- Количество слайдов: 20
Mathematics in Finance Introduction to financial markets
What to do with money? • spend it – car – gifts – holiday –. . . • invest it – savings book – bonds – shares – derivatives – real estate –. . .
I Savings book • Lending K€, getting K(1+r)€ after a year • bank hopes to earn a higher return on K than r • (for example by lending it) • practically no risk
Risk free interest rate r • can be obtained by investing with no risk • USA: often interest which the government pays • Europe: EURIBOR (European Interbank Offered Rate) • positive. • discount factor – 100 today 100(1+r) in one year – 100 in one year 100/(1+r) today
II Bonds An IOU from a government or company. In exchange for lending them money they issue a bond that promises to pay you back in the future plus interest. • • (IOU = investor owned utilities) Fixed-interest bonds Floating bonds Zero bonds
III Shares Certificate representing one unit of ownership in a company. • • Shareholder = owner Particular part of nominal capital Traded on stock exchange No fixed payments Earnings per share: EPS = +
IV Derivatives A derivated financing tool. Its value is derivated from an underlying. • • Underlyings: shares, bonds, weather, pork bellies, football scores, . . . Different derivatives: 1. Forwards 2. Futures 3. Options
IV Derivatives - Forwards Agreement to buy or sell an asset at a certain future time for a certain price. Not normally traded on exchange. • • Over the counter (OTC) Value at begin: Zero Agree to buy long position Agree to sell short position
IV Derivatives - Futures Agreement to buy or sell an asset at a certain time in future for a certain price. Normally traded on exchange. • • Standardized features Agree to buy long position Agree to sell short position Exchanges: CBOT, CME, . . .
IV Derivatives - Options Give the holder the right to buy or sell the underlying at a certain date for a certain price. (European options) • • • Right to buy call option Right to sell put option Payoff function Cash settlement Exchanges: AMEX, CBOT, Eurex, LIFFE, EOE, . . .
IV Derivatives - Options Denotations: • • Strike Maturity Buy option Sell option you can buy or sell for that price date when the option expires long position (holder) short position (writer) Exercising. . . only at maturity possible. . . at any date up to maturity possible European American
IV Derivatives - Options Example 1: Long Call on stock S with strike K=32, maturity T, price P=2. Payoff function: f(S) = max(0, S(T) – K)
IV Derivatives - Options Example 2 (how to use options): 1. 1. : 100 shares of S, each 80 € 30. 6: must pay 7500€ (by selling the shares) Problem: price of shares could fall under 75€ Solution: buy 100 puts with strike 77 each option costs 2 Result: S(T) > 77 S(T) < 77 you have > 7700€ -200€ you have = 7700€ -200€
IV Derivatives - Options Example 3 (how to use options): Situation: You think the prices of S will raise & want to profit from that. One share costs 100€. You have 10000€. Solution 1: you buy 100 shares. Solution 2: you buy calls (10€) with strike 100. Result if the prices raise to 120: Case 1: your profit 100*20€ Case 2: your profit 1000*20€-1000*10€ = 2000€ = 10000€
IV Derivatives - Options Example 4 (how to use options): Call with strike 105 costs 2€ each Put with strike 110 costs 2€ each (same maturity) Action: Buy 100 calls and 100 puts. Result at T: Costs 200*2€ Income (110€-105€)*100 Riskless profit (arbitrage) = 400€ = 500€
IV Derivatives - Options Other options: • Spreads f(S)=max(0, K-S)+max(0, S-K) • Strangles f(S)=max(0, K-S)+max(0, S-L) • Pathdependant options: – Floating rate options F(S) = max(0, S(T)-mean(S)) –. . . • Options on options • . . .
underlying maturity strike Option value dividends volatility Interest rate
II Derivatives - Options
Summary Assets: • Savings book (risk free) • Bonds • Shares • Derivatives Futures Forwards Options
Problem: How can options be priced? – Modelling – Black-Scholes – Solving partial differential equations – Monte-Carlo simulation –. . .