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Definition of Options and origin of options

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An option is the right, but not the obligation, to buy or sell something at a predetermined price at anytime within a specified time period.

Origin of Options:

Chicago Board of Options Exchange (CBOE) was created in 1973 and CBOE standardized the option contracts, improving the liquidity and enabling the general public to participate in option trading for the first time. It is interesting to note that the option pricing theory was developed around the same time by Fischer Black and Myron Scholes. The much acclaimed Black-Scholes model uses the various parameters of an option viz., strike price, price of the underlying asset, time to expiration, interest rate and the volatility of the underlying asset to compute the theoretical price of an option contract. The American, Philadelphia and the Pacific stock exchanges began trading call options by 1975-76. Put options were introduced in 1977 and by then all US stock exchanges started trading in options with gradual increase in volumes.

The Cox-Rubinstein formula was developed in 1979 which is a binomial model for pricing the options. Today almost all stock exchanges around the world trade in equity options and the volumes are phenomenally high.

Salary Income under Indian Income Tax Laws – some points

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  1. Employment is must for income to be assessed under the head ‘Salaries’. For employer – employee relationship to exist, the employer must have the powers to – (i) define the terms of the employment, and (ii) terminate the employment. However, what is relevant is employment and not so much the presence of an employer, as held by the Supreme Court in Justice D.N.Agarwal v. UOI (1999) 237 ITR 872. As long as the money received is remuneration for services, it is salary.
  2. All receipts with respect to the employment are ‘salary’. The term ‘employer’ takes in not only the past and present employers but a future employer as well and any receipt in contemplation of employment from such employer is assessable as salary. S.17(1)(v) says: ‘Salary’ includes ‘any advance of salary’.
  3. ‘Salary’ includes commission when such commission is received with reference to the employment. Where it is independent of the employment, it is assessable either as other sources income or business income depending on the facts of the case. See P.Ramajeyam 133 Taxman 33 (Mad).
  4. Salary due though not received is assessable in the year it falls due. See s.15(a).
  5. Salary earned in India is taxable in India no matter it has been received as salary proper or as leave salary. See Explanation to s.9(1)(ii).
  6. Salary income including leave salary is earned where the service is rendered.
  7. Family pension is assessable as ‘other sources’ income, subject to the deduction under s.57(iia).
  8. For non-government employee cases, the exemption is restricted to 10 months’ salary based on the average salary for the last 10 months immediately preceding the date of retirement, in turn restricted to Rs.3,00,000.
  9. Gratuity received by a government employee is fully exempt. The limit of Rs.3,50,000 is applicable only to other cases. In such a case, the exemption varies depending on whether the payment is covered by the Payment of Gratuity Act, or not, subject to the ceiling of Rs.3,50,000.
  10. The language of s.89 is clear that the relief is available in cases of both arrears and advance receipts. S.89 comes into operation ‘where an assessee is in receipt of a sum in the nature of salary, being paid in arrears or in advance or is in receipt, in any one financial year, of salary for more than 12 months…
  11. The employer’s contribution is taxable on withdrawal at the time of retirement/ termination of service of the employee, much in the same way as interest referred to in (c).
  12. The entire transferred balance of PF would not be taxed; only that part of the transferred balance that would have been taxable, had the fund been an RPF from the date of institution, would be taxed.
  13. The hire charges borne by the employer are reckoned if they are lower than the prescribed percentage of salary. The perquisite value then is: actual hire charges as reduced by the rent recovered from the employee.
  14. That the employee is an interested person like a promoter or director holding more than 10 share in the equity disentitles him from participation in the scheme.
  15. Exemption is available in respect of any other medical reimbursement to the extent of Rs.15,000.
  16. Overseas travel expenses are excludible from perquisites only in the case of an employee whose gross total income before including this perquisite does not exceed Rs.2,00,000.
  17. Tax is deductible from salary only at the time when the salary is received, though salary is assessable to tax when it is due or received, whichever occurs earlier.
  18. For TDS only the loss under house property can be considered.
  19. The employer does not have the power of investigation. If reasonable proof is given, it is not open to the employer to insist on cast iron evidence.

What are Passive investments?

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Investments in equity securities that are purchased and held for short term principally for the purpose of generating gains on resale are classified as trading securities and reported at fair value with unrealized gains and losses included in the earnings.

Investments in equity securities that are not trading securities are classified as available-for-sale (AFS) securities and reported at fair value with unrealized gains and losses excluded from earnings and reported in a separate component of shareholders’ equity. These are typically short term investments made by the investor. It is generally presumed that if the investor holds less than 20% voting rights of the investee company then no significant influence can be exercised on the investee company unless there is evidence to prove otherwise. Cost method is adopted for investments of this nature.

