Options and Derivatives

For 9.220, Term 1, 2002/0302_Lecture17 & 18.pptStudent Version

Outline

1. Introduction2. Option Definitions3. Option Payoffs4. Intuitive Option Valuation5. Put-Call Parity6. Summary and Conclusions

Introduction We now turn our attention to options. Almost every decision can be framed in terms

of an option. The more we understand options, the more we can understand the valuation implications of decisions, flexibility and information.

An understanding of options will also help us understand more about what affects the values of equity and debt.

We will start our analysis with financial options, and then later turn to real options and corporate securities as options.

Option Definitions

Derivative Security: a derivative security is a financial

security that is a claim on another security or underlying asset.

Examples include options, warrants, rights, futures, swaps, convertibles, etc.

The value of the derivative is dependent on the value of the underlying asset.

Option If you have an option, then you have the

right to do something. I.e., you can make a decision or take some action. The option owner has a choice to make. Usually the choice can be made over time after

more information is known. Having an option is valuable; options never have

a negative value, because, at worst, the option owner can discard the option and not take any action.

Call Option The owner of a call option on an asset has the RIGHT TO

BUY the asset (e.g., share of stock) for a pre-specified price (called the exercise or strike price) during some time period (that ends with the expiration date or maturity date). The call option owner is said to be “long” in the option.

The owner possesses the right to do something (the right to buy).

The call option issuer (called the call option writer) is said to be “short” in the option. If the call option owner exercises his/her option, then the call option writer has an obligation to sell the asset and must fulfill the terms of the contract.

Put Option The owner of the put option has the RIGHT

TO SELL an asset (e.g., share of stock) at a prespecified price (the exercise or strike price) during some time period. As with call options, The option owner is said to

be “long” in the option. The put owner possesses the right to do something (the right to sell).

The put option issuer (called the put option writer) is said to be “short” in the option. If the put option owner exercises his/her option, then the put option writer has an obligation to buy the asset and must fulfill the terms of the contract.

Option Writer The option writer is the person who created (or wrote)

the option and then sold it to someone else. As the writer did not previously own the option, the act of writing and selling the option is equivalent to short-selling the option.

The option writer has sold an option or right to the option owner. Thus the writer has taken on an obligation to act on

the instruction of the option owner as specified by the option contract.

When selling the option to the buyer, the writer receives as compensation the “option premium”= the price the buyer pays for the option.

American and European Options

American options can be exercises at any time up to and including the expiration date.

European options can only be exercised on the expiration date All else equal, an American option must

be worth at least as much as a European Option

I’m in the money An option is said to be “in the money”

if exercising it would produce a positive payoff.

An “at the money” option would generate a zero payoff if exercised.

An “out of the money” option would generate a negative payoff if exercised. (Thus an out of the money option would never be exercised.)

Option Payoffs at Expiration It is useful to examine the payoffs of

options when they are about to expire. At this point in time, the owner is forced to make the decision to exercise or to abandon the option. The option owner will exercise if the option is in the money when it is about to expire.

Consider Call and Put Options…

Stock Price

Should call be exercised? (E=$50)

Payoff

$0

$25

$50

$55

$75

Long Call Option Payoff at Expiration(Option to Buy for Exercise Price = E)

Long Call Option Payoff at Expiration

If the call is in-the-money, it is worth ST - E. If the call is out-of-the-money, it is worthless. Therefore the value of the call when it is about to

expire is as follows:CaT = CeT = Max[ST - E, 0]

This assumes investors are rational and will not exercise to get a negative payoff.

