transcript
- Slide 1
- Casualty Actuarial Society Experienced Practitioner Pathway
Seminar Lecture 5 Advanced Quantitative Analysis Stephen P. DArcy,
FCAS, MAAA, Ph.D. Robitaille Chair of Risk and Insurance California
State University Fullerton DArcy Risk Consulting, Inc.
- Slide 2
- Overview 2EPP Lecture 5: Advanced Quantitative Analysis What
you need to know to be part of the conversation Stochastic
processes Interest Rate Models Regime switching and transition
matrices Copulas Extreme value theory Option pricing models
- Slide 3
- Stochastic Processes A stochastic process is an elaborate term
for a random variable Future values of the process is unknown We
want to model some stochastic process Future interest rates can be
viewed as a stochastic process Basic stochastic processes: Random
walk Brownian motion Wiener process Ornstein-Uhlenbeck process 3EPP
Lecture 5: Advanced Quantitative Analysis
- Slide 4
- Features of a Random Walk Example someone moving one step at a
time north or south, but the direction of each step is random
Memory loss History reveals no information about the future
Expected change in value is zero Over any length of time, the best
predictor of future position is the current position This feature
is termed a martingale Variance increases with time As more time
passes, there is potential for being farther from the initial
position 4EPP Lecture 5: Advanced Quantitative Analysis
- Slide 5
- Brownian Motion A Brownian Motion is the limit of the discrete
case random walk This is a continuous time process The simplest
form of Brownian Motion is a Wiener process (dz) 5EPP Lecture 5:
Advanced Quantitative Analysis
- Slide 6
- Generalizing Pure Brownian Motion Lets interpret the following
expression: First, recall that we are modeling the stochastic
process x Think of x as a stock price, a level of interest rates
and an inflation rate The equation states that the change in
variable x is composed of two parts: A drift term which is
non-random A stochastic or random term that has variance 2 Both
terms are proportional to the time interval 6EPP Lecture 5:
Advanced Quantitative Analysis
- Slide 7
- Ornstein-Uhlenbeck Process A constant drift term does not make
economic sense Most models assume mean reversion For interest rates
and inflation rates there is a long-run average value 7EPP Lecture
5: Advanced Quantitative Analysis
- Slide 8
- Classifications of Interest Rate Models Discrete vs. Continuous
Single Factor vs. Multiple Factors General Equilibrium vs.
Arbitrage Free 8EPP Lecture 5: Advanced Quantitative Analysis
- Slide 9
- Discrete Models Discrete models have interest rates change only
at specified intervals Typical interval is monthly Daily, quarterly
or annually also feasible Discrete models can be illustrated by a
lattice approach 9EPP Lecture 5: Advanced Quantitative
Analysis
- Slide 10
- Continuous Models Interest rates change continuously and
smoothly (no jumps or discontinuities) Mathematically tractable
Accumulated value = e rt Example $1 million invested for 1 year at
r = 5% Accumulated value = 1 million x e.05 = 1,051,271 10EPP
Lecture 5: Advanced Quantitative Analysis
- Slide 11
- Single Factor Models Single factor is the short term interest
rate for discrete models Single factor is the instantaneous short
term rate for continuous time models Entire term structure is based
on the short term rate For every short term interest rate there is
one, and only one, corresponding term structure 11EPP Lecture 5:
Advanced Quantitative Analysis
- Slide 12
- Multiple Factor Models Variety of alternative choices for
additional factors Short term real interest rate and inflation
(CIR) Short term rate and long term rate (Brennan-Schwartz) Short
term rate and volatility parameter (Longstaff-Schwartz) Short term
rate and mean reverting drift (Hull-White) 12EPP Lecture 5:
Advanced Quantitative Analysis
- Slide 13
- General Equilibrium Models Start with assumptions about
economic variables Derive a process for the short term interest
rate Based on expectations of investors in the economy Term
structure of interest rates is an output of model Does not generate
the current term structure Limited usefulness for pricing interest
rate contingent securities More useful for capturing time series
variation in interest rates Often provides closed form solutions
for interest rate movements and prices of securities 13EPP Lecture
5: Advanced Quantitative Analysis
- Slide 14
- Arbitrage Free Models Designed to be exactly consistent with
current term structure of interest rates Current term structure is
an input Useful for valuing interest rate contingent securities
Requires frequent recalibration to use model over any length of
time Difficult to use for time series modeling 14EPP Lecture 5:
Advanced Quantitative Analysis
- Slide 15
