Principles of Option Pricing

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Outline

Minimum values of calls and putsMaximum values of calls and putsValues of calls and puts at expirationEffect of Exercise Price, Time to Maturity,

Interest Rates, VolatilityAmerican versus European Style OptionsPut-Call Parity

Options

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

– An option to sell as asset at a predetermined price (also known as exercise price) is known as put option

Types of Options

American-styleEuropean-styleAsian Options

Options

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

In-Money Calls and Puts– Call is in the money if S > E– Put is in the money if E > E

Out of Money Calls and Puts– Call is out of money if S < E– Put is out of money if E < S

Minimum Value of a Call Option

A call cannot have a negative valueThus, C(S,T,E) 0For American Calls, Ca Max [0,(S-E)]Minimum value also called intrinsic valueIntrinsic value positive for in-the-money

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

intrinsic value—Why?

Maximum Value

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

C S

Lower Bound of a European Call

Ce Max [0,{S – E(1+r)-T}]What if

– Ce < Max [0,{S – 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 Effect of Time to Expiration on a Call Option

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 Effect of Exercise Price and Call Value

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(S, Elow,T) Ce(S, 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(S, Elow,T) Ca(S, 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(S, Elow,T) -Ca(S, 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(S, Elow,T) -Ce(S, Ehigh, T) (Ehigh – Elow)(1+r)-T

Interest Rates and Calls

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

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

American Call versus European Call

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

Early Exercise of American Call

Early Exercise of a call on a dividend paying stock

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

Minimum Value of a Put

A put cannot have a negative value– P(S,T,E) 0

An American put can be exercise early. Therefore, Pa Max [0, (E-S)]

Minimum value also known as intrinsic value

Lower bound of a European Put– Pe Max [0, E(1+r)-T – S]

Maximum Value of a Put

At expiration, the payoff from a Put is Max [0, (E-S)]. What is the best outcome that a put holder can expect at put expiration?

Maximum value of American put EMaximum value of European put E(1+r)-T

Time to Expiration and Put Value

Longer lived American put must always be worth at least as much as a shorter-lived American put with the same terms.

Time and Put Price—more complex

Exercise Price and Put

The price of a European put must be at least as high as the price of otherwise identical European put with a lower exercise price

– What if Pe(S,Ehigh,T) < Pe(S,Elow,T)

The price of an American put must be at least as high as the price of an otherwise identical American put with a lower exercise price.

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

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

Interest Rates and Puts

Put options vary inversely with put interest rates—Why?

A put is like deferring the sale of stock. When you sell the stock, you will receive E dollars. If interest rates rise, the present value of E dollars will be lower.

Early Exercise of American Puts

Does it pay to exercise an American put before maturity?

If the put is deep in-the-money, it should be exercised early even if the stock does not pay dividend

Value of Put at Maturity

No time value leftP = Max (0, E-ST)

Put-Call Parity

The prices of European puts and calls on the same stock with identical exercise prices and expiration dates have a special relationship. The put price, call price, stock price, exercise price, and risk-free rate are all related by a formula called put-call parity