WhereST is the value of the stock at expiry (time T)E is the exercise price.CaT is the value of an American call at expiry

CeT is the value of a European call at expiry

Long Call Option Payoff at Expiration

-20

100908070600 10 20 30 40 50

-40

20

0

-60

40

60

Stock price ($)

Op

tio

n p

ayo

ffs

($)

Exercise price = $50

Stock Price

Should call be exercised? (E=$80)

Payoff

$0

$20

$40

$60

$80

$100

$120

$140

$160

Self Study – fill in the blank cellsLong Call Option Payoff at Expiration(Option to Buy for Exercise Price = E)

Short Call Option Payoff at Expiration

Stock Price

Will Owner exercise the Call? (E=$50)

Payoff to Writer

$0

$25

$50

$55

$75

Short Call Option Payoff at Expiration

-20

100908070600 10 20 30 40 50

-40

20

0

-60

40

60

Stock price ($)

Op

tio

n p

ayo

ffs

($)

Exercise price = $50

Short Call Payoff at Expiration

The short call payoff is just the negative of the long call payoff.

The payoff of the short call when it is about to expire is as follows:

-CaT = -CeT = -Max[ST - E, 0]

= Min[E-ST, 0]

Stock Price

Will Owner exercise the Call? (E=$80)

Payoff to Writer

$0

$20

$40

$60

$80

$100

$120

$140

$160

Self Study – fill in the blank cellsShort Call Option Payoff at Expiration(Option to Buy for Exercise Price = E)

Stock Price

Should put be exercised? (E=$50)

Payoff

$0

$25

$50

$55

$75

Long Put Option Payoff at Expiration(Option to Sell for Exercise Price = E)

Long Put Payoff at Expiration

At expiry, an American put option is worth the same as a European option with the same characteristics.

If the put is in-the-money, it is worth E – ST if exercised.

If the put is out-of-the-money, it is worthless. Thus the long put payoff when the put is about to expire is as follows:

PaT = PeT = Max[E - ST, 0] Where

PaT is the value of American Put at expiryPeT is the value of European Put at expiry

Long Put Option Payoff at Expiration

-20

100908070600 10 20 30 40 50

-40

20

0

-60

40

60

Stock price ($)

Op

tio

n p

ayo

ffs

($)

Exercise price = $50

Stock Price

Should put be exercised? (E=$80)

Payoff

$0

$20

$40

$60

$80

$100

$120

$140

$160

Self Study – fill in the blank cellsLong Put Option Payoff at Expiration(Option to Sell for Exercise Price = E)

Stock Price

Will put owner exercise? (E=$50)

Payoff to Writer

$0

$25

$50

$55

$75

Short Put Option Payoff at Expiration

Short Put Option Payoff at Expiration

-20

100908070600 10 20 30 40 50

-40

20

0

-60

40

60

Op

tio

n p

ayo

ffs

($)

Exercise price = $50

Stock price ($)

Short Put Payoff at Expiration

The short put payoff is just the negative of the long put payoff.

Thus the short put payoff when the put is about to expire is as follows:

-PaT = -PeT = -Max[E - ST, 0]

= Min[ST-E, 0]

Stock Price

Will Owner exercise the Put? (E=$80)

Payoff to Writer

$0

$20

$40

$60

$80

$100

$120

$140

$160

Self Study – fill in the blank cellsShort Put Option Payoff at Expiration(Option to Sell for Exercise Price = E)

Call Option Values with time left until expiration. In the previous slides, we saw the payoffs

for options at the time of expiration. If the option was expiring, these payoffs for

the “long positions” would represent the value of the options at expiration.

If there is significant time left before the option expires, then the value of the option is different than the previous payoff amounts.

The following slides examine call options and explain their valuation when there is time left before expiration.

Boundaries on an American call option’s value

In the best outcome, the result of owning a call option is that you will eventually own the underlying asset.

If the underlying asset is the share of stock, then the most the call option could be worth is the current market price of the share of stock.

This gives us an upper bound on the value of the call option.

Upper Bound on the Value of an American Call

$0

$25

$50

$75

$100

$125

$150

$175

$200

$0 $25

$50

$75

$100

$125

$150

$175

$200

$ Current Market Price of Stock

$ V

alu

e

Lower Bound on the Value of an American Call

Can an American call option sell for a price less than the payoff from exercising?

