Futures and Options Definitions

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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 best-fit 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 risk-free 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 non-dividend paying stock it is r, for a stock index it is r-q, for a currency it is r-rf At-the-money option: An option where the strike price equals the price of the underlying asset In-the-money 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

    

Out-of-the-money 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(S-K,0) for a call option and max(K-S,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 non-standard 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 put-call parity) With non-dividend 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


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


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 (1-p) 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 risk-free rate. Known as risk neutral valuation

Can therefore use the risk-free interest rate to find the future price of the stock and put this equal to E( , to find the risk-neutral probabilities.


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: [


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 non-dividend 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 risk-free 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:


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 risk-free 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 Black-Scholes price equals the market price

An American call on a dividend-paying 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 ex-dividend 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 Stop-Loss 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 non-dividend paying stock is position in one call option and for a short position for a long

  

The delta of a European put on a non-dividend 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


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 delta-neutral 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

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