Definition of Equity Securities

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Equity securities are represented by ownership shares as common stock or preferred stock, rights to acquire ownership shares such as stock warrants or rights or call options. It also includes rights to dispose of ownership in shares by way of put options. It should be noted that equity securities do not include preferred stock that are redeemable at the option of the investor or stock that is redeemed by the issuer, own stock purchased by the company often known as treasury stock and convertible bonds.

Binomial Model of option pricing

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Binomial option pricing model is an options valuation method developed by Cox, et al, in 1979. The binomial option pricing model uses an iterative procedure, allowing for the specification of nodes, or points in time, during the time span between the valuation date and the option’s expiration date.

Like the Black-Scholes model, this model also assumes a perfectly efficient market. The binomial model takes a risk-neutral approach to valuation. It assumes that underlying security prices can only either increase or decrease with time until the option expires worthless.

Advantages of Binomial model:

  • Useful for valuing American options which allow the owner to exercise the option at any point in time until expiration.
  • The model is simple mathematically when compared to the Black-Scholes model, and is relatively easy to build and implement with a computer spreadsheet.
  • In this model it is possible to check at every point in an option’s life for the possibility of early exercise.
  • The Binomial options pricing model approach is widely used as it is able to handle a variety of conditions for which other models cannot easily be applied. This is largely because the BOPM models the underlying instrument over time – as opposed to at a particular point.
  • This model is also used to value Bermudan options which can be exercised at various points.
  • This model is considered to be more accurate, particularly for longer-dated options, and options on securities with dividend payments.

Limitations of Binomial model:

  • The main limitation of the binomial model is its relatively slow speed. Even with the power of computers available today this is not a practical solution for calculation of thousands of prices in a short span of time.

Black Scholes Model of option pricing

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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 viz., stock price, strike price, volatility, time to expiration, and short-term risk free interest rate.

The original formula for calculating the theoretical option price is as follows:

Where:

The variables are:

OP       = theoretical option price
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

ln         = natural logarithm
N(x)     = standard normal cumulative distribution function
e           = the exponential function

Assumptions underlying the above formula:

  • It is possible to short sell the underlying stock.
  • There are no arbitrage opportunities.
  • Trading in the stock is continuous.
  • There are no transaction costs or taxes.
  • All securities are perfectly divisible (e.g. it is possible to buy 1/100th of a share).
  • It is possible to borrow and lend cash at a constant risk-free interest rate.

Except the volatility factor all the other parameters used in this model viz., strike price, time remaining till expiration, the risk-free interest rate, and the current underlying price are objective and are observable. Hence we can conclude that there is a direct relationship between the option price and the volatility. By observing the option price and pegging the other parameters in this formula it is possible to arrive at the volatility that is implied by the market. By applying such derived volatility implied by the market over the other strike prices and expiry we can test the validity of the Black-Scholes option pricing model. It would be observed that the implied volatility tends to be higher for lower strike prices, and lower for higher strike prices.

It is interesting to note that currencies tend to have more symmetrical curves, with implied volatility lowest at-the-money, and higher volatilities for deep in-the-money and out-of-the-money strike prices, while commodities have higher implied volatility for higher strike prices and lower implied volatility for lower strike prices, exactly the opposite of equities.

Lognormal distribution

The Black-Scholes pricing model assumes a lognormal distribution. A lognormal distribution is skewed to the right, which means it has a longer tail towards it right as compared with a normal distribution that is bell-shaped. The lognormal distribution allows for a stock price distribution of between zero and infinity and has an upward bias. This is because while a stock price can only drop 100%, it can rise by more than 100%.

Advantages of Black-Scholes model:

  • It enables one to calculate a very large number of option prices in a very short time.

Limitations of Black-Scholes model:

  • Black-Scholes model cannot be used to accurately price options with an American-style exercise as it calculates the option price at expiration only. Early exercise as in the case of American option cannot be priced correctly using this model which is a major limitation of this model.
  • All exchange traded equity options (ETO) have American-style exercise as against the European options which can only be exercised at expiration. That means this model cannot be used for pricing most ERO options. The exception to this is an American call on a non-dividend paying asset as the call is always worth the same as its European equivalent since there is never any advantage in exercising early.

