Contents

1. Options : Introduction

11.1 KEY ELEMENTS OF OPTIONS

11.2 OPTIONS STYLES

11.3 TYPES OF OPTIONS

11. Call Option

2. Put Option

1.4 Option Positions

21.5 Underlying Assets of Options

21.Exchange Traded Options

2. Over- the- counter Options

1.6 Option Strategies

31.Protective Put

2. Covered Call

3. Straddle

4. Spread

5. Collar

1.7 Put- Call Relationship

71.Put Parity Theorem

1.8 Valuation Models

71.Binomial Option Pricing

2.Black-Scholes Option Pricing

1.9 Implied Volatility

OPTIONS: Introduction

An option is a derivative financial instrument. It is a contract between two parties for a future transaction on an asset at a reference price. The buyer of the options gains the right, but not the obligation, to engage in that transaction, while the seller incurs the corresponding obligation to fulfill the transaction.

1.1 KEY ELEMENTS OF OPTIONS1) The date specified in the contract is known as the Expiration date, the exercise date, the strike date or the maturity. If the option is not exercised by the expiration date, it becomes void and worthless2) The reference price at which the underlying asset may be traded specified in the contract is known as the exercise price or strike price.

3) The process of activating an option and thereby trading the underlying at the agreed-upon price is known as exercising the option.

4) Premium is the purchase price of option (paid when the option is purchased, regardless of whether the option is exercised or not)

5) The writer is the person who initially sells the option1.2 OPTIONS STYLES: 1) American options : they can be exercised at any time upto the expiration date

2) European options: they can be exercised only on the expiration date.

1.3 TYPES OF OPTIONS

There are two basic types of options.

1) Call Options : A contract whose holder (buyer) has the right, but not the obligation, to buy

the underlying asset at a fixed price (exercise price E), on or before the expiration date.

For the holder of the call option,

The net profit is Profit = Value at expiration Premium (C)

Lets take an example:An investor buys a European call option with Strike price of $100 to purchase 100 shares Current stock price = $98 Expiration date of option = four months. The price of an option to purchase on share is $5. The initial investment is $500. Since the option is European, the investor can exercise only on expiration date. If the stofk price on this date is less that $100, the investor will choose not to exercise it. So, the investor loses the whle of initial investment of $500. If the stock price is $115. By exercising the option, the investor is able to buy 100 shares for $100 per share. If shares are sold, the investor makes a gain of $15 per share or $1500, ignoring transaction cost. So, the net profit of the investor is $1000.

But if the Microsofts stock price is $102 at the expiration date. The investor would exercise the option at a gain of 100 X ($102 - $100) = $200. And there is a loss of $300. If investor didnt exercise the option, there would be an overall loss of $500, which is worse than the $300 loss. So, call options should always be exercised at the expiration date if the stock price is above the strike price.

Fig. 1. CALL OPTIONS : FOR HOLDER (2) Put Option: A contract whose holder (buyer) has the right to sell the underlying asset at a price before expiration time. The purchaser of a put option hopes that the stock price will decrease. Profits for the holder of the put option,Tthe net profit is Profit = Value at expiration Premium (P)Lets take an example:

A European put option is to sell 100 shares in Oracle with a strike price of $70. Suppose the current stock price is $65, the expiration date of the option is in three moths, and the price of an option to sell one share is $7. The initial investment in $700. Suppose the stock price on expiration date is $55. The investor can buy 100 shares for $55 per share and sell the same shares for $70 to realize a gain of $15 per share or, $1500. So, the net profit is $800. If the final stock price is above $70, the put option is worthless, and the investor loses $700.

