07 Financial Option Contracts

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November 19, 2012Petra Andrlkov

[email protected]

Financial option contracts


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Dates: 7.1.2013 21.1.2013 4.2.2013

Time: 18:30 20:00

Location: Opletalova, room O109

General information about the structure and the content of thewritten examination will be uploaded on course website inadvance.

Written examination

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Theory


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An option provides the holder with the right to buy orsell a specified quantity of an underlying asset at a fixedprice (called a strike price or an exercise price) at orbefore the expiration date of the option.

Since it is a right and not an obligation, the holder canchoose not to exercise the right and allow the option toexpire.

There are two types of options call options (right to buy) and put options (right to sell).

What is an option?

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A call option gives the buyer of the option the right to buy theunderlying asset at a fixed price (strike price or K) at any time priorto the expiration date of the option. The buyer pays a price forthis right.

At expiration (at time T), If the value of the underlying asset (S T) > Strike Price (K), buyer does

exercise the option and makes the difference: S T K If the value of the underlying asset (S T) < Strike Price (K), buyer does not

exercise

More generally, the value of a call increases as the value of the underlying asset increases the value of a call decreases as the value of the underlying asset

decreases

Call Options

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Payoff at time T = max ( STK , 0 )


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Payoff Diagram on a Call Option

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Stock price

Net Payoff on Call

0

Exercise price

Breakeven point

- Optionpremium


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A put option gives the buyer of the option the right to sell theunderlying asset at a fixed price at any time prior to the expirationdate of the option. The buyer pays a price for this right.

At expiration, If the value of the underlying asset (ST) < Strike Price(K), buyer does

exercise the option and makes the difference: K-S T If the value of the underlying asset (S T) > Strike Price (K), buyer does not

exercise

More generally, the value of a put decreases as the value of the underlying asset increases

the value of a put increases as the value of the underlying asset decreases

Put Options

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Payoff at time T = max ( K - ST , 0 )


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Payoff Diagram on a Put Option

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Stock price

Net Payoff on Put

0

Exercise price

Breakeven point

- Optionpremium


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Variables Relating to Underlying Asset Value of Underlying Asset

as this value increases, the right to buy at a fixed price (calls) will becomemore valuable and the right to sell at a fixed price (puts) will become lessvaluable.

Variance in that value as the variance increases, both calls and puts will become more valuable

because all options have limited downside. Expected dividends on the asset

which reduce the price appreciation of the asset, reduce the value of calls andincreasing the value of puts.

Variables Relating to Option Strike Price of Options

the right to buy (sell) at a fixed price becomes more (less) valuable at a lowerprice.

Life of the Option both calls and puts benefit from a longer life.

Level of Interest Rates as rates increase, the right to buy (sell) at a fixed price in the future becomes more

(less) valuable because of the present value effect.

Determinants of option value

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A Summary of the Determinants of Option Value

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Factor Call Value Put Value

in stock price

in strike price in variance of underlying asset

in time to expiration

interest rates

in dividends paid


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American

An American option can beexercised at any time prior toits expiration.

The possibility of earlyexercise makes American

options more valuable thanotherwise similar European

options.

European

European option can beexercised only at expiration.

American vs. European options

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However, in most cases, the time premium associated with the remaining life ofan option makes early exercise sub-optimal.


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Call options If exercised early at time t

underlying asset bought for strike price K at time t Receive any potential dividends paid after exercise

Only convenient if large dividend payments expected Since the price of an underlying asset is assumed to grow over time

and therefore also the payoff from option exercise

Put options If exercised early at time t

Underlying asset sold for strike price K at time t

Never optimal to exercise a put ifP KSt

Nevertheless, it does not imply that under the condition that P < KSt, it will always be optimal to exercisethe put option!

Early Exercise of American Options

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Option premium: Paid by the buyer to the seller at time of option writing

Intrinsic value: The intrinsic value of an option is the difference between the actual price

of the underlying security and the strike price of the option. The intrinsic value of an option reflects the effective financial advantage

which would result from the immediate exercise of that option. Equal to zero for out-of-the-money options

Time value: It is determined by the remaining lifespan of the option, the volatility and

the cost of refinancing the underlying asset (interest rates).

Option premium or price of an option

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OptionPremium

Intrinsic

Value

TimeValue

Intrinsic Value

Call Options max (St K , 0 )

Put Options max (K St , 0 )


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Exercises

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A 2-years CZK-denominated call option on euros witha strike price of 25CZK has a payoff of

max(0, ER - 25),

where ER is the exchange rate in CZK per EUR twoyears from now.

Determine the payoff of a 2-year EUR-denominated put

option on CZK.

Problem 1

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The price of a stock is 50 USD at time t = 0. It is estimatedthat the price will be either 25 USD or 100 USD at t = 1 withno dividends paid. A European call with an exercise price of50 USD is worth C at time t = 0. This call will expire at time t= 1. The market interest rate is 25%.

What return can the owner of the following hedge portfolio

expect at t = 1 for the following actions:

Sell 3 calls for C each, buy 2 stocks for 50 USD each and borrow 40 USD atthe market interest rate.

Calculate the price C of the call option.

Problem 2

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Company A buys a Call on ABC stock at a strike price of 75 EUR,exercisable on June 25th. The call requires an up-front premium paymentof 15 EUR.

This call will be exercised if the price for ABC stock on June 25th is:(A) Greater than 75 EUR

(B) Greater than 90 EUR(C) Less than 75 EUR(D) Less than 90 EUR

Plotthe company As payoff from owning this call.

The call makes the company A:(A) Long ABC stock(B) Short ABC stock

What is the maximum loss the company A could incur from the call?(A) Limited to the 15 EUR premium(B) Limited to 60 EUR (75 EUR strike price less 15 EUR premium)(C) Unlimited

Problem 3

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Company A buys a Call on ABC stock at a strike price of 75 EUR,exercisable on June 25th. The call requires an up-front premium paymentof 15 EUR.

What is the maximum gain possible to the company A from the call?(A) Limited to the 15 EUR premium

(B) Limited to 60 EUR (75 EUR strike price less 15 EUR premium)(C) Unlimited

At what price does the company A break even (i.e. zero profit) on the call?(A) 60 EUR(B) 75 EUR(C) 90 EUR

On June 1st, ABC stock is trading at 69.5 EUR making the call:(A) In-the-money(B) Out-of-the-money

On June 25th, ABC stock trades at 85.5 EUR. Does the company A exercise theoption? Is the call in-the-money or out-of-the-money? What is the utilitys netgainor loss realized from the call?

Problem 4 cont d

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The price for American put options on FMI Co. is currently79 per option. We know that it not optimal to exercisethese options early. Which of these are possible values ofthe strike price K on these options and the stock price S of

FMI Co. stock? More than one answer may be correct.

a) S = 131, K = 152b) S = 209, K = 478c) S = 1503, K = 1375d) S = 512, K = 634e) S = 750, K = 600f) S = 405, K = 204

Problem 5

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Consider an American call option with a 40 USD strikeprice on a specific stock. Assume that the stock sells

for 45 USD a share without dividends. The option sells

for 5 USD one year before expiration.

Describe an arbitrage opportunity, assuming the

annual interest rate is 10%.

Problem 6

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Documents

07 Financial Option Contracts