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 ”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 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: 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 (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 have to. (Why? )
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 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
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 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 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)
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