black schole project

EXECUTIVE SUMMARY

Modern option pricing techniques are often considered among the most

mathematically complex of all applied areas of finance. Financial analysts have reached the

point where they are able to calculate, with alarming accuracy, the value of a stock option.

Most of the models and techniques employed by today's analysts are rooted in a model

developed by Fischer Black and Myron Scholes in 1973. This study examines the evolution

of option pricing models leading up to and beyond Black and Scholes' model.

The use of the Black-Scholes formula is pervasive in the markets. In fact the model

has become such an integral part of market conventions that it is common practice for the

implied volatility rather than the price of an instrument to be quoted. (All the parameters in

the model other than the volatility - that is the time to maturity, the strike, the risk-free

rate, and the current underlying price - are unequivocally observable. This means there is

one-to-one relationship between the option price and the volatility.) Traders prefer to

think in terms of volatility as it allows them to evaluate and compare options of different

maturities, strikes, and so on.

The work done by Black & Scholes in the 70's made way for further pricing of

derivatives and in particular, exotic options. The Black-Scholes partial differential equation

also enabled derivation of the 'Greeks’ of option pricing. The Black-Scholes model today is

used in everyday pricing of options and futures and almost all formulas for pricing of exotic

options such as barriers, compounds and Asian options take their foundation from the Black-

Scholes model.

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TABLE OF CONTENTS

Abstract 1

Introduction 3

Option Pricing 5

The Binomial Option Pricing Model 9

The Black and Scholes Model 16

S&P CNX Nifty Options 25

Option Prices for S&P CNX Nifty Contracts

Option Prices based on Historical Volatility

Option Prices based on Implied Volatility

Analysis 42

Conclusion 43

References 44

Appendix

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Introduction:

It was an ordinary autumn afternoon in Belmont, Mass. 1969, when Fischer Black, a 31 year

old independent finance contractor, and Myron Scholes a 28 year old assistant professor of

finance, at MIT hit upon an idea that would change financial history. Black had been

working for Arthur D. Little in Cambridge, Mass., when he met a colleague who had devised

a model for pricing securities and other assets. With his Harvard Ph.D. in applied

mathematics just five years old, Black's interest was sparked. His colleague's model focused

on stocks, so Black turned his attention to options, which were not widely traded at the time.

By 1973, the tandem team of Fischer Black and Myron Scholes had written the first draft of a

paper that outlined an analytic model that would determine the fair market value for

European type call options on non-payout assets. They submitted their work to the Journal of

Political Economy for publication, who promptly responded by rejecting their paper.

Convinced that their ideas had merit, they sent a copy to the Review of Economics and

Statistics, where it elicited the same response. After making some revisions based on

extensive comments from Merton Miller (Nobel Laureate from the University of Chicago)

and Eugene Fama, of the University of Chicago, they resubmitted their paper to the Journal

of Political Economy, who finally accepted it. From the moment of its publication in 1973,

the Black and Scholes Option Pricing Model has earned a position among the most widely

accepted of all financial models.

What Is an Option?

The idea of options is certainly not new. Ancient Romans, Grecians, and Phoenicians traded

options against outgoing cargoes from their local seaports. When used in relation to financial

instruments, options are generally defined as a "contract between two parties in which one

party has the right but not the obligation to do something, usually to buy or sell some

underlying asset". Having rights without obligations has financial value, so option holders

must purchase these rights, making them assets. This asset derives their value from some

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other asset, so they are called derivative assets. Call options are contracts giving the option

holder the right to buy something, while put options, and conversely entitle the holder to sell

something. Payment for call and put options, takes the form of a flat, up-front sum called a

premium. Options can also be associated with bonds (i.e. convertible bonds and callable

bonds), where payment occurs in installments over the entire life of the bond, but this paper

is only concerned with traditional put and call options.

Origins of Option Pricing Techniques:

Modern option pricing techniques, with roots in stochastic calculus, are often considered

among the most mathematically complex of all applied areas of finance. These modern

techniques derive their impetus from a formal history dating back to 1877, when Charles

Castelli wrote a book entitled The Theory of Options in Stocks and Shares. Castelli's book

introduced the public to the hedging and speculation aspects of options, but lacked any

monumental theoretical base. Twenty three years later, Louis Bachelier offered the earliest

known analytical valuation for options in his mathematics dissertation "Theorie de la

Speculation" at the Sorbonne. He was on the right track, but he used a process to generate

share price that allowed both negative security prices and option prices that exceeded the

price of the underlying asset. Bachelier's work interested a professor at MIT named Paul

Samuelson, who in 1955, wrote an unpublished paper entitled "Brownian Motion in the

Stock Market". During that same year, Richard Kruizenga, one of Samuelson's students, cited

Bachelier's work in his dissertation entitled "Put and Call Options: A Theoretical and Market

Analysis". In 1962, another dissertation, this time by A. James Boness, focused on options.

