derivatives pricing strategies

8/7/2019 Derivatives pricing strategies

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Derivatives pricing strategies-

y An investment asset is an asset that is held for investment purposes by significant numbers ofinvestors, for example-stocks, bonds, gold, silver, etc. However investment assets do not need

to be held for investment only but for other purposes as well but a majority of the holders

must hold it for investment purposes only. A consumption asset is an asset that is held

primarily for consumption, for example-oil, iron, copper etc. Usually it is held forconsumption but can be used as investment asset too.

y Short-selling is referred as shorting, which involves selling an asset that is not owned.Suppose an investor instructs a broker to short 200 Reliance shares. The broker will carry out

the instructions by borrowing the shares from another client and selling them in the market inthe usual way. The investor can maintain the short position for as long as he desires, providedthere are always shares for the broker to borrow. At some stage, however, the investor will

close out the position by purchasing 200 Reliance shares. These are then replaced in theaccount of the client from which the shares were borrowed. The investor takes a profit if the

price has declined and loss if prices have gone up. If at any time while the contract is open the

broker is not able to borrow shares, the investor is forced to close out the position; even it is

not ready to do so. An investor with a short position must pay to the broker any income, suchas dividend or interest that would normally be received on the securities that have been

shorted. The broker will transfer this income to the account of the client from whom thesecurities have been borrowed. Short-squeeze If a stock starts to rise rapidly, the trend maycontinue to escalate because the short sellers will likely want out. For example, say a stock

rises 15% in one day; those with short positions may be forced to liquidate and cover their

position by purchasing the stock. If enough short sellers buy back the stock, the price ispushed even higher.

y Relationship between forward and spot price- suppose a forward contract onan investment asset with spot price S0 that provides no income. T is the time to maturity, r is

the risk-free rate, and F0 is the forward price. The relationship between F0 and S0 is-

F0=S0ert

IfF0>S0ert, arbitrageurs can buy the asset and short forward contracts on the asset. If

F0S0ert

, there can be following strategy:

1. Borrow S0 dollar at an interest r for T years.2. Buy one ounce of gold3. Short a forward contract on 1 ounce of gold.At time T, 1 ounce of gold is sold for F0. An amount S0e

rtis required to repay the loan

this time and the investor makes a profit ofF0 -S0ert

. In the opposite case when F0


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y Relationship between forward and future prices- Theoretically a forwardprice for a contract with a certain delivery date and a future price of a contract with that

delivery date are same, but the relationship doesnt remain same when the interest rate is aknown function of time. Suppose the underlying asset S, is highly correlated with the interestrate, which mean if S increases so will the interest rate. An investor having a long position in

the future market can deposit the gain to a higher interest rate which further infers that daily

settlement is benefitting the investor. In opposite case when S decreases the investor willoccur immediate loss and the investor can finance that loss at lower interest rate. Where as an

investor with a forward contract will not be affected by it as there is no daily settlements.Hence, when S is strongly positively correlated with interest rate the price of the long position

in the future contract is slightly higher than the forward contract. In the same time when S, isstrongly negatively correlated with interest rate then the price of the long forward contract is

slightly higher than the price of the prices.

y Forward foreign exchange rate- Suppose the investor is an US investor and theunderlying asset is one unit of the foreign currency. So spot price S0 will be price of one unitof the foreign currency in dollar term and F0 will be the forward price of 1 unit of the foreigncurrency in dollar term.

A foreign currency has the property that the holder of the currency can earn interest at the

risk-free-interest rate prevailing in the foreign country. For example the investor can invest

that amount in the T-bills. We define rfas the value of the foreign risk-free interest rate whenmoney is invested for time T. The variable r is the US dollar risk-free rate when money isinvested for this period of time.The relationship between F0 and S0 is

F0=S0eRT

Where R= r-rf

Suppose that an individual starts with 1000 units of foreign currency. There are two ways it

can be converted to dollars at time T. One is in investing it for t years at rfentering into aforward contract to sell the proceeds for dollars at time T. This generates 1000e

rfTF0 dollars.

The other is by exchanging the foreign currency for dollars in the spot market and investingthe proceeds for T years at rate r. This generates 1000ert dollars. If there is no arbitrage

opportunities, the two strategies must give the same result.

1000erftF0 = 1000e

rt

So that F0=1000eRT

where R=r-rfy Income, storage costs, and convenience yield- The hedging strategy of gold

producers leads to a requirement on the part of investment bank to borrow gold. Gold ownerssuch as central banks charge interest in the form of what is known as the gold lease rate when

they lend gold. Gold can therefore provide income to the holder. Like other commodities it

also has storage cost.

