math 476 / 567 actuarial risk theory fall 2015 university of illinois at urbana-champaign professor...

Math 476 / 567 Actuarial Risk Theory

Fall 2015University of Illinois at Urbana-Champaign

Professor Rick Gorvett

Options and Put-Call Parity

August 27, 2015

Why is this Option Stuff So Important?

PayoffOf CallOption

ST (Value of Underlying Asset)

X(Exercise price)

Insurance is an Option

PaymentUnderInsurancePolicy

ST (Size of Loss)X(Deductible)

Insurance is an Option

PaymentUnderInsurancePolicy

ST (Size of Loss)Ded.

Policy Limit

A Sampling ofOptions

and Other Derivatives through History

Ancient Greece

“There is the anecdote of Thales the Milesian and his financial device… He was reproached for his poverty, which was supposed to show that philosophy was of no use. According to the story, he knew by his skill in the stars while it was yet winter that there would be a great harvest of olives in the coming year; so, having a little money, he gave deposits for the use of all the olive-presses in Chios and Miletus, which he hired at a low price because no one bid against him. When the harvest-time came, and many were wanted all at once and of a sudden, he let them out at any rate which he pleased, and made a quantity of money. Thus he showed the world that philosophers can easily be rich if they like, but that their ambition is of another sort…”

- Aristotle, Politics, Book One, Part XI

Phoenician Shipping

Merchants and ship-owners used options to hedge their ships and cargoes

Mesopotamia

Mercantile forward contracts, written in cuneiform on clay tablets, circa 1700 BC

China

Forward contracts on rice, entered into prior to planting, circa 2000 BC

Belgium and The Netherlands

• Antwerp and Amsterdam

• Grain

• Herring

• Tulips

Tulip Bubble

• Mid-1630s

• Tulip demand exploded and prices skyrocketed

• Options and futures were used to ensure price and supply

• Bubble burst in 1637

America

• 19th century– “Privileges”

– Non-standardized / over-the-counter

• Synthetic loans– Financier Russell Sage

– Put-call parity

– Get around usury laws

Chicago Board Options Exchange

• Began trading standardized options on April 26, 1973

• 911 contracts traded on first day (options on 16 different “underlying” companies)

CBOE and Options

“…any history of the excitement in finance in the 1960s and 1970s must mention the options pricing work of Black and Scholes (1973) and Merton (1973b). These are the most successful papers in economics – ever – in terms of academic and applied impact. Every Ph.D. student in economics is exposed to this work, and the papers are the foundation of a massive industry in financial derivatives.”

- Eugene F. Fama, “My Life in Finance,” arXiv

Modeling Underlying Assets

“…distributions of stock returns are fat-tailed: there are far more outliers than would be expected from normal distributions – a fact reconfirmed in subsequent market episodes, including the most recent. Given the accusations of ignorance on this score recently thrown our way in the popular media, it is worth emphasizing that academics in finance have been aware of the fat tails phenomenon in asset returns for about 50 years.”

- Eugene F. Fama, “My Life in Finance,” arXiv

Optionsand their

Characteristics

A Type of Derivative

• A forward is the obligation to buy or sell something at a pre-specified time and at a pre-specified price

• An option is the right to buy or sell something at a pre-specified time (or during a pre-specified time-period) and at a pre-specified price

Types of Options

• Call option: the holder has a right to buy the underlying asset

• Put option: the holder has a right to sell the underlying asset

• Counterparties: parties to the option agreement

• One can buy (long) or sell (short) an option

Question # 1

• Abby agrees to buy Ben’s car for $1,000 three months from now

Abby’s position: ________ _________

Ben’s position: ________ _________

(long or short)

(forward or option)(long or short)

(forward or option)

Question # 2

• Abby agrees to sell Ben her car for $500 three months from now, if it is worth less than $500

Abby’s position: ________ _________

Ben’s position: ________ _________

(long or short)

(forward or option)(long or short)

(forward or option)

“Parameters” of Options

• Exercise price = strike price = price at which the holder of the option can exercise the option (and thus buy or sell the underlying asset)

• Premium = amount paid for the option

• Expiration date

• “Style”– American option: can exercise any time up to and

including expiration date

– European option: can exercise only on expiration date

Examples of Options -They’re Everywhere

• Traded options– On stocks, indices, FX, interest rates, futures,

swaps, options,...

• Warrants

• Convertible bonds

• Call provisions on bonds

• On projects– To expand, abandon, postpone

• Insurance

Value Of Options At Expiration

C = Max [ST - X, 0]

C = Call option value (or payoff) at expiration

ST = Price of underlying asset at expiration

X = Exercise price

P = Max [X - ST, 0]

P = Put option value (or payoff) at expiration

Question # 3

• Abby sells Matthew a January European call option on one share of ABC stock

• Suppose ABC stock is initially trading at 32.5

• Exercise price = 35

• Premium = 3

• In January, suppose: ST=30 ST=40

(Total payoff [profit/loss])

Abby: ___ [ __ ] ___ [ __ ]

Matthew: ___ [ __ ] ___ [ __ ]

Option Values

• Prior to expiration: Call Put

– In-the-money St > X St < X

– At-the-money St = X St = X

– Out-of-the-money St < X St > X

• Intrinsic value: profit that could be made if the option was immediately exercised– Call: stock price - exercise price

