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: CT1 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 8.19b). 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 put-options 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 $1.
The investor decides to hedge by buying 1,000 put options (“10 contracts”)
Portfolio Value in Two Months with and without Hedging
20,000
25,000
30,000
35,000
40,000
20 25 30 35 40
Stock Price ($)
Value of Holding ($)
No Hedging
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 $2,000 into • 100 shares of Amazon.com stock• 2,000 strike-22.5, expiry-2M call-options
Profit or loss from speculating on the Amazon.com stock price
-3000
-2000
-1000
0
1000
2000
3000
4000
5000
6000
7000
8000
15 20 25 30
Stock price ($)
Val
ue
of
ho
ldin
gs
($)
Buy stock
Buy 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. CT1 Unit 12, Sec 1
Base-case:
Fwd(t,T) = exp(r*(T-t))*Spot(t)
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)