MathematicalFinance: past,presentandfuture … · 2011-02-10 · Theory(APT) ⇐⇒ Practice...
Transcript of MathematicalFinance: past,presentandfuture … · 2011-02-10 · Theory(APT) ⇐⇒ Practice...
Mathematical Finance: past, present and future
University of Heidelberg
January 2011
Thaleia Zariphopoulou
University of Oxford
and
The University of Texas at Austin
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“Mathematics is the mother of all Sciences...
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... and Finance their mother-in-law”(anonymous mathematician)
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Today, I will talk to you about another member of the family:
“Mathematical Finance”
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Mathematical Finance
An interdisciplinary field on the interface of Mathematics, Economics, Finance
and Statistics
Fundamental problems
Valuation of contracts, Investment and, more broadly, Risk Management of
Financial Risks
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What is needed to work in this field?
• Mathematical skills
Probability and Stochastics
Linear and Non-linear PDE
Stochastic Optimization
Numerical Analysis
Statistics
• Modeling skills
• Knowledge of finance practice
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Mathematical Finance: the past (mid 70’s–late 90’s)
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Time line of development of the field
• How and when did all start?
Bachelier (Ph.D. Thesis) 1900
First stochastic model for the evolution of the stock price
Work and ideas ahead of their time; not appreciated; dormant progress
• Sporadic work in between
• Black-Scholes-Merton (1973)
Valuation of contingent claims
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Contract Valuation
Contract: a fair coin is tossed and you get $60 in case of “Head” and lose $40
in case of “Tail”.
Will you accept the bet? And if yes, at what price?
• Expectation is positive
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2$60 +
1
2(−$40) = $10
• A possible “reasonable” price is $10
• But, how about changing the bet to winning $600 ($6000) and losing $400
($4000)?
• Same model uncertainty
• A possible “reasonable” price is $100 ($1000)
• But, how do you feel about it?
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Linear pricing
C = p× Gain + (1− p)× Loss
• Easy and intuitive, but wrong
• After the coin is tossed, only one event will be realized: you either win
or you lose
• Can you afford the loss?
• Do you fear the loss?
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The idea that brought a revolution
Now you are in a casino playing roulette (virtual).
A person next to you proposes an equivalent contract:
you get $60 in case of “Red” and lose $40 in case of “Black”
(Assume no zero, only “Red” and “Black” for simplicity)
• Model uncertainty and contract values are the same
• But, how about you take the following action:
Bet $50 on the table on “Black”
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The idea that brought a revolution (cont’d)
• This is what happens:
Bet/contract Table/“action” Net payoff
Red outcome +60 −50 +10
Black outcome −40 +50 +10
• The price of the contract is the same: $10
• This amount is what to need to “cover all risk” and, thus, this is what you
should charge/accept
Uncertainty has been completely eliminated!
The capital needed to run a strategy to achieve this yields the fair
(and unique) price!
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Golden era of Mathematical Finance
mid-80’s to late 90’s
Arbitrage-free Pricing Theory and Stochastic Calculus
A perfect match!
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Derivative Pricing and Stochastics
• Contract ⇐⇒ Random variable
Example: European option expiring at T : payoff PT = (ST (ω)−K)+
• Replication ⇐⇒ Stochastic representation of this random variable
PT = (ST (ω)−K)+ = Ct (ω) +∫ T
0ht (ω)
dStSt
, 0 ≤ t ≤ T
• Stochastic model for the stochastic stock returns
dStSt
= μtdt + σtdWt
• Integrand ht (ω) ⇐⇒ hedging strategy
• Initial condition Ct (ω) ⇐⇒ derivative security price
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Arbitrage free pricing theory and Mathematics
Universal theory in a model independent way
The ability to replicate yields the derivative’s price!
Stochastics
Stochastic Calculus yields the representation of the price
Ct (ω) = EQ
(e−r(T−t)PT (ω)
∣∣∣Ft
)
Important ingredients: Q, Ft, and linear pricing rules (conditional expectation)
• Pricing measure: Q (turns the stock price into a martingale)
Characterizing their set is a challenging probabilistic question
• Information: Ft (what the market reveals)
Incomplete information, filtering problems
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Derivatives and Partial Differential Equations
Under Markovian assumptions on the stock price dynamics and on the payoff
( CT = H (ST ) ), the price solves a linear second order pde
Ct (ω) = h (St, t)
⎧⎪⎨⎪⎩
ht +12σ
2x2hxx + rxh = rh
h (x, T ) = H (x)
Moreover, the hedging strategies and sensitivities (greeks) are represented through
the “derivatives” of the above equation
• Computations (high dimensionality)
• Malliavin Calculus for the calculation of sensitivities
• Infinite dimensional analysis (SPDE) for term-structure models
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Creation of Mathematical Finance
• A well founded economic theory together with the appropriate powerful
mathematical tools led to a spectacular growth of the derivatives industry
and the development of the academic field of Mathematical Finance
• More and more complex products were constructed (European, American,
Exotics, Foreign Exchange, Bonds,...)
