Why Bankers Should Learn Convex Analysis

A tale of two financial economistsSummary

Model Financial MarketA discrete model for the financial markets

No Arbitrage and MartingalePreference

Example: Markowitz Portfolio TheoryA Question

Why Bankers Should Learn Convex AnalysisJim Zhu

Western Michigan UniversityKalamazoo, Michigan, USA

Part 1: Stochastic market model

March 3, 2011

Part 1: Stochastic market model Why Bankers Should Learn Convex Analysis





A tale of two financial economists

Edward O. Thorp and Myron Scholes

Influential works:

“Beat the Dealer(1962)” and “Beat the Market(1967)”The Black-Scholes formula(1973).








Influential works:


Place the ideas were conceived: MIT








Influential works:



Investment practice:

Managing partner of Princeton/Newport Partners and thePresident of Edward O. Thorp & Associates. Annualizedreturn of 20% over 28.5 yearsPartner of Long Term Capital Management: essentiallybankrupted in less than two years and almost causing a crisis.








Influential works:



Investment practice:

Managing partner of Princeton/Newport Partners and thePresident of Edward O. Thorp & Associates. Annualizedreturn of 20% over 28.5 yearsPartner of Long Term Capital Management: essentiallybankrupted in less than two years and almost causing a crisis.








Recognition:

One of the story in Poundstone’s 2005 book “Fortune’sformula: The untold story....”Noble Price in Econ 1997 and main stream financial economics.

What is going on?Bankers don’t know convex analysis!








Recognition:

One of the story in Poundstone’s 2005 book “Fortune’sformula: The untold story....”Noble Price in Econ 1997 and main stream financial economics.

What is going on?Bankers don’t know convex analysis!






Summary

Convex analysis used to play a central role in economics andfinance via (concave) utility functions.

A ‘new’ paradigm was emerged since the 1970’s afterBlack-Scholes introduced the replicating portfolio pricingmethod for option pricing, and






Summary



Cox and Ross developed the risk neutral measure pricingformula.






Summary




This new paradigm marginalized many time tested empiricalrules.






Summary





It brought about unprecedented prosperity in financialindustry yet also led to the 2008 crisis.






Summary





It brought about unprecedented prosperity in financialindustry yet also led to the 2008 crisis.






Summary

We will show that the ‘new paradigm’ is a special case of thetraditional utility maximization and its dual.

Overlooking sensitivity analysis in the ‘new paradigm’ is oneof the main problem.






Summary



The recent financial crisis is a wake up call that it is timeagain for bankers to learn convex analysis.






Summary



The recent financial crisis is a wake up call that it is timeagain for bankers to learn convex analysis.






Outline

The talk is divided into two parts. In the first part we discuss

A discrete model for financial markets.

Arbitrage and martingale (risk neutral) measure.

Fundamental theorem of asset pricing.

Utility functions and risk measures.

Markowitz portfolio theory






Outline

The second part focuses on the financial derivatives.

The new paradigm of financial derivative pricing.

A Convex Analysis Perspective.

Sensitivity and Financial Crisis.

Alternative methods and an illustrative example using realhistorical market data.






UncertaintyModel UncertaintyAn exampleA random variableA discrete stochastic processInformationExample of FiltrationDefinition of filtration

Uncertainty

Uncertainty is ubiquitous in the financial world

Stock price is unpredictable.

Financial derivatives can bring about prosperity and disaster.

Bond is considered safe but that is when interest rate is stable.

Cash is better if only there is no inflation.

To model financial markets one has to model uncertainty.







Uncertainty

Uncertainty is ubiquitous in the financial world

Stock price is unpredictable.

Financial derivatives can bring about prosperity and disaster.

Bond is considered safe but that is when interest rate is stable.

Cash is better if only there is no inflation.

To model financial markets one has to model uncertainty.







Model Uncertainty

For problem involving only one decision such as analyzing aportfolio we need random variables.

For problem involving multiple decisions such as trading weneed stochastic process







Model Uncertainty



The process of information becomes available also need to bemodeled.







Model Uncertainty



The process of information becomes available also need to bemodeled.







The game of tossing a coin

Bet on flipping a fair coin.

Head the house will double your bet.Tail you lose your bet to the house.







A random variable

Suppose we play the game only once and bet 1.

