1. October 21, 2010MATH 2510: Fin. Math. 2 1 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

2. October 19, 2010MATH 2510: Fin. Math. 2 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

3. October 19, 2010MATH 2510: Fin. Math. 2 3 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.

4. October 19, 2010MATH 2510: Fin. Math. 2 4 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

5. October 19, 2010MATH 2510: Fin. Math. 2 5 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 This has interesting consequences.

6. October 19, 2010MATH 2510: Fin. Math. 2 6 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”.

7. October 19, 2010MATH 2510: Fin. Math. 2 7 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”):

8. October 19, 2010MATH 2510: Fin. Math. 2 8 This formula/relationship is called the (base- case) put-call parity. It is surprisingly useful. Dividends: See Workshop #4.

9. October 19, 2010MATH 2510: Fin. Math. 2 9 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”)

10. October 19, 2010MATH 2510: Fin. Math. 2 10 Set-up: Cash-flows at t k Yield curve flat at i (or continuously compounded/on force form: ) Present value of cash-flows:

11. October 19, 2010MATH 2510: Fin. Math. 2 11 Macauley Duration The Macauley duration (or: discounted mean term) is defined by 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.

12. October 19, 2010MATH 2510: Fin. Math. 2 12 Some Duration Formulas The duration of a zero coupon bond is its maturity. The duration of an n-year annuity making payments D is where as usual with v=1/(1+i),

13. October 19, 2010MATH 2510: Fin. Math. 2 13 and IA is the value of an increasing annuity 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

14. October 19, 2010MATH 2510: Fin. Math. 2 14 Duration w/ a Non-Flat Yield Curve If the yield curve is not flat, it is the natural to define duration as 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.)

15. October 19, 2010MATH 2510: Fin. Math. 2 15 The Macauley duration can also calculated with a non- flat 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.