Fig 2. Put Options : Holder

1.4 OPTION POSITIONSThere are two sides to every option contract. On one side is the investor who has taken the long position (i.e has bought the option) and other side is the investor who has taken the short position (i.e. has sold or written the option). The writer of an option receives cash up front. The writers profit or loss is exactly opposite of that for the purchaser of the option. There are four types of option positions:

(i) A long position in a call option (ii) a long position in a put option

(ii) A short position in call option (iv) A short position in put option

The terminal value or the payoff to the investor at maturity:

If X is the strike price and S(T) is the final price of the underlying asset,

The payoff from a long position in a European call option is

Payoff to call holder = S(T) X , if S(T) > X

= 0 , if S(T) < X

The option will be exercised if S(T) > 0 and will not be exercised if S(T) < 0

The payoff to the holder of a short position in a European Put Option is

Payoff to put holder = 0 , if S(T) > X

= X S(T) , if S(T) < X

1.5 THE UNDERLYING ASSETS OF OPTIONSThe Options can be classified into following types :

(1) Exchange- traded Options: These are also called listed Options. They are a class of exchange-traded derivatives. Exchange-traded Options have standardized contracts, and are settled through a clearing house with fulfillment guaranteed by the credit of exchange. They also have accurate pricing models. These include:

(i) Stock options

(ii) Bond options and other interest rate options

(iii) Stock market index options

(iv) Options on futures contracts

(v) Callable bull/bear contract

(2) Over-the counter Options: They are also called dealer Options. They are traded between two private parties that are not listed on an exchange. The terms of these options are unrestricted and may be individually tailored to meet any business need. Some of the commonly traded over the counter options include: (i) Interest rate Options (ii) Currency cross rate Options (iii) Options on Swaps1.6 OPTION STRATEGIES(1) PROTECTIVE PUT :

An investor who purchases a put option while holding shares of the underlying stock from a previous purchase is employing a Protective Put.

The investor employing the protective put strategy owns shares of underlying stock from a previous purchase, has unrealized profits accrued from an increase in value of those shares. The main concern is about unknown, downside market risks in the near term and wants some protection for the gains in share value. Purchasing puts while holding shares of underlying stock is a bullish directional strategy.The value of protective put at expiration:

S(T) < X

S(T) > X

Payoff of Stock

S(T)

S(T)

Payoff of Put

X S(T)

0

Total Payoff

X

S(T)

Fig 3. Protective Put Profit of Holder(2) COVERED CALL

The covered call is a strategy in which an investor writes a call option contract while at the same time owning an equivalent number of shares of the underlying stock. If this stock is purchased simultaneously with writing the call contract, the strategy is referred to as a BUY WRITE. If the shares are already held from a previous purchase, it is referred as OVERWRITE. This strategy combines the flexibility of listed options with stock ownership.This strategy is most often employed when the investor, while bullish on the underlying stock, feels that its market value will experience little range over the lifetime of the call contract. The investor desires to either generate additional income from shares of underlying stock, or provide a limited amount of protection against a decline in underlying stock value.

Value of Covered Call at Expiration:

S(T) < X

S(T) > X

Payoff of Stock

S(T)

S(T)

Payoff of Call

0

- (S(T) X)

Total Payoff

S(T)

X

Fig 4 .Covered Call Profit of Holder(3) STRADDLE :

Straddle is the simultaneous purchase of a call and a put on the same underlying security with both options having the same expirations and same strike price. Since straddle includes both a call and a put, the investor should have a complete understanding of the risks and rewards associated with both calls and puts. This strategy may prove beneficial when the investor feels large price movement, either up or down, is imminent but is uncertain of the direction. Value of Straddle at Expiration

S(T) < X

S(T) > X

Payoff of Call

0

S(T) - X

Payoff of Put

- ( S(T) X)

0

Total Payoff X S(T)

S(T)- X

Fig. 5. Straddle Profit of Holder

(4) SPREADS

This involves the purchase of combinations of two or more call options (or Put options) on the same stock, with different exercise prices or expiration dates. The money spread is equal to the difference between options exercise price. The time spread is the difference between options is expiration date. Value of Spread at Maturity:

S(T) < X1

X1 < S(T) < X2

S(T) > X2Payoff of on Call 1

0

S(T) X1

S(T) X1Payoff of on Call 2

0

0

- (S(T) X2)Total Payoff 0

S(T) X1

X2 X1

Fig. 6 Spread Calls Profits and Payoffs Fig 7. Profit of Holder

(5) COLLAR A collar can be established by holding shares of a stock, purchasing a protective put and writing a covered call on that stock. The option portion of this strategy is referred to as combination. The primary concern in employing a collar is protection of profits accrued from underlying shares rather than increasing returns on the upside. Investors use this strategy after accruing unrealized profits from the underlying shares, and want to protect these gains with the purchase of a protective put.Value of Collar at Maturity:

