derivativespricing-2013-L7_8

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INTRODUCTION TO DERIVATIVES PRICING

LECTURES: 7, 8& 9

DERIVATIVES

Derivatives are financial securities whose payoffs are explicitly given bythe value of some other underlying asset.

Important examples are Forward contracts, Futures contracts, Options,Swaps.

1. Forward contracts

Forward contract on an asset is a contract to buy or sell a specific amountof the asset at a specific price and at a specific time in the future.

Example. Consider a Forward contract to buy 100, 000 kg of sugar at 40Rs per kg on 15th December.

Some features of Forward contracts.

The contract is between two parties. The asset involved in the forward contract is called the underlying

asset.

Buyer is said to be in long position of the Forward contract. Seller is said to be in short position. The agreed price of the asset in the contract is called the Forward

price. The time of trading of the asset in the contract is called maturity The open market for immediate delivery of the asset is called the

Spot market. The price of the asset in the open market is called Spot price.

Forward price formula To find the forward price, we need to introducea trading mechanism called Short selling. By Short selling we meanborrow the asset and sell it in the spot market.

Suppose you are short selling a stock for the period [0, T]. Assume that thestock will pay a dividend some time before T. What is your cash flow?

We make the following assumptions about the Market.

There is an ideal bank with a spot rate curve {st|t = 1, 2, } Market contains the underlying asset. No arbitrage. Short selling of the underlying asset is allowed.

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2 INTRODUCTION TO DERIVATIVES PRICING LECTURES: 7, 8& 9

Theorem 1.1. Let F denote the forward price, T maturity time, d(0, T)the discount factor for the period [0, T] and S current spot price of the

underlying asset. Also assume that there is no storage cost or dividend forthe underlying asset. Then

F =S

d(0, T).

Proof. The proof is based on the no arbitrage argument.

Suppose

F >S

d(0, T).

Borrow S from Bank. Buy unit asset from the spot market and store.

Short unit forward contract.This creates an arbitrage.

Initial value of the above investment is zero and it has a final payoff ofF S

d(0,T) , an arbitrage opportunity.

Note F is the amount the investor receives at T from the forward contractand S

d(0,T) is the liability to the bank at time T.

Hence

F S

d(0, T).

Now show that (exercise)

F S

d(0, T).

Now we assume the underlying asset has carrying cost c(k) for the period[k, k + 1) , k = 0, 1, , M payable at the beginning of the period.

M is the number of periods until maturity T of the forward contract.

If underlying is a commodity, then carrying cost (storage cost) canbe cost to store or insurance fee etc. which in general positive.

If the underlying is a financial security such as stock, carrying is costis dividend and hence is negative.

Theorem 1.2. The forward price assuming carrying cost is given by

F =S

d(0, M)+

M1

k=0

c(k)

d(k, M).


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INTRODUCTION TO DERIVATIVES PRICING LECTURES: 7, 8& 9 3

Proof. If

F 0 .

Therefore, the above investment strategy is an arbitrage.

Hence

F S

d(0, M)+

M1

k=0

c(k)

d(k, M).

Similarly, we have (exercise)

F S

d(0, M)+

M1

k=0

c(k)

d(k, M).

Suppose short selling of the asset is not allowed or a situation where noone is willing to lend you the asset.

This can arise, if the asset is in short supply and it is anticipated thatasset will be in short supply in future also, i.e. a tight market. Thishappens mainly for commodities. See the illustration based on soybeangiven in Luenberger, page 272.

The the forward price only satisfies the inequality (Why?)

F S

d(0, M)

+M1

k=0

c(k)

d(k, M)

.

Now introducing the notion of convenience yield y, the forward priceformula becomes

F =S

d(0, M)+

M1

k=0

c(k)

d(k, M)

M1

k=0

y

d(k, M).

Convenience yield represents the value of holding the asset. If short sellingis possible, then y = 0.


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Value of a Forward contract. Forward price varies with time. i.e. it is afunction of time. So let Ft denote the forward price at time t. Thus we have

the following question. Suppose you are holding a long forward contract attime 0 and current time is t. What is the value of your contract today?i.e. suppose you would like to transfer your contract to a second party, howmuch she will pay(+ or -) for it.

Answer : Value of the contract, ft = (Ft F0)d(t, T).Solution: Hint: Use arbitarge argument.

2. Swaps

Swap is a contract to exchange one cash flow for the other.

For example(vanilla swap): Consider a contract which allows partyA

toreceives spot price of a unit of an asset for each period(end of) while payingthe party B a fixed amount X at each period for a fixed number of periodsM.

What is right choice of X?

Note A is at risk, since A receives random cash flow. How can A eliminatethis risk.

If A takes short position on forwards with underlying as the asset in theswap and maturity as end of each period. Then at the end of ith period hisreceipt will be Fi = Si + Fi Si where Fi denote the forward price for theforward contract with maturity end of ith period.

Now choice of X is such that the cash flows

(0, X, X, X), (0, F1, F2, , FM)

have same present value. i.e.

M

i=1

d(0, i)X =M

i=1

d(0, i)Fi

Hence

X = 1Mi=1 d(0, i)

M

i=1

d(0, i)Fi .

