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Princeton

RTG Summer Schoolin Financial Mathematics (2013)

Travis Craig Johnson1

2013-06-17

1Dept. of Eng. Sci. and App. Math., Northwestern University, Evanston, IL 60208, USA. email: [email protected]


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Chapter 1

Administrivia

1.1 Lecturers

1. Rene Carmona (Princeton University)http://www.princeton.edu/ rcarmona/

2. Rama Cont (Imperial College London)3. Michael Coulon (Princeton University)4. Jean-Pierre Fouque (UC Santa Barbara)5. Johannes Muhle-Karbe (ETH Zurich)6. Alexander Schied (University of Mannheim)7. Ronnie Sircar (Princeton University)8. Glen Swindle (Scoville Risk Partners LLC & NYU)

1.2 Times

1. 9-9:50 - Lecture

2. 10-10:50 - Lecture3. 11-11:30 - Break4. 11:30-12:20 - Lecture5. 12:30-14:00 - Lunch6. ?? 13:30-14:00 - Special Q&A Session ??7. 14:00-14:50 - Lecture8. 15:00-15:50 - Guest Lecture

1.3 Locations

1. Week 1: Friend Center room 0062. Week 2: Computer Science Building room 104

1.4 Links

1. https://orfe.princeton.edu/rtg/fmsummer/reading

2. https://github.com/traviscj/fmsummer

2


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Part I

Systemic Risk

3


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Chapter 2

Systemic Risk 1 - Fouque

2.1 History

1. 60s and 70s:

(a) Problem of Portfolio Allocation (Mean-Variance/Markovitz/)

(b) Option Pricing(1973) (Black-Scholes/etc)

2. 90s:

(a) Local Volatility(b) Stochastic Volatility

3. 2000s:

(a) Credit - Taking into account the possibility of default(ofcompany, counterparty, etc)

(b) Credit Basket - structuring risk(Mortgages)(c) Credit Default Swap(d) Collatorized Default Options - Huge Huge Market.(e) Financial Crisis - Mortgages were completely mispriced.

4. 2008-2010:

(a) NIF: National Institute of Finance (for doing researchon the banking system)

(b) Handbook on Systemic Risk: Cambridge(c) Dodd-Frank: Created Office of Financial Research (under

the treasury department)

2.2 Systemic Risk

Can consider from many views: mathematics, statistics, etc.

Two main approaches:1. Coupled Diffusions: Continuous time2. Networks

The starting point is Brownian Motion. Suppose we start withan asset:

dSt= Stdt (2.1)

which we can solve by

St= S0et (2.2)

Okay, but usually we dont know the , and there is some noise!How can we add noise? Since is the return, maybe by addingsome white noise

dSt= St(dt + Noise) (2.3)

where the Noise term is given by Brownian Motion, which hasthe form

dWt (2.4)

with >0, and (Wt)t0 fort T. The properties of BrownianMotion:

1. W0= 02. Wt is continuous3. Independent, increments

4. If we consider0< t1< ... < tn T (2.5)

then it gives rise to many differences:

(Wt1 Wt0, Wt2 Wt1,...Wtn Wtn1) (2.6)such that

D(wt ws) =N(0, t s) (2.7)A bit of Brownian Motion History:

1. Brown2. Buchelier (1900)3. Albert Einstein (1905): Heat Equation/Brownian Motion tie-in4. Weiner (1930s): Constructed Brownian Motion: Construct

a measure over all continuous trajectories5. Ito (1940s): Figures out chain rule for brownian motion6. Samuelson (1960s): Generalized Brownian Motion

If we have a bounded function f(t) where

T0

(f(ti+1) f(ti))< (2.8)

If we have a brownian motion instead,

T0

Wti+1 Wti (2.9)

Book Reference: Carson & Tree???

2.3 Hitting Times

Draw a plot:

1. time on horizontal axis2. y = brownian motion on vertical axis3. liney = .

We are interested in the time, defined by

a= inf{t > 0 : Wt a} (2.10)(We are really looking for Wt= a)

We can instead think

{a t} = { max0stWs a} (2.11)Then

Wt= Wtt+ (2a Wt)t> (2.12)is a Brownian Motion. RP.

Now suppose we want to find the probability:

P( t, |Wt a| > b) (2.13)=P( t, Wt> a + b) + P( t, Wt< a b) (2.14)=2P( t, Wt> a + b) (2.15)=2N(0,t)(a b) (2.16)

4


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2.3. HITTING TIMES 5

Figure 2.1: caption

But we also know that

P( t) = 2N(0,t)(a) (2.17)

A few remarks:1. is predictable. We say that isannouncedby a sequence

n, the hitting time ofa 1n .2. Large Deviations: Consider the probability of a very large

deviation (a )

P(a t) ea2

2t (2.18)

where we mean this in term of the log.


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Chapter 3

Systemic Risk 2 - Fouque

3.1 Joint Distributions

Consider the joint distribution of (, Wt). We can think of this as

P( t, Wt= v) =P(Wt= v) ifv > a

P(Wt= 2a v) ifv < a =

g(v)

g(2a v)(3.1)

For nonstandard Brownian Motions,

Xt= x + t + Wt (3.2)

And then= inf{t > 0, xt a} (3.3)

So then,

xt a (x + t + Wt a) (3.4) t + Wt a x (3.5)

t + wt a x

(3.6)

Define = , and a= ax .

Then we have that

Wt+ t Wt (3.7)Then Girsanovs Thm is

Mt =eWt

12

2t (3.8)

ThenE(Mt) = 1 (3.9)

anddP

dP|fx = Mt (3.10)

We can show thatWt+ t Wt is a brownian motion underP. So next,

E (x/Fs) = 1

MsE(XMt/Fs) (3.11)

Then we will be able to show

E eiu(Wt W

s)

=e

2

2 (ts) (3.12)

Why do we care about all of that? Well, we can do the following:

= inf{t > 0, Wt a} (3.13)Then

P( t) = E(t) = E

tdP

dP

(3.14)

We can use the joint distribution of (, Wt).

=N(a x t

t) + e

2(ax))

2 N(

a x + t

t

) (3.15)

3.2 Geometric Brownian Motion

Now considerdSt= St(dt + dWt) (3.16)

which is a stochastic differential equation. We should think of thisas a convient notation for something more complicated. Basically,

StdWt t0

SadWa (3.17)

We will use Itos lemma which lets us do the chain rule witha brownian motion:

dg(Wt) = g(Wt)dWt+

1

2g(Wt)dt (3.18)

whereg(t, Wt) is once differentiable in t and twice differentiableinWt. Then if we have

dXt= tdt + tdWt (3.19)

then

dg(Xt) = g(Xt)dXt+

1

2g(Xt)dxt (3.20)

Then an SDE like

dXt= b(t, Xt)dt + (t, Xt)dWt (3.21)

has a solution like

St= S0e(2

2 )t + Wt (3.22)

3.3 Passage Time

Now consider

dSt= St(dt + dWt) = St(rdt + (dWt+ r

dt)) (3.23)

well then, P: ertSt is a martingale.

