003 futures and forwards

SKOOL COMPUTER EDUCATION

The DerivativesFUTURES AND FORWARDS



COMPOUNDING ISSUES

• In reality interest rates underlying fixed income securities, bank deposits, bank loans etc. are compounded using a variety of ways.

• They could be compounded annually, semiannually, daily and so on.

• Here we are going to use continuously compounded interest rates, just because they help us derive closed form solutions, not only for forwards and futures, but for a whole set of derivative securities.


COMPOUNDING ISSUES


SHORT SELLING

• Some of the strategies exploiting arbitrage opportunities require short selling.

• This trade involves selling assets that the investor does not own.

• It is carried out using the following steps:• The customer’s broker borrows the securities from

another investor and sells them in the usual way in the market.

• At some point in the future, the short seller has to buy the securities at the market price of that time, in order to replace them in the other clients account.

• The short seller has the obligation to pay all dividends and other benefits to the owner of the securities that have been borrowed.


SHORT SELLING

• Short selling requires an agreement --formal or informal, between an investor and her broker.

• What happens if short selling is not possible?

• It follows that it does not make any difference• There is a fairly large number of people that hold the

assets for investment purposes. • If there is an arbitrage opportunity, then the holder of

the asset will exploit it himself and make a riskless profit, rather than lending the securities to someone else to do so.


THE FORWARD PRICE

• In the remaining of this chapter we will maintain a number of assumptions:

• There are no transaction costs;• The underlying security is traded in a market --

thus excluding for example weather derivatives;

• All investors are subject to the same tax rates;• The interest rate is them same for borrowing

and lending; and• All arbitrage opportunities are immediately

exploited.


THE FORWARD PRICE


COUPONS, DIVIDENDS &FX FORWARDS

• Suppose now that the asset provides a known income in cash.

• This could well be a stream of coupons when valuing a T-bond forward. It is easy to show that the arbitrage free value of the forward will be


COUPONS, DIVIDENDS &FX FORWARDS


FORWARDS VS FUTURES

• It can be shown that when interest rates are constant and the same for all maturities, then the futures and forward prices are the same.

• If the interest rates are stochastic, this relationship does not hold.

• Whether the forward price is lower than the futures price or higher will depend on the correlation of the underlying asset with the interest rates.

• This situation arises from the daily settlement procedure that takes place in the futures market.


EXAMPLE-001SPOT-INTEREST CORRELATION

• Suppose that the interest rates and the underlying asset are negatively correlated.

– When the interest rates fall the price of the underlying asset increases, something that is true in the stock markets.

• Consider an investor that holds a long futures position. When the asset price increases, because of the marking-the-market procedure, the investor is making an immediate gain –the basis increases.

• This extra gain will be invested at an interest rate which is lower than average, due to the negative correlation.

• In a similar fashion, when the price of the underlying falls, the immediate loss will have to be financed at a rate which is above the average.

• Forwards are not subject to daily settlements, and therefore not affected by the spot-interest correlation.

• This makes forward contracts more attractive; in an efficient market when the spot interest correlation is negative we expect forward prices to be higher than the futures ones.


STOCK INDEX FUTURES

• Stock index tracks the change in the value of a hypothetical portfolio of stocks.– S&P 500 Index is based on a portfolio of 500

stocks (400 industrial, 40 utilities, 20 transportation, and 40 financial companies.)

– Nikkei 225 Stock Average is based on a portfolio of 225 of the largest stocks trading on the Tokyo Stock Exchange

• Stock index futures: futures contract on the stock index– S&P 500 Futures is on 250 times the index– Nikkei 225 Stock Average is on 5 times the index


STOCK INDEX FUTURES

• Can be viewed as an investment asset paying a dividend yield

• The futures price and spot price relationship is therefore

F0 = S0 e(r–q )T where q is the dividend yield on the portfolio

represented by the index• For the formula to be true it is important that the

index represent an investment asset• In other words, changes in the index must correspond

to changes in the value of a tradable portfolio• The Nikkei index viewed as a dollar number does not

represent an investment asset


HEDGING USING INDEX FUTURES


PERFECT HEDGE

• A company wants to perfectly hedge a well diversified portfolio worth £1.2m for two months using FTSE100 futures with four months to maturity.

• The beta of the portfolio is 1.5 and the level of the FTSE100 index is 6000 points.

• The FTSE100 contract is valued as £10 per point. • This means that the value of the assets underlying one

futures contract is


EXAMPLE PERFECT HEDGE

• Suppose that over the course of the next two months the interest rate is 6%p.a., or 1% over the two month period.

• Suppose that the market collapses in these two months --perhaps what the company's fears were!!-- and offers a return of -9%.

• The CAPM will therefore dictate that the return of the portfolio is



• If the dividend yield is 3%p.a. or 0.5% per two months, it is implied that the ftse100 index has declined by 9.5% over these two months, down to 5430 points. The initial and final futures prices are respectively

The total income of the company over these two months due to the shorting of the futures is therefore



• The differences in the above example occur because we have ignored the distinction between continuously and discretely compounding returns.

• We did not take into account the daily settlements --tailing the hedge.


COST OF CARRY

• The cost of carry,C , summarizes the relationship between the futures price and the spot price.

• It is defined as cost of carry=interest+ storage cost-income earned. Therefore the cost of carry would be

• The cost of carry allows one to write the futures price for an investment asset as


COST OF CARRY

• For a consumption asset as

• Where y is the convenience yield, a measure of the benefits from ownership of an asset that are not obtained by the holder of a long futures contract on the asset.


FUTURES PRICES AND FUTURE SPOT PRICES

• Is the futures price an unbiased estimator of the future spot price?

• The answer is no in general.• Two types of risk in the economy

– The systematic – the Nonsystematic risk.

• The nonsystematic risk can be eliminated by holding a well diversified portfolio, which is perfectly correlated with the market.

• The systematic risk cannot be eliminated, since it is the risk of the portfolio that is inherited from the market as a whole and it cannot be diversified away.

a risklessinvestment will grow in value with the risk free rate of return


EXAMPLE -FUTURES RISK

• An investor takes a long futures position. She puts the present value of the futures position into a risk free investment, to meet the requirements when the contract matures, in order to buy the asset on the delivery date.

• The cash flows of the speculator are


EXAMPLE -FUTURES RISK

• Et is the conditional expectations operator,

• and rI is the discount rate appropriate for the investment --meaning the expected return required from investors in order to compensate for the risks that are beard.

• The fact that the present value of all investment opportunities is equal to zero will give

It is straightforward to observe that the relationship of the futures with the expected spot price will depend on the relationship between the two returns, which in turn depends on the correlation of the Investment with the market due to the CAPM.

003 futures and forwards

Economy & Finance