- Which Type of Model is Best? There is no single ideal term
structure model useful for all purposes Single factor models are
simpler to use, but may not be as accurate as multiple factor
models General equilibrium models are useful for modeling term
structure behavior over time Arbitrage free models are useful for
pricing interest rate contingent securities How the model will be
used determines which interest rate model would be most appropriate
15EPP Lecture 5: Advanced Quantitative Analysis
- Slide 16
- Regime Switching Models Many real life processes are too
dispersed to reflect a single regime model One way to deal with
this dispersion is to allow different regimes Data have too many
outliers (leptokurtic) to fit common distributions If volatility is
set high enough to generate the outliers, there are not enough
mid-range observations Solution is to use a regime switching model
with a transition matrix Examples Stock returns Mary Hardy
Inflation Ahlgrim and DArcy 16EPP Lecture 5: Advanced Quantitative
Analysis
- Slide 17
- Copulas Copulas are conditional joint distribution functions
Early applications Joint life estimations by life actuaries
Mortality is generally independent, but the likelihood of a spouse
dying increases if their partner has died recently Common event
Adverse health reaction to stress of losing spouse Stock and bond
prices for a single company Prices generally move independently If
stock price declines significantly, bond prices decline as well to
reflect increased risk of bankruptcy Copulas now widely used in
quantitative risk models 17EPP Lecture 5: Advanced Quantitative
Analysis
- Slide 18
- Commonly used copulas Archimedean copulas Joint probability
distribution can be expressed in a closed form They are defined
using only one or two parameters Examples One parameter Gumbel
(upper tail dependence only) Frank (neither upper nor lower tail
dependence) Clayton (if >0 lower tail dependence only, otherwise
none) Two parameters Generalized Clayton (upper and lower tail
dependence) Gaussian (or normal) copula Linear correlation
(Pearsons Rho) Every pair of variables can have different
correlation No tail dependence 18EPP Lecture 5: Advanced
Quantitative Analysis
- Slide 19
- Dependency Structure Comparison - 1 19EPP Lecture 5: Advanced
Quantitative Analysis Normal Copula Gumbel Copula Correlation =.50
Gumbel parameter 1.5
- Slide 20
- Dependency Structure Comparison - 2 20EPP Lecture 5: Advanced
Quantitative Analysis Frank Copula Clayton Copula Frank parameter =
3.7 Clayton parameter 1.0
- Slide 21
- Drawbacks of Copulas Commonly used copulas may be
mathematically elegant, but not appropriate for specific
application Asymmetry issues The relationship between two variables
may not be the same for large increases as for large decreases Need
to consider tail dependency and select appropriate copula One
variable may depend on another, but the relationship may not be
reciprocal Each of two variables could depend on a third variable,
but not on each other Try to determine and model key variable and
relate other variables to it 21EPP Lecture 5: Advanced Quantitative
Analysis
- Slide 22
- Extreme Value Theory In many cases the extreme scenarios are
what is important Extreme value theory focuses on the maximum value
from a set of independent observations (eg. the largest hurricane
each year) Regardless of the distributions generating the
observations, the maximum values will follow a known distribution,
the Generalized Extreme Value (GEV) 22EPP Lecture 5: Advanced
Quantitative Analysis
- Slide 23
- Generalized Extreme Value Distribution Advantage By determining
the GEV parameters, the shape of the tail is determined Power law
Exponential Finite endpoint Disadvantages Ignores most of the
observations Extreme value is not necessarily the important value
Extreme value might be too large to be relevant All values above a
particular level (solvency) might be more important 23EPP Lecture
5: Advanced Quantitative Analysis
- Slide 24
- Generalized Pareto Distribution Considers all observations
above a chosen hurdle level As the hurdle increases, the
conditional loss distribution converges to a Generalized Pareto
distribution, regardless of the underlying distributions Key to
select the right hurdle (or threshold) Too high and not enough
observations to calculate parameters Too low and it does not focus
on the tail 24EPP Lecture 5: Advanced Quantitative Analysis
- Slide 25
- What is an Option Contract? Options provide the right, but not
the obligation, to buy or sell an asset at a fixed price Call
option is right to buy Put option is right to sell Only option
sellers (writers) are required to perform under the contract (if
exercised) After paying the premium, option owner has no duties
under the contract 25EPP Lecture 5: Advanced Quantitative
Analysis
- Slide 26