3.4 Defaultable Bond

Now we want to have a defaultable bond. The bond starts atS0 and has a default value ofD(if it touches D before maturitytimeTthen we have no payout, otherwise get 1.) So then theprice of the bond is

PD(0,T)= E(>T) = P

( > T). (3.24)

So now we consider

{ inf00} (3.25)

By taking the logarithm, we get a nonstandard brownian motion,which we can evaluate. By some computation, we can get

P(0, T) = erT

N(d+2)

S0D

1 2r2

N(d2)

(3.26)

6


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3.6. SYSTEMIC RISK 7

with

d2 = logS0D +

r 22

t

t(3.27)

The main ingredients here were reflection principle and changeof measure. Alternatively, one could do this completely withpartial differential equations.

3.5 Yield

We can represent the yield by

y(0, t) = 1T

logPD(0, T)

P(0, T) (3.28)

3.5.1 Example

Consider some bond B1 with a 10% default rate. This is toorisky for index funds, and not risky enough for hedge fund.

A financial engineer will do this: Buy two bonds, B1 andB2(both with 10% default rate), then stack them. Now

1. pays if 2 defaults: 0.992. pays if 1 default: .8

Whats the problem? We assumed independence.In credit, the main risk was correlation of default. This is not

really computable.Consider two processes W

(1)t ,W

(2)t and the hitting times

(1)

and(2) try and figure the probability P(z(1) > T; z(2) > T),well, its hard to quantify. If

3.6 Systemic Risk

Usually we will consider the log-capitalization of Banks, and inparticular multiple banks:

dXit = itdt +

idWit . (3.29)

3.6.1 Toy Model

We can consider the drift of capitalization, which includes somecoupling of the systems.

dXit = aN

j=i

(XjtXit)dt + i(dWt +

1 2dWit ) (3.30)

where the Wit are independent. What is the intuition here?That borrowing and lending goes on between the banks. Thisis related to flocking and swarming models.

What other mathematical model behaves this way(where wehave randomness but also attraction)? ornstein uhlenbeck process:

dYt= a(m yt)dt + dWt, (3.31)which we know how to solve:

Yt= m + (y

m)eat + eat

t

0

easdWp (3.32)

And we know that this isN(m,22a).Next step is:

d(1

N

Ni=1

Xit) = 0dt + dW0t +

1 2N

Ni=1

dWit . (3.33)

Now consider ifXi0= xi0= 0, which gives

xt= 1

N

Ni=1

Xit =W0t +

1 2N

Ni=1

Wit (3.34)

If we take = 0 for a second, then

xT N

Wt (3.35)


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Chapter 4

Systemic Risk Day 3

We will continue considering the equation governing thelog-capitalization of the banks

dXit = a

N

Nj=1

(xjt xit)dt + (dW0t +

1 2dWit ) (4.1)

where W0, ..., WN are independent brownian motions and arepresents the speed of trading. We can further consider themean over all these,

Xt= 1

N

N

i=1Xit (4.2)

which lets us write as

dXit = a(Xt Xit)dt + Noise (4.3)where the mean is governed by

dXt= dW0t +

1 2N

Ni=1

dWit (4.4)

Which is equivalent to [ed: is this true?]

=2 1 2

N

dBt (4.5)

This gives rise to a flocking behaviorall the banks will followthe meanat least as long as the mean is large.

Consider the case where = 0 which gives

N

dBt. (4.6)

We will count a bank as defaulting when it hits some defaultamount D 0,b


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4.2. GAMES 9

some other cost at the very end, and also take the expectation,which gives:

Ji = E

T0

((ti)

2

2 qit(Xt Xit) +

2(Xt Xit)2)dt +

c

2(XT XiT)2

(4.17)We can not take too small, we must have q2.

This looks like a Mean Field Game(MFG), pioneeredLions-Lasry for the N

case. This is interestingwe pass

to the limit to get an easier problem. Another approach is fromCarmona-Delarue. But it turns out that we can solve this gameforNfinite, via a dynamic programming approach.

We start with it = i(t, Xt). The dynamic programming

approach means also introducing the quantities

Vi(t, x) = minEx

Tt

as above + (terminal cond)|Ft

(4.18)This is governed by the Hamilton-Jacobi-Bellman (?) equation:

TVi +N

j=1

(a(x

xj) + i)xjV

i +2

2jk2 + (1 )

2j,k+(i)2

2 qi(x

xi) +

2

(x

xi)2 = 0

(4.19)[Ed: there was a randomxj,xkV

i floating around here.. I thinkit goes after the jk) part?] Also jk is the Kronecker delta.Because we want the infimum, we just take the gradient. Andwe want it with respect toi so

xiVi+iq(xi) = 0 = i = xiVi+q(xi), i= 1,..,N

(4.20)But this assumes that we know Vi, which we do not! So nextwe consider

tVi+

N

j=1(a + q)(x x

j)

xjV

jxjVi+2

2jk2 + (1

2)jkxjxkVi+12

(

q2)(x

xi)2+

1

2

(xiVi)2

(4.21)Well try an Ansatz here:

Vi(t, x) =t

2(x xi)2 + t (4.22)

and impose that

Vi(T, x) T =C, T = 0 (4.23)which gives

xjVi =t(

1

Nij)(x xi) (4.24)

and

xj,xkVit =t(

1

N ij)(1

N ik) (4.25)

which gives

t=2(a + q)t+ (1 1

N2)2t ( q2), T =C

(4.26)

t= 2(1 2)(1 1

N)t, T= 0 (4.27)

which are x-independent terms! Hooray!

Lets solve it:

t=( q2)(exp((T t)) 1) C(+ exp((T t))

( exp((T t)) +) C(1 1N2 )(exp((T t)) 1(4.28)

where

=+ (4.29) =

(a + q)

R (4.30)

R=(a + q)2 + (1 1N2

)( q2)>0 (4.31)

Where are we at here? First, we have

it= (q+ (1 + 1

N)t)(X Xit) (4.32)

and also

dXit = a

N

Nj=1

(Xjt Xit)dt + (above...)dt + Noise (4.33)

[ed] I missed one here.Could doT because the terminal time is annoying. Oneoption is to take c = 0, and optimizing 1TJ

iT. TakingT

mean the effective rate of borrowing and lending is

( q2)

(4.34)

We could also take N , but this will be on Friday.Tomorrow will be a different approach and compare the two.


