Derivative Securities

1
Derivative Securities Forwards and Options

381 Computational Finance
Imperial College London
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Topics Covered

• Derivatives
• Forward Contracts, Options
• Valuation techniques
• Option Pricing Models
• Binomial Option Pricing

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Introduction to Derivatives

• security
• whose payoff is explicitly tied to value or price
  of other financial security
• that determines value of derivative is called
  underlying security
• derivatives
• arise when individuals or companies wish to buy
  asset or commodity in advance to insure against
  adverse market movements
• effective tools for hedging risks designed to
  enable market participants to eliminate risk.
• business dealing with a good faces risk
  associated with price fluctuations.
• control that risk through use of derivative
  securities.
• Example
• farmer can fix price for crop even before
  planting, eliminating price risk
• an exporter can fix a foreign exchange rate even
  before beginning to manufacture product,
  eliminating foreign exchange risk.

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Example 1 Derivatives

• A forward contract to purchase 2000 pounds of
  sugar at 12 cents
• per pound in 6 weeks.
• The contract is a derivative security because
  its value is derived from the price of sugar.
• No reference to payoff - contract only
  guarantees purchase of sugar.
• The payoff is implied and determined by the
  price of sugar in 6 weeks.
• If price of sugar was 13 cents per pound, then
  contract would have a value of 1 cent per pound,
• Strategy the owner of contract could
• buy sugar at 12 cents according to the contract
  
• then sell that sugar in the sugar market at 13
  cents.

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Example 2 Derivatives

• Assume that a contract gives one the right, but
  not the obligation to purchase 100 shares of GM
  stock for 60 per share in exactly 3 months.
• This is an option to buy GM.
• Payoff of option will be determined in 3 months
  by the price of GM stock at that time.
• If GM is selling then for 70, the option will
  be worth 1000
• The owner of option could at that time
• purchase 100 shares of GM for 60 per share
  according to option contract,
• immediately sell those shares for 70 each

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Forward Contracts

• Forward contract is specified by a legal
  document, the terms of which bind two parties
  involved to a specific transaction in the future.
• on a priced asset is a financial instrument,
  since it has an intrinsic value determined by the
  market for underlying asset
• on a commodity is a contract to purchase or sell
  a specific amount of commodity at specific time
  in future at a specific price agreed upon today
• Contract is between two parties, buyer and
  seller.
• buyer (long ) obligated to take delivery of
  asset pay agreed-upon price at maturity
• seller (short) obligated to deliver asset
  accept agreed-upon price at maturity
• Claims are settled at defined future date both
  parties must carry out their side of agreement at
  that time.
• Forward price applies at delivery, negotiated so
  that initial payment is zero.

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Replicating Portfolio

• used to find the value of derivatives
• derivatives can be replicated using other
  securities
• portfolio that replicates a forward contract
  is obtained
• price of the portfolio is the forward
  contract's price
• Notation

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Standard Formulation Discrete Compounding

• Assumptions
• buy one unit commodity at price S0 with no
  dividend payment
• enter a forward contract to deliver at T one
  unit at price F
• store until T with no cost, deliver to meet our
  obligation obtain F
• Cash flow sequence in two market operations is (
  - S0 , F ) fully determined at t 0 consistent
  with interest rate between t 0 and T
• For asset with zero storage cost, current spot
  price S0 , forward price F is calculated as
• Buying the commodity at price S0 lending
  amount S0 of cash for which we will receive an
  amount F at time T since storage costless.