- Some Option Terminology The exercise or strike price is the
agreed on fixed price at which the option holder can buy or sell
the underlying asset Exercising the option means to force the
writer to perform Make option writer sell if a call, or force
writer to buy if a put Expiration date is the date at which the
option ceases to exist American option can be exercised anytime
prior to expiration European option can only be exercised on the
expiration date 26EPP Lecture 5: Advanced Quantitative
Analysis
- Slide 27
- Option Valuation Basics Two components of option value
Intrinsic value Time value Intrinsic value is based on the
difference between the exercise price and the current asset value
(from the owners point of view) For calls, max(S-X,0) X= exercise
price For puts, max(X-S,0) S=current asset value Time value
reflects the possibility that the intrinsic value may increase over
time Longer time to maturity, the higher the time value 27EPP
Lecture 5: Advanced Quantitative Analysis
- Slide 28
- In-the-Moneyness If the intrinsic value is greater than zero,
the option is called in-the- money It is better to exercise than to
let expire If the asset value is near the exercise price, it is
called near-the- money or at-the-money If the exercise price is
unfavorable to the option owner, it is out-of-the- money 28EPP
Lecture 5: Advanced Quantitative Analysis
- Slide 29
- Determinants of Call Value Value must be positive Increasing
maturity increases value Increasing exercise price, decreases value
American call value must be at least the value of European call
Value must be at least intrinsic value For non-dividend paying
stock, value exceeds S-PV(X) Can be seen by assuming European style
call 29EPP Lecture 5: Advanced Quantitative Analysis
- Slide 30
- Determinants of Call Value (p.2) As interest rates increase,
call value increases This is true even if there are dividends As
the volatility of the price of the underlying asset increases, the
probability that the option ends up in-the-money increases 30EPP
Lecture 5: Advanced Quantitative Analysis
- Slide 31
- Put-Call Parity Consider two portfolios One European call
option plus cash of PV(X) One share of stock plus a European put
Note that at maturity, these portfolios are equivalent regardless
of value of S Since the options are European, these portfolios
always have the same value If not, there is an arbitrage
opportunity (Why?) 31EPP Lecture 5: Advanced Quantitative
Analysis
- Slide 32
- Fisher Black and Myron Scholes Developed a model to value
European options on stock Assumptions No dividends No taxes or
transaction costs One constant interest rate for borrowing or
lending Unlimited short selling allowed Continuous markets
Distribution of terminal stock returns is lognormal Based on
arbitrage portfolio containing stock and call options Required
continuous rebalancing 32EPP Lecture 5: Advanced Quantitative
Analysis
- Slide 33
- Black-Scholes Option Pricing Model C= Price of a call option S=
Current price of the asset X= Exercise price r= Risk free interest
rate t= Time to expiration of the option = Volatility of the stock
price N= Normal distribution function 33EPP Lecture 5: Advanced
Quantitative Analysis
- Slide 34
- Using the Black-Scholes Model Only variables required
Underlying stock price Exercise price Time to expiration Volatility
of stock price Risk-free interest rate 34EPP Lecture 5: Advanced
Quantitative Analysis
- Slide 35
- Use of Options Options give users the ability to hedge downside
risk but still allow them to keep upside potential This is done by
combining the underlying asset with the option strategies Net
position puts a floor on asset values or a ceiling on expenses
35EPP Lecture 5: Advanced Quantitative Analysis
- Slide 36
- Common Uses of Options Interest rate risk Currency risk Equity
risk Market risk Individual securities Catastrophe risk 36EPP
Lecture 5: Advanced Quantitative Analysis
- Slide 37
- Summary 37EPP Lecture 5: Advanced Quantitative Analysis If you
dont understand terminology or techniques being used in a model,
then ask for an explanation You should have a basic understanding
of the key terms to grasp the most important issues if they are
explained well Basic topics covered Stochastic processes Interest
Rate Models Regime switching and transition matrices Copulas
Extreme value theory Option pricing models
- Slide 38
- References Copulas and extreme value theory Paul Sweeting,
Financial Enterprise Risk Management Regime switching Ahlgrim and
DArcy, The Effect of High Inflation or Deflation on the Insurance
Industry Hardy, A Regime Switching Model for Long-Term Stock
Returns, North American Actuarial Journal Interest rate models
DArcy and Gorvett, Hacking a Path through the Thickets, Global
Reinsurance, 2000. 38EPP Lecture 5: Advanced Quantitative
Analysis