In his work, entitled "A Theory and Measurement of Stock Option Value", Boness developed

a pricing model that made a significant theoretical jump from that of his predecessors. More

significantly, his work served as a precursor to that of Fischer Black and Myron Scholes,

who in 1973 introduced their landmark option pricing model.

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OPTION PRICING

In general, the value of any asset is the present value of the expected cash flows on that asset.

In this section, we will consider an exception to that rule when we will look at assets with

two specific characteristics:

They derive their value from the values of other assets.

The cash flows on the assets are contingent on the occurrence of specific events.

These assets are called options, and the present value of the expected cash flows on these

assets will understate their true value. In this section, we will describe the cash flow

characteristics of options, consider the factors that determine their value and examine how

best to value them.

Cash Flows on Options

There are two types of options. A call option gives the buyer of the option the right to buy the

underlying asset at a fixed price, whereas a put option gives the buyer the right to sell the

underlying asset at a fixed price. In both cases, the fixed price at which the underlying asset

can be bought or sold is called the strike or exercise price.

To look at the payoffs on an option, consider first the case of a call option. When you buy the

right to sell an asset at a fixed price, you want the price of the asset to increase above that

fixed price. If it does, you make a profit, since you can buy at the fixed price and then sell at

the much higher price; this profit has to be netted against the cost initially paid for the option.

However, if the price of the asset decreases below the strike price, it does not make sense to

exercise your right to buy the asset at a higher price. In this scenario, you lose what you

originally paid for the option. Figure 1 summarizes the cash payoff at expiration to the buyer

of a call option.

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With a put option, you get the right to sell at a fixed price, and you want the price of the asset

to decrease below the exercise price. If it does, you buy the asset at the exercise price and

then sell it back at the current price, claiming the difference as a gross profit. When the initial

cost of buying the option is netted against the gross profit, you arrive at an estimate of the net

profit. If the value of the asset rises above the exercise price, you will not exercise the right to

sell at a lower price. Instead, the option will be allowed to expire without being exercised,

resulting in a net loss of the original price paid for the put option. Figure 2 summarizes the

net payoff on buying a put option.

With both call and put options, the potential for profit to the buyer is significant, but the

potential for loss is limited to the price paid for the option.

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Determinants of Option Value

What is it that determines the value of an option? At one level, options have expected cash

flows just like all other assets, and that may seem like good candidates for discounted cash

flow valuation. The two key characteristics of options -- that they derive their value from

some other traded asset, and the fact that their cash flows are contingent on the occurrence of

a specific event -- does suggest an easier alternative. We can create a portfolio that has the

same cash flows as the option being valued, by combining a position in the underlying asset

with borrowing or lending. This portfolio is called a replicating portfolio and should cost

the same amount as the option. The principle that two assets (the option and the replicating

portfolio) with identical cash flows cannot sell at different prices is called the arbitrage

principle.

Options are assets that derive value from an underlying asset; increases in the value of the

underlying asset will increase the value of the right to buy at a fixed price and reduce the

value to sell that asset at a fixed price. On the other hand, increasing the strike price will

reduce the value of calls and increase the value of puts. While calls and puts move in

opposite directions when stock prices and strike prices are varied, they both increase in value

as the life of the option and the variance in the underlying asset’s value increases. The reason

for this is the fact that options have limited losses. Unlike traditional assets that tend to get

less valuable as risk is increased, options become more valuable as the underlying asset

becomes more volatile. This is so because the added variance cannot worsen the downside

risk (you still cannot lose more than what you paid for the option) while making potential

profits much higher. In addition, a longer life for the options just allows more time for both

call and put options to appreciate in value. Since calls provide the right to buy the underlying

asset at a fixed price, an increase in the value of the asset will increase the value of the calls.

Puts, on the other hand, become less valuable as the value of the asset increase.

