Futures and Options – Definitions and Formulas Lecture 1 and 2 A Derivative is a financial instrument whose value depends on the value of some underlying asset. The derivatives value is derived from the underlying asset A Forward contract is the agreement between a buyer and seller to trade specified quantity of an asset at a specified price (forward/delivery price) at a specified time (maturity/delivery date) and place. A Spot contract is an agreement to buy or sell an asset today Payoff from long position: Payoff from short position: A Futures contract is an agreement to buy or sell a certain asset at a certain price at a certain time in the future. Whereas a forward contract is traded OTC, a futures contract is traded on an exchange
A Call option is a contract which indicates the right to buy a certain asset by a certain maturity date for an American option and on the maturity date for a European option
A Put option is a contract which indicates the right to sell a certain asset by a certain maturity date for an American option and on the maturity date for a European option
Hedgers use derivatives to reduce risk from potential future market movements (however there is no guarantee that the outcome with hedging will be better than the outcome without hedging)
Speculators bet on the future direction of market variables
Arbitrageurs aim to lock in a profit by taking offsetting positions
A Margin is cash or marketable securities deposited by an investor with his/her broker  The balance in the margin account is adjusted to reflect daily settlement Margins minimise the possibility of a loss through a default on a contract
Open interest: the total number of contracts outstanding Equal to the number of long positions or short positions
Settlement Price: The price just before the final bell each day  Used for daily settlement process
Volume of Trading: the number of trades in one day
Forwards Private contract between two parties Not standardised Usually one specific delivery date Settled at the end of the contract Delivery or final cash settlement usually occurs Some credit risk
Futures Exchange Traded Standardised contract Range of delivery dates Settled daily Contract usually closed out prior to maturity Virtually no credit risk
Basis risk arises because of the uncertainty about the basis when the hedge is closed out. It is a measure of the risk involved when using futures to hedge the risk of an asset.  The underlying asset of the futures contract may not be the same as the asset to be hedged  The hedger may be uncertain as to the exact date when the asset will be bought or sold  The hedge may require the futures contract to be closed out before expiration
F1 = Futures Price at time hedge is set up (t1) F2 = Futures Price at time asset is purchased S2 = Asset Price at time of purchase b2 = Basis at time of purchase Cost of Asset = S2 Loss on futures = F1 – F2
(long hedge example)
Net amount paid: S2 + (F1 – F2) = F1 + b2
When there is no futures contract on the asset that you wish to hedge, choose the futures contract whose price is most highly correlated with the asset price. This is known as Cross hedging
Hedge Ratio: The size of the position taken in the futures contract relative to the size of the exposure
The hedge ratio that minimises the variance of the hedger’s position is the optimal hedge ratio (OHR)
Proportion of the exposure that should be optimally hedged is is the standard deviation of DS, the change in the spot price during the hedging period standard deviation of DF, the change in the futures price during the hedging period 

is the
is the coefficient of correlation between DS and DF. The OHR is the slope of the bestfit line when we regress DS on DF
To hedge the risk in the portfolio the number of contracts that should be shorted is

Where P is the value of the portfolio, futures contract.
is its beta, and F is the value of one
Short position in: ( Long position in: ( ,
,
To change beta
Treasury rates: The rates an investor earns on treasury bills and bonds (these are used by government to borrow on its own currency)
Libor rates (London Interbank Offered Rate): LIBOR is the rate offered by banks on Eurocurrency deposits (i.e. the rate at which a bank is willing to lend to other banks)
REPO Rates: a repo or repurchase agreement involves borrowing by selling securities and buying them back later at a slightly higher price. The rate of interest in a repo transaction is the repo rate
 Compound interest conversion formula
A Zero rate (or spot rate), for maturity T is the rate of interest earned on an investment that provides a payoff only at time T (no immediate payments occur)
The bond yield is the discount rate that makes the PV of the cash flows on the bond equal to the market price
The par yield for a certain maturity is the coupon rate that causes the bond price to equal face value
Investment assets are assets held by significant numbers of people purely for investment purposes
Consumption assets are assets held primarily for consumption Short selling involves selling securities you do not own  Your broker borrows the securities from another client and sells them in the market, at some stage you must buy the securities back so they can be replaced in the account of the client, you must pay benefits such as dividends the owner of the securities receives
If the spot price of an investment asset is S0 and the futures price for a contract deliverable in T years is F0, then:
– Where I is the PV of the income during life of a forward contract – Where q is the average yield during the life of a contract
( )
– Where rf is the foreign riskfree interest rate – Where U is the PV of all storage costs
The value of a long forward contract f, is
The value of a short forward contract is
The cost of carry, c, is the storage cost plus the interest cost less the income earned  For a nondividend paying stock it is r, for a stock index it is rq, for a currency it is rrf Atthemoney option: An option where the strike price equals the price of the underlying asset Inthemoney option: Either a call option where the asset price is greater than the strike price or a put option where asset price is less than the strike price
Outofthemoney option: A call option where the asset price < strike price for call and where Asset price>strike price for a put option Option class: all options of the same type on a particular stock (call or put) Option series: All options of a certain class with the same strike price and expiration date Intrinsic value: max(SK,0) for a call option and max(KS,0) for a put option Time value: The value of an option arising from time left to maturity (=an option’s price minus its intrinsic value)
Flex option: traders agree to nonstandard terms (e.g., different strike prices or maturities from what is usually offered by the exchange)
Position limit: Defined the maximum number of option contracts that an investor can hold on one side of the market
Stock splits: Occur when the existing shares are split into more shares Executive stock options: are a form of remuneration issued by a company to its executives. They are usually at the money when issued. When options are exercised the company issues more stock and sells it to the option holder for the strike price.