Denote the outcome of the game by X .

Then X is a random variable taking only 1 or −1 as its valueand P(X = 1) = P(X = −1) = 1/2.







A discrete stochastic process

Play the game i times and always bet 1.

Denote the outcome of the ith game by Xi .










Then Xi is a random variable andP(Xi = 1) = P(Xi = −1) = 1/2.











If we start with an initial endowment of w0 then our totalwealth after the ith game is

wi = w0 + X1 + . . .+ Xi . (1)












wi = w0 + X1 + . . .+ Xi . (1)

Now (wi )ni=1 is an example of a discrete stochastic process.












wi = w0 + X1 + . . .+ Xi . (1)

Now (wi )ni=1 is an example of a discrete stochastic process.







Information

Will knowing X1, . . . ,Xi , help us to play the (i + 1)th game?

The answer should be NO but how do we clearly describe thisconclusion?







Information



Let us look at the game with n = 3 to get some feeling.

We use H to represent a head and T , tail.

The information we can get at each stage can be illustratedwith the following binary tree.







Information



Let us look at the game with n = 3 to get some feeling.

We use H to represent a head and T , tail.

The information we can get at each stage can be illustratedwith the following binary tree.







F0 F1 F2 F3

HHH

HH

HHT

H

HTH

HT

HTT

ΩTHH

TH

THT

T

TTH

TT

TTT







Filtration for 3 coin tosses

All the information are represented by F3 = 2Ω,Ω =HHH,HHT ,HTH,HTT ,THH ,THT ,TTH,TTT.

Similarly, after 2 tosses F2 = 2HH,HT ,TH,TT, whereHH,HT ,TH,TT =HHH,HHT, HTH,HTT, THH ,THT, TTH ,TTT.

F2 has less information than F3.

Similarly, F1 = 2H,T, where H,T =HHH,HHT ,HTH,HTT, THH ,THT ,TTH ,TTT.

F0 = ∅, Ω.







Filtration for 3 coin tosses

The sequenceF : F0 ⊂ F1 ⊂ F2 ⊂ F3

is a filtration for (wi )3i=0.

For each i , Fi is a set algebra, i.e., its elements as sets areclosed under union, intersection and compliment.







General filtration

Let Ω be a sample space (representing possible states of achance event).

A sequence of algebra (σ-algebra when Ω is infinite)F : Fi , i = 0, 1, . . . , n satisfying

F0 ⊂ F1 ⊂ F2 ⊂ . . . ⊂ Fn (2)

is called a filtration.

If F0 = Ω and Fn = Ω then F is called an informationstructure.






Random variableInformation systemMarketPortfolioTrading strategy

Random variable

All possible economic states is represented by a finite set Ω.

Probability of each state is described by a probability measureP on 2Ω.

Let RV (Ω) be the space of all random variables on Ω, withinner product

〈ξ, η〉 = E[ξη] =

∫

Ω

ξηdP =∑

ω∈Ω

ξ(ω)η(ω)P(ω).

0 < ξ ∈ RV (Ω) means ξ(ω) ≥ 0 for all ω ∈ Ω and at leastone of the inequality is strict.







Information system

Suppose that actions can only take place at t = 0, 1, 2, . . ..

Use F = Ft | t = 0, 1, . . . to represent an information

system of subsets of Ω, that is,

σ(Ω) = F0 ⊂ F1 ⊂ . . . ⊂ Ft ⊂ . . . and ∪∞t=0 Ft = σ(Ω).

Here, algebra Ft , represents available information at time t.

Implied in the definition is that we never loss any informationand our knowledge increases with time t.

If action is finite t = 0, 1, . . . ,T , we assume FT = σ(Ω).

The triple (Ω,F ,P) models the gradually availableinformation.







Market

Let A = a0, a1, . . . , aM be M + 1 assets.

a0 is reserved for the risk free assets.

The prices of these assets are represented by vector stochasticprocess S := Stt=0,1,...,

where St := (S0t ,S

1t , . . . ,S

Mt ) is the discounted price vector of

the M + 1 assets at time t.

Using the discounted price, we have S0t = 1 for all t.

Assume St is Ft -measurable, i.e. determined up to theavailable information.

We say such an S is F-adapted.

