using derivatives

8/8/2019 Using Derivatives

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Using derivatives

Susan Thomas

26 February, 2009

Susan Thomas Using derivatives
http://goforward/http://find/http://goback/


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Recap: Types of derivatives

Linear derivatives where the payoff is linearly related to

the price of the underlying asset.When the spot price, the price of the derivative also rises,and when the price falls, the derivatives price falls.

1 Forwards Contracts that give the right to buy/sell an assetat a future date (maturity or exercise date), but at a price

that is fixed today (futures price).2 Futures Forwards contracts with contract features that are

standardised and is traded only at an exchange.

Nonlinear derivatives where the payoff is nonlinearlyrelated to the price of the underlying.

1 Options Contracts which gives the buyer right topurchase/sell an asset at a predetermined price (strikeprice) at or before a predetermined time (maturity orexercise date).



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Recap: Payoffs

Graph of profits/losses that accrue (y-axis) as the

underlying price changes (x-axis).

Forwards/futures have a linear payoff.Since the (long) forwards contract is a future claim on the

underlying, as the price of the underlying goes up, the

value of the futures go up.

The payoff diagram looks like:

4000 4200 4400 4600

Futures rice on ex iration (Rs.)

-200

-100

0

100

200

Payofftofu

turesbuyer(Rs.)



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Recap: Payoffs to holding options

Options have a non-linear payoff structure.

The options contract is a future claim on the underlying

only under certain conditions about the price of the

underlying.

These conditions can be favourable to the owner of theoption under different price movements.

Call options give the buyer benefits if the price of the

underlying goes up.

Put options give the buyer benefits if the price of the

underlying goes down.



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Recap: Payoff to a call option buyer

The payoff diagram of a call option looks like:

1000 1100 1200 1300 1400 1500

S ot rice on ex iration Rs.)

-200

-100

0

100

200

ayo

toca

opton

uyer

s.



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Recap: Payoff to a put option buyer

The payoff diagram of a put option looks like:

1000 1100 1200 1300 1400 1500

S ot rice on ex iration Rs.)

-200

-100

0

100

200

ayo

toputopton

uyer

s.



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Recap: Payoff diagrams of options

1000 1100 1200 1300 1400 1500

Spot price on expiration (Rs.)

-200

-100

0

100

200

Payofftocalloptionbuyer(Rs.)

1000 1100 1200 1300 1400 1500


-200

-100

0

100

200

Payofftoputoptionbuye

r(Rs.)

1000 1100 1200 1300 1400 1500


-200

-100

0

100

200

Payofftocalloptionseller(Rs.)

1000 1100 1200 1300 1400 1500


-200

-100

0

100

200

Payofftoputoptionseller(Rs.)



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Uses of derivatives


Wh d i ti f l?


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Why are derivatives so useful?

The advantage of derivatives are two: leverage and

liquidity.The uses are three: speculation, hedging and arbitrage.


L ith d i ti


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Leverage with derivatives

Leverage using a small base of capital to take exposure on a

larger base of capital.Example, when a group starts with Rs.100 of

equity capital and can borrow Rs.900 of debt from

a bank to create a firm worth Rs.1000. The base

capital of Rs.100 access an exposure of Rs.1000.With derivatives:

No payment is required upfront, when

buying/selling futures,

The price of an option is typically a fraction of

the underlying value.Little capital is required to trade derivatives: this

makes it accessible to a much larger audience.


Li idit ith d i ti


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Liquidity with derivatives

Liquidity : trading derivatives is cheaper than trading the

underlying asset.

This typically leads to higher liquidity in derivativeson underlyings.


Uses of derivatives


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Uses of derivatives

Speculation Putting money on knowledge/forecasts of future

price behaviour. Typically, this is a short-term

trade.Long term speculative behaviour becomes more

like investment.

Hedging Modifying the risk (and therefore, return) profile of

a portfolio. Once again, this tends to be temporary.

Arbitrage By definition, derivatives price ought to be strongly

linked with the underlying price. Any unexpected

discrepancy between the two requires trades in

both.

Beyond these three, there is also the investment motivation to use

derivatives but this is rare. For example, an index fund management

firm will typically use index futures when fresh investments come into

their fund. However, it is cheaper to eventually trade the underlying

equity itself.Susan Thomas Using derivatives


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Speculation


Speculation using derivatives


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Speculation using derivatives

Speculation is putting money on your knowledge about

future movements in an asset price.For example, you are convinced that Nifty will rise by the

end of March. What do you do to prove your conviction?

You want to take a position of 100 units of Nifty.

Different ways:1 Go long Nifty spot, with the intention of selling on 29th

March.Pay on (T + 2) how much?