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Chapter 5

Systemic Risk Day 4

Consider

dXit =a(xt xit)dt + (dW0t +

1 2dWit ) + idt (5.1)for i = 1,..,N. One way to handle this is the open-loopfeedback approach, which introduces

Ji = E

T0

(i

2) qi(xt xit) +

(xt xit)2)dt +

c

2(xT xt)2

(5.2)Another way is the dynamic programming approach wherewe introduce Vi(t, x). A more different way is the closedloop approach due to Pontugigin (spelling?). This is also a

Forward-Backward DE approach. It works by introducing aHamiltonian for each player i:

Hi(x,yi, ) =

Nk=1

((xxk)+k)yik+12

(i)2qi(xxi)+ 2

(xxi)2

(5.3)Then imposing that fork =i, we havek =k(t, x). All of thisbecomes the stochastic differential equations

dXi=Hi

y dt (5.4)

dYi,jt = H

xjdt + ZtdWtY

i,jT = C(xT xiT)(

1

N ij)

(5.5)

So now we must calculate these Hamiltonian parts:

Hi

xj =

Nk=1

(a + q)(

1

N kj)

yikqi(1

Nij)+(xxi)(1

Nij)

(5.6)and

dYijt = Nk=1

(a + y)(1

N kj)Yikt qi(

1

N ij) +

(xt xit)(

1

N ij)

dt+ZtdWt

(5.7)To solve, we take the anstaz

Yijt =t(1N ij)(xt xit) (5.8)

Plug that in, then we get some expression for dYijt . Eventually,we get the differential equation from

t= 2(a + q)t+ (1 1

n2)2t ( q2), T=C (5.9)

Okay, lets switch gears a bit and consider N . ThenXt mt is related to E(Xit/w0) Then the one-player modelends up being

dXt= a(mt Xt)dt + tdt + (dW0t +

1 2dWt) (5.10)

So we will consider

infE

T0

(t)

2

2 qt(mt Xt) +

2(mt Xt)2

dt+(otherst

(5.11)Then the Hamiltonian is

H= [] y+2

2 q(mt x) +

2(t x)2 (5.12)

and again well set up the hamiltonians

dXt=H

y

dt + (noise) X0= x0 (5.13)

dYt=H

xdt + (noise) YT = (0orTC) (5.14)

(5.15)

Now then, because E(mt Xt) = 0, we have E(Yt) = 0. Usingall of that, we can find E(Xt) = mt (no surprise) Then we getsome equation like

t= tXtdt (5.16)(but this equation is wrong, there is a qsomewhere)

Two references:

1. Carmona-Delarue2. Mean Field Games: Lions-Cassry Cassy

10


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Chapter 6

Systemic Risk Day 5???

6.1 Diversification vs Systemic Risk

This will be an endogenous approach.Consider two stocks S1, S2 which are geometric brownian

motions. Suppose they have mean growth and random part and independentW1,W2. Well consider two banks with initialwealthX10 =X

20 = 1. Consider the equations:

dX1t =X1t dt + X

1t

(1 )dW1t + dW2t

(6.1)

dX2t =X2t dt + X

2t

dW1t + (1 )dW2t

(6.2)

Now consider T >0 with 1, 2 and default levels D1, D2.

Then U(t, x1, x2) = P(1> t,2> t) (6.3)

Then the probability of a systemic event is

P(Systemic Event) = 1 + u u1, u2,. (6.4)There is a joint distribution of 1, 2. Then we can find aplot: PutP(systemic) vs(the diversification parameter) Theprobability looks something like

(1 u1)(1 u2) (6.5)Then it is clear that there is some critical valuec.

Switching gears a bit, it is clear that u(t, x1, x2) satisfies a

PDE: tu = 0 + T.C. + B.C. (6.6)

(Insert plot: horizontal: (0, T) , x1from some value to infinity, etc.)

6.2 T. Ichibu Paper

This is known as Fellers Diffusion (CIR) model:

dXit = idt +nj=1

(xjt xit)pij(Xt)dt + 2

XitdWit (6.7)

withXt is the capitalization.

6.2.1 Quick review

If we have dYt = (m Yt)dt +

2YtdWt (6.8)

then ifY0 = 0 and if2 m (with m > 0) then Yt > 0t.(That is, the force of the process outweighs the volatility.) Thenice part of this is that it never hits zero.

We will need to solve an equation Lu= 0 of the form(m y)u + 2yu = 0 (6.9)

which has a solution of the form

u(y) =

y1

e z

2 zm

2 dz (6.10)

Now consider a bound [, M]. We are interested in the hittingtime. We can caluclate

u(Yt) = u(y) +

t0

u(ys)

2YsdWs (6.11)

The variance is given by

E((u(y))2) = u(y)2 +E

t0

u(Ys)222Ysds (6.12)

Then we have that

E() 0 and Pij bounded, regular enoguh, then there

exists a solution (According to Bass-Perkins 2003).WeSt=

ni=1 X

it, andPij= Pji, and we have the equation

dSt = (ni=1

i)dt + 2ni=1

XtdW

it (6.15)

Equation:D: 2StdBt.Then we have

dSt=

ni=1

i

dt + 0 + 2

ni=1

xitdW

it (6.16)

Which is equivalent to

StdBt. Also, define

=ni=1

i (6.17)

This gives rise to a square Bessel process: 2; thenP( < +) = 0. If = 2 still P( < +) = 0 but alsoP(limsupt St= ) = 1 andP(inft St= 0) = 1. Next, consider0 < < 2; Then P( < +) = 1 via a reflecting principle.Finally, if = 0, then 0 is absorbing.

6.3 Multiple Defaults

If for (1,...,k) {1, 2,...,n},

supx

|xi xj|pi,j(x)


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12 CHAPTER 6. SYSTEMIC RISK DAY 5 ???

Can also show the converse.We can showPij=

1n . We can then consider the mean field

model:

dXit =

i+ m xit

dt + 2

XitdW

it (6.20)

where

limn

1

n

ni=1

Xi0 (6.21)

Next, we could consider the health of a group of banks:nj=1

ki=1

Xit Xkt

pk,j(Xt) (6.22)

We could also create networks: Create a link betweeni andj if there is a certain amount of flow. Now we have a graph bylooking at these quantities. Then we can start asking questionsabout the size of the system, path lengths between banks,statistics of the system, etc.


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Chapter 7

Rama Cont

Bank balance sheets:

1. Assets

(a) Liquid Assets(b) Interbank claims(c) Other assets

2. Liabilities

(a) Deposits(b) Interbank liabilities(c) Short-term debt(d) Capital

Key parts: Solvency vs Liquidity.

7.1 Microprudential approachTraditional approach to risk management and bank regulation is tofocus on failure/non-failure solvency, liquidity) of individual banks.It focuses on the balance sheet structure of the individual banks.

They assume that losses arise due to exogenous randomfluctuations in risk factors. The main tool for stabilizationof system is the capital requirements. But it ignores links orinteractions between market participants.

7.2 Liability Chain

Due to Hellwig. Consider a finite set of banks i = 1..N, where

1. iborrows $100 fromi 1 overi years2. ilends $100 to i + 1 overi + 1 years

13


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Chapter 8

Contagion and systemic risk in financial networksRama Cont

Exposure networks: LetEij denote exposure ofi toj.

14


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Part II

Commodities and Energy

15


16/46

Chapter 9

Commodities & Energy Markets 1 - Coulon

What are commodities? Commodities are goods (eithernatural resources or processed) with little or no variation inquality across supply sources.

