Financial Mathematics

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```					    Forward and futures contracts

(Long position) Boths are agreements to buy
an underlying asset at future (delivery) date
for a specific price.
institutional features – most noticably daily
settlement.
Material: Hull Chapters 2,3 and 5. We’ll skip
and jump. (Thus: no need to read Hull like
CT1.)
1                November 23, 2010    MATH 2510: Fin. Math. 2
Futures contracts

Available on a wide range of underlyings
Specifications need to be defined:
–   What can be delivered,
–   Where it can be delivered
–   When it can be delivered (often a period)
Settled daily: On long position you receive the
day-to-day change in the futures price.

2                  November 23, 2010        MATH 2510: Fin. Math. 2
Convergence of Futures to Spot

If the futures price is above the spot price
during the delivery period, arbitrage is
possible
–   Short futures contract
–   Make delivery
And vice versa if the futures price is below the
spot price.

3                   November 23, 2010   MATH 2510: Fin. Math. 2
Futures
Spot Price
Price
Spot Price                       Futures
Price

Time                              Time

(a)                              (b)
4                November 23, 2010     MATH 2510: Fin. Math. 2
Futures vs. forward prices

If you think that the definition of the futures
contract/price seems circular --- I’m inclined
to agree.

Let’s look a Appendix A in Hull’s chapter 5.
This shows that if interest rates are constant,
then futures and forward prices are equal.

5                November 23, 2010      MATH 2510: Fin. Math. 2
If interest rates are stochastic, then forward
and futures prices are not (necessarily)
equal.
If the price of the underlying is positively
correlated w/ interest rates, then we receive
futures payments in high interest rate
scenarios -> futures above forward.
But differences are typically small.
6                November 23, 2010      MATH 2510: Fin. Math. 2
Margins

A margin is cash or marketable securities
deposited by an investor with his or her
broker; serves as collateral.
The balance in the margin account is adjusted
to reflect daily settlement.
Margins minimize the possibility of a loss
through a default on a contract.

7               November 23, 2010   MATH 2510: Fin. Math. 2
Collateral

It is becoming increasingly common (post
Subprime Crisis/Credit Crunch/Financial
Meltdown in particular) for contracts to be
collateralized/have margins.

They are then similar to futures contracts in
that they are settled regularly (e.g. every day
or every week)
8                November 23, 2010      MATH 2510: Fin. Math. 2
Delivery

Closing out a futures position involves entering
into an offsetting trade; most contracts are
closed out before maturity.

Though sometimes spectacularly not:
 Business Snapshot 2.1: Live cattle
 Amajaro summer 2010: Cocoa

9               November 23, 2010     MATH 2510: Fin. Math. 2
When there are alternatives about what is
delivered, where it is delivered, and when it
is delivered, the party with the short position
chooses.

A few contracts (for example, those on stock
indices and Eurodollars) are settled in cash.

10                November 23, 2010     MATH 2510: Fin. Math. 2
Accounting and Taxation

This is really tricky stuff.
 Ideally hedging profits (losses) should be
recognized at the same time as the losses (profits)
on the item being hedged
 Ideally profits and losses from speculation should be
recognized on a mark-to-market basis
 Roughly speaking, this is what the accounting and
tax treatment of futures in the U.S.and many other
countries attempts to achieve.

11                 November 23, 2010        MATH 2510: Fin. Math. 2
Long & Short Futures Hedges

A long futures hedge is appropriate when you
know you will purchase an asset in the
future and want to lock in the price

A short futures hedge is appropriate when
you know you will sell an asset in the future
& want to lock in the price

12               November 23, 2010     MATH 2510: Fin. Math. 2
Arguments in Favor of Hedging

Companies should focus on the main
business they are in and take steps to
minimize risks arising from interest rates,
exchange rates, and other market variables.

13              November 23, 2010   MATH 2510: Fin. Math. 2
Arguments Against Hedging

   Shareholders are usually well diversified and
can make their own hedging decisions
   It may increase risk to hedge when
competitors do not
   Explaining a situation where there is a loss
on the hedge and a gain on the underlying
can be difficult

14                 November 23, 2010    MATH 2510: Fin. Math. 2
Basis Risk

Basis is the difference between spot and
futures prices.

Basis risk arises because of the uncertainty
about the basis when the hedge is closed
out. (Say you can’t match w/ exact delivery
date and/or underlying asset for futures.)

15              November 23, 2010    MATH 2510: Fin. Math. 2
Long Hedge

Suppose that
F1 : Initial Futures Price
F2 : Final Futures Price
S2 : Final Asset Price
You hedge the future purchase of an asset by
entering into a long futures contract
Cost of Asset=S2 – (F2 – F1) = F1 + Basis

16               November 23, 2010    MATH 2510: Fin. Math. 2
Choice of Contract

Choose a delivery month that is as close as
possible to, but later than, the end of the life
of the hedge
When there is no futures contract on the asset
being hedged, choose the contract whose
futures price is most highly correlated with
the asset price. This is known as cross
hedging.

17                November 23, 2010     MATH 2510: Fin. Math. 2
Optimal Hedge Ratio

Proportion of the exposure that should optimally be hedged is

sS
r
sF
where
sS is the standard deviation of DS, the change in the spot price
during the hedging period,
sF is the standard deviation of DF, the change in the futures
price during the hedging period
r is the coefficient of correlation between DS and DF.

18                   November 23, 2010              MATH 2510: Fin. Math. 2

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