• More and more challenging problems for academics (Free-boundary prob-
lems, high-dimensionality, path-dependence, SPDE,...)
Valuable and fruitful interplay between industry and academia
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On the other (darker) side of the Finance Practice
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Investments
Given market opportunities, maximize returns on investments!
Key ingredients
• Risk premium (measures asset performance): λt =μtσt
(average return per
unit of risk)
dSt = μtStdt + σtStdWt = σtSt (λtdt + dWt)
• Individual/Corporate Risk aversion (measures attitude towards uncertainty)
U (x) ≥ E (U (x + Z (ω))) with E (Z) = 0, x certain
• Target and investment types (long-short term, equity, mutual funds, hedge
funds, pensions,...)
The objectives are different from the ones in derivatives where the
goal is to eliminate uncertainty in order to price; rather, in portfolio
management one desires uncertainty in order to invest!19
The core theoretical investment problem
• Investment horizon: T
• Criterion-utility: UT (x)
• State variable: wealth process: Xt
• Information on market returns: Ft
• Admissible investment strategies: πt ∈ ATrading constraints: no shortselling, leverage, discrete trading, etc.
Measurability; partial information
Objective – Value function process
V (x, t) = supA
E (U (XT )|Xt = x,Ft)
Controlled (diffusion) wealth process
dXt = μtπtdt + σtπtdWt
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Theoretical advances
• In Markovian models (local dependence), the expected utility problem is
directly related to fully non-linear pdes (Hamilton-Jacobi-Bellman eqn); still,
many open questions due to high-dimensionality and degeneracies
• In general semi-martingale models, very important mathematical results were
produced for the dual problem (structure of the dual space, martingale rep-
resentation theorem, multiplicity of martingale measures)
• Deep connection between optimal investment problem and derivatives prob-
lem in complete markets; important application of duality theory
While this is a beautiful and rich part in stochastic optimization,
it has found little relevance in industry applications
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Modeling and technical difficulties
• Utility is a difficult, if not elusive concept
• Normative and not descriptive theory
• Estimating the average rate of return is difficult
• Multiple asset classes, different horizons and objectives
• Transaction costs
• Trading constraints
• The underlying optimization problems are very difficult (dimensionality, trad-
ing constraints, etc.)
• In particular, transaction costs give rise to multi-dim. variational inequalities
with gradient constraints
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The core practical investment problem
Markowitz (1950 – Nobel Prize in Economics in 1990)
Minimize the risk (variance) of a portfolio given
a desired expected performance
Novel concepts of diversification, correlation and other fundamental ideas
for portfolio management
• A constrained quadratic optimization problem
• Generates the so-called efficient frontier which provides a transparent picture
of optimal investment choice!
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Expe
cted
Retu
rn
Standard Deviation
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Successful elements
• Great appeal due to the universality and simplicity of the objective
• Intuitively clean and powerful results
• Connection with micro-economic theories
• Implementable quadratic optimization problems
• Transparency of the trade-off between risk and return
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Difficulties
• Essentially, this is a single-period framework
• Limited understanding how to extend it to multi-periods in a consistent way
• Discrete-time framework; not clear what the continuous-time limit is
• Total disconnection with the dynamic utility theory
• Serious challenges on estimation market parameters
• Parameter estimation is the primary focus and not the (static) criterion per se
• Solution is very unstable with regards to market inputs
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Summary of first developments
Derivatives
Theory (APT) ⇐⇒ Practice (linear pricing)
Strong and fruitful feedback
Unified approach
Phenomenal growth
Investments
Theory (expected utility) ⇐= =⇒ Practice (mean-variance)
Slow progress
No unified theory
Stagnation in progress
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Mathematical Finance: the middle years
mid–late 90’s–present
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Main lines of development
• Derivatives
Credit Risk
Commodities
Energy / Electricity
Emissions...
• Risk measures
• Behavioral Finance
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Derivatives
“Haunting” difficulties gradually emerge from the Black and Scholes idealized set-up
• Perfect hedging is not always viable
• Therefore, the price as an expectation does not follow from the theory
• However, expectation as “the” linear pricing rule prevails
• Need to identify the “correct” pricing measure, which is “revealed” by
the (liquid) derivatives market
• However, correct calibration is not always possible
Difficulties and methodological inconsistencies are emerging as products
become more and more complex, and more and more obscure
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Credit Risk
Risk from defaulting an obligation
Credit Risk products
Asset-backed Securities built around Credit Risk
• CDO, CLN, CBO, CDO2,...
Mathematics and Credit Risk
• Models for defaults (jump-diffusion processes, Levy processes)
• Models for correlation (copula, need for dynamic models)
• Models for contagion, mitigation etc.