The final two inputs that affect the value of the call and put options are the riskless interest

rate and the expected dividends on the underlying asset. The buyers of call and put options

usually pay the price of the option up front, and wait for the expiration day to exercise. There

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is a present value effect associated with the fact that the promise to buy an asset for $ 1

million in 10 years is less onerous than paying it now. Thus, higher interest rates will

generally increase the value of call options (by reducing the present value of the price on

exercise) and decrease the value of put options (by decreasing the present value of the price

received on exercise). The expected dividends paid by assets make them less valuable; thus,

the call option on a stock that does not pay a dividend should be worth more than a call

option on a stock that does pay a dividend. The reverse should be true for put options.

A Simple Model for Valuing Options

Almost all models developed to value options in the last three decades are based upon the

notion of a replicating portfolio. The earliest derivation, by Black and Scholes, is

mathematically complex, In this section; we consider the simplest replication model for

valuing options – the binomial model.

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The Binomial Model

The binomial option pricing model is based upon a simple formulation for the asset price

process in which the asset, in any time period, can move to one of two possible prices. The

general formulation of a stock price process that follows the binomial is shown in Figure 3.

Figure 3: General Formulation for Binomial Price Path

In this figure, S is the current stock price; the price moves up to Su with probability p and

down to Sd with probability 1-p in any time period. For instance, if the stock price today is $

100, u is 1.1 and d is 0.9, the stock price in the next period can either be $ 110 (if u is the

outcome) and $ 90 (if d is the outcome).

Creating a Replicating Portfolio

The objective in creating a replicating portfolio is to use a combination of riskfree

borrowing/lending and the underlying asset to create the same cash flows as the option being

valued. In the case of the general formulation above, where stock prices can either move up

to Su or down to Sd in any time period, the replicating portfolio for a call with a given strike

price will involve borrowing $B and acquiring Δ of the underlying asset. Of course, this

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formulation is of no use if we cannot determine how much we need to borrow and what

is. There is a way, however, of identifying both variables. To do this, note that the value

of this position has to be same as the value of the call no matter what the stock price does.

Let us assume that the value of the call is Cu if the stock price goes to Su, and Cd if the stock

price goes down to Sd. If we had borrowed $B and bought Δ shares of stock with the money,

the value of this position under the two scenarios would have been as follows:

Note that, in either case, we have to pay back the borrowing with interest. Since the position

has to have the same cash flows as the call, we get

Su - $ B (1+r) = Cu

Sd - $ B (1+r) = Cd

Solving for Δ,

We get Δ = Number of units of the underlying asset bought = (Cu - Cd)/(Su - Sd)

Where, Cu = Value of the call if the stock price is Su

Cd = Value of the call if the stock price is Sd

When there are multiple periods involved, we have to begin with the last period, where we

know what the cash flows on the call will be, solve for the replicating portfolio and then

estimate how much it would cost us to create this portfolio. We then use this value as the

estimated value of the call and estimate the replicating portfolio in the previous period. We

continue to do this until we get to the present. The replicating portfolio we obtain for the

present can t be priced to yield a current value for the call.

Value of the call = Current value of underlying asset * Option Delta - Borrowing needed to

replicate the option

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An Example of Binomial valuation

Assume that the objective is to value a call with a strike price of 50, which is expected to

expire in two time periods, on an underlying asset whose price currently is 50 and is expected

to follow a binomial process. Figure 4 illustrates the path of underlying asset prices and the

value of the call (with a strike price of 50) at the expiration.

Figure 4: Binomial Price Path

Note that since the call has a strike price of $ 50, the gross cash flows at expiration are as

follows:

If the stock price moves to $ 100: Cash flow on call = $ 100 - $ 50 = $ 50

If the stock price moves to $ 50: Cash flow on call = $ 50 - $ 50 = $ 0

If the stock price moves to $ 25: Cash flow on call = $ 0 (Option is not exercised).

Now assume that the interest rate is 11%. In addition, define

Δ = Number of shares in the replicating portfolio

B = Dollars of borrowing in replicating portfolio

The objective, in this analysis, is to combine shares of stock and B dollars of borrowing

to replicate the cash flows from the call with a strike price of $ 50.

