Agenda
A lot of hand-outs
Comments to Course Work #1
The put-call parity (Hull Ch. 9, Sec. 4)
Duration (CT1, Unit 13, Sec. 5.3)
 A few explicit formulas
 Generalization to non-flat yield curves
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October 21, 2010
MATH 2510: Fin. Math. 2
Today’s Hand-Outs: Collect ’em All
Graded Course Works #1
Solution to Course Work #1
Course Work #2
Exercises for Workshop #4
Today’s slides w/ Hull Ch. 9, Sec. 4 attached
Updated course plan (blue paper)
Solutions to Workshops #2 & #3
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October 19, 2010
MATH 2510: Fin. Math. 2
The Final Exam
Earlier I may have been fuzzy, but now: The final exam
will be closed book.
Reason, I: ”Local customs.”
Reason, II: Exemptions from having to take the
”professional” CT1-exams – which are closed book.
You will get ”very life-like” exam papers to practice on.
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October 19, 2010
MATH 2510: Fin. Math. 2
Course Work #1
Generally: Good work.
Common errors and/or what to do about them:
1.26: Borrow money too
1.27:Tell customer he may loose all w/ options
1.29-1.30: Draw the pay-off profile
1.32: Short put
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October 19, 2010
MATH 2510: Fin. Math. 2
The Put-Call Parity
As seen in Hull Assignment 1.32, a position that is long
one call and short one put pays off S(T) – K.
Algebraically because


(S (T )  K )  (K  S (T ))  S (T )  K
This has interesting consequences.
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October 19, 2010
MATH 2510: Fin. Math. 2
If we happen to come across a strike-price, say K*,
such that the call and the put cost the same, then the
forward price must (”or else arbitrage”) equal K*,
irrespective of any dividends.
This follows ”from first principles”.
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October 19, 2010
MATH 2510: Fin. Math. 2
If the underlying pays no dividends (during the life
of the options), then the (long call, short put)
payoff is replicated by being long the underlying
and short K zero coupon bonds bonds w/
maturity T.
Thus (”or else aribtrage”):
Call(0)  Put(0)  S (0)  P(T ) K
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October 19, 2010
MATH 2510: Fin. Math. 2
This formula/relationship is called the (basecase) put-call parity.
It is surprisingly useful.
Dividends: See Workshop #4.
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October 19, 2010
MATH 2510: Fin. Math. 2
Duration
Measures the sensitivity of present
values/prices to changes in the interest rate.
It has ”dual meaning”:
 A derivative wrt. the interest rate
 A value-weighted average of payment times
(so: its unit is ”years”)
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October 19, 2010
MATH 2510: Fin. Math. 2
Set-up:
 Cash-flows
at tk
C
 Yield curve flat at i (or continuously
compounded/on force form: )

 Present value of cash-flows:
tk
A  k Ct k (1  i) tk  k Ct k P(tk )
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October 19, 2010
MATH 2510: Fin. Math. 2
Macauley Duration
The Macauley duration (or: discounted mean term) is
defined by
 :
k tk Ctk (1  i)
tk
A
Clearly a weighted average of payment dates.
But also: Sensitivity to changes in the force of interest.
Or put differently: To parallel shifts in the (continuously
compounded) yield curve.
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October 19, 2010
MATH 2510: Fin. Math. 2
Some Duration Formulas
The duration of a zero coupon bond is its maturity.
The duration of an n-year annuity making payments
D is
( Ia) n|
a n|
where as usual
12
,
an|  (1  v ) / i with v=1/(1+i),
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October 19, 2010
MATH 2510: Fin. Math. 2
and IA is the value of an increasing annuity
n
(
1

i
)
a

nv
n|
v  2v 2  3v3  ... nvn : ( IA)n| 
,
i
as shown in CT1, Unit 6, Sec. 3.2.
Using exactly similar reasoning, the duration of a bullet
bond w/ coupon payments D and notional R is
D( Ia) n|  Rnvn
Dan|  Rvn
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October 19, 2010
MATH 2510: Fin. Math. 2
Duration w/ a Non-Flat Yield Curve
If the yield curve is not flat, it is the natural to define
duration as
t C P(tk )

F W
k k tk

:
.
A
This is called the Fisher-Weil duration.
Can still be intepreted as sensitivity to parallel shifts in
the yield curve. (But reminds us that other
deformations may occur.)
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October 19, 2010
MATH 2510: Fin. Math. 2
The Macauley duration can also calculated with a nonflat yield curve.
In that case the Macauley and Fisher-Weil durations
are not equal.
The Fisher-Weil duration of a portfolios is a
”straightforward” weighted average of its
constituents; Macauley duration is not.
To calculated F-W duration we must know the yield
curve; not so for Macauley.
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MATH 2510: Fin. Math. 2