# Financial Mathematics

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These slides

Repeat appearences
 Hull’s Chapter 7 on swaps
 Mock Exam #1

1              November 30, 2010   MATH 2510: Fin. Math. 2
Today’s Topics: A Mixed Bag

Hull Section 3.3-4: Hedging w/ futures
especially w/ less-than-perfect correlation
(“cross hedging”.)
 Course Work 3
 Student surveys
Hull Chapter 7 on swaps.

2                November 30, 2010    MATH 2510: Fin. Math. 2
Futures and forward contracts

Very briefly: A bet that the underlying goes up
(long) or down (short).

To hedge: To bet against your own assets. Win
some, lose some. In the same way, fire
insurance is a bet that your house burns
down.

3               November 30, 2010     MATH 2510: Fin. Math. 2
Futures Hedges

Futures contracts are suitable for hedging i.e.
for “covering you your bets”. When/what
you lose 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.
4               November 30, 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.)

5              November 30, 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

6                 November 30, 2010     MATH 2510: Fin. Math. 2
Hull’s Example 3.2

A company knows it needs to buy 20,000
barrels of crude oil. Doesn’t know exactly
when.
Now is June 8. Futures contracts are for 1,000
barrels. December-futures price is \$18 (per
barrel).
Company goes long 20 Dec.-futures.

7               November 30, 2010    MATH 2510: Fin. Math. 2
On Now. 10, the company buys the oil. Spot
price is, say, \$20 and Dec.-futures price is
\$19.10.
It closes the futures contract.
Gain on futures (ignoring time value of money)
is \$1.10, and the effective price paid for oil is
\$20-1.1 = \$18.90 (per barrel;
20,000*\$18.90= \$378,000 is paid in total).
8                 November 30, 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.

9                   November 30, 2010              MATH 2510: Fin. Math. 2
Less-than-perfectly futures-spot
correlation?

If the futures contact is on a (slightly) different
 Jet fuel vs. crude oil
 Profit of a tire manufactures vs. oil prices
 The price of your house vs. an index
 …

10                 November 30, 2010      MATH 2510: Fin. Math. 2
If the ”buy the underlying for borrowed money
and hold on to it”-argument does not work
 Storage not possible (electricity) or costly
 No spot market; the underlying good does
not exist (corn not harvested yet)
 Convinience yield (more or less tangible)
from holding the underlying

11               November 30, 2010     MATH 2510: Fin. Math. 2
Course Work #3; Stochastic Rates of
Return

Q1+3: Standard CT1, Unit 14, Table 1.3.1. This
will be on the exam. For the annuity case,
you will only be required to use recursive
formulas. Careful w/ expectations and non-
linear functions.
Q2: Pascal’s triangle.
Q4: Non-identical distributions can be handled
too. Could come at the exam.
12               November 30, 2010   MATH 2510: Fin. Math. 2
Q5:
 Promises may sound alike but be very
different.
 Based on a true story.
 If you think this was difficult to understand,
imagine having it written by lawers.
 Things don’t always go as easlily as in CT1,
Unit 14. Simulation is then a powerful tool.

13                November 30, 2010     MATH 2510: Fin. Math. 2
Student Surveys

I’n not angry, I’m disappointed. And so are you.
(Or some combination.)

The first time the course in given. (So the lack of exam
papers will solve itself over time.)

Mix of slides and whiteboard: Completely intentional,
but culture chock. Hard-to-read handwriting: Not so
much.

14                  November 30, 2010        MATH 2510: Fin. Math. 2
CT1 Exam exemption imposes contraints.

Course Works
 Yes, they are demanding. (I.e. not the
easiest way to get 5%.)
 No, they are not unrelated to lectures,
workshops, text-books - or even Math 1510.
 And you’re doing fine.

15               November 30, 2010   MATH 2510: Fin. Math. 2

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