Lecutre 12 Future

TOC

Introduction to derivatives Features of derivatives Necessarity of financial derivatives Financial Dderivatives in China Futures specification Forwars v.s. Futures Forward Futures Future Pircing: no arbitrage Case I: Stock Future Pricing without Dividend Case II: a lump-sum payment of dividend D Case III: underlying asset pays a dividend yield of q Case IV: When the Underlying Asset is a Currency

Introduction to derivatives

Features of derivatives

Derived from something else — underlying asset

All financial assets could be taken as derivatives

Stocks, bonds, exchange

Usually, only those with financial assets as underlyings are financial derivatives

Necessarity of financial derivatives

Hedger: To fix prices and reduce risk

Speculator: To enlarge and to facilitate speculation

Arbitrageur: To make free profit without taking risk

there are supposed to be rational relationships between underling asset and derivatives. Once the relationships are breaked, there existss arbitrage opportunities

Financial Dderivatives in China

Forward, Future

Option

Swap

Structured product: structured fund, MBS, CDO

Credit derivative: CDS

Futures specification

Forwars v.s. Futures

Fowards

Agreement to buy or sell the underlying assets (commodity or financial security) at a predetermined price at a future date

Futures

Agreement to buy or sell the underlying assets (commodity or financial security) at a predetermined price at a future date

Both is opposed to a spot transaction where payment and delivery are immediate

Differences

Forward contract ⇒ OTC, not standardized

Future ⇒ Exchange, standardized

Forward

Underlying assets are traded on over-the-count market with dealers

The contract is not standardized

What, how much, when and where are all decided by traders

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Delivery usually takes place

Futures

Underlying assets are traded on a standardized future exhcange (Auction market: CME, CBOT)

The contract is standardized by the exchange:

What assets

Quality of assets

How much assets to be traded within one contract

Where to deliver

Range of delivery dates

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Contract closed prior to maturity

Margin accounts in case of defalut.

is settled daily

the balance of account is to be settled at the end of contract

Marking to market: margin account adjusted with gains/losses at the end of each trading day, features:

initial margin (opening balance), margin rate

maintenance margin, for instance, of the initial margin

triggers margin calls

variation margin

Index Future

The underlying assets are index.

Settled in cash, not by delivery of the underlying asset

Contract size is decided by Exchange

CME: 250 times, S&P500
CBT: 10 times, Dow Jones
SSE: 300 times, CSI 300 (hs 300)

Future Pircing: no arbitrage

When pricing derivatives, we cannot use discounted future cash flow. The prices are derived relative to the price of the underlying assets

The relations have to hold to eliminate arbitrage opportunities

Case I: Stock Future Pricing without Dividend

Assuming no payment from the underlying assets

At time 0

Portfolio A:

To sign on future contract to buy a stock at a price of at maturity of T
Save an amount of cash equal to the present value of the future price in bank for the next T terms
use to buy stock at , and have when sell the stock at

Portfolio B:

To buy one unit of the underlying security with current price of
T: prce

At time T

both portfolio worth one unit of the underlying security, thus they must also have the same value today

Example Index Future Price without Dividend

Scenario

Suppose to buy a 2-year future contract on the S&P 500 index. If the index currently stands at 400 then the futures price in the contract would be?

Assume that:

No dividend is paid

the risk free interest rate is 8%

if , Construct an arbitrage strategy:

intuition: the contract is undervalued, thus we should long it, then think about how to realize that (i.e. where and how much money should we get etc)

Term 0

Take long position in future contract at

Borrow a stock and sell at 400. Save 400 at 8% for two years

Term 2

Withdraw 466.56 and spend of it to buy the stock as predetermined in future contract

Return the stock

net profit = 466.56 -

if , Construct an abitrage strategy:

intuition: the contract is overvalued, thus we should short it and then think about how to realize that

Term 0

Borrow 400 from bank at rate of 8%

use the money buy a stock

Take short position in future contract at

Term 2

sell the stock at future price of

use 466.56 of it to return to the bank

net profit is - 466.56

Case II: a lump-sum payment of dividend D

Assuming a lump-sum payment of dividend D by the end of the maturity from the underlying assets. The present value of dividend is therefore

At time 0

Portfolio A:

Taking a long position in one future contract promising to buy a stock at a price of at maturity of T
Save an amount of cash equal to for T terms

Portfolio B:

To buy 1 unit of the underlying security with current price of
get at T term

At time T

both will be worth on unit of the underlying security

thus, they must have the same value today

Example lump-sum payment of dividend

Scenario

Assuming a 2-year future contract on stock A, assuming the current price of the stock is 40, what would be the future price in the contract?

A’s lamp-sum dividend whose present value is 5 will be paid during the life of the contract; Risk-free rate is 8%

if < 40.82, construct the arbitrage strategy:

Term 0

take long position of the future contract

Borrow a share of stock and sell it at $40. Save the money in the bank at the rate of 8%

Term T

withdraw from bank.

buy a share of stock at future price

return the stock and pay the lender

net profit is

if > 40.82, construct the arbitrage strategy:

Term 0

borrow 40 at the rate of 8%

buy a share of stock A in spot market

write a future contract with a short position at price of

Term T

Receive the dividend of

sell the stock A at the future price of

return bank with

net profit is

Case III: underlying asset pays a dividend yield of q

Assuming the underlying asset pays a dividend yield of q (i.e. gain more share of stocks) and all dividends are re-invested to the same assets before maturity

The percentage of the asset price when the dividend is paid:

By the end of T, the total value would be

Portfolio A

To sign one future contract to buy a stock at a price of at maturity of T

Save in bank for T terms

Portfolio B

Term 0: Hold units of the security, thus value

Term T: value is

thus,

if continuously compound rate is assumed,

if , construct the arbitrage strategy:

Term 0

Take short position of the future contract

borrow from bank at the rate of , and buy shares of the stock at price of

Term T

sell the stock (one share now) at the future price

return of the money to the bank

net profit is

if , construct the arbitrage strategy:

Term 0

Take long position of the future contract

borrow shares of stock and sell it at price . Save the money in bank at the rate of

Term T

withdraw from the bank

buy a stock at future price and return the stock

net profit is

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Since we only borrow the stock, thus when we return it, we should include the dividend yield during the period. In other word, we borrow share at term 0, but we should return 1 share at term T, including the dividend yield.

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Trick: We can directly know how much can we arbitrage. Note that is the price in contract and the other side of ge/le is money related to bank. So if is larger, it means at term T, should be minused, and we should borrow money from bank at term 0.

Case IV: When the Underlying Asset is a Currency

Spot foreign exchange contract: an agreement to exchange one currency for another currency today.

Buy 1 AUD with 0.6620 EUR

Commodity currency: asset being bought or sold
Term currency: the other currency in which the commodity currency is expressed

The same with Case III when underlying asset is an equity with constant dividend yield q.

Assume the interest rate of the term currency is , the interest rate of the commodity currency is

Example

Assumption:

The EUR per 1 AUD future price is

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Basic Rule for Future Pricing Future price = spot price + cost of funds - benefit of carry

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A problem with no arbitrage pricing Only local equilibrium instead of global equilibrium: + The method is okay when the market if efficient + But mispriced when the underlying assets are mispriced + Lesson: 2008 financial crisis