Now one can use no-arbitrage argument to show that that

1M

i=1 d(0, i)

M

i=1

d(0, i)Fi

is the right choice for X, i.e., anything else will create arbitrage.


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To see this, suppose

X > 1Mi=1 d(0, i)

Mi=1

d(0, i)Fi .

Then

Enter into the fixed leg of the swap, i.e. receive X while paying Siat end of period i, i = 1, 2, cdots, M.

Take long i-forward on the asset (whose price at the end of periodi is Si), i = 1, 2, M. (Here i-forward contract means forwardcontract with maturity as end of period i.

At the end of period i put X Fi in risk-free asset.

This is an arbitrage.

The reverse inequality is similar.

3. Futures contract

Futures contractis like forward contract with the difference of being tradedin an organized exchange.

Futures contract is designed to eliminate the default risk associated withforward contract.

It is achieved through the process called marking to the market.

While entering into the futures contract both parties need to depositan amount with the broker called the margin account.

At the end of each day, the net receipt will be marked. i.e. at the endof each day, the delivery price will be revised to the current delivery(futures price) and the net receipt from this process is added to themarket.

For example, at the end of day one, for the party holding the longposition in the futures contract F1 F0 will be deposited in her/hismargin account, where F0 is the futures price at the time of entry ofthe contract and F1 is the futures price at the end of day one.

when the margin value falls below a certain level, the contract holderneed to pay money to makeup the margin account.

4. Options: basic facts.

Definition: Option is a financial contract where the buyer of the contracthas the right to buy/sell specific number of units of the underlying asset ata specific price by a specific time in the future.

Some terminologies

Buyer of the option is called holder and seller is called writer.


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If the holder has the right to buy, then it is a call option. If the holder has the right to sell, it called put option.

Agreed price to buy/sell is called strike price. The specific time till the option is valid is called the maturity. If the holder can buy/sell the underlying only at maturity, then it is

called European option. If the holder can buy/sell at any time till maturity, it is called Amer-

ican option. A call option with strike K is said to be in the money, at the money

and out of the money if S0 > K, S0 = K and S0 < K respectively,where S0 denotes the current price of the underlying.

A put option with strike K is said to be in the money, at the moneyand out of the money if S0 < K, S0 = K and S0 > K respectively.

The payoff at maturity of a European call option(Holder) is given bymax{ST K, 0}

where ST price of the underlying at maturity T and K strike price.

Exercise. What is the payoff at maturity of the above option for thewriter (i.e. the party which is selling the option)?

Exercise. What about the payoff of the European put option ?

Holder of the option always has a non negative payoff. Need to pay to buy an option: the option price.

How to find the option price? We will try to answer this question.

Factors which affect option price.

Current price of the underlying: Bigger the current price bigger thecall option price. What about put option?

Strike price: Bigger the strike price smaller the call option price.What about put option?

High volatility (i.e. variance of the underlying per annum) implieshigh price (both call and put).

Large time to maturity imply high option price (both call and put).

Risk-free rate : More complex


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4.1. Trading strategies involving options. In principle, one can con-struct any payoff pattern at maturity( which is L2-integrable) by combining

options of various strikes and the underlying asset. We will see some verycommon trading strategies below.

Trading strategies involving a single option and a stock

1 Covered call: Portfolio of a long position in the underlying asset anda short position in the call option. Covered call protects the writerof the call option from the upward movement of the stock price.

2 Protective put: Portfolio of a long position in put option and a longposition in the underlying stock.

To see the profit patterns of the above, see page 246 of [Hull].

Trading strategies involving two or more options of the same type (Spreads)All options are assumed to be with same maturity and same underlying.

1 Bull Spread: Portfolio of a long position in call option with strike K1and a short position in call option with strike K2, where K1 < K2.Payoff at maturity T is given by

0 if ST K1ST K1 if K1 < ST K2K2 K1 if ST > K2 .

Looking at the payoff pattern, one can observe that an investorwho anticipates the up movement of the underlying will prefer Bullspread.

If both the calls are out of the money, then(a) Cost of buying the bull spread is low.(b) Small probability of receiving the maximum payoff K2 K1.

If Both calls are in the money, then(a) Cost of buying the bull spread is high.(b) High probability of receiving the maximum payoff K2 K1.

2 Bear spread: Portfolio of a short position in put option with strikeK1 and a long position in put option with strike K2, where K1 < K2.Payoff at maturity T is given by

K2 K1 if ST K1K2 ST if K1 < ST K20 if ST > K2 .

Looking at the payoff pattern, one can observe that an investor whoanticipates down movement of the underlying will prefer Bear spread.

Exercise. Analysis what happens when options are in the moneyand out of the money.


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3 Butterfly spread: Portfolio of a long position in a call option withstrikes K1, K3 and two short positions in a call option with strike

K2, where 2K2 = K1 + K3. Payoff at maturity is0 if ST K1ST K1 if K1 < ST K22K2 K1 ST if K2 < ST K32K2 K1 K3 = 0 if ST > K3 .

To see the profit patterns of the above, see pages 247-253 of [Hull]. Studentis advised to read pages 254-259 of [Hull] to see other trading strategiesinvolving options.

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