Question of Derivatives

8/13/2019 Question of Derivatives

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Path independent OptionAnoptionwhose payoff depends solely on the events specified to take place upon

expiration, rather than the path taken by the underlying variable (price, rate, index, etc). A

prime example is aEuropean optionwhere the price for the buyer depends exclusively onthe terminal price. If an investor buys a call option with a strike price of USD 50, he would

benefit if the underlying rises above 50 upon expiration. If not, he would not exercise and

the premium paid to buy the option is lost. It is irrelevant which route the underlying price

(the sport price of the option's underlying asset) takes within time to expiration.

This option, opposite ofpath-dependent option,is also referred to asnon-path-

dependent option.
http://www.investment-and-finance.net/derivatives/o/option.htmlhttp://www.investment-and-finance.net/derivatives/o/option.htmlhttp://www.investment-and-finance.net/derivatives/o/option.htmlhttp://www.investment-and-finance.net/derivatives/e/european-option.htmlhttp://www.investment-and-finance.net/derivatives/e/european-option.htmlhttp://www.investment-and-finance.net/derivatives/e/european-option.htmlhttp://www.investment-and-finance.net/derivatives/p/path-dependent-option.htmlhttp://www.investment-and-finance.net/derivatives/p/path-dependent-option.htmlhttp://www.investment-and-finance.net/derivatives/p/path-dependent-option.htmlhttp://www.investment-and-finance.net/derivatives/n/non-path-dependent-option.htmlhttp://www.investment-and-finance.net/derivatives/n/non-path-dependent-option.htmlhttp://www.investment-and-finance.net/derivatives/n/non-path-dependent-option.htmlhttp://www.investment-and-finance.net/derivatives/n/non-path-dependent-option.htmlhttp://www.investment-and-finance.net/derivatives/n/non-path-dependent-option.htmlhttp://www.investment-and-finance.net/derivatives/n/non-path-dependent-option.htmlhttp://www.investment-and-finance.net/derivatives/p/path-dependent-option.htmlhttp://www.investment-and-finance.net/derivatives/e/european-option.htmlhttp://www.investment-and-finance.net/derivatives/o/option.html


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'Path Dependent Option'

The right, but not the obligation, to buy or sell an

underlying asset at a predetermined price duringa specified time period, where the price is based

on the fluctuations in the underlying's value

during all or part of the contract term. A path

dependent option's payoff is determined by the

path of the underlying asset's price.

A basic American option is one type of path dependent

option. Because it can be exercised at any time prior to

expiration, its value will change as the underlying asset's

value changes. An Asian option, also called an average

option, is another type of path dependent option, because

its payoff is based on the average price of the underlying

asset during the contract term. Similarly, a barrier option

would be considered a path dependent option because its

value changes if the underlying asset reaches or surpasses a

specified price. The lookback option and Russian option are

also path-dependent options.

The cost of carry or carrying chargeis the cost of storing a physicalcommodity,such

asgrainormetals,over a period of time. The carrying charge includes insurance,storage andintereston

theinvestedfunds as well as other incidental costs. In interest ratefutures markets,it refers to the

differential between the yield on acashinstrument and the cost of the funds necessary to buy the

instrument.[1][2]

If long, the cost of carry is the cost of interest paid on a margin account. Conversely, if short, the cost of

carry is the cost of paying dividends, or rather the opportunity cost;the cost of purchasing a

particularsecurityrather than an alternative. For most investments, the cost of carry generally refers to

the risk-free interest rate that could be earned by investing currency in a theoretically safe investment
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vehicle such as amoney market accountminus any future cash-flows that are expected from holding an

equivalent instrument with the same risk (generally expressed in percentage terms and called

theconvenience yield). Storage costs (generally expressed as a percentage of the spot price)should be

added to the cost of carry for physical commodities such as corn, wheat, or gold.

The cost of carry model expresses theforward price(or, as an approximation, thefutures price)as a

function of thespot priceand the cost of carry.

where

is theforward price,

is thespot price,

is the base of thenatural logarithms,

is therisk-free interest rate,

is the storage cost,

is theconvenience yield,and

is the time to delivery of theforward contract(expressed as a fraction of 1 year).