Lecture 3
o
D denotes the PV of dividends during the life of the option
o – Because it may be optimal to exercise it early
o American Options; D=0 European Options; D>0 American Options; D>0 (lower bound for putcall parity) With nondividend paying stocks and
Lecture 4 Bull Spread with calls: Buy a call with strike price which and sell a call with strike price ,in
Bull Spread with puts: Buy a put with strike price which
and sell a put with strike price
,in
Bear Spread with calls: Sell a call with strike price which o Buy a call with strike price
and buy a call with strike price
, in
and sell a call with strike price
.
Bear spread with puts: Sell a put with strike price which o Buy a put with strike price
and buy a put with strike price
, in
and sell a put with strike price
Box spread: A combination of a bull call spread and bear put spread. The payoff is always . If all the options are European, a box spread is worth the present value of the difference between the strike prices.
Long Butterfly spread of calls/puts: Long 1 call/put at call/put at . In which and
, short 2 calls/puts at
and long 1
Short Butterfly spread of calls/puts: Short 1 call/put at 1 call/put at . In which and
, long 2 calls/puts at
and short
Calendar spread using calls: Short a call with strike price K and maturity with strike price K and maturity where .
and long a call
Calendar spread using puts: Short a put with strike price K and maturity with strike price K and maturity where
and long a put
Straddle combination: Buy a call and a put with the same strike price and expiration date Strangle combination: Buy a call and a put with the same expirations but different strike prices:
Strip: Long position in 1 call and 2 puts with strike price K and maturity T Strap: Long position in 2 calls and 1 put with strike price K and maturity T
Lecture 5 A portfolio is riskless if the value of portfolio at time t is the same if the price of the stock moves up or down
Consider the portfolio; long
shares and short in 1 option
is the value of the portfolio when the stock price goes up by a proportion u>1. is the value of the portfolio when the stock price goes down by a proportion d<1.
The portfolio is riskless when
or
The value of the portfolio today is
which is the cost of setting it up
Substitute “ [
” = Expected present value of derivative Where
P is the probability of an up movement and (1p) is the probability of a down movement. P is also called the risk neutral probability
The value of a derivative is then its expected payoff in a risk neutral world discounted at the riskfree rate. Known as risk neutral valuation
Can therefore use the riskfree interest rate to find the future price of the stock and put this equal to E( , to find the riskneutral probabilities.
Let
years denote a time step. Then binomial trees with more than one step can be
considered. In this case if follows: [
Repeated application in the above equation gives: o o [ [
Substituting the last 2 equations into the initial equation leads to: [
Delta
is the ratio of the change in price of a stock option to the change in the price of the
underlying stock
The construction of a riskless hedge is referred to as delta hedging One way of matching the volatility is to set
Lecture 6
– for a nondividend paying stock – for a stock index where q is the dividend yield on the index – for a currency where – for a futures contract is the foreign riskfree rate
Discrete time – The value of a variable can change only at certain fixed points in time Continuous time – Changes can take place at any time
Markov process – Only the present value of a variable is relevant to predict the future, past information is irrelevant (this is consistent with weak form of market efficiency).