They are widely traded.Recent changes:

1. gas/electricity deregulated2. number of market participants has rapidly expanded3. banks actively trade derivative products4. investment volume

Main participants in price models: Traditionally:

1. Producers (farmers, mines, power plants, refineries)2. Consumers (large industrials, utilities, airlines)

3. Storage/Delivery (gas pipelines, logistics companies)Recently:

1. Financial institutions, speculators, investors, regulators2. everyone else (if you read the news, drive a car, eat food, etc)

What are the main modeling challenges:

1. Adapt financial math to very different markets2. Capturing features of price dynamics and fundamen-

tals(inventory levels, weather )3. Strike balance between supply/demand, calibration, pricing

derivatives, etc.

Differences:

1. Exhibit mean reversion, possibly seasonal, very high volatility

2. Dynamics closely linked to economic factors3. Spot commodities are not traded assets in the same sense

as stocks or options (due to storage/delivery issues) (mainly,cant assume no arbitrage)

4. Critically: Same commodity delivered at two slightly differentlocation or times can behave as separate assets.

5. Set of liquidly traded products often significantly smaller thanthe risks needed to be hedged( specialness )

6. Correlations are very important: companies interested inhedging multi-commodity exposure.

Spot power prices: craziest of all.Spot price history: more regular.Categories of commodities:

1. Storable vs non-storable2. continuous vs seasonal production (or demand)3. local vs regional vs global4. elasticity of supply or demand to price

Modeling categories:

1. Reduced-form: direct price modeling, traditional financial math2. econometic: reduced form elements combined with regressions

to find relationship between price and key factors3. structural: capture key features of supply and demand and

approximate market mechanisms while retaining tractability4. full equilibrium: detailed matching of supply and demand, op-

timization over participants behavior, physical constraints, etc.

Forward curve behavior(Samuelson effect): Forward contractsbecome more volatile as they become closer to expiration.

Cost of carry relationship: For a financial asset with no storagecost and interest rates r constant, by a simple no-arbitrageargument:

F(t, T) = Ster(Tt) (9.1)

But since storing commodities is expensive:

F(t, T) = Ste(r+c)(Tt) (9.2)

However, since arbitrage argument only holds in one direction,so instead

F(t, T)

Ste(r+c)(Tt) (9.3)

Why? We cannot actually short physical quantities. (You cantborrow from future harvests, for example.)

9.0.1 Theory of storage

IntroduceF(t, T) = Ste

(r+c)(Tt) (9.4)

whereis the convenience yield. Very artificial, but still someintuitions. Basically, we get some dividend from being the holderof the asset. Frequently roll the costs into the convenience yield.

9.0.2 Reduced-Form Model

Simplest spot price model consistent with cost-of-carry:

F=Ser

(T t) (9.5)We get from the Generalized Brownian Motion as

dSt= (r )Stdt + StdWt (9.6)underQ, wherer,, and are constant. Then

F(t, T) = EQt [St] (9.7)

which gives

F(t, T) = Ste(r 1

22)(T t)EQt

e(wTw0)

(9.8)

soF(t, T) = Ste

(r)(Tt) (9.9)

What are the weaknesses here?

1. We cannot obtain both contango and backwardation. Anatural extension is to let t be stochastics. This is goodbecause we are capturing the relationship with inventory(inversely), butt is unobservable

2. No mean reversion in spot price, so all forwards have the samevolatility. (Can see this by applying Ito to F(t, T), giving

dFt=F

tdt +

F

SdS+

1

2

2F

S2dS2 (9.10)

16


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17

which impliesdFtFt

=dWt (9.11)

which has noTdependence, which means no Samuelson effect.An alternative: Schwarz one-factor (97):

dSt= ( log(St))Stdt + StdWt (9.12)underQ. If we apply Ito toyt= log St, then

St= eyt (9.13)

whereyt is ornstein uhlenbeck process

dYt= ( 2

2 yt)dt + dWt (9.14)

which is an exponential ornstein uhlenbeck process. It gives

log St N((log S0)e(Tt)+(1e(Tt)),2

2(1e2(Tt)))

(9.15)We can apply Ito to yte

t. So it is easy to findF(t, T):

F(t, T) = EQt [St] = exp{mean +12variance} (9.16)

Then apply Ito to Ft to see

dF(t, T)

F(t, T) =e(Tt)dWt (9.17)

Weaknesses: Volatility of long forwards is underestimated.Also all forwards are perfectly correlated.


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Chapter 10

Commodities & Energy Markets 2 - Coulon

10.1 Reduced form models

So far, two simple one-factor spot price model. The two mainapproaches are

1. Generalized Brownian Motion (no mean reversion)

dStSt

= (r )dt + dWt = dF(t, T)f(t, T)

=dWt (10.1)

2. Exponential OU (mean reverting)

dStSt

=(log St)dt+dWt = dF(t, T)F(t, T)

=e(Tt)dWt

(10.2)

Here, the coefficients of the stochastic differential equationsforF(t, T) describe the vol term structure.

The key observation here is that:

mean reversion inSt decaying vol ofF(t, T). (10.3)

Picture goes here: time is horizontal axis. GBM is a constantyvalue, exponential OU shows a decay. Typical data ( either impliedor historical) show somewhat slower decay of the forward curves.

Notes:

1. Why the mean reversion of the spot prices? We can makeeconomic arguments about short term shocks vs long term

equilibrium (production/consumption levels).2. Why not mean reversion in F(t, T) itself? Its a traded

derivative, it must satisfy the martingale condition. (Forwardsare martingales, at least for fixed T)

3. What about F(t, t+) (fixed tau)? For a start, its not a tradedcontract. Therefore, it is likely that it could be mean reverting.And of course, as 0, we should get the spot price.

Any exceptions? In theory, we can have something that meanreverts under P but not under Q. But this is quite unusual or arti-ficial. Why? Drift under P with a mean reverting process gives amarket price at risk . That is, drift under P to drift under Q gives

(

Yt)dt

(

Yt)dt (10.4)

where is the market price of risk. We could actually chooseto be a function ofYt to kill mean reversion, but again, thisis very artificial.

Related question: How does EPt [St] compare to F(t, T)?Clearly, it depends on the sign of;

1. Normal backwardation: > 0 (this is hedging pressurefrom risk producers pushing F(t, T) downwards). See

Inconvenience Yield and The Theory of Normal Contango- Bouchuev, 2012 for an argument that recent change towardcontango change driven by contango.

2.

10.2 Two Factor Models

These try to correct some of the weaknesses of the volatilitystructures, etc. See Schwortz (97) for two factors. We need shortterm shocks which are not mean reverting but long term which is.

dSt=(r t)Stdt + StdWt (10.5)dt=( t)dt + dWt (10.6)

dWtdWt=dt (10.7)

The advantage is that this is a combination of mean revertingand non-mean reverting(which is basically just saying short termand long term.)

10.2.1 Schwartz & Smith (00)

Instead:St= exp{Xt+ Yt} (10.8)

whereXt is arithmetic brownian motion andYt is O.U.Both models are lognormal and find find

F(t, T) = EQt[St] (10.9)

We can show the equivalence with Schwartz (97).

