# Financial Mathematics 2

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A template exam-paper (pink.)
Hull’s Chapter 7 on swaps (or next week.)
Course plan (blue) and these slides.

1               November 25, 2010    MATH 2510: Fin. Math. 2
The Template Exam-Paper

17 numbered questions each worth 5%.
The exam-paper in January will be quite similar
in content and organization.
There will be one more template in the last
week. Do you want it “simulation style” w/

2               November 25, 2010    MATH 2510: Fin. Math. 2
General – But Useful – Exam Advice

   Be quick.
   Questions are quite independent, so if you
get stuck, go on.
   Partial credit is given (a lot.) Give it the old
college try.
   And, oh: Solving two linear equations w/ two
unknowns will come up --- without warning.

3                  November 25, 2010      MATH 2510: Fin. Math. 2
The Cash-Flows of a Futures Contract

   Entering into a futures contract costs nothing
(when you do it.)
   When you enter a long position in a futures
contract at time t-dt (think dt~1 day), you
recieve Fut(t) – Fut(t-dt) at time t, where
Fut(t) is the co-called futures price.
   At the delivery date T: Fut(T) = S(T)

4                 November 25, 2010     MATH 2510: Fin. Math. 2
When interest rates are constant, futures and
foward prices are equal – or else arbitrage.
(And we/Hull then just write F). That’s a
theorem. (Hull Ch. 5; proof in Appendix.)

Futures and fowards do not have the same
5.2) – but we can transform on into the other
(proof of theorem.)
5               November 25, 2010     MATH 2510: Fin. Math. 2
We can think of a futures contract as a forward
contract where gain/loss are settled each
day. (This happens via a margin account,
where collateral is posted --- lower credit
risk.)

Or in a tight spot: ”futures is forward”.

6                November 25, 2010       MATH 2510: Fin. Math. 2
Remember our theoretical expression for arbitrage-free
forward prices

F (0)  e  ( S (0) corrected for dividends)
rT             e  qT

where ”to correct” means to:
 Subtract i  Dt e rt in case of cash dividends.
i
i

 qT
 Multiply by e         in case of a dividend yield (or a foreign
interest rate.)

7                   November 25, 2010           MATH 2510: Fin. Math. 2
Futures Hedges

Futures contracts are suitable for hedging .e.
for “covering you your bets”. When/what
you loose one thing, you gain on another.

A long (short) futures hedge is appropriate
when you know you will purchase (sell) an
asset in the future and want to lock in the
price.
8               November 25, 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.

9              November 25, 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

10                 November 25, 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.)

11              November 25, 2010    MATH 2510: Fin. Math. 2
Long Hedge

Suppose that
F(1): Initial Futures Price
F(2): Final Futures Price
S(2) : Final Asset Price
You hedge the future purchase of an asset by
entering into a long futures contract
Cost of Asset= S(2) – (F(2)– F(1)) = F(1) + Basis

12                 November 25, 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 and
r is the coefficient of correlation between DS and DF.

13                   November 25, 2010              MATH 2510: Fin. Math. 2
On Hull’s Futures Chapters

This week we’ve covered more pages than in
all of CT1.
We skip and jump in Hull. Same level of
detailed knowledge as for CT1 is not
required on the exam.
Skip pages 65-66 on ”changing beta”. (It’s not
deep, but one needs to know what the
”Capital Asset Pricing Model” is for it to make
sense.)
14                November 25, 2010     MATH 2510: Fin. Math. 2

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