Consider, for example, if E=100, St=$120, and Cat=$5 then

Upper and Lower Bounds on the American Call Value

$0

$25

$50

$75

$100

$125

$150

$175

$200

$0 $25

$50

$75

$100

$125

$150

$175

$200

$ Current Market Price of Stock

$ V

alu

e

What is the actual value of the American Call?

The actual value of the American Call depends on how much time there is until it expires, the underlying stock’s total risk (σ), the risk-free interest rate over the life of the call, and obviously the stock price and exercise price.

Actual Value of the American Call

$0

$25

$50

$75

$100

$125

$150

$175

$200

$0 $25

$50

$75

$100

$125

$150

$175

$200

$ Current Market Price of Stock

$ V

alu

e

Understanding the Value of the American Call

$0

$25

$50

$75

$100

$125

$150

$175

$200

$0 $25

$50

$75

$100

$125

$150

$175

$200

$ Current Market Price of Stock

$ V

alu

e

1st Consider this area

Next, consider this area

Finally, consider this

area

$0

$25

$50

$75

$100

$125

$0 $25

$50

$75

$100

$125

$150

$175

$200

$ Current Market Price of Stock

$ V

alu

e

Change in Call Option Value as the Time until Expiration Decreases

$0

$25

$50

$75

$100

$125

$0 $25

$50

$75

$100

$125

$150

$175

$200

$ Current Market Price of Stock

$ V

alu

e

Change in Call Option Value as the Risk of the Underlying Asset Decreases

Change in Call Option Value as the Risk-free Interest Rate Decreases

$0

$25

$50

$75

$100

$125

$0 $25

$50

$75

$100

$125

$150

$175

$200

$ Current Market Price of Stock

$ V

alu

e

Summary of Factors that Affect the American Call Option Value

Factor Effect on value of the American call option

A stock price increase Value rises

An increase in the exercise price

Value decreases

An increase in the stock’s risk Value increases

A decrease in the time left until expiration

Value decreases

A decrease in the risk-free rate

Value decreases

Summary of Factors that Affect the American Put Option Value

Factor Effect on value of the American put option

A stock price increase Value decreases

An increase in the exercise price

Value increases

An increase in the stock’s risk

Value increases (same as for call)

A decrease in the time left until expiration

Value decreases(same as for call)

A decrease in the risk-free rate

Value increases (PV of exercise price received is higher)

Questions for Discussion

1. Assuming there are no dividends paid, would it ever be optimal to exercise an American call option early (i.e., before the expiration date)?

What does this imply about the values of American and European call options?

2. Assuming there are no dividends paid, would it ever be optimal to exercise an American put option early?

What does this imply about the values of American and European put options?

Put-call Parity Given the following securities: a call option, a put

option, the underlying stock, and risk-free borrowing or lending; it is possible to replicate the payoffs from one security by holding a combination of the other three.

To prevent arbitrage, it must hold that the value of the replicated security equals the sum of the values of the components used to replicate it.

This leads to a relationship between put and call options (which holds precisely for European options). The relationship is called Put-Call Parity

Synthetic Security

Definition: A synthetic security is a portfolio of other securities which will have the same cash flows as the “original” security being copied.

To determine the relationship between put and call options, we can construct a synthetic call and equate its value to a real call option.

Construction of a Synthetic European Call: initial transactions at date t

where St is the stock price at time t Pet is the price at time t of the European put option

Initial Transactions: buy 1 share of stock (long the stock) -St

borrow the present value of the exercise price (E) at the risk free rate (or short the risk-free asset)

+(PV of E)

buy (long) the put option with the same underlying stock, exercise price and expiration date as the call option

-Pet

Initial net cash flow (will be an outflow): ‑St + (PV of E) ‑ Pet

Synthetic European Call: transactions on the expiration date (T)

Final cash flows given the different relevant states of nature (which depend on whether ST is less than or greater than E):

ST < E ST ≥ E

liquidate the long stock position (sell the stock)

+ST +ST

liquidate the short-risk free asset position (repay the loan)

‑E ‑E

liquidate the long put option position (discard or exercise depending upon which is optimal)

+(E‑ST) 0

Net cash flow at the expiration date T: 0 +(ST‑E)

Potential arbitrage strategies: If the “real” call is more expensive than the synthetic call, we

could go long a synthetic call and write a real call. If the real call is less expensive, we could short a synthetic call and buy a real call. These strategies create a positive cash flow at date t.