Pricing of Option

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Option pricing is based on some key parameters as discussed before in this chapter. To recapitulate essentially the following factors play a key role in determining the theoretical value of an option:

  • Underlying price of the stock
  • Strike price of the option contract
  • Time to maturity
  • Interest Rate
  • Anticipated volatility of the underlying
  • Dividends – declared and anticipated

Greeks in Option pricing

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Option pricing is based on some key parameters as discussed before in this chapter. Essentially the following factors, known as ‘Greeks’ should be grasped to understand the option pricing:

  1. Delta
  2. Gamma
  3. Theta
  4. Vega
  5. Rho
  6. Omega

These factors are discussed below in detail.

Delta

Delta is a measure that reflects the relationship between a change in the price of the underlying and the price of the option. Delta plays a vital role in determining market strategies for several analysts. Delta is a measure that determines how much the price of an option will change consequent to a change in the price of the underlying stock. Also this measure can be thought of as indicating the probability that the option will finish in-the-money.

A call option at the strike price that equals the market rate, the probability that the option will finish in-the-money is exactly a 50% chance. Hence the delta for a call option equals 0.5. The sign for a long call option is positive as the price of the call option would increase corresponding to an increase in the price of the underlying.

A put option at the strike price that equals the market rate, again the probability that the option will finish in-the-money is exactly a 50% chance. Hence the delta for a put option equals 0.5. However the sign for a long put option is negative since the price of a put option would decrease corresponding to an increase in the price of the underlying.

Delta is also referred to as the hedge ratio. Delta is a key factor that is usually relied upon to implement the delta-neutral strategies for several fund managers. Delta-neutral strategy creates a risk less position which means that a portfolio would be worth the same whether the stock price rose by a small amount or fell by a small amount. However, wide fluctuation in the stock price will not protect the position.

If a fund manager has a portfolio that contains short position of x number of option contracts in a stock then x multiplied by the delta gives the fund manager the number of shares that would be needed to create a risk-less position.

Type of holding

Delta

Long underlying stock

Positive

Short underlying stock

Negative

Long call

Positive

Short call

Negative

Long put

Negative

Short put

Positive

For a stock that trades at $100, the following table illustrates the likely delta value for both call and put options at various strike prices. As the strike price gets higher than the market price of the underlying, the call delta approaches zero value and as the strike price gets lower and lower than the market price of the underlying, the put delta approaches zero value. When the option is deep in the money the value of delta approaches 100, indicating that for every dollar increase in the price of the underlying, the option price would also increase by a dollar.

Market Price: $100

Option Strike Price

Call Delta

Put Delta

$25

+100

0

$50

+90

-10

$75

+75

-25

$100

+50

-50

$125

+25

-75

$150

+10

-90

$175

0

-100

  • $100 strike price option which is at-the-money option has a delta of +50 for calls and -50 for puts
  • For strike price $150 and above, the call option delta drops substantially and approaches zero for calls of strike price $175; correspondingly the put option delta approaches -100 for strike price $175 and above
  • For strike prices $50 and below, the put option delta drops substantially and approaches zero for calls of strike price $25; correspondingly the put option delta approaches -100 for strike price $25 and below

There is an interesting relationship between the delta of an option product and its time to expiry. As mentioned earlier, delta can be thought of as the probability that the option will finish in-the-money on expiry of the contract. Hence the delta for an in-the-money option will approach 100 as time to expiry decreases. The reason is that the probability of the option finishing in-the-money increases as the time to expiry decreases.

The delta of an at-the-money option will always tend to remain at 50, irrespective of the time to expiry as there is always a 50% chance of the option finishing in-the-money.

Similarly for an out-of-the-money option, the delta will approach zero as the time to expiry decreases, because the probability of the option finishing in-the-money decreases as the time to expiry decreases.

Gamma

While delta is the first derivative, gamma is the second derivative of option price. Gamma is the acceleration of the option’s price to its underlying stock. Gamma is the ratio of the change of an option’s delta to a small change in the price of the underlying stock. Like delta, gamma is also expressed as a number between zero and one. It is also expressed as a number between 100 and zero when the gamma is multiplied by 100 which is normally the number of stock represented by one contract.

Gamma can be thought of as a measure that determines how much an option’s delta changes for every $1 change in the price of the underlying stock. Like delta, gamma value can be either positive or negative. A positive gamma indicates that the option’s delta increases when the price of the underlying stock increases in value and decreases as the price of the underlying stock decreases. A negative gamma similarly indicates that the option’s delta decreases when the price of the underlying stock decreases in value and increases as the price of the underlying stock increases.

Unlike delta, both short call and short put have negative gamma and long call and long put have positive gamma.

