the option depends on the spot price of the underlying. Whatever is the buyer's profit is the seller's loss. If upon expiration, the spot price exceeds the strike price, the buyer will exercise the option on the writer. Hence as the spot price increases the writer of the option starts making losses. Higher the spot price, more is the loss he makes. If upon expiration the spot price of the underlying is less than the strike price, the buyer lets his option expire un-exercised and the writer gets to keep the premium.gives the payoff for the writer of a three month call option (often referred to as short call) with a strike of 2250 sold at a premium of 86.60.

#### Payoff for writer of call option

The figure shows the profits/losses for the seller of a three-month Nifty 2250 call option. As the spot Nifty rises, the call option is in-the-money and the writer starts making losses. If upon expiration, Nifty closes above the strike of 2250, the buyer would exercise his option on the writer who would suffer a loss to the extent of the difference between the Nifty-close and the strike price. The loss that can be incurred by the writer of the option is potentially unlimited, whereas the maximum profit is limited to the extent of the up-front option premium of Rs.86.60 charged by him.

### Payoff profile for buyer of put options: Long put

A put option gives the buyer the right to sell the underlying asset at the strike price specified in the option. The profit/loss that the buyer makes on the option depends on the spot price of the underlying. If upon expiration, the spot price is below the strike price, he makes a profit. Lower the spot price, more is the profit he makes. If the spot price of the underlying is higher than the strike price, he lets his option expire un-exercised. His loss in this case is the premium he paid for buying the option. gives the payoff for the buyer of a three month put option (often referred to as long put) with a strike of 2250 bought at a premium of 61.70.

#### Payoff for buyer of put option

The figure shows the profits/losses for the buyer of a three-month Nifty 2250 put option. As can be seen, as the spot Nifty falls, the put option is in-the-money. If upon expiration, Nifty closes below the strike of 2250, the buyer would exercise his option and profit to the extent of the difference between the strike price and Nifty-close. The profits possible on this option can be as high as the strike price. However if Nifty rises above the strike of 2250, he lets the option expire. His losses are limited to the extent of the premium he paid for buying the option.

### Payoff profile for writer (seller) of put options: Short put

A put option gives the buyer the right to sell the underlying asset at the strike price specified in the option. For selling the option, the writer of the option charges a premium. The profit/loss that the buyer makes on the option depends on the spot price of the underlying. Whatever is the buyer's profit is the seller's loss. If upon expiration, the spot price happens to be below the strike price, the buyer will exercise the option on the writer. If upon expiration the spot price of the underlying is more than the strike price, the buyer lets his option un-exercised and the writer gets to keep the premium. Figure 1.6 gives the payoff for the writer of a three month put option (often referred to as short put) with a strike of 2250 sold at a premium of 61.70

#### Payoff for writer of put option

The figure shows the profits/losses for the seller of a three-month Nifty 2250 put option. As the spot Nifty falls, the put option is in-the-money and the writer starts making losses. If upon expiration, Nifty closes below the strike of 2250, the buyer would exercise his option on the writer who would suffer a loss to the extent of the difference between the strike price and Nifty- close. The loss that can be incurred by the writer of the option is a maximum extent of the strike price (Since the worst that can happen is that the asset price can fall to zero) whereas the maximum profit is limited to the extent of the up-front option premium of Rs.61.70 charged by him.

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