10.2.2 Extension: 3 factor

We can also add a Vasicek model for rt. But this one doesntreally matter because interest rates have only minuscule amounts

of volatility relative to the volatility of the commodity.10.3 Calibration

The first priority is typically to match the observed forwardcurve. (That is, find ,,). But of course, we cant do this.So we need to let be time dependent.

We take the Hull-White/Ho-Lee approach: Now we try to take{,,(t)}. We can either take continuous time or some numberof piecewise linear intervals to get the right number of parameters.

One thing to be careful of: Can easily overfit the modelbecause of the amount of freedom in this calibration. One thingto look out for is if the model calibrations come out completelydifferently each day; we should exercise caution.

10.4 Forward Curve Models

Instead, we can start with the forward curve directly (especiallyif we do not care about St). The general model under Q is:

dF(t, T)

F(t, T) =

Ni=1

i(t)dW(i)t (10.10)

where W(i)t are independent brownian motions for simplicity.

Now we must choose:

1. The number of factorsN2. Shape of1(t, T), ...,N(t, T).

Notes on this:18


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10.4. FORWARD CURVE MODELS 19

1. All of the previousSt models are special cases of this. IE:ifN= 1, 1(t, T) = e(Tt) gives the Schwarz 1-factor

2. Calibration toF(0, T) Tis immediate, because it is just theinitial condition for the stochastic differential equationthereis no calibration.

3. Can show thatF(t, T) is lognormal. Apply Ito to log F(t, T),which implies

F(t, T) = F(0, T)exp{

N

i=1

1

2 t

0

2

i(u, T)du +

t

0

i(u, t)dWi

u}

(10.11)

So we went from the St dynamics to the F(t, T) dynamics.Can we go from the F(t, T) dynamics to the St dynamics?Recall thatSt= F(t, t), so

log St= log(F(0, t) +

ni=1

[...]) (10.12)

If we apply Ito again tolog St and do the calculations, then wehave

dStSt =logF(0, t)

t Ni=1

{ t

0 i(u, t)

i(u, t)

t du + t0

i(u, t)

t dW(i)u }dt+

Ni=1

i(t, t)dW(i)

t

(10.13)which actually implies that St is non-Markovian! (Bad news!)Unless you have just the right form that the terms cancel.

How can we estimate the volatility functions? 1) Fromhistorical returns, we do Principal Component Analysis. 2) Fromoptions directly. People usually propose a particular parametricform. One option (which Glen probably uses):

dF(t, T)

F(t, T) =

Ni=1

vi(t)i(T)ei(Tt)dW

(i)t (10.14)

where the termsvi(t), i(T), andei(Tt)

represent the affectof the whole curve, maturity specific, and steepness of the volterm structure. (That is, a Gaussian Exponential Factor Model.)


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Chapter 11

Commodities - Day 3

Recall the general forward curve model

dF(t, T)

F(t, T) =

Ni=1

i(t, T)dW(i)t (11.1)

11.1 Option Pricing

Most commodity markets options traded on futures, not spot,and maturity just before the futures maturity. That gives threecritical times:

1. t: current time2. T1: option maturity3. T2: forward maturity

The price we want is the

Vt= exp(r(T1 t))EQt(F(T1, T2) K)+

(11.2)

We can use Blacks Formula on any lognormal E [(ex K)+]andx N(x, 2x). Many people are familiar doing somethinglike

=

logKxx

(ex+xx K) 12

e12x

2

dx (11.3)

ad after a few lines we get

=ex+12

2x

x+ 2x log Kx K

x log Kx (11.4)

But we have that

x= EQ[log(F(T1, T2))] = log(F(t, T2)) 1

22x (11.5)

EQt [F(T1, T2)] =F(t, T2) = ex+12

2x (11.6)

Hence, we have

Vt= er(T1t) [F(t, T2)(d+) k(d)] (11.7)

whered= log(F(t,T2)/K)

2x

xand2x= Var

Q [log F(T1, T2)].Fine. Now the general forward model is

logF(T1, T2) N(log F(t, T2)12

Ni=1

T1

t

2i (u, T2)du,Ni=1

T1

t

2i (u, T2)du)

(11.8)We just need to integrate the squared forward vol over the lifeof the option. For example, the Schwartz 1-factor model gives

2x=

T1t

(e(T2u))2du= 2

2

e2(T2T1) e2(T2t)

(11.9)

Plot vs maturity: Exponential decay fromT2 T1 untilT2 t.(Or, alternatively, integrating the exponential increase from ttoT1 (and avoidingT1 to T2)).

11.2 Spread Options

General spread option payoff at time Thas the form

(aXT bYT K)+ (11.10)where XT and YT are different commodity prices (spot orforward):

1. Input/Output (e.g. dark if between electricity/coal, spark)2. Input/Output (e.g. crack if between refined product, crude)3. Calendar (e.g. Dec13 Forward vs Jun13 forward) (storage)4. Locational (e.g. Henry Hub vs NorthEast gas) (transport)

Spread options are critically important, due to the strong link

with physical assetsthey are useful as hedging and valuationtools. There are some tricky aspects here, because these reflectoptimal(unconstrained) operation, which is not usually available.

11.2.1 Classical Spread Option Pricing

Margrabe proposed:

dS(1)t =rS

(1)t dt + 1S

(1)t dWt (11.11)

dS(2)t =rS

(2)t dt + 2S

(2)t dWt (11.12)

dS(1)t dS

(2)t =dt (11.13)

which are correlated geometric brownian motions.

11.2.2 Power Plants

We can approximate a power plant value via a string of spreadoptions

Plant Value jJ

exp(rTj)EQ(PTj hgGTj egATj K)+

(11.14)The main challenge is to capture the multi-commodity dependencestructure and the link with demand in a mathematically tractablemodel.

Main approaches:

1. Reduced-Form: Correlated Lognormals, etc (Carmona &Durrleman)

2. Full Fundamental: Via production cost optimization problem

3. Structurally: Embedded into a model for spot power:

Power =f(gas, coal, carbon,...) (11.15)

Some issues using Margrabe for power valuation:

1. prices negative2. not brownian motion3. etc...

11.2.3 Electricity Markets

Main point: Electricity is not storable, so we require hourlymatching of supply and demand for market clearing. The pricevaries widely across different locations and is

20


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11.2. SPREAD OPTIONS 21

Huge huge huge spikes. The prices might even go negative.Some pretty weird markets, because of subsidies, etc.

Well focus on the correlation between natural gas andelectricity.

11.2.4 Structural Models for Power

Were shooting for a middle ground which lets us use forward-looking models and soforth. Also, using historical data isproblematic.

1. DemandDt: Barlow (2002)2. Capacityt: Burger et all(2004), Cartea et al (2007)3. Fuel PricesGt: Pirrong and Jermakyan(2005) , Aid (2011)

11.2.5 Bid Stack function

Day-ahead generator bids arranged by price to form the bidstack. The spot pricePt is highest bid needed to match inelasticdemand Dt. The merit order of production costs drives thedynamics of the stack.

We can model some of these ideas which captureWe can get some closed form solutions via the assumption

of an exponential bid stack.A better way is to look at the power price vs gas price curves.


