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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