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The first step in doing this is to start with the last period and work backwards. Consider, for

instance, one possible outcome at t =1. The stock price has jumped to $ 70, and is poised to

change again, either to $ 100 or $ 50. We know the cash flows on the call under either

scenario, and we also have a replicating portfolio composed of Δ shares of the underlying

stock and $ B of borrowing. Writing out the cash flows on the replicating portfolio under

both scenarios (stock price of $ 100 and $ 50), we get the replicating portfolios in figure 5:

Figure 5: Replicating Portfolios when Price is $ 70

In other words, if the stock price is $70 at t=1, borrowing $45 and buying one share of the

stock will give the same cash flows as buying the call. The value of the call at t=1, if the

stock price is $70, should therefore be the cash flow associated with creating this replicating

position and it can be estimated as follows:

70 Δ - B = 70-45 = 25

The cost of creating this position is only $ 25, since $ 45 of the $ 70 is borrowed. This should

also be the price of the call at t=1, if the stock price is $ 70. Consider now the other possible

outcome at t=1, where the stock price is $ 35 and is poised to jump to either $ 50 or $ 25.

Here again, the cash flows on the call can be estimated, as can the cash flows on the

replicating portfolio composed of Δ shares of stock and $B of borrowing. Figure 6 illustrates

the replicating portfolio:

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Figure 6: Replicating Portfolio when Price is $ 35

Since the call is worth nothing, under either scenario, the replicating portfolio also is empty.

The cash flow associated with creating this position is obviously zero, which becomes the

value of the call at t=1, if the stock price is $ 35. We now have the value of the call under

both outcomes at t=1; it is worth $ 25 if the stock price goes to $ 70 and $0 if it goes to $ 35.

We now move back to today (t=0), and look at the cash flows on the replicating portfolio.

Figure 7 summarizes the replicating portfolios as viewed from today:

Figure 7: Replicating Portfolios for Call Value

Using the same process that we used in the previous step, we find that borrowing $22.5 and

buying 5/7 of a share will provide the same cash flows as a call with a strike price of $50.

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The cost, to the investor, of borrowing $ 22.5 and buying 5/7 of a share at the current stock

price of $ 50 yields:

Cost of replicating position = 5/7 X $ 50 - $ 22.5 = $ 13.20

This should also be the value of the call.

More on the Determinants of Option Value

The binomial model provides insight into the determinants of option value. The value of an

option is determined not by the expected price of the asset but by its current price, which, of

course, reflects expectations about the future. In fact, the probabilities that we provided in the

description of the binomial process of up and down movements do not enter the option

valuation process, though they do affect the underlying asset’s value. The reason for this is

the fact that options derive their value from other assets, which are often traded.

Consequently, the capacity investors possess to create positions that have the same cash

flows as the call operates as a powerful mechanism controlling option prices. If the option

value deviates from the value of the replicating portfolio, investors can create an arbitrage

position, i.e., one that requires no investment, involves no risk, and delivers positive returns.

The option value increases as the time to expiration is extended, as the price movements (u

and d) increase, and as the interest rate increases.

The second insight is that the greater the variance in prices in the underlying asset in this

example, the more valuable the option becomes. Thus, increasing the up and down

movements, in the illustration above, makes options more valuable. This occurs because of

the fact that options do not have to be exercised if it is not in the holder’s best interests to do

so. Thus, lowering the price in the worst case scenario to $ 10 from $ 25 does not, by itself,

affect the gross cash flows on this call. On the other hand, increasing the price in the best

case scenario to $ 150 from $ 100 benefits the call holder and makes the call more valuable.

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The binomial model is a useful model for illustrating the replicating portfolio and the effect

of the different variables on call value. It is, however, a restrictive model, since asset prices

in the real world seldom follow a binomial process. Even if they did, estimating all possible

outcomes and drawing a binomial tree, as we have, can be an extraordinarily tedious

exercise.

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The Black & Scholes Model

The Black and Scholes Option Pricing Model didn't appear overnight, in fact, Fisher Black

started out working to create a valuation model for stock warrants. This work involved

calculating a derivative to measure how the discount rate of a warrant varies with time and

stock price. The result of this calculation held a striking resemblance to a well-known heat

transfer equation. Soon after this discovery, Myron Scholes joined Black and the result of

their work is a startlingly accurate option pricing model. Black and Scholes can't take all

credit for their work, in fact their model is actually an improved version of a previous model

developed by A. James Boness in his Ph.D. dissertation at the University of Chicago. Black

and Scholes' improvements on the Boness model come in the form of a proof that the risk-

free interest rate is the correct discount factor, and with the absence of assumptions regarding

investor's risk preferences.

In order to understand the model itself, we divide it into two parts. The first part, SN(d1),

derives the expected benefit from acquiring a stock outright. This is found by multiplying

stock price [S] by the change in the call premium with respect to a change in the underlying

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stock price [N(d1)]. The second part of the model, Ke(-rt)N(d2), gives the present value of

paying the exercise price on the expiration day. The fair market value of the call option is

then calculated by taking the difference between these two parts.

