금융수학 101
송성주, 고려대학교 통계학과
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What is Math Finance/ Financial Engineering?
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History of Math Finance
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Issues
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Basics of Derivative Pricing and Hedging
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Black-Scholes Model
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Risk Measure
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Mathematical Finance/Financial Engineering
From Wikipedia:
Mathematical finance is the branch of applied mathematics
concerned with the financial markets.
Mathematical finance will derive and extend the mathematical or
numerical models suggested by financial economics.
While a financial economist might study the structural reasons why a
company may have a certain share price, a financial mathematician
may take the share price as a given, and attempt to use stochastic
calculus to obtain the fair value of derivatives of the stock.
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Mathematical Finance/Financial Engineering
Mathematical finance also overlaps heavily with the fields of financial
engineering and computational finance. Arguably, all three are largely
synonymous, although the latter two focus on application, while the
former focuses on modeling and derivation.
Financial engineering applies mathematical finance results, along
with techniques such as statistical data analysis, mathematical
optimization, stochastic process modeling, and MC simulation.
Many universities around the world now offer degree and research
programs in mathematical finance.
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Origin of Mathematical Finance
Louis Bachelier's doctoral dissertation:
“Theorie de la speculation”(1900)
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Application of probability to stock markets
Main objective: valuation of options
Historically the first attempt to use advanced mathematics in
finance
Witnessed the introduction of Brownian motion: Wiener-Bachelier
process (Feller, 1966)
Rediscovered in 50s (Savage, 1954; Samuelson, 1965)
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Subsequent research in Math. Finance

Wiener, Kolmogorov, Ito, Doob, Meyer, Kunita and
Watanabe...
Construction of BM, Stochastic integration, Ito formula,
Doob-Meyer decomposition, martingale representation
theorem, ...

Markowitz, Sharpe (1990 Nobel Prize)
Mean-variance analysis, Capital Asset Pricing Model

Savage, Samuelson , Fama, McKean…
Martingale, Efficient market hypothesis, Geometric BM,
Option valuation
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Subsequent research in Math Finance
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Merton, Black, Scholes (1973; 1997 Nobel prize)
Option pricing formula, Perfect replication, Arbitrage-free
pricing
Harrison, Kreps, Pliska (1979, 1981)
Close link with martingale theory has been established/ Fair
price of the an option is the expected discounted payoff of the
option under an equivalent martingale measure
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Issues
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Derivative Pricing and Hedging
Portfolio Optimization
Risk Management
Others: Modeling, Volatility, Inference, Computation...
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Mathematical Tools: Stochastic Calculus, Martingale theory,
Stochastic control, PDE, Numerical methods …
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Statistical contribution: Model fitting, Time series (ARCH,
GARCH), Inference for stochastic processes, Measuring/Modeling
the risk, Statistical data analysis …
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Portfolio Optimization
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An economic agent decides how much to consume, how much to
save, and how to allocate his capital between different assets in
order to maximize his expected utility.
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Optimal solution depends on the mean return rates of the risky
assets, which are difficult to estimate, and also depends heavily
on the kind of utility function we choose.
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Portfolio Optimization
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Optimal Mean-variance portfolio (Markowitz)
Minimize the variance with given mean value or maximize the
mean value with given variance, usually considered in a
single-period framework
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Optimal consumption/asset allocation for utility maximization
Lagrange Multipliers (Single-period)
Dynamic Programming (Multiperiod, Backward Algorithm)
Hamilton-Jacobi-Bellman PDE (Continuous time)
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Basics of Derivative Pricing
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Derivative Securities: Forward, Future, Swap, Option (Call, Put)…
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Forward: an agreement to buy or sell an asset at a certain future
time for a certain price.
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Future: an agreement to buy or sell an asset at a certain time in
the future for a certain price, traded on an exchange.
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Swap: an agreement to exchange two cash flows with different
features. eg: fixed interest rate vs. floating interest rate
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Basics of Derivative Pricing
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Option: gives the owner the right to buy or sell a certain asset at a
pre-specified price, on or before a pre-specified date.
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Arbitrage-free market
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Forward Price ( F  S 0 e ) : delivery price
r
(i) F  S 0 e : long forward, shortsell stock, and invest S 0 to
moneymarket
r
(ii) F  S 0 e
moneymarket
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r
: short
forward, buy stock, and borrow S 0 from
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European call option price: single-period binomial
model (Cox, Ross, and Rubinstein 1979)
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Conditions: S1d  K  S1u , S1d  S0 e r  S1u ,
S1u  K
a
 0  a 1
S1u  S1d
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v  a( S0  S1d e  r ) : price
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Arbitrage-free pricing
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If V0  v, buy a call, shortsell a shares of stock, and invest
aS0  V0 in the moneymarket.
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If V0  v, S1  S1u
( S1u  K )  aS1u  (aS0  V0 )e r  ( S1u  K )  aS1u  (aS0  v)e r
 ( S1u  K )  aS1u  aS1d
 ( S1u  K )  a( S1u  S1d )  0

