Math 476/567:
Actuarial Risk Theory
Professor Rick Gorvett
374 Altgeld Hall
University of Illinois
at Urbana-Champaign
Fall 2015
Syllabus
• Office Hours: 3-4 pm Tuesdays, 3-4 pm
Wednesdays, or by appointment
• Textbook: McDonald, Derivatives Markets,
3rd edition
• Exam dates: 3 exams, per syllabus
• Grades: Exams, homeworks, project,
possible other assignments
Me
• Director of the UIUC Actuarial Science Program
• MBA (University of Chicago)
• Ph.D. in Finance (UIUC)
• FCAS: Fellow of the CAS
• ASA: Associate of the SOA
(I wish)
• CERA: Chartered Enterprise Risk Analyst
• Actuarial corporate / consulting experience
Class Objectives
• Understand the mathematical foundations of
stochastic processes and financial options
• Learn Exam MFE material
• Appreciate this material in a broad, crossdisciplinary framework
Class Plan
•
Option pricing theory
•
Stochastic processes
–
Brownian motion
•
Stochastic simulation
•
Interest rate modeling
Stochastic Processes
• Stochastic process: collection of random
variables over time; X(t), where t is typically
a time index; X(t) is the “state” of the
process at time t.
• Examples:
– “The drunk”
– Coin tosses:
HTTHTHTHHTHT vs HHHHTTTTHHHT
– Financial / economic series
S&P 500 Index Monthly Values
(per Yahoo Finance)
2500
S&P 500 Index
2000
1500
1000
500
0
1/3/50
1/3/55
1/3/60
1/3/65
1/3/70
1/3/75
1/3/80
1/3/85
1/3/90
1/3/95
1/3/00
1/3/05
1/3/10
1/3/15
Data per FRED, St. Louis FRB, for 3-Month T-Bills, Secondary Market
Simulation
• Scenario testing
• Stochastic simulation
– Example: Simulate series X ten times
– Let X(2006) = 30
– Simulate for 30 years according to:
dX / X = 0.03 dt + 0.10 dZ
Sample Simulation of Series X
120
Value of X
100
80
60
40
20
0
2006
2011
2016
2021
Year
2026
2031
2036
Figure 1
1 Month Real Interest Rate (B4 to B30)
0.12
0.10
0.08
0.06
0.04
0.02
0.00
-0.02
-0.04
-0.06
B4
B6
B8
B10
B12
B14
B16
B18
Cell
B20
B22
B24
B26
B28
B30
The Actuarial Science
Research Triangle
Mathematics
Fuzzy Set
Theory
Markov Chain
Monte Carlo
Stochastic Calculus /
Ito’s Lemma
Theory
of Risk
Interest
Theory
Financial Mathematics
Chaos and
Complexity
Actuarial
Science
Dynamic
Financial
Analysis
Portfolio
Theory
Interest
Rate
Modeling
Contingent
Claims
Analysis
Finance
Actuarial Science and Finance
• “Coaching is not rocket science.”
- Theresa Grentz, University of Illinois
Women’s Basketball Coach
• Are actuarial science and financial
mathematics “rocket science”?
• Certainly, lots of quantitative Ph.D.s are on
Wall Street and doing actuarialor finance-related work
• But….
Actuarial Science and Finance (cont.)
• Actuarial science and finance are not rocket
science – they’re harder
• Rocket science:
– Test a theory or design
– Learn and re-test until successful
• Actuarial science and finance
– Things continually change – behaviors, attitudes,….
– Can’t hold other variables constant
– Limited data with which to test theories
Why is this Option Stuff So Important?
Payoff
Of Call
Option
X
(Exercise price)
ST (Value of
Underlying Asset)
Insurance is an Option
Payment
Under
Insurance
Policy
X
(Deductible)
ST (Size of Loss)
A Sampling of
Options
and Other Derivatives
through History
Ancient Greece
“There is the anecdote of Thales the Milesian and his
financial device… He was reproached for his poverty,
which was supposed to show that philosophy was of no
use. According to the story, he knew by his skill in the
stars while it was yet winter that there would be a great
harvest of olives in the coming year; so, having a little
money, he gave deposits for the use of all the olivepresses in Chios and Miletus, which he hired at a low
price because no one bid against him. When the
harvest-time came, and many were wanted all at once
and of a sudden, he let them out at any rate which he
pleased, and made a quantity of money. Thus he
showed the world that philosophers can easily be rich if
they like, but that their ambition is of another sort…”
- Aristotle, Politics, Book One, Part XI
Phoenician Shipping
Merchants and ship-owners used options to
hedge their ships and cargoes
Mesopotamia
Mercantile forward contracts, written in
cuneiform on clay tablets, circa 1700 BC
China
Forward contracts on rice, entered into prior
to planting, circa 2000 BC
Belgium and The Netherlands
• Antwerp and Amsterdam
• Grain
• Herring
• Tulips
Tulip Bubble
• Mid-1630s
• Tulip demand exploded and prices skyrocketed
• Options and futures were used to ensure price
and supply
• Bubble burst in 1637
America
• 19th century
– “Privileges”
– Non-standardized / over-the-counter
• Synthetic loans
– Financier Russell Sage
– Put-call parity
– Get around usury laws
Chicago Board Options Exchange
• Began trading standardized options on April 26,
1973
• 911 contracts traded on first day (options on 16
different “underlying” companies)
CBOE and Options
“…any history of the excitement in finance in the
1960s and 1970s must mention the options pricing
work of Black and Scholes (1973) and Merton
(1973b). These are the most successful papers in
economics – ever – in terms of academic and
applied impact. Every Ph.D. student in economics is
exposed to this work, and the papers are the
foundation of a massive industry in financial
derivatives.”
- Eugene F. Fama, “My Life in Finance,” arXiv
Modeling Underlying Assets
“…distributions of stock returns are fat-tailed: there
are far more outliers than would be expected from
normal distributions – a fact reconfirmed in
subsequent market episodes, including the most
recent. Given the accusations of ignorance on this
score recently thrown our way in the popular media,
it is worth emphasizing that academics in finance
have been aware of the fat tails phenomenon in asset
returns for about 50 years.”
- Eugene F. Fama, “My Life in Finance,” arXiv
Next Time
• Review of financial options
• Begin binomial option pricing theory