In the absence of storage cost and income F0=S0ert

Storage costs can be treated as negative income. If U is the present value of all the storagecosts, net of income, net of income, during the life time of a forward contract, then

F0= (S0+U)ert

If the storage costs, net of income incurred at any time are proportional to the price of thecommodity, they can be treated as negative yield. In this case


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F0= S0e(r+u)T

where u denotes the storage cost as per annum as aproportion of the spot price net of any yield earned on the asset.

The benefits from holding the physical assets are sometimes referred as the

convenience yield. It is basically for the consumption assets; for the investment assets

the convenience yield must be zero; otherwise it will create arbitrage opportunity. Ifthe dollar amount of storage costs is known and has a present value U, then the

convenience yield y is defined as-

F0eyT

= (S0+U)erT

If the storage costs per unit are constantly proportionate (u) of the spot price, then y isdefined so that

F0eyT

= S0e(r+u)T

Or F0= S0e(r+u-y)T

The convenience yield reflects the markets expectations concerning the future availability ofthe commodity. The greater the possibility that shortage will occur, the higher the value of the

convenience yield.

y Forward prices using cost-of-carry model- Cost-of-carry model measure thestorage cost plus the interest paid to finance the asset less the income earned on the asset. For

a non-dividend paying stock, the cost-of-carry is r, because there is no storage cost and no

income is earned; for a stock index, it is r-q, because income is earned at rate q on the asset.For a currency it is r-rf; for a commodity that provides income at rate q and requires storage

costs at rate u, it is r -q+u; and so on. As we know it is beneficial to take a long position in thecase of investment assets and for a consumption asset it is more feasible to and appropriate tohave a forward contract, so the forward price of the consumption asset is

F0=S0e(c-y)T

Where c is the cost-of-carry and y is the convenience yield.

y Delivery options- Unlike forward contracts, a future contract allows a party with theshort position to choose to deliver any time during a certain period. It further leads to

complication into the determination of the future price (should the maturity of the futurecontracts are assumed to be the beginning, middle or the end of the delivery period?). If thefuture price is an increasing functions of the time to maturity, so that the benefits from

holding the asset are less than the risk-free rate. it is usually optimal in such a case for the

party with the short position to deliver as early as possible, because the interest earned on thecash outweighs the benefits of the holding the asset. As a rule future price in these

circumstances should be calculated on the basis that delivery will take place at the beginning

of the delivery period. If future price are decreasing as time to maturity increases, the reverseis true. It is then usually optimal for the party with the short position to deliver as late aspossible, and future prices should, as a rule, be calculated on this assumption.

y Relationship between current future price and expected future spotprice- The spot price of an asset at a future time is referred as expected future spot price.Suppose it is March and the July future price of wheat is Rs.400. so what should be theexpected future spot price of the wheat, should it be less than Rs400 or more than Rs400 or

exactly Rs400? As we know that the future price converges to the spot price. If the futureprice of the wheat is more than the spot price, it means the investor with long position will


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gain and the investors with short position will lose; reverse will be the composition if the

future price is less than the spot price.Suppose a speculator who takes a long position in a future contracts that lasts for T years in

the hope that the future prices will be above that of the future contract (it is supposed thatthere is no daily settlement). Suppose that speculator puts the present value of the future price

into a risk-free investment while simultaneously taking a long future position. The proceedsfrom the risk-free investment are use to buy the asset on the delivery date. The asset is then

sold for its market price. The cash flow to the speculator is as follows:

Today: -F0erT

End of the futures contract: +ST(expected future spot price)In order to evaluate this investment, the discount rate which needs to be used for the expectedcash flow at time T equals an investors required return on the investment. Suppose k is aninvestors return for this investment. The present value of this investment is

-F0e-rT+ E(ST)e

-kT where E denotes the expected value.We suppose that the all investment in the market securities is priced in the way that they havezero net present value. This means

F0e

-rT

= E(ST)e

-kT

Or F0 = E(ST)e(r-k)T

The returns investors require on an investment depend on its systematic risk (Interest rates,

recession and wars all represent sources of systematic risk because they affect the entiremarket and cannot be avoided through diversification). If the returns from the a future

contract are uncorrelated with the stock market, the discount rate to use is the risk-free rate r,

then k = r then

F0=E(ST)If the return from the asset is positively correlated with the stock market, k>rand

F0< E(ST)it shows a positive systematic risk and future price should understate the futurespot price. If the return from the asset is negatively correlated with the stock market, k< r

and F0> E(ST)it shows a negative correlation and future price should overstate the futurespot price.

y Contango and backwardation- when future price is below the expected future spotprice, the situation is known as normal backwardation; and when the future price is above

the expected future spot price, the situation is known as contango. Contango compensates thestockholders (commodity) for the interest cost associated with buying and carrying

inventories.

derivatives pricing strategies

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