2 Go long Nifty 29th March futures.Zero money upfront, other than Initial Margin for NSCCL.

3 Go long Nifty 29th March call at a strike of 3650.Pay the call option price.

Principle: choose the strategy that is the cheapest.


LOB for Nifty 29th March futures


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LOB for Nifty 29th March futures


LOB for Nifty 29th March call at 3650


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LOB for Nifty 29th March call at 3650


The underlying value involved in the speculation


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The underlying value involved in the speculation

The position we want to take is 100 units of Nifty, current value

of 3650. Underlying value = Rs.365,000.

Assume the brokerage fees are the same for all trades.


Cost of each speculative strategy


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Cost of each speculative strategy

Spot transaction:1 impact cost = 9 bps of value on buy+sell side = Rs.401.5

2 STT = 12.5 bps of value on buy+sell side = Rs.456.253 Interest on borrowing to buy Nifty = 365,000*0.005 = 18254 Total = Rs.2,682.75

Futures transaction:1 impact cost = 3.67 bps of value on buy side = 134.97

2 STT = 1.33 bps of value = 48.913 Interest on IM = 29,420.60*0.005 = 147.104 Need to make arrangement for paying mark-to-market

margins, x, if the position becomes negative.5 Total = Rs.330.98 + x

Options transaction:1 impact cost = 9.28% of call value = 1,150.722 STT = 1.33 bps of call value = 1.653 Interest on call value = 12,400*0.005 = 624 Total = Rs.1,224.37


Analysing the speculative strategies


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Analysing the speculative strategies

Both futures and options are cheaper in total cost than the

spot.

However, the potential loss in the options payoff is capped.

Therefore, the different strategies are not readily

compareable on the cost dimension alone.

Fundamental reasoning : Each person has his own U(),his own speculative view about the pdf of future returns,

and based on that, chooses the position which gives the

highest EU.

A risk averse individual would choose the options (cheap

and with known loss).A risk loving individual would choose the futures (cheap at

the cost of a potentially large loss).



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Hedging


Defining hedging


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Defining hedging

Defn: A perfectly hedged portfolio is one where the price

risk of the portfolio is zero.

If the hedging is done using (say) a futures contract, ittypically involves having a portfolio of the spot asset, and

an equal and opposite position in a related futures contract.


Alternative hedging methods


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Alternative hedging methods

The traditional way of reducing price risk was to reduceprice exposure.

What is new about derivatives are that these are a way of

reducing risk, while retaining the inherent underlying

exposure.

Derivatives are useful to hedge risk1 over short periods of time, or2 of a specific magnitude.

There is more control on managing risk.

Derivatives also have the advantage of leverage: thehedge can be acheived with little capital.


Types of hedges


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yp g

The principle of hedging rests on the idea that risk is of two

types: systematic and unsystematic.

Therefore, hedges can be differentiated by whether youare trying to reduce (or enhance!) systematic orunsystematic risk.

Systematic risk is best captured by the index. Therefore,these are implemented using the index futures.Unsystematic risk is captured by a portfolio of (long assetand short index) futures.

The more sophisticated type of hedges are structured

hedges: reducing the loss in a particular direction (buying

price insurance) or to a certain fixed amount (limiting

prices to move within a pre-determined range).Structured hedges are more easily done by options.

But in either of the above, the final hedging strategy you

choose depends upon the liquidity of the contracts in the

market.


Operationalising systematic risk hedging using futures


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p g y g g g

Given the linearity of spot and future payoffs, hedgingusing futures is simple: for every unit of the spot held, sell

a unit of the futures. This becomes like a portfolio, where

the characteristics of the portfolio, rp, is given by:

rp = rj

rfut

2rp =

2rj 2cov(rj, rfut) +

2rfut

The reduction in the risk of the spot-futures portfolio

depends upon the correlation between the spot and the

futures contract.If correlation is 1, then 2p = 0 because cov(rj, rfut) = jfut.


Operationalising unsystematic risk hedging using


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p g y g g g

futures

If an asset has a futures contract trading on it, then the

correlation is 1 and there is perfect risk hedging in the

combined portfolio of J F.

Typically, the underlying and the asset with the futures

contract need not be not the same.

For example, SAIL does not have futures trading on it.

To remove systematic risk in SAIL we short Nifty futures.

To remove unsystematic risk in SAIL we try (say) short

Tata Steel futures.We can do this using futures that has the highest

correlation with the risk component in the asset.


Operationalising unsystematic risk hedging using


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p g y g g g

futures

In the above case, the risk is reduced but cannot be

eliminated.

The amount by which the hedge is imperfect is called the

basis and is defined as:

basis = futures returns spot returns

The extent to which the hedge is imperfect is measured by

basis.