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Commodities / Energy / Electricity/ Water management /Emissions Markets
Difficulties
Very different characteristics than mainstream financial derivatives
• Seasonality
• Irreversibility (storage etc.)
• Much smaller markets
• Different concepts of quality of product
• Different “currency” (e.g. certificates in emissions markets)
• Rather limited theoretical understanding
• Pricing through game-analysis / supply and demand ?
• Fundamental / core problems not well formulated yet
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Risk measures
Axiomatic framework to define and quantify risks of positions/portfolios
Coherent risk measures
• Monotonicity: if X2 > X1, then ρ (X1) ≤ ρ (X2)
(if portfolio X2 always performs better than X1, then the risk of X2 is less
than the risk of X1)
• Sub-additivity - diversification: ρ (X1 +X2) ≤ ρ (X1) + ρ (X2)
(the risk of two portfolios together cannot get any worse than adding the
two risks separately)
• Positive homogeneity - linear pricing: ρ (αX1) = αρ (X1) , for α > 0
(if you double the portfolio then the risk is doubled)
• Translation invariance - insurance: ρ (X1 + a) = ρ (X1) + a
(adding riskless amounts acts like insurance)
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Convex risk measures
• The properties of sub-additivity and positive homogeneity are replaced by
the notion of convexity,
ρ (λX1 + (1− λ)X2) ≤ λρ (X1) + (1− λ) ρ (X2)
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Risk measures (cont’d)
• The theory has reached phenomenal growth
• Beautiful results from probability, stochastics, real and functional analysis
• Representation results (BSDEs, Choquet integrals, etc.)
• Popular in the industry
• Value at Risk (VAR), MinMax, Entropic risk measures, ...
This is a very active line of research with huge potential for the
correct quantification of risk and development of risk metrics with
practical significance
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Behavioral Finance
This is a new field in financial economics which extends the classical utility
theory and investigates much more complex features of human behavior towards
risk
Prospect Theory
Daniel Kahneman (Nobel 2002) and Amos Tversky
Couple of the main ideas:
We react differently to losses than to gains (!)
We distort probabilities of extremely bad events (!)
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S-shaped utilities
Utilit
y
Gains/Losses
Mathematical problem
V (x, t) = supEP (U (XT )|Xt = x,Ft)
• Many difficulties arise due to non-concavity of the criterion and lack ofsmoothness
• Beautiful duality theory between expected utility and distorted probabilities
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Mathematical Finance: the present
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Moving beyond pricing of derivatives
• Price formationExecution of trades and liquidityMarket microstructurePercolation of information
Singular stochastic control, Filtering, Boltzman equation, ...
• Financial bubblesFormationBurstingArbitrage
Semimartingales, irreversibility, free-boundary problems
• Contract theoryRisk transferProduct design
Game theory, zero and non-zero (stochastic) differential games
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In a bigger scale of questions and applications
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Valuable lessons from the Financial Crisis
Think about “Risk” and “Risk Management” in a much more
broader perspective!
• So far, the only well quantified risk was the one coming from not “perfectly
hedging” a derivative - this risk was remedied by an ad-hoc, but universally
accepted, linear pricing rule
• However, risk aggregates and goes through phases of metalaxis as it con-
stantly moves from one desk to the other
• Need to develop methodologies for integrated risk management
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How do we quantify and manage risk?
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Kinds of risk
Inconvenient truth: it is impossible to manage what cannot be measured!
• Market risk
Equity, Interest rate, Currency, Commodity, Mortgage, Volatility risk,
Liquidity risk
• Credit risk
Counterparty risk
Sovereign risk
Agglomeration/concentration risk, contagion
• Operational risk
• Systemic risk
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Systemic risk
Risk associated with the dynamic interaction and co-dependence of financial
units, institutions and sectors
A plethora of difficulties
The main difficulty stems from the fact that the industry is used to manage risks
within the specific instituition and not as a coherent and risk-efficient whole
• Interaction and aggregation of risks
Frequently, even good prospects super-impose to unfavorable situations that
would lead to a crisis and bubbles
• Transparency
• Differences in technical sophistication, complexity and innovation
• Differences in objectives per sector (e.g. derivatives and investments)
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Systemic Risk (cont’d)
• Accounting rules (marked-to-market, etc.)
• Legal aspects
• Data collection and interpretation
• Theoretical foundations less than primitive
First mathematical models for
Systemic Risk
• Mean-field games
• Particle systems
• Networks
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Looking ahead in Mathematical Finance!
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Integrated Risk Management =⇒ Quantitative Business
Challenges and needs
• Modeling
• Mathematical Tools
• Education and training
• Much more intense interdiscplinary effort
Despite the challenges and primitive stages of developement, nobody can deny:
• The need of quantitative knowledge, training and expertise
• The changes that Mathematics brought to the industry in terms of
analysis, thinking and expectations!
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Vielen Dank furIhre Aufmerksamkeit!
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