22/46

Chapter 12

Commodities & Energy Markets 4Inventory and Implications for Valuation : Glen Swindle

12.1 Main Themes

1. Liquidity vs Dimensionality: Liquidity is concentrated inbenchmarks. Risk can be spread over hundreds of locationsand many delivery months. Price decoupling (specialness)occurs routinely. (Dodd-Frank)

2. Covariance Structures vs Options Markets: The term structureof volatility and correlation is nontrivial. Liquidity in optionsis concentrated in vanilla products with mechanics with limitinformation about (limit utility as hedges of)

3. Broad ranges of time scales: Price dynamics exhibits structureover years (infrared) to hours (ultraviolet)

4. Uncommoditized risks

Carry formalism The forward curves can be viewed as yieldcurves. We can do the forward yield via

y(t,T,T+ S) = 1

Slog

f(t, T+ S)

f(T, T)

(12.1)

Key points:

1. All the you can get from market data is q 2. The cost of storage is not exogenous. Storage owners will

charge what the market will bear.

One approach: Fit a fourier series, then look at the residual.Storage: Types of storage: Aquifer storage, reservoir, salt

caverns, Actually 8.7 tcf of storage, but 4.2 tcf is base gascantget it back out.

Dynamic Optimization:

V[0, S(0), F(0, )] = sups()A

E{ T0

d(0, t) [s(t)F(t, t) (s(t), S(t), F(t, t))]dt}(12.2)

1. dandFare discount factor2. Sis the current inventory level, s =S

3. denotes costs associated with injection and withdrawal. Eg:=k|s(t)|F(t, t)

4.A denotes allowed controlsThis is a super hard problem.

Lets figure out a calendar spread option: time spread option.Consider options with the following payoff

maxF(, T+ U) + erUF(, T), 0

(12.3)

22


23/46

Chapter 13

Commodities & Energy : Day 5Non Standard Expiry

Vanilla options markets in commodities always have expirationdates very near the delivery. Options structures arise in whichexpiration varies substantially from the standard market conven-tion. Modeling/estimation is required to infer implied vols overnonstandard time interval from vanilla implied vols. Examples:

1. CSOs2. Price holds (one type: Swaptions)3. Options on ETFs

(a) These involve the vol of the current nearby, which requiresinference from term vols for each contract to the vol perti-nent to the time when the contract is active in the index.

(b) Recursive dynamics ofVt is due to the contract rolls:

Vt=

N(t)n=1

F(Tn, Tn)

F(Tn, Tn+1)

F(t, Tn+1) (13.1)

Working problem: On 18Feb2009 you are asked by the salesdesk to price WTI Dec11 at-the-money European straddles withexpiration on 17Dec2009. This is the simplest incarnation ofnonstandard expiration. Most occurences are much more complex.

What you know: Dec11 is a liquid contract with standardoptions expiration date 16Nov2011. NYMEX options marketare visible on this horizon.

13.0.1 Non-Stationarity

Empirical analysis of price dynamics in commodities constantlyis encumbered by a technical annoyance: contracts expire.

Two basic approaches:

1. Construct rolling nearby series;2. construct constant-maturity forwards.

Two methods:

1. Nearby contracts refer to a sequence of live contracts (1stnearby is first, second nearby is next live contract)

2. Constant maturity.

Keypoint: Do analysis on concatenated RETURNS not prices.

13.1 Volatility Backwardation

Volatility backwardates.

23


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Part III

Portfolio Optimization,Transaction Costs, Dynamic Games

24


25/46

Chapter 14

Portfolio Optimization, Transaction Costs & Dynamic Games 1 - Muhle-Karbe

1. Setting2. Frictionless Control Heuristics (no transaction costs)3. Verification via Convex Duality4. Control Heuristics with Transaction Costs5. Verification via Shadow Prices

Well consider the Black-Merton-Scholes model with two assets:

1. Safe asset normalized toS0t = 12. Risky asset following geometric Brownian Motion:

dStSt

=dt + dWt (14.1)

for a standard Brownian motion (Wt)t0 defined on a filteredprobability space (, F, (Ft)t0, P).

3. Returns have constant expectations >0, vol >04. simplest benchmark5. no tx costs

We start with a positive cash endowmentx >0. An investor chooses numbert of risky shares. Corresponding wealth process:

Xt =x +

t0

sdSs (14.2)

The safe position implicitly determined by self-financing condition:

0t =x + t

0

sdSs tSt (14.3)

To make everything well-defined, (t)t0needs to beS-integrable:

t0

ssds < t 0 (14.4)

Preferences: Investor maximize expected utility from terminalwealth at some time horizon T >0:

E [U(XT)] max (14.5)

over all trading strategies t.The utility function U should:

1. be increasing: more is better than less.2. Concave: risky payoff is worth than its expectation3. Inada conditions: Extra dollar matters a lot when poor,

irrelevant when rich. (Codify in math by:

limx

U(x) 0 (14.6)limx0

U(x) (14.7)

)

One messiness here is this: The units are goofy. An alternativeis to use a different unit. A better interpretation is: maximizecertainty equivalent:

CE() = U1(E [U(XT)]) max (14.8)How can we determine optimal strategy and utility? Theres a

complex dependence on market parameters, investment horizon,and preferences! How can we tackle an infinite dimensionalstochastic control problem? Today, well do a heuristic derivationand tomorrow, well do a rigorous verification.

We first make the problem harder: via a value function like

v(t, x) := sup(s)s[t,T]

E

U(x + Tt

sdSs)|Ft (14.9), which evaluates along any wealth processXt (a supermartingale

with nonpositive drift) and along optimal wealthXt (martingalewith zero drift.) This boils down to: Martingale optimalityprinciple of stochastic control.

What do we mean by all of this? We start with

E [v(t, Xt )|Fs] (14.10)

=E

E

U(Xt +

Tt

sdSs)|Ft

|Fs

(14.11)

=E

U(Xt + T

t

sdSs)|Fs

(14.12)

=E

U(Xs +

ts

sdSs) +

Tt

sdSs)|Fs

(14.13)

E

U(Xs +

Ts

sdSs)|Fs

(14.14)

=v(s, xs ) (14.15)

How can we use the martingale optimality princeiple? Thevalue function v(t, x) generally can depend on many statevariables. Here, the wealth dynamics

dXt =tStdt + tStdWt (14.16)

=tdt + tdWt (14.17)

for some risky position t only depend on the control t. Oneguess is a value function only depends on the initial wealth xand time t. This is an assumption!! Not a result. This needsto be verified eventually.

Suppose that the value function is smooth. Then Itos formulayields

dv(t, Xt) = (vt+ tvx+2

22t vxx)dt + (...)dWt (14.18)

25


26/46

26 CHAPTER 14. PORTFOLIO OPTIMIZATION, TRANSACTION COSTS & DYNAMIC GAMES 1 - MUHLE-KARBE

This should be a supermartingale for any t: drift 0. It willalso be a martingale for the optimizer t: with drift = 0. Thenmaximize drift pointwise int. Plug in maximizer. Set to zero.This yields equations for optimal stategry and value functions.