A Wiener process or Brownian motion considers a variable, Z, whose value changes continuously with time o o o o where
A wiener process has a drift rate (i.e. average change per unit time) of 0 and variance rate of 1 In a generalised wiener process the drift rate and the variance rate can be set equal to any constants o The variable x follows a generalised wiener process with a drift rate of a and a variance of b2 if
In an itô process the drift rate and the variance rate are functions of the stochastic variables underlying the derivative and the time (i.e. of x and t)
An itô process for stock prices (Geometric Brownian motion)
Where µ is the expected rate of return (or the drift rate) and
is the volatility of the stock
price. Can simulate random values for and predict future movements given current price of stock
If a variable X follows the itô process: o Itô’s lemma shows that a function ( follows the process: )
Suppose we have daily data for a period of several months o o is the average of the returns in each day [ ]
is the expected return over the whole period covered by the data measured with continuous compounding (or daily compounding).
Volatility is the standard deviation of the continuously compounded rate of return in one year o The standard deviation of the return in rime is
Estimating volatility from historical data o o Take observations at intervals of years [ ]
Calculate the continuously compounded return in each interval as:
o
Calculate the standard deviation, s, of the
. The usual estimate of s is:
√
∑
The historical estimate is
Time is usually measured in trading days in which there are 252 trading days in a year Black –Scholes – Merton model 1. The stock price follows a geometric Brownian motion with constant drift and volatility 2. There are no transaction costs or taxes 3. All securities are perfectly divisible 4. There are no dividends during the life of the security 5. The short selling of securities with full use of proceeds is permitted 6. There are no riskless arbitrage opportunities 7. Security trading is continuous 8. The riskfree interest rate is constant and the same for all maturities
Any security whose price is dependent on the stock price satisfies the BSM differential equation (Refer to equations for the BSM as they are provided and not required to be remembered)
N(x) is the cumulative probability distribution function for a standardised normal distribution  The probability that a normally distributed variable with a mean of zero and a standard deviation of 1 is less than x
The implied volatility of an option is the volatility for which the BlackScholes price equals the market price
An American call on a dividendpaying stock should immediately be exercised prior to an exdividend date
Suppose dividend dates are at times if the dividend at that time is greater than
. Early exercise is sometimes optimal at time
When dealing with dividends in American call options, set the price equal to the maximum of the two European price of an option maturing at the same time and an option maturing just before the final exdividend date.
Lecture 7 Naked position: Take no action to cover exposure Covered position: Buy the amount of stocks necessary to cover obligation of option. A StopLoss strategy: Hold a naked position whenever the stock price is less than K and a covered position whenever the stock price is above K.
Delta ( )  The rate of change of the option price with respect to the underlying stock price If you short a call for some number of shares you must long shares to hedge your position
A position with a delta of 0 is referred to as being delta neutral The delta of a European call on a nondividend paying stock is position in one call option and for a short position for a long
The delta of a European put on a nondividend paying stock is The hedge must be readjusted periodically to remain delta neutral this is known as rebalancing The delta of a portfolio of options dependent on a single asset whose price is S is
The delta of the portfolio can be calculated from the deltas of the individual options:
∑ Where is the delta of the ith option and denotes the quantity of the option I (i=1,…,n)
The delta of a call is positive and the delta of a put is negative (calls gain value at the price of the underlying increases but puts lose value in this case) and
The theta of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to time
The theta of a call or put option is usually negative. This means that, if time passes with the price of the underlying asset and its volatility remaining the same, the value of the option tends to become less valuable
Gamma ( ) is the rate of change of delta ( ) with respect to the price of the underlying
( o
)
If the absolute value of gamma is large, delta is highly sensitive to the price of the underlying and it is risky to leave a delta neutral portfolio unchanged
o
The curvature of the relationship between the option price and the stock price leads to a hedging error. Gamma measures this curvature.
Suppose that a deltaneutral portfolio has a gamma equal to and a traded option has a gamma of o If the number of traded options added to the portfolio is is , the gamma of the portfolio
Hence the position in the traded option necessary to make the portfolio neutral is 
The Vega (V) of a portfolio of derivatives is the rate of change of the value of the portfolio with respect to the volatility of the underlying asset:
If the absolute value of vega is high, the portfolio’s value is sensitive to small changes in volatility
can be changed by taking a position in the underlying To adjust and V it is necessary to take a position in an option or other derivative
Rho ( ) is the rate of change of the value of the portfolio with respect to the interest rate
The delta of long forward contract is always 1.0, when the stock price changes by S, with all else remaining the same, the value of the forward contract on the stock also changes by For an asset that provides a known yield, q; = .
The delta of a futures contract is is
The required position in futures contract for delta hedging