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Arbitrage Portfolio

• Assume that
• borrow S0 cash and buy one unit of the
  underlying asset
• take one-unit short position (sell) in forward
  market
• at T, deliver asset receiving cash amount F
  repay our loan in amount
• obtain positive profit of
  for zero net investment
• Assume that
• shorting one unit of underlying asset borrow
  asset from s.o who plans to store it during this
  period, then sell borrowed asset and replace
  borrowed asset at T
• take one-unit long position (buy) in forward
  market
• at T, receive from loan and pay F
  one-unit of asset and return
• this to lender who made the short possible
• profit is

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Dividend Payment with Discrete Compounding

• stock pays dividend with total cumulative value
  for T1 year
• two strategies for constructing portfolios A and
  B
• buy a share for S0 and sell share forward in T
  for forward price F
• invest S0 at risk free interest rate of r
• Both portfolios have the same payoff values, the
  forward price is

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Example

• Consider a stock is trading at 145 today and
  pays no dividend during the next 3 months. Annual
  interest rate is 8. What is forward price under
  monthly compounding?
• Portfolio A buy a share for 145 and sell share
  forward in 3 months for forward price F
• Portfolio B invest 145 in a bank account at
  risk free interest rate of 8
• Payoff of portfolio A is certain equal to F
  although we do not know price of
• stock after 3 months.
• We invest 145 today in a risk-less bank account
  and receive
• Considering no arbitrage rule two portfolios
  must have the same payoff F 147.9193

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Example Continued Forward Arbitrage

• No-arbitrage prices must adjust so that no
  market participant can make a riskless profit
• Case 1 Forward contract is overpriced as F 149
• Case 2 Forward contract is under priced as F
  143
• RESULT Only price in the arbitrage free market F
  147.9193

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Dividend Payment-Continuous Compounding

• If stock pays dividends we need to buy
  units of stock smaller than 1 unit
• obtain dividends while holding the stock,
  reinvesting the dividends enables us to purchase
  another units of the stock
• At maturity we own exactly 1 unit of the stock
• Arbitrage free markets require that total payoff
  of the portfolio is zero at maturity

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Example

• Consider a six-month forward contract on a stock
  that is currently trading at 95 and has a
  dividend yield of 2. The risk free rate is 7.
  Show that the 6-month forward should be priced at
  97.40.
• If you buy
  units of stock, you invest 0.99x95 94.05
• You also reinvest all dividends, so in 6 months
  you own 1 unit of stock
• sell this unit forward so return on your
  portfolio is riskless
• invest your 94.05 at the risk free rate, and
  obtain a payoff
• An arbitrage profit can be obtained
• selling stock and buying it back forward,
  investing proceeds in bonds if F lt 97.40
• buying stock and selling it forward, where we
  would borrow the money
• to purchasing the stock, if F gt 97.40

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Commodity Forwards

• owner of commodities has to maintain their
  value,
• requires storage (wheat, gold), feeding (live
  hogs), or security (gold)
• cost is called cost of carry
• expressed as an annual percentage rate q
• It is treated as a negative dividend.
• the valuation formula for commodity forwards is
  obtained as

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Options

• Holder of forward contract is obliged to trade at
  maturity of contract
• Unless the position is closed before maturity,
  the holder must take possession of the commodity,
  currency or whatever is the subject of the
  contract, regardless of whether the price of the
  underlying asset has risen or fallen.
• An option gives holder a right to trade in the
  future at a previously agreed price but takes
  away the obligations. If stock falls, we do not
  have to buy it after all.
• An option is a privilege sold by one party to
  another that offers the buyer the right to buy or
  sell a security at an agreed-upon price during a
  certain period of time or on a specific date.
• Option holder has the right to chose to purchase
  a stock at a set-price within a certain period
• Option writer has the obligation to fulfil the
  choice of the holder
• deliver the asset (for call option ) OR buy the
  asset (for put option )
• receives the premium