We can consider a special case: Take the exponentialutility function U(x) =eax, and factor the wealth out viav(t, x) =eaxv(t, 0), with the constant absolute risk-tolerance1/a. This gives the constant risky position

t=

a2 (14.19)

This is independent of horizon. It is independent of wealth. Thevalue function and certainty-equivalent are the same.

v(t, x) = eax exp( 2

22(t T)) = CE() = x + 2

2a2T

(14.20)Here, the assets are ranked by the Sharpe ratio /. The sameSharpe ratio leads to the same payoffs.

An alternative: If we have twice as much money, we are willingto take twice as much risk. This is debatable, but why not??

U(x) = x1

1 (14.21)

rederive v(t,x), risk tolerance, proportion, value function, certaintyequivalent.

v(t, x) = x1v(t, 1) (14.22)

Constant relative risk-tolerance x/. Constant risky propor-

tion t := t/Xt = /

2, which is independent of horizon.Investment scales with wealth.

Value function and certainty equivalent:


27/46

Chapter 15

Lecture 2

We can verify that all of this actually works.

15.1 Verification via Convex Duality

Need to rule out doubling strategies. Usual notion: Xt C(we have a bounded credit line). The valueC= 0 works well forutilities defined onR+; optimal wealth process for power utilityfollows geometric Brownian motion:

dXtXt

=

2(dt + dWt) (15.1)

This is not compatible with utilities on R: optimal wealth pro-cesses for exponential utility follows brownian motion with drift:

dXt =

2(dt + dWt) (15.2)

How can we verify?

15.1.1 Convex Duality

For a concave function U, we can define the conjugate:

V(y) = supx>0

{U(x) xy} (15.3)

Then

E [U(X

T)] E [V(yYT)] +E [XTYT] = E [V(yYT)] +EQ [X

T] y(15.4)fory >0 and density process Yt of an EMM Q.

ThenXt =x +t0

sdSs is a local Q-martingale.

1. Supermartingale if bounded from below for utilities on R+2. Martingale if risky position is sufficiently integrable for utilities

onR (eg, bounded). This implies admissibility.

In either case, E[U(XT)] E[V(yYT)] + xy.This upper bound is for any trading strategy , any t, and

anyy. Candidate is optimal if bound is tight for

1. U(Xt ) =yYt2. E[YT(U

)1(yYT)] =x

For exponential utilities U(x) = ex

, we getV(y) = ylog y 1. The second condition gives y =

exp(xE[YTlog YT]). This yields a simplified upper bound:

E[eXT ] exE[YtlogYT] (15.5)

In complete market with unique YT, we can verify equalityfor candidate by direct computation! Sweet!

We can do analogous results for power utilitiesU(x) = x1/(1 ). The conjugate comes out toV(y) = 1y

11/. Then we look at the expectation, and look

at the upper duality bound simplifies.

15.2 Markets with Transaction Costs

Before, we bought and sold at the same price St. Now, wewill buy at the ask price St and sell at the bid price (1 )St.(clearly, is the width of the relative bid-ask spread.) We onlyconsider finite variation strategiest =

ut dt . These track

the number of shares bought and sold by time t, respectively.Infinite costs otherwise.

We can do integration by parts on the frictionless self-financingcondition, which then reads as

d0t =tdSt d(tSt) = Stdt (15.6)The counterpart with transaction costs is

d0t = Stdut + (1 )Stddt . (15.7)Why is all of this important? Non-trivial spreads are common

even in the most liquid markets. Constant position/weightrequires infinite turnover, which causes infinite costs, whichis not feasible. How can we adapt optimal trading strategies?How much welfare is lost? What are the endogenous spreadsin equilibrium between market makers and rebalancing investors?(This is a sort of static game which can determine the bid-askspread as an output of a model such as: Find a bid-ask spread byexamining market maker profit from optimal strategy of tradersgiven a bid-ask spread.) There is also a new difficulty: complexhorizon dependence. Do not trade if horizon is close. Way out?

Two alternatives:

1. We could look at the expected utility of the final value of ourportfolio:

E[U(XT)] (15.8)

but this is a bit fishy...2. Another option is to look at

E[

0

etU(ct)dt] (15.9)

which is good, except that it is basically intractable.

So, as before, well consider the exponential utility

U(x) =ex or power utility U(x) = x11. But infinitehorion to reduce stationary problem: For exponential utilities,

we maximize the equivalent annuity:

limsupT

1T

logE

eXT

max! (15.10)

For power utilities, we maximize the equivalent safe rate:

limsupT

1

(1 )T logE(XT)

1 max! (15.11)

This keeps scaling properties in the wealth, and also gets ridof the horizon dependence.

Open questions: What are the control heuristics? How canwe verify it?

27


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Chapter 16

Lecture 5

[tcj: Missing lectures 3/4 because notes online.]Idea: Directly tackle asymptotic optimality equations.General model: General asset prices, costs, preferences. Nor-

malize the safe asset to one. The risky asset will be traded withproportional costst= t> 0. The mid price will be given by

dSt= bStdt +

cStdWt (16.1)

This gives the general diffusive dynamics. We can includeheteroskedasticity and predictable returns leading to markettiming. We do not need any markovian structure.

Why doe we need general asset prices? Transaction costs havesmall effect in Black-Scholes model, because investors essentially

should buy and hold, and then profit from the market growth.On the other hand, Mnay strategies are market neutral. Theymake a profit by active trading. For example: Buy low, sell highfor a mean-reverting instrument, which might be governed by

dSt= Stdt + dWt (16.2)or a momentum strategy, which says: go long in good times,short in bad times. That gives:

dSt=tdt + dWt, (16.3)

dt= tdt + dWt (16.4)In these cases, it is much harder to avoid the transaction costs.We will need to find a direct tradeoff between the gains from

trading and costs incurred by trading.Here, the investor will solve

E

T0

U1(t, t)dt + U2(X

T(

, ))

max (16.5)

over all policies (t, t).

Motivations:

1. Individual household: Receive labour income stream, consumesover lfetime

2. Retirement Fund: Maximize long-run expected utility fromterminal wealth

3. Option trading board: Hedge derivative until maturity.4. Hedge Fund: time the market on short horizon.

Here, the general model incorporates all these asset prices andoptimization problems.

The optimal policy is to consume at the rate:t = t +

t(X

t Xt). We will trade to keep the risky shares

in some no-trade regionNTt NTt, NTt+ NTt

. Clearly,

the midpoint is given

NT

=t+ t(X

t Xt) (16.6)

We can determine the halfwidth to be

NTt=

3Rt2

dtdSt t

13

(16.7)

This is determined by a type of portfolio gamma dtdSt . This

implies that Active strategies require a wide buffer and turbulentmarkets call for closer tracking.

Note that only the current spread t matters here; futuredynamics are hedged at higher orders, but not considered here.