The same model in currency markets is known asinterest rate parity.

For example, a US investor buying a Standard and Poor's 500 e-

minifutures contracton theChicago Mercantile Exchangecould

expect the cost of carry to be the prevailing risk-free interest rate

(around 5% as of November, 2007) minus the expected dividends that

one could earn from buying each of thestocksin theS&P 500and

receiving anydividendsthat they might pay, since thee-minifuturescontract is a proxy for the underlying stocks in the S&P 500. Since the

contract is a futures contract and settles at some forward date, the

actual values of the dividends may not yet be known so the cost of

carry must be estimated.

Cross hedgingis when you hedge a position by investing in two

positivelycorrelatedsecurities or securities that have similar price movements.

The investor takes opposing positions in each investment in an attempt to

reduce the risk of holding just one of the securities. The success of cross-

hedging depends completely on how strongly correlated the instrument being

hedged is with the instrument underlying the derivatives contract. When cross
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hedging, thematurityof the two securities has to be equal. In other words, you

cannot hedge a long-term instrument with a short-term security. Both financial

instruments have to have the same maturity.

Hedging is a form of investment insurance that is meant to reduce risk. Hedging

does not eliminate the amount of risk involved in an investment; it just softens

the negative effect on the hedger. Typically, hedging involves investing in two

securities that have a negative correlation. A negative correlation means that

the two securities move in opposite directions. When one security looses value,

the other gains value.

For example, you could have a long position in a gold company then take a

short position in a goldETF.Because the price of the gold company's stock

would move in tandem with the price of gold, it would create a cross hedge.

There wouldn't be a perfect correlation, so this example would not provide aperfect hedge.

Definition of 'Hedge Ratio'

1. A ratio comparing the value of a position protected via a hedge with the size

of the entire position itself.

2. A ratio comparing the value of futures contracts purchased or sold to the

value of the cash commodity being hedged.

1. Say you are holding $10,000 in foreign equity, which exposes you to

currency risk. If you hedge $5,000 worth of the equity with a currency position,

your hedge ratio is 0.5 (50 / 100). This means that 50% of your equity position

is sheltered from exchange rate risk.

2. The hedge ratio is important for investors in futures contracts, as it will help

to identify and minimize basis risk
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Optimal hedge ratio

In hedging, you can hedge your whole portfolio or some portion of it. The hedge ratio is

the size of the futures contract relative to the cash transaction. In a previous example involving a

trader with oil en route from the Gulf, the hedge ratio was one, since she sold a futures contract

representing each barrel of oil. When using turnips to hedge, the ratio was still 1, since the value

of the crude was hedged with an equal value of turnips. If the hedger had sold _ barrel for each

barrel in transit, the hedge ratio would have been _. It turns out that it is not always optimal to

hedge your whole product and Hull (2000) works out what the optimal hedge ratio is to minimize

risk. In his exposition, he defines

S = ST - St = the change in the spot price over the life of the contract,

F = FT - Ft = the change in the futures price over the life of the contract,

S = the standard deviation of S,

F = the standard deviation of F,

= SF/(SF) = the correlation coefficient between S and F, which is the covariance

of S and F divided by the standard deviations of S and F, and

h = the hedge ratio.

If the hedger owns the products and sells the future, his portfolio value is (S - hF). The change in

value of the portfolio is

S - hF.

Alternatively, if the hedger buys the future and is short the product, his portfolio value is hF - S.

The change in value of the portfolio is

hF - S.

The variance (2

) for the above two portfolios is


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2

=

S

2

+ h2

F

2

- 2hSF =

S

2

+ h2

F

2

- 2hSF.

To find the optimal hedge ratio, which minimizes risk or variance, minimize the above expression

with respect to h.

2

/h = 2h

F

2

- 2SF = 0.

Checking second order conditions

2


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2

/h2

= 2

F

2

> 0.

So h is a minimum. Solving for h, we get

h = S/F.Now if S and F are for the same products it is likely that S and F are close to the same value

and

is close to 1. Then the optimal hedge ratio is near 1. Where this becomes more interesting is

where you are hedging one product with a diffe

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