Type of holding

Delta

Gamma

Long underlying stock

Positive

No Gamma

Short underlying stock

Negative

No Gamma

Long call

Positive

Positive

Short call

Negative

Negative

Long put

Negative

Positive

Short put

Positive

Negative

Delta of at-the-money option contracts which are close to 50 are susceptible to small changes in the price of the underlying stock and thus ATM calls have the greatest gamma. Deep in-the-money options or far out-of-the-money options have a gamma value close to zero.

Theta

Theta is the measure of time value of the option. At the time of expiry of the option contract, both the at-the-money and out-of-the-money options become worthless as there is no intrinsic value for these options. Only the intrinsic value would remain for in-the-money option contracts, leaving no time value.

For an option seller the value of the time premium that decays as time passes is a benefit while the same is a loss for someone who is long in an option.

Theta represents how much value of an option’s price is attributable to time value. Theta is expressed as the dollar value that is lost every day, including holidays, known as the time decay, while the other factors remain constant.

It is interesting note the relationship of gamma and theta. Options that have a positive theta have a negative gamma and the options contracts that have a negative theta have a positive gamma. As we have observed earlier, at-the-money options have the maximum time premium or maximum theta. For at-the-money option contracts, the theta accelerates towards expiry. For in-the-money and out-of-the-money option contracts, the decay is linear towards the expiry.

Rho

Rho is a measure of the sensitivity of option prices to changes in the interest rates. The rate of interest is another factor in the determination of the price of an option, but considered less significant when compared with other factors like delta, vega and theta. Higher the interest, higher will be the call option premium and lower the put option premium. Lower the interest, lower will be the call option premium and higher the put option premium. Rho is expressed as a positive number for calls and negative number for puts. Rho indicates the theoretical change in the option premium for every one percent change in the rate of interest.

Vega

Vega or volatility as discussed earlier, is a measure of how fast the underlying futures prices are moving. It is a measure of the speed and magnitude at which the underlying stock’s prices change. This is a key factor in the determination of the price of an option. This is also known as Lambda.

Omega

Omega is a measure of the change in an option’s value with respect to the percentage change in the underlying price. The omega gives option investors an idea of how the option price and the stock price that underlies it move together. Omega is the third derivative of the option price, and the derivative of gamma.

Volatility & pricing of options

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Volatility forms a very key factor in the pricing of an option. As we discuss later in this chapter this component known as ‘vega’ is a factor that is used in the mathematical computation of arriving at the theoretical value of an option price. Increased volatility means higher option prices and vice versa.

Historical Volatility & Implied Volatility in Options

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Volatility is a measure of how fast the underlying futures prices are moving. It is a measure of the speed and magnitude at which the underlying stock’s prices change. This is expressed in a percentage. The two basic types of volatility are historical volatility and implied volatility. Historical volatility is a measure of how much the prices have been fluctuating in the past. Implied volatility is the option market’s assessment of how volatile the prices would be in the future. Implied volatility is a barometer of the collective thought process about the price fluctuation of the underlying. If the market is highly volatile then, the premium on options would surge because the traders become skeptical that price of the underlying may move in a different direction. Therefore, they increase option premiums to compensate for this perceived risk. If the investors think that the volatility will go up, then they should buy options irrespective of the direction of the movement of the market. On the other hand if the investors think that the volatility will go down, then they should sell options.

Historical Volatility

Historical volatility is the measure of actual price changes to an underlying asset over a specific period in the past. Stated in simple terms, historical volatility is the standard deviation worked out for the underlying for any defined period of time which could be a week, a month, a quarter or even a year. Historical volatility can be computed for the closing prices, weighted average prices of the underlying, etc.

Implied Volatility

As the name suggests, this is the volatility implied by the market based on the market rates of the underlying and the rates of the option contracts. Given the various parameters that go into the pricing of the options, the volatility that is assumed in the pricing model can be easily derived. The theoretical volatility derived from such a computation is the volatility that the market implies or other wise known as implied volatility. This is bound to be different from the historical volatility as the historical volatility is based on the past data and the implied volatility is based on the present and perceived future conditions of the market.

Factors influencing implied volatility

The price of option like any other product is predominantly determined by the supply and demand for the product. The market makers who play a vital role in maintaining the balance between the supply and demand adjust the spread for the option product. When the demand for an option is more than the supply, then the market makers increase the price of the product and vice versa. The increase or decrease in the price of the product abruptly ceteris paribus, reflects the change in the implied volatility of the market for that product.