Under either strategy, the final net cash flows at date T (or final payoffs) are guaranteed to be zero under all relevant states of nature.

The above strategies would yield a positive initial cash inflow followed by zero cash flows with no risk. All investors would try to exploit the arbitrage opportunity. As they did so, the prices of the synthetic and real calls would adjust (until they became equal) and the arbitrage opportunity would disappear.

The concept of no‑arbitrage can be used to show the relationship between the prices of European put and call options by equating the price of a synthetic call to the price of a real call.

Put-Call Parity ConditionReal Call Synthetic Call

Cash flows from purchasing (or constructing) the option

-Cet ‑St + (PV of E) ‑ Pet

Equating the cash flows and rearranging terms, we get the following relationships:

-Cet=‑St + (PV of E) ‑ Pet

Cet=St - (PV of E) + Pet

Pet = Cet-St + (PV of E)

Where Cet is the cost of the European call option at time t

St is the stock price at time t

Pet is the cost of the European put option at time t

PV of E is the present value at time t of the exercise priceNote: The put-call parity condition holds precisely for European options. However, because it may be optimal to exercise an American put early, the value of the American put will be higher than what is determined above.

Self Study – fill in the blank cellsConstruction of a Synthetic European Put: initial transactions at date t

where St is the stock price at time t Cet is the price at time t of the European call option

Initial Transactions: fill in the empty cells short 1 share of stock Invest the present value of the exercise

price (E) at the risk free rate (or long the risk-free asset)

Buy a call option on the stock with same exercise price and same expiration date

Initial net cash flow (will be an outflow):

Self Study – fill in the blank cellsSynthetic European Put: transactions on the expiration date (T)

Final cash flows given the different relevant states of nature (which depend on whether ST is less than or greater than E):

ST < E ST ≥ E

liquidate the short stock position (buy the stock)

liquidate the long risk-free asset position (collect the proceeds from the investment)

liquidate the long call option position (discard or exercise depending upon which is optimal)

Net cash flow at the expiration date T: E-ST 0

Put-Call Parity Example Suppose a call option on Macrosoft is

currently selling for $9.50. The call has E=$75 and 6 months until expiration. The risk-free rate of interest is 4% per year effective and Macrosoft is currently trading for $76 per share.

What is the value of a European put option on Macrosoft (with the same terms as the call: E=$75 and 6 months until expiration)?

If a European put with E=$80 and 6 months until expiration is selling for $9.95, what is the equivalent call value?

Self Study – calculate the bolded cellsPut-Call Parity Examples

Cet Pet S E

$9.30 $5.01 $92 $90

$7.54 $7.25 $88 $90

$9.84 $4.80 $83 $80

$6.83 $7.79 $77 $80

$11.30 $4.51 $75.00 $70

$5.03 $8.25 $65 $70.00

Notes:

•Rf=3.5% per year effective

•Expiration is in 9 months for each option

Summary and Conclusions The two main types of financial options are calls (option to buy)

and puts (option to sell). The factors that affect the value of options are the underlying

asset price, the volatility of the underlying asset returns, the exercise price, the risk-free rate, and the time until expiration of the option.

We can construct synthetic options, and from these we can relate the prices of put and call options (the Put-Call Parity Condition).

As all options have value, understanding what impacts the value of an option is important and will help us understand other finance concepts that contain option characteristics. Including the option to continue, expand, contract or abandon a

capital budgeting project. Special option-like features that are part of other financial

securities Including call and conversion features of bonds and preferred stock,

the limited liability feature of common stock, etc.