Download Financial Derivatives - William & Mary Mathematics

Financial Derivatives
Daniel Thaler
December 1, 2009
What are financial derivatives?
• They are financial instruments whose
value is derived from some other asset,
index, event, value, or condition.
– Those from which it is
derived is known as an
underlying asset.
Conceptual Example
• It’s Super Bowl XLII between
the Giants and the Patriots and
the Patriots are a 4-1 favorite.
Your friend places a $1,000 bet
on the Giants to win the game.
How much would you pay your
friend to have the option to
purchase his bet? (Question 1)
Scenarios
• Would you pay him more than $1,000 for this option at the start
of the game?
– No, you could make the bet yourself
•
The Giants are winning 3-0 at the end of the first quarter how
would the price of the option change?
– The price of the option would increase
• The Patriots are winning 14-10 with 2:42 left in the game how
would the price of the option change?
– The price of the option would decrease
• The Giants score a late touchdown to make it 17-14 with 0:35
left in the game would you pay him more than $1,000 for this
option?
– Yes, considering if the Giants win the payout is $4,000.
Background information
• Rather than trade or exchange the underlying
asset itself, derivative traders enter into an
agreement to exchange cash or assets over time
based on the underlying asset.
• Derivates are often highly levered, so a small
change in the underlying asset can cause a
large change in the value of the derivative.
More background
• Derivatives can be used by investors to
speculate and to make a profit if the value of the
underlying moves the way they expect
• Traders can use derivatives to hedge or mitigate
risk in the underlying, by entering into a
derivative contract whose value moves in the
opposite direction to their underlying position
and cancels part or all of it out.
Back to the Super Bowl
• You are a Patriots fan and bet $4,000 on
them to win the game (remember the odds
are 1-4 so the payout is $1,000). It’s the
end of the 3rd Quarter and the Patriots are
only up 4pts. You want to hedge your risk
so you find someone to sell you a $500
option on a $1,000 bet that the Giants win.
Answer Question 2.
Hedging Solutions
How much will you win/lose if the Patriots
win/lose?
– Pats win, you win $1,000 - $500 = $500
– Pats lose, you lose $4,000 – $4,000 - $500 = -$500
• How much would have had to wager without
options if you wanted to win $500
– $2,000
Categories
• The type of the underlying
– Stocks, Bonds, Commodity
• The market which they trade
– Over-the-counter (OTC), Exchange-traded derivates (ETD)
• The relationship between the underlying and the
derivative
– Options, Futures/Forwards, Swaps
Options
• Contracts that give the owner the right, but not
the obligation to buy or sell an asset
• The strike price is the price at which the
transaction would take place
• The option must also have a maturity date
• 1 Options contract usually represents the right to
buy 100 shares of the underlying security
Types of Options Trades
• Long Call
• Long Put
• Short Call (“Write a Call”)
• Short Put (“Write a Put”)
Long Call
• Buy the right to purchase the stock at the strike
price.
• Will only exercise if the stocks price is higher
than the strike price plus the price paid for the
option
• Believe the price will INCREASE
• For the same amount of money you can obtain a
larger amount of options than shares
Long Call
Long Put
• Buy the right to sell the stock at the strike
price
• Will exercise only if the stock price plus
the premium is below the strike
• Believe the stock price will DECREASE
Long Put
Write a Call
• Selling a call option to a buyer and had the
obligation to fulfill the contract at a strike price
• Will profit only if the stock price remains below
that of the strike price plus the premium
• Potential loss is unlimited
• Believe the stock price will DECREASE
Write a Call
Write a put
• An obligation to buy the stock from the put
buyer at the strike price
• Will profit if stock price plus the premium is
above the strike price.
• Loss is capped at the full value of the stock
• Believe the stock price will INCREASE
Write a put
Identifying options
• Google’s stock price is $570 and Bill has
bought 3 option contracts for $15($5 per
contract) with a Strike Price of $580.
– If the Stock price goes above $600 Bill will
exercise the option, What has he bought?
• He has a call option
– How much will he make?
• ($600-$580)*300 – $15 = $5,985
Identifying options
• Google’s stock price is $570 and Bill has
written 3 option contracts for $9,000 ($3,000
per contract) with a Strike Price of $600.
– If the Stock price goes down to $560 Bill will
have to exercise the option, What has he
written?
• He has written a put
– How much will he lose?
• ($560-$600)*300 + $9,000 = -$3,000
How Risky are Options?
• They can expire worthless and they increase leverage
• Example: Stock A is selling at 100 and its options are
selling at $2.50 with a strike price of $120
– You want to invest $1,000
– So you can buy 10 shares of stock or….
– 4 options contracts
– In a week the price of the stock is now at 110 so your
profit with just the stocks is 10*10 = 100 but lets say the
value of the option went up to $4.50(very reasonable)
your profit is $2 * 400 = 800
Option Strategies
• Combine any of the four basic options
trades (possibly with different exercise
prices)
• Can also use the two basic kinds of stock
trades (long and short)
• Used to engineer a particular risk profile to
movements in the underlying security.