A described above is a financial market model.Part 1: Stochastic market model Why Bankers Should Learn Convex Analysis






Portfolio

Portfolio

A portfolio Θt on the time interval [t − 1, t) is a Ft−1 measurablerandom vector Θt = (Θ0

t ,Θ1t , . . . ,Θ

Mt ) where Θm

t indicates theweight of asset am in the portfolio.

A portfolio Θt is always purchased at t − 1 and liquidated at t.The acquisition price is Θt ·St−1 and the liquidation price is Θt ·St .







Trading strategy

Trading strategy

A trading strategy is a F-predictable process of portfoliosΘ = (Θ1,Θ2, . . .), where Θt denotes the portfolio in the timeinterval [t − 1, t). A trading strategy is self-financing if at any t

Θt · St = Θt+1 · St ,∀t = 1, 2, . . . .

We use T (A) to denote all the self-financing trading strategies formarket A.

Θ = (Θ1,Θ2, . . .) is F-predictable means that Θt is Ft−1

measurable.







Gain

For a trading strategy Θ, the initial wealth is

w0 = Θ1 · S0. (3)

The net wealth at time t = T is

wT =T∑

t=1

Θt · (St − St−1) + w0. (4)

Random variable

GT (Θ) =T∑

t=1

Θt · (St − St−1) = wT − w0 (5)

is the net gain.Part 1: Stochastic market model Why Bankers Should Learn Convex Analysis





ArbitrageFair game and martingaleExamples of martingalesMartingale for a financial marketMartingale characterization of no arbitrage

Arbitrage

Arbitrage

A self-financing trading strategy is called an arbitrage if Gt(Θ) ≥ 0for all t and at least one of them is strictly positive.

Intuitively, an arbitrage trading strategy is a risk free way ofmaking money. We note that we may always assume GT (Θ) > 0.







No Arbitrage Principle

No Arbitrage Principle

There is no arbitrage in a competitive financial market.







Fair game and martingale

Toss a fair coin is a fair game in the sense that no player hasan advantage.

In other words, restricted to information at (i − 1)th game,the expectation of wi and wi−1 are the same.

Mathematically,

EP [wi | Fi−1] = wi−1. (6)

A F-adapted stochastic process satisfying (6) is called aF-martingale.

We will omit P and/or F if it is clear in the context.







Examples

1 Let Xi be independent with E[Xi ] = 0 for all i . Then, S0 = 0,Si = X1 + . . . + Xi defines a martingale.

2 Let Xi be independent with E[Xi ] = 0 and Var[Xi ] = σ2 for alli . Then, M0 = 0, Mi = S2

i − iσ2 gives a martingale.

3 Let Xi be independent random variables with E[Xi ] = 1 for alli . Then, M0 = 0,

Mi = X1 × . . .× Xi

gives a martingale with respect to Fi .







Martingale for a financial market

1 Let A be a financial market.

2 We say that a probability measure P is a martingale of A ifP(ω) > 0 for all ω ∈ Ω and all the price processSmt ,m = 0, 1, . . . ,M are martingales with respect to P .

3 We use M(A) to denote the set of all martingale measures ofA.







No Arbitrage and Martingale

No Arbitrage and Martingale

Let A be a financial market model with finite period T . Then thefollowing are equivalent

(i) there are no arbitrage trading strategies;

(ii) M(A) 6= ∅.







Proof (ii)→(i)

Let Q ∈ M(A). If Θ ∈ T (A) is an arbitrage, then, for some t,Gt(Θ) > 0 and consequently EQ(Gt(Θ)) > 0, a contradiction.







Proof (i)→(ii)

Observe that, GT (T (A)) ∩ intRV (Ω)+ = ∅. Since GT (T (A)) is asubspace, by the convex set separation theorem GT (T (A))⊥

contains a vector q with all components positive. We can scale q

to a probability measure Q. Then it is easy to check Q ∈ M(A).







Remark

(i) No arbitrage principle does not say one cannot make morethan the risk free rate.

(ii) It says to do that one has to take risk.

(iii) Martingale probability measure is not the same as the realprobability of economic events.






PreferenceUtility functionsRisk Measure

Preference

To discuss beating the risk free rate by taking risks we needmeasures for risk and reward. The preference of different marketparticipants are different. Common way of modeling the preferenceare

(i) Utility functions;

(ii) Risk measures; and

(iii) The combination of the two.