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Example Real life

• You have seen a sale on a TV for 120 in a
  newspaper. You go to shop to purchase it at the
  advertised price. Unfortunately at that time the
  TV is already out-of stock. But the manager gives
  you a rain-check entitling you to buy the same TV
  for the advertised price of 120 anytime within
  the next 2 months.
• You have just received a call option
• gives you the right, but not the obligation, to
  buy the TV in the future
• at the guaranteed strike price of 120
• until the expiration date of 2 months
• Scenario 1 A few weeks later you go to exercise
  your rain check -
• TV is now in stock and priced at 150. Since
  you have a rain check the store manager
• agrees to issue the rain check and
• sells you TV at 120. SAVED 30
• TV is now in stock but on sale for 100. Your
  rain check is worthless since you can buy TV at
  the reduced price. You can let your option expire
  worthless have no obligation to exercise it.
• Scenario 2 Your friend phoned you and told you
  that he needs a new TV. You mentioned your rain
  check and agreed to sell it to him for 10.
• the option premium is 10, the same strike price
  of 120 and expiration date of 2 months.
• your friend is taking risk TV might be cheaper
  than 120 (rain check is worthless lose 10)

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Vanilla Options Call and Put

• Call option right to buy particular asset for
  an agreed amount at specified time in future
• Put option right to sell a particular asset for
  an agreed amount at a specified time in future
• Example Consider a call option on IBM stock
  which gives the holder the right to buy IBM
  stock for an amount of 25 in one month. Today's
  stock price is 24.5.
• amount 25 which we can pay for stock is called
  exercise or strike price
• date on which we must exercise our option, if we
  decide to, is called expiry or expiration date
• stock (IBM ) on which option is based is known as
  underlying asset
• premium is the amount paid for the contract
  initially
• Lets see what may happen over the next month
  until expiry!

Case 1 Suppose that nothing happens stock
price remains at 24.5. What do we do at
expiry? - exercise the option,
handing over 25 to receive the stock.
- !!!! This is not a sensible decision since the
stock is only worth 24.5. - not
exercise option or if really wanted the stock
we would buy it in the stock market for the
24.5. Case 2 What happens if the stock price
rises to 29? - exercise the
option, paying 25 for a stock, worth 29, and
get a profit of 4
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Example How do Options Work?

• Suppose today is 1st of May. Consider Microsoft
  (MS) stock with current price of 67. Premium is
  3.15 for a July 70 Call.
• July 70 Call indicates that the expiration is
  July and strike price is 70 for call
• stock option contract is an option to buy 100
  shares
• multiply contract premium by 100 to get total
  price of 1 call option contract will cost
• 3.15 x 100 (for the underlying shares)
  315
• strike price of 70 means that the MS stock
  price must rise above 70 before the option is
  worth anything. Since the contract is 3.15 per
  share, the break-even price would be 73.15.

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Example how do options work?

• May 1st stock price 67, (lt strike price of 70)
  we paid 315 for option theoretically
  worthless.
• But you might not lose the entire 315 because
  you are allowed to trade the options contract
  like a stock as long as it hasn't expired.
• 3 weeks later, the stock price is 78.
• options contract has increased along with the
  stock price worth 8.25 x 100 825
• Profit is (8.25 - 3.15) x 100 510 ---
  doubled your money in just three weeks.
• If you wanted, you could sell your options
  closing your position take your profits.
• If you think the stock price will continue to
  rise, you can let it ride.
• On the expiration date, the MS stock price tanks,
  and is now 62.
• This is less than strike price, and there is no
  time left option contract is worthless.
• We are now down the original investment 315

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How to Read an Option Table?

• 1 Strike price (exercise) the stated price
  per share for which underlying stock may be
  purchased (for a call) or sold (for a put) by
  the option holder upon
• exercise of the option contract.
• 2 Expiry Date shows end of life of options
  contract.
• 3 Call or Put refers to whether option is
  call or put.
• 4 Volume the total number of options contracts
  
• traded for the day.
• 5 Bid price which someone is willing to pay
  for the
• options contract.
• 6 Ask price which someone is willing to sell
  an options contract for.
• 7 Open Interest number of options contracts
  that are open.
• These are contracts which have not expired or
  have not been exercised.
• Total open interest is given at the bottom of the
  table.

