Agenda

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 MATH 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 MATH 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. MATH 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 MATH 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. MATH 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”. MATH 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”): MATH 2510: Fin. Math. 2

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

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