Slide1 : Derivative Securities
Forwards and Options 381 Computational Finance
Imperial College
London
Topics Covered : Topics Covered Derivatives:
Forward Contracts, Options
Valuation techniques
Option Pricing Models
Binomial Option Pricing
Introduction to Derivatives : 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.
Example 1: Derivatives : 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.
Example 2: Derivatives : 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
Forward Contracts : 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.
Replicating Portfolio : 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:
Standard Formulation: Discrete Compounding : 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.
Arbitrage Portfolio : 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
Dividend Payment with Discrete Compounding : Dividend Payment with Discrete Compounding stock pays dividend with total cumulative value for T=1 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
Example : 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
Example Continued: Forward Arbitrage : 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
Dividend Payment-Continuous Compounding : 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
Example : 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 < 97.40
buying stock and selling it forward, where we would borrow the money
to purchasing the stock, if F > 97.40
Commodity Forwards : 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
Options : 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
Example: Real life : 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)
Vanilla Options: Call and Put : 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
Let’s 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
Example: How do Options Work? : 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.
Example: how do options work? : Example: how do options work? May 1st: stock price $67, (< 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
How to Read an Option Table? : 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.
Types of Options : 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
Options Valuation : 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
Payoff Diagram : 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
Call Option Value at Expiry : 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
Put Option Value at Expiry : 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
Example : 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?
Example : 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
Put-Call Parity : 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 today’s date is t.
At T, payoff of portfolio of call and put options is sum of individual payoffs.
Put-Call Parity at T : Put-Call Parity at T
payoff of portfolio of call & put options
Put-Call Parity: Before Expiry (t : Put-Call Parity: Before Expiry (t
Example 1 : 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
Example 2 : 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
Option Pricing Models : 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
Binomial Lattice Model : Binomial Lattice Model N trading periods and N+1 trading dates,
invest on a risky security with price of Sn (n=0,1,…,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
Single Period Binomial Lattice : 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 > d > 0 )
borrow or lend at risk free interest rate r and R = r+1
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
Risk –Neutral Probability : Risk –Neutral Probability Based on discounting expected value of option using risk-free rate
For risk-neutral probabilities q and p= 1-q ( 0 < q,p < 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
Replicating Portfolio : 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
Parameters: Binomial Lattice Model : 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 >0 - 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
Multi-period Option Pricing : 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
Multi-period Option Pricing: Risk Neutral Discounting : 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
Multi-period Option Pricing : Risk Neutral Discounting : Multi-period Option Pricing : Risk Neutral Discounting At time period 2, the option value
Risk neutral probability
Replicating Portfolio Method : Replicating Portfolio Method Let V be the option value. x units of stocks and y amount of cash investment
Replicating Portfolio Method : Replicating Portfolio Method £1cash investment at each node
Replicating Portfolio Method : Replicating Portfolio Method
Example:Multi-period Binomial Lattice : Example:Multi-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
Example: continued : Example: continued Entry at the top node is computed as
Stock Price Evaluation Option Price