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Types of Options

• Vanilla Options simplest ones
• Call and Put
• European Options exercise only at expiry
• American Options exercise at any time before
  expiry
• Asian Options payoff depend on average price
  of underlying asset over a certain period of
  time
• Bermudan options exercise on specific days,
  periods
• Exotic Options more complex cash flow
  structures Barrier, Digital, Lookback so on

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Options Valuation

• procedure for assigning a market value to an
  option
• market value of an asset is the value for which
  it could be sold in the market today.
• how much is the contract worth now, at expiry,
  before expiry?
• no idea on stock price is between now expiry
  but contract has value
• at least there is no downside to owning option
  contract gives you specific rights but no
  obligations
• value of contract before expiry depends on 2
  things
• how high asset price is today the higher asset
  today the higher we expect the asset to be at
  expiry, more valuable we expect a call option
• how long there is before expiry the longer
  time to expiry, the more time for the asset to
  rise or fall

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Payoff Diagram

• value of an option at expiry as function of
  underlying stock price
• explains what happens at expiry, how much money
  option contract is worth

• right to buy asset at certain price within
  specific time
• buyers of calls hope that stock will increase
  before expiry
• buy and then sell amount of stock specified in
  contract

• right to sell asset at certain price within
  specific time
• buyers of puts hope that stock will decrease
  before expiry
• sell it at a price higher than its current market
  value

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Call Option Value at Expiry

• Consider a call option with stock price and
  the exercise price at the expiry date T
• Value of a call option is zero or the difference
  between the value of the underlying and strike
  price, whichever is greater.
• If holder can purchase a share
  more cheaply in market than by exercising option
• If holder receives one share
  from writer of the call option for price of
  
• then make a profit of

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Put Option Value at Expiry

• Consider a put option with stock price and
  the exercise price at expiry date T
• Value of a put option is zero or the difference
  between strike price and value of the underlying,
  whichever is greater.
• If holder sells share to the
  writer of the put option at price E and
  makes a profit of
• If holder prefers not to exercise
  the option

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Example
What are the payoffs of a call and put option at
expiry if the exercise price is 50 and the stock
prices are 20, 40, 60, 80?
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Example

• Suppose the price of IBM is 666 now. The cost of
  a 680 call option with expiry in 3 months is 39.
  You expect the stock to rise between now and
  expiry. How can you profit if your prediction is
  right?
• Suppose that you buy the stock for 666.
• Assume that just before expiry, the stock has
  risen to 730.
• Profit is 64 and the investment rises by
• Suppose that you buy the call option for 39.
• At expiry, you can exercise the call pay 680
  to receive something worth 730. You have paid
  39 and gain 50.
• Profit is 11 per option. In percentage the
  profit is

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Put-Call Parity

• Suppose that you buy one European call option
  with strike price of E and you write one European
  put option with the same strike. Both options
  expire at T and todays date is t.
• At T, payoff of portfolio of call and put
  options is sum of individual payoffs.

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Put-Call Parity at T
payoff of portfolio of call put
options
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Put-Call Parity Before Expiry (tltT)

• If you buy the asset today, then it costs
  worth at expiry
• uncertain but the amount can be guaranteed
  by buying the asset
• Locking in payment E at T involves a cash flow
  of at t
• A portfolio of a long call and a short put gives
  same payoff as a long asset
• and short cash position

the same strategies considered today and at T
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Example 1

• Suppose that European call and put options on
  stock A with the same exercise price of 40 and
  six months to maturity are selling for 5 and 3,
  respectively. The current stock price is 40 and
  the annual interest rate is 8 . Show whether
  put-call parity is satisfied under annual
  compounding?
• Put-call parity is not satisfied the violation
  might be because of 3 reasons call option is
  over-priced - put option is under-priced - stock
  is under-priced
• Arbitrage portfolio