– Put: exercise price - stock price

• Time value: the difference between the option price and the intrinsic value

Option Values: Payoff Charts

• Call -- long position:

• Call -- short position:

• Put -- long position:

• Put -- short position:

Payoff

XST

X

X

X

ST

ST

ST

Payoff vs. Profit/Loss:Long a Call Option

ST

Profit/Loss

Payoff

X

CallPremium

Purposes of Derivatives

• Speculative– Highly risky– Highly leveraged– Very volatile

• Hedging– Combine with other securities– Hedge (minimize) risk from other securities

Hedging

• “Hedge”: Take a position that offsets a

risk

• “Risk”: “Uncertainty” regarding the value of

the underlying asset

• By hedging, one changes the risk inherent in owning the underlying asset

• The return distribution of the underlying asset is not changed

“Risk” vs “Uncertainty”

• “The term ‘risk,’ as loosely used in everyday speech and in economic discussion, really covers two things which, functionally at least, in their causal relations to the phenomena of economic organization, are categorically different…. The essential fact is that ‘risk’ means in some cases a quantity susceptible of measurement, while at other times it is something distinctly not of this character…. It will appear that a measurable uncertainty, or ‘risk’ proper, as we shall use the term, is so far different from an unmeasurable one that it is not in effect an uncertainty at all. We shall accordingly restrict the term “uncertainty” to cases of the non‐quantit(at)ive type.”

- Frank Knight, Risk, Uncertainty, and Profit, 1921

Using Options to Hedge

• Combine the underlying asset with an

option or options

• Can reduce or eliminate downside risk

while retaining upside potential

• Can protect against falls in held asset

values, or against increases in input prices

Option Strategies

• Protective put– Own stock (long position)– Own put (long position)

• Covered call– Own stock (long position)– Sell call (short position)

• Straddle

• Spread

Protective Put

• Investor owns asset

• Investor also buys (holds) a put on the asset

• Guarantees investment portfolio proceeds are at least equal to the exercise price of the put

+ =

Question # 4

• Suppose you own a share of stock, and you purchase a put option with an exercise price of 22.5 on that stock, for a premium of $ 0.75

ST : 30 25 20 15

Premium: ____ ____ ____ ____ Put Payoff: ____ ____ ____ ____ ===== ===== ===== =====Overall: ____ ____ ____ ____

Covered Call

• Investor purchases stock• Investor also sells (writes) a call option on the

stock

• Option position is “covered” by owning the underlying stock itself

• (vs. “naked option”)• Provides additional (premium) income

+ =

Question # 5

• Suppose you own a share of stock, and you write a call option with an exercise price of 35 on that stock, for a premium of $ 2.00

ST : 30 35 40 45

Premium: ____ ____ ____ ____ Call Payoff: ____ ____ ____ ____ ===== ===== ===== =====

Overall: ____ ____ ____ ____

Straddle

• (Long) Straddle: buy both a call and a put on a stock

• Each option has the same exercise price and expiration date

• Believe stock will be relatively volatile

• Worst-case: no movement in stock price

Spread• Combination of options

– Two or more calls, or– Two or more puts

• Horizontal spread: sale and purchase of options with different expiration dates

• Vertical spread: simultaneous sale and purchase of options with different exercise prices -- e.g.,

+ =X1

X2

X1 X2

A “Spread” in the Context of Insurance

Ded. Ded. +PolicyLimit

Size of Loss

InsuranceRecoveryFrom Loss

Pol

icy

Lim

it

“Exotic” Options

• Certain characteristics of “plain vanilla” options are adjusted to produce “exotic options”

• Some characteristics of plain vanilla options:– American or European

– Linear payoff

– Does not “disappear”

– Value of underlying at exercise

Put-Call ParityGeneral Concept

• Arbitrage implies a certain relationship between put, call, and underlying asset prices

• Assuming same exercise prices and expiration dates, and non-dividend-paying stock, two portfolios have, at payoff, identical values:– One European call option + cash of PV(X)

– One European put option + one share of stock

• C + PV(X) = P + S

Put-Call ParitySpecific Relationships

• McDonald text terminology:

Call – put = PV(forward price – strike price)

• For non-dividend-paying stock:

)(),(),( ,0 KFeTKPTKC TrT

KeSTKPTKC rT 0),(),(

Put-Call ParitySpecific Relationships (cont.)

• For dividend-paying stock:

• For index options:

KeDivPVSTKPTKC rTT

)(),(),( ,00

KeSeTKPTKC rTT 0),(),(

Put-Call Parity Example

• Find the value of the one-year (T = 1) put with exercise price (X) of 110 when

C(110,1) = 10.16

S0 = 100 = 0

r = 0.10

P(110,1) = 10.16 + 110 e -.10 × 1 - 100 = 9.69

Put-Call Parity Example (cont.)

• Find the value of the one-year (T = 1) put with exercise price (X) of 110 when

C(110,1) = 10.16

S0 = 100 = 0.02 (2% div yield)

r = 0.10

P(110,1) = 10.16 + 110 e -.10 × 1 - 100 e -.02 × 1

= 11.67