Assumptions of the Black and Scholes Model:

1) The stock pays no dividends during the option's life

Most companies pay dividends to their share holders, so this might seem a serious limitation

to the model considering the observation that higher dividend yields elicit lower call

premiums. A common way of adjusting the model for this situation is to subtract the

discounted value of a future dividend from the stock price.

2) European exercise terms are used

European exercise terms dictate that the option can only be exercised on the expiration date.

American exercise term allow the option to be exercised at any time during the life of the

option, making american options more valuable due to their greater flexibility. This

limitation is not a major concern because very few calls are ever exercised before the last few

days of their life. This is true because when you exercise a call early, you forfeit the

remaining time value on the call and collect the intrinsic value. Towards the end of the life of

a call, the remaining time value is very small, but the intrinsic value is the same.

3) Markets are efficient

This assumption suggests that people cannot consistently predict the direction of the market

or an individual stock. The market operates continuously with share prices following a

continuous Itô process. To understand what a continuous Itô process is, you must first know

that a Markov process is "one where the observation in time period t depends only on the

preceding observation." An Itô process is simply a Markov process in continuous time. If you

were to draw a continuous process you would do so without picking the pen up from the

piece of paper.

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4) No commissions are charged

Usually market participants do have to pay a commission to buy or sell options. Even floor

traders pay some kind of fee, but it is usually very small. The fees that Individual investor's

pay is more substantial and can often distort the output of the model.

5) Interest rates remain constant and known

The Black and Scholes model uses the risk-free rate to represent this constant and known

rate. In reality there is no such thing as the risk-free rate, but the discount rate on U.S.

Government Treasury Bills with 30 days left until maturity is usually used to represent it.

During periods of rapidly changing interest rates, these 30 day rates are often subject to

change, thereby violating one of the assumptions of the model.

6) Returns are log normally distributed

This assumption suggests, returns on the underlying stock are normally distributed, which is

reasonable for most assets that offer options.

Term Used In Black and Scholes Model:

Delta:

Delta is a measure of the sensitivity the calculated option value has to small changes in the

share price.

Gamma:

Gamma is a measure of the calculated delta's sensitivity to small changes in share price.

Theta:

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Theta measures the calcualted option value's sensitivity to small changes in time till maturity.

Vega:

Vega measures the calculated option value's sensitivity to small changes in volatility.

Rho:

The work done by Black & Scholes in the 70's made way for further pricing of derivatives

and in particular, exotic options. The Black-Scholes partial differential equation also enabled

derivation of the 'Greeks’ of option pricing. The Black-Scholes model today is used in

everyday pricing of options and futures and almost all formulas for pricing of exotic options

such as barriers, compounds and Asian options take their foundation from the Black-Scholes

model.

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Graphs of the Black and Scholes Model:

This following graphs show the relationship between a call's premium and the underlying stock's price.

The first graph identifies the Intrinsic Value, Speculative Value, Maximum Value, and the Actual premium for a call.

The following 5 graphs show the impact of diminishing time remaining on a call with:S = $48E = $50r = 6%sigma = 40%

Graph # 1, t = 3 monthsGraph # 2, t = 2 monthsGraph # 3, t = 1 monthGraph # 4, t = .5 monthsGraph # 5, t = .25 months

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Graph #1

Graph #2

Graph #3

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Graph #4

Graph #5

Graphs # 6 - 9, show the effects of a changing Sigma on the relationship between Call premium and Security Price

S = $48E = $50r = 6%sigma = 40%

Graph # 6, sigma = 80%Graph # 7, sigma = 40%Graph # 8, sigma = 20%Graph # 9, sigma = 10%

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Graph #6

Graph #7

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Graph #8

Graph #9

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http://bradley.bradley.edu/~arr/bsm/pg05.html

S&P CNX Nifty Options

An option gives a person the right but not the obligation to buy or sell something. An option

is a contract between two parties wherein the buyer receives a privilege for which he pays a

fee (premium) and the seller accepts an obligation for which he receives a fee. The premium

is the price negotiated and set when the option is bought or sold. A person who buys an

option is said to be long in the option. A person who sells (or writes) an option is said to be

short n the option.

NSE introduced trading in index options on June 4, 2001. The options contracts are European

style and cash settled and are based on the popular market benchmark S&P CNX Nifty index.