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Example 2

• Consider a stock, a European put option, a
  European call option and T-bill.The stock is
  currently selling for 100. Both put and call
  options have maturity of 3 months and the same
  exercise price of 90. A call option has a price
  of 12 and a put 2. The annual interest rate is
  5. Is there an arbitrage opportunity available
  at these prices under continuous compounding?
• Put-call parity Not satisfied call option is
  under-priced, put stock are over-priced

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Option Pricing Models

• Approaches to option pricing problem based on
  different assumptions about market, dynamics of
  stock price behaviour
• Theories based on the arbitrage principle,
• applied when dynamics of underlying stock take
  certain forms
• The simplest of these theories is based on
  binomial model of stock price fluctuations
• widely used in practice since it is simple and
  easy to calculate
• approximation to movement of real prices
• generalizes one period up-down model to
  multi-period setting

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Binomial Lattice Model

• N trading periods and N1 trading dates,
• invest on a risky security with price of Sn
  (n0,1,,N)
• a risk-less bond with annual interest rate of r
• If price is known at beginning of period, then
  price at next period is
• one of only two possible values
• increases with factor of u
• decreases with a factor of d

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Single Period Binomial Lattice

• Assumptions
• the initial price of the stock is S
• up move u with probability q and down move d
  with probability p ( u gt d gt 0 )
• borrow or lend at risk free interest rate r and R
  r1
• Call option on the stock with exercise price E
  and expiration at the end of period
• lattices have common arcs stock price and value
  of risk-free loan and value of call option all
  move together on a common lattice
• risk free value is deterministic

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Risk Neutral Probability

• Based on discounting expected value of option
  using risk-free rate
• For risk-neutral probabilities q and p 1-q
  ( 0 lt q,p lt 1 ) value of one-period call option
  on a stock governed by a binomial lattice is
  found by
• taking expected value of option using the
  probability
• discounting this value according to risk free
  rate
• risk neutral formula holds for underlying stock

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Replicating Portfolio

• portfolio (made up of stock and risk free-asset
  duplicates the outcome of option
• Cu and Cd are values of a call option after a
  single time period.
• purchase ws and wa pounds or dollars worth of
  stock and risk free asset
• portfolio will have payoffs depending on which
  path is taken
• Value of portfolio
• No-arbitrage rule

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Parameters Binomial Lattice Model

• In order to specify the model completely, chose
  values of u, d and probabilities p, q such a way
  that stochastic nature of stock is captured as
  much as possible
• multiplicative in nature and u, d gt0 - Stock
  price never becomes negative
• Expected yearly growth rate
• In deterministic process, exponential growth rate
• Other parameters
• Binomial model match when period of length is
  smaller and large number of steps is considered

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Multi-period Option Pricing

• Single period option pricing model can be
  extended to multistage option pricing
• Find the stock price evaluation through time
  periods
• Find the option values at expiry using the payoff
  function.
• To find option price, use either
• Risk Neutral Discounting Method
• or
• Replicating Portfolio Method

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Multi-period Option Pricing Risk Neutral
Discounting

• Two-stage lattice representing 2-period call
  option stock price
• Stock price S is modified by up u and down d
  factors
• Call option has strike price E expiration
  corresponds to final point in lattice
• Starting from the final period and working
  backward
• Single period risk-free discounting is applied
  at each node of lattice

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Multi-period Option Pricing Risk Neutral
Discounting

• At time period 2, the option value
• Risk neutral probability

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Replicating Portfolio Method
Let V be the option value. x units of stocks and
y amount of cash investment
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Replicating Portfolio Method
1cash investment at each node
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Replicating Portfolio Method
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ExampleMulti-period Binomial Lattice

• Consider a stock with a volatility of
  The current price of the stock is 62 pays no
  dividends. A call option on this stock has an
  expiration date 3 months from now and strike
  price is 60. Current interest rate is 10
  compounded monthly. Determine price of call
  option by binomial lattice approach.
• Time period length is 1 month
  Risk Neutral Probabilities

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Example continued

• Entry at the top node is computed as
• Stock Price Evaluation
  Option Price