Utility functions

Experience tells us that mathematical expectation is often notwhat people use to compare payoffs with uncertainty.

Lottery and insurance are typical examples.

Economists explain this using utility functions: people areusually comparing the expected utility.

Utility function is increasing reflecting the more the better and

Concave: the marginal utility decreases as the quantityincreases.

Concavity is also interpreted as the tendency of risk aversion:the more we have the less we are willing to risk.







Examples of Utility functions

Several frequently used utility functions are

Log utility u(x) = ln(x) goes back to Bernoulli and the St.Petersburg wager problem.

Power utility functions (x1−γ − 1)/(1 − γ), γ > 0 and

Exponential utility functions −e−αx , α > 0.

We note that ln(x) = limγ→1(x1−γ − 1)/(1 − γ).







Common properties

The following is a collection of conditions that are often imposedin financial models:

(u1) (Risk aversion) u is strictly concave,

(u2) (Profit seeking) u is strictly increasing andlimt→+∞ u(t) = +∞,

(u3) (Bankruptcy forbidden) For any t < 0, u(t) = −∞ andlimt→0+ u(t) = −∞,

(u4) (Standardized) u(1) = 0 and u is differentiable at t = 1.







Risk Measure

An alternative to maximizing utility functions is to minimizerisks.

Pioneering work: Markowitz’s portfolio theory measures therisks using the variation.

Modeling the risk control of market makers of exchanges,Artzner, Delbaen, Eber and Heath introduced the influentialconcept of coherent risk measure.







Common properties

Here are some common properties of risk measures

(r1) (Convexity) for X1,X2 ∈ RV (Ω) and λ ∈ [0, 1],

ρ(λX1 + (1− λ)X2) ≤ λρ(X1) + (1− λ)ρ(X2),

diversification reduces the risk.

(r2) (Monotone) X1 − X2 ∈ RV (Ω)+ implies ρ(X1) ≤ ρ(X2). adominate random variable has a smaller risk.

(r3) (Translation property) ρ(Y + c~1) = ρ(Y )− c for anyY ∈ RV (Ω) and c ∈ R , one may measure the risk by theminimum amount of additional capital to ensure not tobankrupt.

(r4) (Standardized) ρ(0) = 0.







Convexity and diversity

Convexity is essential in characterizing the preference.

Diverse in choosing particular preference is intrinsic.






Portfolio problemDual problemMarkowitz bullet

Portfolio problem

Use S and Θ to denote risky part of the price process and theportfolio. Giving the expected payoff r0 and an initial wealth w0,Markowitz’s problem is

minimize Var(Θ · S1)

subject to E[Θ · S1] = r0 (7)

Θ · S0 = w0.







Equivalent form

The portfolio problem is equivalent to the entropy maximizationproblem

minimize f (x) :=1

2x>Σx

subject to Ax = b. (8)

Here x = Θ>, Σ = (E[(S i1 − E[S i

1])(Sj1 − E[S j

1])])i ,j=1,...,M and

A =

[

E[S1]

S0

]

, and b =

[

r0w0

]

.







Dual problem

Assuming Σ positive definite the dual problem is

maximize b>y −1

2y>AΣ−1A>y . (9)







Dual problem

Solving the dual problem we derive the following relationship

σ(r0,w0) =

√

γr20 − 2βr0w0 + αw20

αγ − β2, (10)

where α = E[S1]Σ−1

E[S1]>, β = E[S1]Σ

−1S>0 and γ = S0Σ

−1S>0 .

The corresponding minimum risk portfolio is

Θ(r0,w0) =E[S1](γr0 − βw0) + S0(αw0 − βr0)

αγ − β2Σ−1 (11)







Markowitz bullet

Draw this function on the σµ-plan we get

u

1.0

0.0750.025

1.02

0.98

0.05 0.1

1.03

1.01

0.99

0.0

s

which is commonly known as a Markowitz bullet for its shape.






A Question

Markowitz portfolio theory became popular largely due to itssimple linear -quadratic problem with explicit solutions. Wellknown extensions and applications include Capital Asset PricingModel and Sharpe ratio for mutual fund performances.Are there other risk - utility function pairings can lead toconvenient explicit solutions?


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Why Bankers Should Learn Convex Analysis