Option Strategies
• Bull Call Spread
– Combines a short call and a call
Option Strategies
• Long Strangle
– Combines a call and a put
Other Option Strategies
Bullish Strategies
Bull Put Spread
Bull Call Spread
Covered Straddle
Call Time Spread
Ratio Call Spread
Bearish Strategies
Bear Put Spread
Bear Call Spread
Put Time Spread
Ratio Put Spread
Neutral Strategies
Short Straddle
Long Straddle
Short Combo
Guts
Condor
Strangle
Long Butterfly
Excel
How are options priced?
• Want to find a way to quantify the expected
payoffs that would occur if the stock price
goes up or goes down.
• Also, it must incorporate the length for which
the option is available
How are options priced?
• Binomial Options Pricing Model
– Uses a “discrete-time” model of the varying
price over time of the underlying financial
instrument
• Black-Scholes Model
– A continuous extension of the binomial model
Binomial Model
• Provides a general numerical model
• Process is iterative
• Each node represents a possible price at a
particular point in time
Binomial Model
• Steps:
– Price tree generation
– Calculation of option value at each final node
– Calculation of option value at each earlier node
– The value at the first node is the price of the
option
Price Tree Generation
• It assumed that at point in the tree the
underlying instrument will move either up or
down.
– Let S = Current Price
– Let Su = S * u = Price when stock moves up (u >1)
– Let Sd = S * d = Price when stock moves down (0<d<1)
Price Tree Generation
• To determine d and u we will use the volatility
of the underlying asset which is σ
– u = e ^ σrad(t)
– d = e ^ - σrad(t) = 1/u
• t is the time between periods measured in years
• This ensure that the tree is recombining which
accelerates the computation of the option
price
Price Tree Generation
• Answer Question 5 part (a)
Suu = 128.40
Su = 113.31
Sud = 100
S = 100
Sd = 88.25
Sdd = 77.88
Value at Final Nodes
• The final node is expiration, there if it is
profitable to exercise the option you will if
not you will let it expire.
For a call option
Max [ (K – S), 0 ]
For a put option
Max [ (S – K), 0 ]
K is the strike price, S is price of underlying asset
Value at Final Nodes
• Answer Question 5 part (b)
Vuu = 28.40
Vu =
Vud = 0
V=
Vd =
Vdd = 0
Value of Option at earlier nodes
• Use the value of the option at an
intermediate node using the value of the
option at the following nodes.
• First: need to assign a probability to the
price will increase by u or decrease by d
(We will use 50/50 chance to keep it simple)
Value of Option at earlier nodes
• The value of an option at an earlier node is then
equal to the following:
(- r × t)
Max [ (S – Strike), p × Vu+ (1-p) × Vd] × e
Value of Option at earlier nodes
• Answer question 5 part (c), calculate the value
of the option
Vuu = 28.40
Vu = 14.02
Vud = 0
V = 6.92
Vd = 0
Vdd = 0
Excel
Black-Scholes Model
• A continuous continuation of the binomial
model
• The binomial model assumes that
movements in the price follow a binomial
distribution; for many trials, this binomial
distribution approaches the normal
distribution assumed by Black-Scholes.
Black-Scholes Model
• Developed by Fischer Black and Myron
Scholes in a 1973 paper.
• They received the 1997 Nobel Prize in
economics for this and related work.
Black-Scholes Model
• It assumes the underlying asset follows a
geometric Brownian Motion and using partial
differential equations to get the BlackScholes PDE:
Black-Scholes Model
• The value of a call option is found by solving
the PDE and the result is
N(•) is the standard normal distribution
T - t is the time to maturity
S is the price of the underlying asset
K is the strike price
r is the risk free rate
σ is the standard deviation
Excel
Futures/Forwards
• Futures contract is a standardized contract
to buy or sell a specified commodity at a
certain date in the future for a certain price.
• Forwards are similar to futures except that
they are traded OTC and as such are more
customizable
Futures
• A futures contract gives the holder the
obligation to make or take delivery under the
terms of the contract
• Differs from an options contract in that both
parties must fulfill the contract at the
settlement date.
Who Buys Them?
• Speculators who seek to make a profit by
predicting market moves.
• Producers and consumers purchase futures
contracts to guarantee a certain price.
Types of Futures
•
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•
•
•
•
•
•
Crude Oil
Corn
Soybean
Sugar
Wool
Cotton
Coffee
Cocoa
•
•
•
•
•
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Wheat
Lumber
Orange Juice
Silver
Gold
Copper
Trading Places
Trading Places Explanation
• Standard contract size is 15,000 pounds
• The Dukes got a fake report and think that
FCOJ is going to be valuable and cause the
price to rise
• V and W wait until the price gets to $1.42
per pound and then sell contracts that they
don’t own.
Trading Places Explanation
• When the real crop report is announced it is
obvious that the crop will be good and prices
begin to fall all the way down to 46.
• Since V and W don’t own any FCOJ they
start to buy back contracts at this price to
cover the ones that were sold.
Trading Places Explanation
• Just some rough numbers:
($1.42 - $0.46) * 15,000pounds/contract * 20,000 contracts =
= 288,000,000
Conclusions
• Derivatives can offer a way to:
Hedge portfolio risk
Lock in a specific price for a commodity
Provide investing leverage
Cheap form of speculating