financial mathematics 2
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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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
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• 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
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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.
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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.
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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.
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Pay-off-diagrams: Hockey-sticks.
Unlike forward contacts, call- and put-options cost money up front. Clearly, they have to. (Why?)
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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.
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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 $1.
The investor decides to hedge by buying 1,000 put options (“10 contracts”)
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Portfolio Value in Two Months with and without Hedging
20,000
25,000
30,000
35,000
40,000
20 25 30 35 40
Stock Price ($)
Value of Holding ($)
No Hedging
Hedging
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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 $2,000 into • 100 shares of Amazon.com stock• 2,000 strike-22.5, expiry-2M call-options
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Profit or loss from speculating on the Amazon.com stock price
-3000
-2000
-1000
0
1000
2000
3000
4000
5000
6000
7000
8000
15 20 25 30
Stock price ($)
Val
ue
of
ho
ldin
gs
($)
Buy stock
Buy options
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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)
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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)