Contract Specifications

Security descriptor

The security descriptor for the S&P CNX Nifty options contracts is:

Market type: N

Instrument Type: OPTIDX

Underlying: NIFTY

Expiry date: Date of contract expiry

Option Type: CE/ PE

Strike Price: Strike price for the contract

Instrument type represents the instrument i.e. Options on Index.

Underlying symbol denotes the underlying index, which is S&P CNX Nifty

Expiry date identifies the date of expiry of the contract

Option type identifies whether it is a call or a put option, CE - Call European, PE – Put

European

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Underlying Instrument

The underlying index is S&P CNX NIFTY.

Trading cycle

S&P CNX Nifty options contracts have a maximum of 3-month trading cycle - the near

month (one), the next month (two) and the far month (three). On expiry of the near month

contract, new contracts are introduced at new strike prices for both call and put options, on

the trading day following the expiry of the near month contract. The new contracts are

introduced for three month duration.

Expiry day

S&P CNX Nifty options contracts expire on the last Thursday of the expiry month. If the last

Thursday is a trading holiday, the contracts expire on the previous trading day.

Strike Price Intervals

The number of contracts provided in options on NIFTY is related to the range in which

previous day’s closing value of NIFTY falls as per the following table:

NIFTY Index Level Strike IntervalScheme of strikes to be introduced

(ITM-ATM-OTM)

upto 1500 10 3-1-3

>1500 upto 2000 10 5-1-5

>2000 upto 2500 10 7-1-7

>2500 10 9-1-9

New contracts with new strike prices for existing expiration date are introduced for trading

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on the next working day based on the previous day's close Nifty values, as and when

required. In order to decide upon the at-the-money strike price, the Nifty closing value is

rounded off to the nearest 10. The in-the-money strike price and the out-of-the-money strike

price are based on the at-the-money strike price interval.

Trading Parameters

Contract size

The value of the option contracts on Nifty may not be less than Rs. 2 lakhs at the time of

introduction. The permitted lot size for futures contracts & options contracts shall be the

same for a given underlying or such lot size as may be stipulated by the Exchange from time

to time.

Price steps

The price step in respect of S&P CNX Nifty options contracts is Re.0.05.

Base Prices

Base price of the options contracts, on introduction of new contracts, would be the theoretical

value of the options contract arrived at based on Black-Scholes model of calculation of

options premiums.

The options price for a Call, computed as per the following Black Scholes formula:

C = S * N (d1) - X * e- rt * N (d2) and

The price for a Put is: P = X * e- rt * N (-d2) - S * N (-d1)

Where:

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d1 = [ln (S / X) + (r + σ2 / 2) * t] / σ * sqrt(t)

d2 = [ln (S / X) + (r - σ2 / 2) * t] / σ * sqrt(t)

= d1 - σ * sqrt(t)

C = price of a call option

P = price of a put option

S = price of the underlying asset

X = Strike price of the option

r = rate of interest

t = time to expiration

σ = volatility of the underlying

N represents a standard normal distribution with mean = 0 and standard deviation = 1

ln represents the natural logarithm of a number. Natural logarithms are based on the constant

e.

Rate of interest may be the relevant MIBOR rate or such other rate as may be specified.

The base price of the contracts on subsequent trading days will be the daily close price of the

options contracts. The closing price shall be calculated as follows:

If the contract is traded in the last half an hour, the closing price shall be the last half an hour

weighted average price.

If the contract is not traded in the last half an hour, but traded during any time of the day, then the

closing price will be the last traded price (LTP) of the contract.

If the contract is not traded for the day, the base price of the contract for the next trading day

shall be the theoretical price of the options contract arrived at based on Black-Scholes model

of calculation of options premiums.

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Price bands

There are no day minimum/maximum price ranges applicable for options contracts.

However, in order to prevent erroneous order entry, operating ranges and day

minimum/maximum ranges for options contract are kept at 99% of the base price. In view of

this, members will not be able to place orders at prices which are beyond 99% of the base

price. Members desiring to place orders in option contracts beyond the day min-max range

would be required to send a request to the Exchange. The base prices for option contracts

may be modified, at the discretion of the Exchange, based on the request received from

trading members.

