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  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...

strike underlyingmaturity volatility Interest rate Option value 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 –...