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Financial Mathematics 2 The plan for Tuesday October 5, 2010 • Practical matters • 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

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

Practical matters The admin’ does not want us to move Workshops around ”willy-nilly”. Those 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. 19 b). Until further notice we stick to the orginal plan. Hand-out: Course Work #1. Due at lectures on Thursday October 14.

Who would want to use/trade in forward contracts? • Hedgers. Hull’s p. 10 example: 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.

Options Call-option: The right, but not the obligation, to buy the underlying for the 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.

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

Why Study Options? Used by • Hedgers (put ~ portfolio insurance) • Speculators Embedded 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.

Hedging w/ Put-Options An investor owns 1, 000 Microsoft shares currently worth $28 per 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”)

Portfolio Value in Two Months with and without Hedging Portfolio Value in Two Months with and without Hedging

Speculating with Call-Options An investor with $2, 000 to invest feels that Amazon. com’s 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-2 M call-options

Valuation of Forward Contracts How are spot and forward prices related? A simple yet 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. CT 1 Unit 12, Sec 1 Base-case: Fwd(t, T) = exp(r*(T-t))*Spot(t)

Extensions of Forward Valuation CT 1 Unit 12 • Sec. 2. 3: Fixed intermediate Extensions of Forward Valuation CT 1 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)