Quantity freeze

Order which may come to the exchange as a quantity freeze shall be based on the notional

value of the contract of around Rs. 5 crores. In respect of orders which have come under

quantity freeze, members would be required to confirm to the Exchange that there is no

inadvertent error in the order entry and that the order is genuine. On such confirmation, the

Exchange may approve such order. However, in exceptional cases, the Exchange may, at its

discretion, not allow the orders that have come under quantity freeze for execution for any

reason whatsoever including non-availability of turnover / exposure limits

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Order type/Order book/Order attributes

· Regular lot order

· Stop loss order

· Immediate or cancel

· Spread order

Option Prices Based on Historical Volatility

As on 17th Oct' 05

Current Index Value = 2485.15

Strike Price = 2400

Risk free rate = 6.70%

Historical Volatility = 16%

Expire on 27th Oct' 05

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Index Analysis

IndexOption Price

Time Value

Snapshot Calls Puts

2,323.73 3.879 3.879 Price 91.996 2.4452,336.44 5.799 5.799 Delta 0.919 -0.0812,349.15 8.394 8.394 Gamma 0.002 0.0022,361.86 11.784 11.784 Theta -0.888 -0.4482,374.58 16.07 16.07 Vega 0.653 0.6532,387.29 21.328 21.328 Rho 0.601 -0.0552,400.00 27.594 27.594 Elasticity 24.83 -82.252,412.71 34.869 22.157 Position ITM OTM2,425.42 43.11 17.686 Probability of closing ITM 91.50% 8.50%2,438.14 52.243 14.1072,450.85 62.168 11.322,463.56 72.771 9.2112,476.27 83.932 7.66

Call Option Price & Time Value by Index

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Strike Price = 2400 Expire on 24th Nov' 05

Index Analysis

Index Option Price Time Value Snapshot Calls Puts2,267.89 7.118 7.118 Price 109.963 11.6332,289.91 10.697 10.697 Delta 0.817 -0.1832,311.93 15.524 15.524 Gamma 0.002 0.0022,333.95 21.803 21.803 Theta -0.859 -0.4212,355.96 29.698 29.698 Vega 1.934 1.9342,377.98 39.317 39.317 Rho 1.582 -0.3792,400.00 50.7 50.7 Elasticity 18.46 -39.152,422.02 63.817 41.799 Position ITM OTM2,444.04 78.571 34.535 Probability of closing ITM 80.40% 19.60%2,466.05 94.813 28.7592,488.07 112.359 24.2882,510.09 131.006 20.9172,532.11 150.551 18.444


Strike Price = 2500 Expire on 27th Oct' 05

Index AnalysisIndex Option Price Time Value Snapshot Calls Puts

2,420.55 4.041 4.041 Price 21.495 31.762,433.79 6.041 6.041 Delta 0.443 -0.5572,447.03 8.744 8.744 Gamma 0.006 0.0062,460.27 12.275 12.275 Theta -1.529 -1.0712,473.52 16.74 16.74 Vega 1.626 1.6262,486.76 22.216 22.216 Rho 0.297 -0.3862,500.00 28.744 28.744 Elasticity 51.26 -43.552,513.24 36.322 23.08 Position OTM ITM

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2,526.48 44.906 18.423 Probability of closing ITM 43.30% 56.70%2,539.73 54.42 14.6952,552.97 64.759 11.7922,566.21 75.803 9.5952,579.45 87.429 7.979





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Strike Price = 2600 Expire on 27th Oct' 05


2,517.37 4.202 4.202 Price 1.433 111.5152,531.14 6.282 6.282 Delta 0.052 -0.9482,544.91 9.093 9.093 Gamma 0.002 0.0022,558.69 12.765 12.765 Theta -0.355 0.1212,572.46 17.409 17.409 Vega 0.476 0.4762,586.23 23.105 23.105 Rho 0.036 -0.6752,600.00 29.894 29.894 Elasticity 90.62 -21.122,613.77 37.774 24.003 Position OTM ITM2,627.54 46.702 19.16 Probability of closing ITM 4.90% 95.10%2,641.31 56.596 15.2822,655.09 67.349 12.2642,668.86 78.835 9.9792,682.63 90.926 8.298


34






Strike Price = 2600 Expire on 29th Dec' 05


2,415.24 10.34 10.34 Price 24.944 116.042,446.03 15.598 15.598 Delta 0.282 -0.7182,476.83 22.679 22.679 Gamma 0.002 0.0022,507.62 31.863 31.863 Theta -0.621 -0.1482,538.41 43.366 43.366 Vega 3.138 3.138

35

2,569.21 57.316 57.316 Rho 0.937 -2.592,600.00 73.744 73.744 Elasticity 28.05 -15.392,630.79 92.581 61.787 Position OTM ITM







36

Strike Price = 2700 Expire on 29th Dec' 05




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Implied Volatility

The Implied Volatility study uses the value of options on an underlying instrument to

estimate the underlying instrument's volatility. To calculate the implied volatility you need to

know the following information:

The price of the underlying instrument

The market price of an option

The strike price of an option

The expiration date of an option

The interest rate, if applicable

Finding out the Value of Call Option based on Implied Volatility

Strike Price

Expiry Month

Market Price of Call Options

Implied VolatilityValue of Call Option

based on IV

Rs. % Rs.

