1. Financial Mathematics 2 The plan for Tuesday October 5, 2010 Practical matters Forwards: Hull Sec. 1.6-8 Options: Hull Sec. 1.5, 1.8

2. The rest of Hull Ch. 1 is self-reading. (We’ll get back to ”futures”.) Valuing forward contracts by (no-)arbitrage arguments: CT1 Unit 12

3. Practical matters The admin’ does not want us to move Workshops around ”willy-nilly”. Those of you with time-table conflicts contact Louise Feaviour (room 8.19b). Until further notice we stick to the orginal plan. Hand-out: Course Work #1. Due at lectures on Thursday October 14.

4. Who would want to use/trade in forward contracts? Hedgers. Hull’s p. 10 example: A US company will pay £10 million for imports from Britain in 3 months and decides to hedge using a long position in a forward contract. Speculators. Hull’s example p. 12 (For ”futures” read ”forward”.) But clearer in a minute w/ options. Arbitrageurs: people who attempt to make risk-free profits by exploiting relative mis-pricing between assets/products/contracts. More on these shortly.

5. Options Call-option: The right, but not the obligation, to buy the underlying for the (strike- or exercise-)price K at the future (expiry-)date T. Put-option: Right, not obligation, to sell.

6. Pay-off-diagrams: Hockey-sticks. Unlike forward contacts, call- and put- options cost money up front. Clearly, they have to. (Why?)

7. Why Study Options? Used by Hedgers (put ~ portfolio insurance) Speculators Embedded in many other financial contratcs (pensions, mortgages, …) We will not study how options are priced, i.e. why they cost, what they cost.

8. Hedging w/ Put-Options An investor owns 1,000 Microsoft shares currently worth $28 per share. A two-month put-option with a strike price of $27.50 costs $1. The investor decides to hedge by buying 1,000 put options (“10 contracts”)

9. Portfolio Value in Two Months with and without Hedging

10. Speculating with Call-Options An investor with $2,000 to invest feels that Amazon.com’s stock price will increase over the next 2 months. The current stock price is $20 and the price of a 2-month call option with a strike of 22.5 is $1 He can put his $2,000 into 100 shares of Amazon.com stock 2,000 strike-22.5, expiry-2M call-options

12. Valuation of Forward Contracts How are spot and forward prices related? A simple yet powerful principle: Absence of arbitrage. Or: There is no such thing as a free lunch. CT1 Unit 12, Sec 1 Base-case: Fwd(t,T) = exp(r*(T-t))*Spot(t)

13. Extensions of Forward Valuation CT1 Unit 12 Sec. 2.3: Fixed intermediate cash-flows on the underlying (~ fwd on coupon bond) Sec. 2.4: Dividend yield (~ currency underlying; ~commodities w/ storage costs) Sec. 2.6: Value between initiation (t) and expiry (T) (motivates introduction of futures contracts)