S(T) < X1

X1 < S(T) < X2

S(T) > X2

Payoff of on Stock

S(T)

S(T)

S(T)

Payoff on Put

X1 S(T)

0

0Payoff of on Call 2

0

0

- (S(T) X2)

Total Payoff X1

S(T)

X2

The Payoff and the profit of the holder are equal since the premiums cancel out.

Fig 8. Collar Calls Profits and Payoffs1.7 PUT CALL PARITY RELATIONSHIPThis is an alternative strategy that provides the same type of protection as a protective put is a call with same expiration date and strike price X and a riskless bond with face value equal to X.

Value of Investment at Maturity :

S(T) < X

S(T) > X

Payoff of Stock

0

S(T) - X

Payoff of Put

X

X

Total Payoff

X

S(T)

This is exactly the same payoff pattern as the protective put.

Put Call Parity Theorem: This is applicable only to European Options because they are exercised only at maturity. Arbitrage Argument : According to this argument, if two investments always have the same value, they should have the same price. The price of the protective put is the sum of put premium and stock price at time 0. The sum of the price of the call and bond investment is the total of the call premium and the present discounted value of the bond (i.e. of X)

So, C + X/ (1+ r(f))^T = S(0) + P

1.8 VALUATION MODELS:

The value of an option can be estimated using a variety of quantitative techniques based on concept of risk neutral pricing and using stochastic calculus.

1) Binomial Option Pricing: This illustrates fundamental issues in option pricing. This shows how option price can be derived from no-arbitrage-profits condition.

Let, S = current stock price

u = 1 + fraction of change in stock price if price goes up

d = 1 + fraction of change in stock price if price goes down

r = risk free interest rate.

C= current price of call option

C(u) = value of call next period if price is up

C(d)= value of call next period if price is down

E = strike price of option

H = hedge ratio, number of shares purchased per call sold.

Investor writes one call and buys H shares of underlying stock.

If price goes up, it will be worth uHS C(u)If the price goes down, it will be worth dHS C(d)

H = C(u) C(d) / (u-d) S

If an investor invested HS C to achieve riskless return, the return must equal to (1+r)(HS C)

(2) Black Scholes Option Pricing:Fischer Black and Myron Scholes derived continuous time analogue of binomial formula, continuours trading for European Options only. The Black-Scholes continuous arbitrage is not really possible. Call T the time to exercise, 2 the variance of one-period price change (as fraction) and N(x) the standard cumulative normal distribution function (sigmoid curve, integral of normal bell-shaped curve) =normdist(x,0,1,1) Excel (x, mean,standard_dev, 0 for density, 1 for cum.)

3. Implied Volatility : Turning around the Black-Scholes formula, one can find out what would generate current stock price. depends on strike price, options smile

References1. Fundamentals of Futures and Options Markets By John C . Hull

2. Options Markets : Introduction

3. Futures, options and swaps By Robert W. Kolb

4. Understanding Options By Michel Sincere

EMBED Equation.3

Payoffs

Payoff = Value at Expiration

Net Profit

C

X

S(T)

Payoff

X

P

Net Profit

Profit = Value at Expiration

S(T)

Protective Put Payoff

Protective Put Profit

Payoff

Stock

Put Payoff

X

X (S(T) + P)

S(T)

Payoff

Stock

Covered Call Payoff

Covered Call Profit

X

X

Call Payoff

(X+C) S(0)

S(T)

Payoff

Straddle Payoff)

Call

Straddle Profits

S(T)

Put

X

-(P+C)

-C

-P

X P = C

X

Call sold

Call Bought

Payoff

C2

-C1

X1

X2

X2 X1

C2 C1

X1

X2

Spread Profit

Spread Payoff

S(T)

-P = - C

Put Bought

X2

X1

S(T)

Call Sold

C = P

Payoff

Stock

Payoff

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