If
V0  v, S1  S1d
aS1d  (aS0  V0 )er  aS1d  (aS0  v)e r
 aS1d  aS1d  0
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Arbitrage-free pricing (cont’d)
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If V0  v, sell a call, buy a shares of stock, and borrow
aS0  V0 from the moneymarket.
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If V0  v, S1  S1u
( K  S1u )  aS1u  (aS0  V0 )e r  ( K  S1u )  aS1u  (aS0  v)e r
 ( K  S1u )  aS1u  aS1d
 ( K  S1u )  a( S1u  S1d )  0

If
V0  v, S1  S1d
aS1d  (aS0  V0 )er  aS1d  (aS0  v)er
 aS1d  aS1d  0
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Risk-Neutral Measure
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Equivalent martingale measure
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Pricing with risk-neutral measure
r


S
e
0  S1d

r
V0  e (S1u  K ) 
 S1u  S1d 


 er ( p(S1u  K )  (1 p)(S1d  K ) )
 E (er (S1  K ) ),

S0 e r  S1d 
p

S

S
1u
1d


S0  e  r ( pS1u  (1  p ) S1d )  E (e  r S1 )
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Martingales
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Fair game, one of main tools in the study of random processes
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Expected fortune of a gambler at time n given all the information
in the first n-1 trials is the same as the actual value of the
gambler’s fortune before the nth round.
S0  E (e rt St ),
e rt St  E (e rT ST | all information up to time t)
V0  E (e rtVt )
e rtVt  E (e rT ( ST  K ) | all information up to time t)
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Hedging
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Reducing the financial risk by trading available securities
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Hedging strategy: how many shares of which securities we should
hold at each time
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How Black and Scholes originally obtained the pricing formula
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Completeness/Incompleteness of markets
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Replication(Perfect Hedge), Quadratic Hedge (Mean-variance,
Local risk minimization), Super(conservative) Hedge, Quantile
Hedge, Efficient Hedge, ...
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Black-Scholes Model
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Options can be completely hedged and their prices can be
uniquely determined under some assumptions.
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Assumptions:
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Underlying asset ~ geometric Brownian Motion.
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Constant volatility, constant expected rate of return
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No transaction costs or taxes, all securities are perfectly
divisible
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No dividends
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No arbitrage opportunities
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Continuous trading
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Short rate is constant
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Black-Scholes formula
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European Call option
C ( x, t )  x( z )  e r (T t ) K ( z   T  t )
x log( x / K )  (r   2 / 2)(T  t )
where z 
 T t
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Needs volatility input
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Perfect hedging strategy: Hold Cx ( St , t ) shares of stock at any
time t  T : Delta hedging
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Not a good fit to the real data: kurtosis, skewness, volatility smile
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Financial risk
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Financial risk: Market risk, Credit risk, Operational risk, Liquidity risk
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Market risk: Risk that the value of an investment will decrease due to
market movements
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Credit risk: Risk of loss due to counter-party defaulting on a contract
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Operational risk(Basel II Accord): Risk of losses resulting from
inadequate or failed internal processes, people and systems, or
external events
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Liquidity risk: Additional risk in the market due to the timing and size
of a trade
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Risk Measure: Value at Risk
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Value at Risk: the most prominent risk measure (J.P. Morgan's
RiskMetrics,1993)
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Given a risk X with cumulative distribution function FX and a
probability level   (0,1) ,
VaR ( X )  FX1 ( )  inf{x  R : FX ( x)   }.

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Does not measure the potential size of a loss, lacks the
property of subadditivity, CDF unknown in practice
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Coherent Risk Measure
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Artzner, Delbaen, Eber and Heath (1999)
 ( X  c)  c   ( X )
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Translation invariance:
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Positive homogeneity:
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Monotonicity:
X  Y a.s.   ( X )   (Y )
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Subadditivity:
 ( X  Y )   ( X )   (Y )
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 (cX )  c  ( X ), c  0
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Expected tail loss
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Expected shortfall, Tail conditional expectation, TailVaR
ES ( X )  E ( X | X  VaR ( X ))
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Well-known in insurance: ETL=Mean excess loss+ VaR ( X )
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For continuous risks, ETL is a coherent risk measure
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Estimation of VaR or ETL
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Assume normality (Variance-Covariance method)
Historical simulation
Monte Carlo simulation
Asymptotic distribution of order statistics
Quantile regression
Econometric approach
Extreme value approach
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