Unit 3(a)

Price of option – Fair price and Market price

Principle of call option pricing:

o Minimum value of a callo Maximum value of a callo Value of a call at expirationo Lower bound of callo European vs. American Call

Factors Affecting Call Price

o Effect of Exercise Priceo Time to Maturityo Interest Rateo Stock Volatility

American versus European Style Options

Early exercise of American call on dividend paying stock

and non-dividend paying stock

A contract between two parties—a buyer and a seller/writer—in which the buyer purchases from the seller/writer the right to buy or sell an asset at a fixed price. The buyer pays the seller a fee called the premium, which is the option’s price.

◦ An option to buy an asset at a predetermined price (also known as exercise price) is known as the call option

American-style:

An option that can be exercised anytime during its life. The majority of exchange-traded options are American.

Since investors have the freedom to exercise their American options at any point during the life of the contract, they are more valuable than European options which can only be exercised at maturity.

European-style:

An option that can only be exercised at the end of its life, at its maturity. European options tend to sometimes trade at a discount to its comparable American option. This is because American options allow investors more opportunities to exercise the contract.

European options normally trade over the counter (OTC), while American options usually trade on standardized exchanges. A buyer of an European option that does not want to wait for maturity to exercise it can sell the option to close the position.

In-the-Money Option◦ One that would lead to positive cash flows to the

holder if it were exercised immediately

At-the-Money Option◦ One that would lead to zero cash flows to the

holder if it were exercised immediately

Out-of-Money Option◦ One that would lead to negative cash flows to the

holder if it were exercised immediately

Thank You.

Fair Price:

It is a concept defined as a rational and unbiased estimate of the potential market price of Option. There are several options pricing models that use to determine the fair market value of the option. Of these, the Black-Scholes model is the most widely used.

Market Price:

The current price at which an option can be bought or sold. Economic theory contends that the market price converges at a point where the forces of supply and demand meet. Shocks to either the supply side and/or demand side can cause the market price to be re-evaluated.

9

For the Disney JUN 22.50 Call buyer:

-$0.25

$22.50

$0

Maximum loss

Breakeven Point = $22.75

Maximum profit

is unlimited

10

For the Disney JUN 22.50 Call writer:

$0.25

$22.50

$0

Maximum profit Breakeven Point = $22.75

Maximum loss

is unlimited

11

For the Disney JUN 22.50 Put buyer:

-$1.05

$22.50

$0

Maximum loss

Breakeven Point = $21.45

Maximum profit = $21.45

Minimum Value of a Call Option

A call cannot have a negative value

Thus, Vc 0 For American Calls, Ca Max [0,(Vs-E)] Minimum value also called intrinsic value Intrinsic value positive for in-the-money calls

and zero for out-of-money calls Usually, call options trade above their

intrinsic value—Why?

Call derives its value from the underlying asset/stock on which it is written. Therefore, it cannot never exceed the value of the underlying asset

Thus, Vc Vs

An option's expiration value is its market value at expiration. In the case of a call, expiration value is either:

zero, or

the difference between the value of the underlier and the strike price, whichever is greater.

If, at expiration, the underlier value is below the strike price, the option expires worthless.

Note: The value from which a derivative derives its value is called its underlier.

Market Value of a Call at Expiration

Vc Max [0,{Vs – E(1+r)-T}] exer What if Vce < Max [0,{Vs – E(1+r)-T}] leads to

arbitrage

Buy call and risk-free bonds and sell short the stock. The portfolio will have positive initial cash flow, because the call price plus the bond price is less than the stock price.

At maturity, the payoff is either E – ST if E > ST and 0 otherwise.

The lower bound for any non-dividend paying call option is: i.e. the current stock price minus the options strike price multiplied by the natural e, to the power of negative risk-free interest rate multiplied by the options time to expiry.Example:Current stock price (Vo) = $20Strike price (E) = $18 Risk-free rate ( r ) = 10% Time to expiry (T)= 1 year (T)Value of (e) = 2.718

Soln. The lower bound for this call option is $3.71

If the market is quoting the European call option at $3.00, such a price is less than the lower bound or “theoretical minimum”.

What would happen is that an arbitrageur would short the stock and then buy the call option.