2400.00 Oct 98.75 24.00 102.07

2400.00 Nov 115.00 18.68 134.10

2500.00 Oct 36.10 24.99 39.40

2500.00 Nov 69.70 24.79 76.00

2600.00 Oct 10.55 27.53 10.00

2600.00 Nov 37.55 26.86 37.94

2600.00 Dec 80.00 32.24 61.55

2700.00 Nov 18.45 28.03 16.60

2700.00 Dec 41.00 29.26 34.34

Implied Volatility % 27.01

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Implied Volatility is arrived based on the formula given below used in Excel Sheet:

39

Analysis

Using the Black-Scholes formula with the option price known from market data, it is

possible to solve for s if all other parameters are known. In estimating the implied volatility,

the solution is done by approximation. Trial and error in the spreadsheet, the bisection

method or the Newton-Raphson method can be used.

The implied volatility seems to be more closely related to the option strike price than

the time to maturity. This illustrates the phenomenon of the “volatility smile” seen in market

pricing of options. For the three option contracts with strike price of 2600 we take the

average volatility. Linear interpolation is used for strike prices between those of the market

priced options.

It appears that estimates of s based on historical data may be less appropriate for use

in the option pricing formula when the strike price is significantly different from the current

stock price. On the other hand, implied volatility values become suspect when extrapolating

beyond the range of strike prices currently being traded in the market. Correct volatility

values are likely to lie somewhere between the two.

It is observed that the implied volatility of the option is rarely equal to and often

persistently higher than the historical volatility of the underlying asset. And the implied

volatilities differ significantly among options on the same underlying asset with different

striking prices and expirations, which are often described as volatility smile, volatility smirk

and volatility term structure. This discrepancy in the implied volatilities among the options

on the same underlying asset is theoretically a risk less arbitrage opportunity. Since the

volatility smile has been persistent with respect to all the solutions arrived from the Black-

Scholes PDE, it is conclusive that the option market is inefficient.

The discrepancy is less in short term options to that of long term options. So, we can

consider the Black-Scholes model as one of the viable models in arriving option prices,

beyond market imperfection.

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CONCLUSION

The use of the Black-Scholes formula is pervasive in the markets. In fact the model

has become such an integral part of market conventions that it is common practice for the

implied volatility rather than the price of an instrument to be quoted. (All the parameters in

the model other than the volatility - that is the time to maturity, the strike, the risk-free rate,

and the current underlying price - are unequivocally observable. This means there is one-to-

one relationship between the option price and the volatility.) Traders prefer to think in terms

of volatility as it allows them to evaluate and compare options of different maturities, strikes,

and so on.

It is observed from the calculations that the implied volatility of the option is rarely

equal to and often persistently higher than the historical volatility of the underlying asset.

And the implied volatilities differ significantly among options on the same underlying asset

with different striking prices and expirations, which are often described as volatility smile,

volatility smirk and volatility term structure. This discrepancy in the implied volatilities

among the options on the same underlying asset is theoretically a risk less arbitrage

opportunity. Since the volatility smile has been persistent with respect to all the solutions

arrived from the Black-Scholes PDE, it is conclusive that the option market is inefficient. So,

we can consider the Black-Scholes model as one of the viable models in arriving option

prices, beyond market imperfection.

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REFERENCES

1. “Black-Scholes Model is Right, Option Market is Inefficient: A Robust

Proof”, by Chen Guo, Division of Banking & Finance, Nayang Business

School, NTU – pdf file

2. A power point presentation file on “Option Pricing and Implied Volatility”

from www.utstat.toronto.edu/pub/sam2/ brov4 .ppt

3. A power point presentation files on “Options Pricing” by Charles J

CORRADO and Bradlord D JORDAN.

4. Option Volatility & Pricing by Sheldon Natenberg

5. www.hoadley.net

6. www.cboe.com

7. www.nseindia.com

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black schole project

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