Creating minimum variance hedges


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g g

The most commonly used method of creating minimum

variance hedges is where the amount of hedging is done

so that the variance of the hedged portfolio is as small aspossible. This is called minimum variance hedging.


Portfolio with minimum variance hedges


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g

If

the value of the spot being hedged is V, andh units of the futures position is taken at F0,

then,

the MTM value of the hedged portfolio at t is

MTMt = V

(Ft

F0)h

2MTMt

= 2V + 2h2(V,Ft)

+ h22Ft

where h =

2(V,Ft)

2Ft

and

2MTMt

= 2V

2(V,Ft)

2Ft


Maximising utility


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Alternatively, we can create optimal hedging portfolios,

where the utility of the spot asset holder is maximised.

Here, the objective function becomes:maximizeh EU[V + h(Ft F0)]

For example, a typical utility function used is the quadratic

utility function of the form:

U(MTMt) = MTMtb

2MTM2t

Often, the preference parameters being maximised aretypically the mean and the variance of the profits/losses of

the hedged portfolio.

Then, the solution could be the same as that reached for

the minimum variance optimisation problem.


Hedging using options


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The linearity of futures makes the risk low and the

expected returns low.

Options offer more directional hedges.

For example: I have a portfolio which has a 95% correlation

with the market index. For every one unit move in the

market, my portfolio moves by 0.95 in the same direction.I want to limit the expected loss on this portfolio till the end

of March to less than 2%.

If my portfolio had options trading on it, I would implementthis protection by

1 long end-March portfolio put options2 with strike of 98% of the value of my portfolio


Hedging using options


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But there are no options available on my portfolio.

I have to use Nifty options.

Suppose the correlation between my portfolio and Nifty is

0.95%.

I could:

Go long 29th March Nifty put options with strike of 3600

(98% of 3650 is 3577; thus, 3600 is a bit better thandropping to 98% of the portfolio value).If Nifty should reach 3600, my portfolio would probablyreach 98% of its original value.For every move down in Nifty, the put option would pay me

0.95% of the move value.This should adjust for the loss in my portfolio!

This is why long a put option is called buying portfolio

insurance.


LOB for 29th March Nifty puts


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Arbitrage using derivatives


Some ground rules for arbitrage


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There ought to be a close link between the price of the

derivative (F/C/P) and the price of the underlying asset (S).

Lets assume that we know the link:

E(F) = f(S)

Lets define the difference between the price of the

derivative and the model as basis.

basis = F

E(F)


Defining arbitrage


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Deviations in the basis between derivative and underlyingprice can lead to arbitrage opportunities as follows:

1 If the basis becomes too small (or negative), the arbitragestrategy would be

Long futures, Short spot

2 If the basis becomes too large, the arbitrage strategy would

be the opposite:

Long spot, Short futures

At maturity of the futures, unwind both positions.

If your arbitrage calculation was correct, you have profit atthe end of the unwind.

Arbitrage can be found in any set of these:

spot/futures/options


Example: Arbitrage using gold futures


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The price of gold is Rs.8000/tola.

What should be the least price at which anyone ought tobe willing to sell a 20 April gold futures contract?

As a conservative short, I take the following approach:1 Start with no capital.2 Buy gold today at Rs.8000.3 Borrow the money from the bank.

Tfhe bank at 6% needs repayment per month on theborrowed Rs.8000.

4 If I have to settle on the 20th April, thats 25 days of interest,which is approximately Rs.40 and exactly Rs.34.58.

Therefore, Id be willing to become a 20th April gold futuresshort if the futures market price was anything above

Rs.8035.


The full arbitrage position


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If the spot market price S = 8000 and

the 20th April futures market price F = 8050this is an arbitrage opportunity.

The arbitrage (riskless/certain) profit is

8050 8035 = Rs.15

The sole position short 20th April futures 8050 is risky.

What if the price on 20th April rises to 8070?

Therefore, position = short 20th April futures 8050 and

long gold 8000 today.This gets the arbitrage profit of Rs.15 regardless of what

happens to the market price of gold on 20th April.


Market price is different from the sellers price


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The market price can be legitimately different from the

E(F) = f(S) price as computed above.It does not take other factors into account (brokerage fees,

STT, impact cost, etc).

However, the entire arbitrage approach gives us an idea of

what to expect as a fair price in the market.

Therefore, arbitrage is the first step to pricing derivatives

using the spotprice.


HW: Hedging puzzle


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Puzzle: What is the payoff for this combined portfolio of

(long portfolio + long Nifty put options)?

What is the combination to adopt for a better hedge?


using derivatives

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