This will provided the arbitrageur with a cash inflow of $20.00 – $3.00 = $17.00.

If this amount is then invested for 1 year at the market risk-free interest rate of 10% per annum, then the $17.00 will grow to 17x1.1 = $18.79.

Therefore at the end of the year, when the option will expire if the stock price is greater than $18.00 (the strike price), then the arbitrageur will be able to exercise the option for $18.00.

By doing this, it will enable them to close out the short position to make a profit of $18.79 –$18.00 = $0.79.

This means that the arbitrageur will buy back the stock at a cheaper rate than what he/she originally shorted and then invested that money at the risk-free rate, and then finally met the shorting obligations by buying the stock back with the aid of the option at the strike price.

What happens if the stock price is less than the strike price after 1 year (at time of option expiry?)

Then the arbitrageur will make an even GREATER profit. This is because he/she will be able to buy back the stock at the market price which will be even cheaper than that of the strike price to close out the shorted stock position.

For example, if the stock price after 1 year = $17.00

The arbitrageur’s profit will then = $18.79 – $17.00 = $1.79

Which is $1 more than the previous example where the stock is greater than the strike price.

The additional difference in profit is the difference between the strike price and the stock price.

An American call must be worth at least as much as a European call with the same terms.◦ Ca Ce

An American call on a non-dividend paying stock will never be exercised early, and we can treat it as if it is a European call

Time to maturity

Exercise price

Interest rate, and

Stock volatility

Two American call options differ only in their times to expirations, one with a higher time to expiration will be worth at least as much as a shorter-lived American call with the same terms

When will the longer-lived call is worth the same as shorter-lived call?

The price of a European call must be at least as high as the price of an otherwise identical European call with a higher exercise price

◦ Ce(Vs, Elow,T) Ce(Vs, Ehigh, T)

The price of an American call must be at least as high as the price of another otherwise identical American call with a higher exercise price

◦ Ca(Vs, Elow,T) Ca(Vs, Ehigh, T)

The difference in the price of two American calls that differ only by their exercise price cannot exceed the difference in their exercise prices

◦ Ca(Vs, Elow,T) –Ca(Vs, Ehigh, T) (Ehigh – Elow)

The difference in the price of two European calls that differ only by their exercise price cannot exceed the present value of the difference in their exercise prices

◦ Ce(Vs, Elow,T) –Ce(Vs, Ehigh, T) (Ehigh – Elow)(1+r)-T

Or,

Ce(Vs, Elow,T) –Ce(Vs, Ehigh, T) (Ehigh – Elow)/(1+r)T

A call option is a deferred substitute for the purchase of the stock

If the stock price is expected to rise, the investor can either choose to buy the stock or buy the call. Buying the call will cost far less than purchasing the stock. Invest the difference in risk-free bonds.

If rates rise, the combination of calls and risk-free bonds will be more attractive

Volatility gives rise to risk and need to buy insurance

Greater volatility increases the gains on the call if the stock price rises big time, and

Zero downside risk if the stock price declines big time

Zero downside risk : this is the cushion against loss, in case of a price decline by the underlying security, that is afforded by the written call option. Alternatively, an price amount equal to the option premium)

Early Exercise of a call on a dividend paying stock

The exercise of an option contract before its expiration date.

Exercise it just before the ex-dividend date if the DPS exceeds Speculative value of the call

Do not exercise it if the DPS is less than the speculative value of the call

If you purchase before the ex-dividend date, you get the dividend.

Declaration Date

Ex-Dividend Date

Record Date Payable Date

7/27 8/6 8/10 9/10

Early Exercise of a call on a non-dividend paying stock

By exercising a call option early, not only do you accept full downside risk in the stock but you also throw away the time value of that call option. To put it in a more distressing way, you throw money away in exchange for accepting more risk!

It does not matter how short of a time period you intend to hold the stock, either. Even if you plan to sell the stock the next day, you’re still at risk of some serious negative news announced before the opening bell.

It’s important to remember that, with a call option, your purchase price is locked in. It doesn’t matter how high the stock’s price may rise; you will always be able to purchase it for the strike price, so there really is no need to take delivery of the stock early.