Mathematics in Finance

Mathematics in Finance Introduction to financial markets

spend it car gifts holiday ... invest it savings book bonds shares derivatives real estate ... What to do with money?

ISavings 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: • Forwards • Futures • 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  you can buy or sell for that price • Maturity  date when the option expires • Buy option  long position (holder) • Sell option  short position (writer) Exercising ... ... only at maturity possible  European ... at any date up to maturity possible  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  you have > 7700€ -200€ S(T) < 77  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€ = 2000€ Case 2: your profit 1000*20€-1000*10€ = 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€ = 400€ Income (110€-105€)*100 = 500€ Riskless profit (arbitrage)

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 volatility Option value Interest rate dividends

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 • ...

Mathematics in Finance