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
Div1
Earnings per share: EPS =
P0
P1  P0
+
P0
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
16
14
12
10
8
6
4
2
0
-2
underlying at T
45
41
37
33
29
25
21
17
13
9
5
-4
1
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
volatility
Interest rate
dividends
II Derivatives - Options
strike
underlying
maturity
volatiliy
interest rates
dividends
up
up
approaching
up
up
are paid
Call Put
down up
up down
down down
up
up
~
~
down up
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
– ...