We can do the following calculation: Given t = (St), thenwe have

dT= (St)dSt+ (...)dt (16.8)

and then we can consider the quadratic variation which is

dTtdS = (

(St))2 = 2(St) (16.9)

28


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Chapter 17

Portfolio Optimization, Transaction Costs, Dynamic Games 6Sircar

17.1 Announcements

Well hold extra discussion sessions as follows:

1. Tuesday 13:30: Schied2. Wednesday 13:30: Swindle3. Thursday 13:30: Cont4. Thursday 15:00: Sircar

17.2 Dynamic Oligopoly Games

17.2.1 Motivation

1. Try to understand the evolution of energy markets2. Competition between different fuels a many dimensional

problem.Concerns: Exhaustibility of fossil fuels(Oil running out); Green En-ergy (renewables, eg Solar); Exploration and Improved technology;

One approach is to consider the Liquid Financial Marketswith a large number of small price takers. The completelyopposite extreme is having a small number of relatively largeplayers, all of whom are competing with eachother, and whichstrongly influence prices. Later in the week well return to thefinancialization of commodities markets.

17.3 Cournot Market (1838)

This is the first example of a Nash equilibrium. Consider a static,1 period, two player game. Key points: This model was set up inthe context of producers of mineral water, which is an essentially

inexhaustible resource. We can choose quantitiesq1 andq2 tobring to market. The market is governed by pricing (inversedemand) functionP: quantity price, withq1+ q2= Q. Firstcharacteristic: This is a decreasing function! (Flooding themarket sents price to zero.) Well also assume that the goods areperfect substitutes, but that the players have different marginal(ie, per bottle) costs of production 0 a1, a2 1.

Later, well consider theseai to be dynamic (ie, shadow costsor scarcity costs). For energy, we note that oil is cheap but windis more expensive.

To set up the competition, we have

1. Player 1 objective:

maxq10

q1(1 q1 q2 a1) (17.1)

2. Player 2 objective:

maxq20

q2(1 q1 q2 a2) (17.2)

We can recast these profits as i(q1, q2). We can solve in thesense of a Nash equilibrium (NE) which is the intersection ofthe best response:

R1(q2) =argmaxq1

1(q1, q2) (17.3)

R2(q2) =argmaxq2

2(q1, q2) (17.4)

A plot of this in theq2 vs q1 shows two triangles, where thehypotenuses intersect is the nash equilibrium.

Definition 1 A point(q1, q2) []2 is a nash equilibrium if

1(q1, q

2) 1(q1, q2)q1 (17.5)

2(q1, q

2) 2(q1, q2)q2 (17.6)

We can verify this by noting that

R1(q2) =1

2(1 q2 a1) (17.7)

R2(q1) =12

(1 q1 a2) (17.8)

which implies that

q1 =1

3(1 2a1+ a2) (17.9)

q2 =1

3(1 2a1+ a1) (17.10)

Observe that this is decreasing in your costs and increasingin your opponents cost.

We can calculate an aggregate quantity Q = q1 +q22) =

13(2

a1

a2). We can calculate the market price

P(Q) = 1 Q = 13(1 + a1+ a2).We can compare this versus the monopoly case, where the

only player will solve

maxq0

q(1 q a) = q =12

(1 a) (17.11)

which implies the monopoly pricePM= 12(1 + a).

We can also calculate the duopoly case (ie, whereN= 2) anda1 =a2=a. Then we get PD =

13(1+2a)< PM=

12(1+a). Key

point: Competition decreases price, which benefits the consumer.

17.4 Bertrand (1883) Competition

Now, consider where firms set prices, not quantities.

17.5 Exhaustible Resource Problem

Hotelling (1931) did this in the monopoly case, while Dasgupta+ Heal (1979).

First: Start with two oil producers with finite resources(x(t), y(t)). In the Cournot setting, choose the quantities(q1(t), q2(t)) with

dx

dt = q1(t){x>0}+ (Noise..) (17.12)

dy

dt = q2(t){y>0}+ (Noise..) (17.13)

29


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30 CHAPTER 17. PORTFOLIO OPTIMIZATION, TRANSACTION COSTS, DYNAMIC GAMES 6 SIRCAR

Now, player 1 will want to solve something like

v(x, y) = maxq1

0

ertq1(t)(1 q1(t) q2(t))dt (17.14)

(this assumes zero cost of production), while player two considers

w(x, y) = maxq2

0

ertq2(t)(1 q1(t) q2(t))dt (17.15)

wherex(0) =x andy(0) =y.

17.6 Dynamic Programming

We can take the two points with two value functions, andnon-zero sum differential games.

17.6.1 Recipe

Introduce

L = q1 x

q2 y

(17.16)

Thenrv= max

q1Lv+ q1(1 q1 q2) (17.17)

and similarly,rw= max

q2Lw + q2(1 q1 q2) (17.18)

Taking this a step ahead, we have

rv= maxq1

q1(1 q1 q2 v

x)

q2 v

y (17.19)

and

rw= maxq2

q2(1 q1 q2 w

y)

q1w

x (17.20)

If we return to the static two player, and recall the optimalquantity

q1 =1

3(1 2a1+ a2) (17.21)

and profit 1(a1, a2) = q1(1 q1 q2 a1) = (a1)2. Then thePDE that we want to deal with is y vs x, on the domainx 0,y 0, and we want to satisfy the equations

1(vx, wy) q2(vx, wy)v

y=rv (17.22)

2(vx, wy) q1(vx, wy)w

x =rw (17.23)

In the case of inexhaustible oil, then v =w=constant. In fact,

v= w=1

r1,2(0, 0) (17.24)

We can interpretvx andwy are shadow costs or scarsity.Next time: Well try to model what happens onx = 0 and

y= 0. (ie, what happens on the exhaustible case?) Either theother player gets a monopoly or we must bring in renewables.


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Part IV

HighFrequency Trading and Limit Order Book

31


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Chapter 18

High-Frequency Trading & Limit Order Book 1 - Carmona: Limit Book Orders

Black-Scholes theory. The price is given by a single number.There is infinite liquidity. One can buy or sell any quantityat this price with no impact on the asset price. This fixes toaccount account for liquidity frictions. This doesnt account fortransactions costs, and we add some liquidity frictions, or put thetransaction cost as a proxy for liquidity. This theory will not besatisfactory for large trades over short periods or high frequencytrading. We need to understand the market microstructure.

Several types of markets:

1. Quote driven Markets: Market makers or dealer centralizesbuy/sell orders and provides liquidity by setting bid and askquotes.

2. Order driven markets: Electronic platforms aggregate all

available orders in a Limit Book Order. (Eg: NYSE,NASDAQ, LSE, etc)

Here, the same stock is traded on several venues. Price discoveryis difficult due to many instruments being traded off the publicmarket book. The competition between markets leads to lowerfees and smaller tick sizes.

Recently, a huge change has been the creation of dark pools(etcLit market). Another issue is the increase in updating frequencyof order books.

18.1 High Frequency Trading

We suspect 60-75% high frequency trading, 10% of which ispredatory. (Amaranth, etc)

Pros: Smaller tick size; HF traders provide extra liquidity; dark

pools reduce trade execution costs from price impact; marketsmore efficient.

Cons: Expensive technological arms race; Dark tradingincentivizes price manipulation, fishing, predatory trading; Littleor no oversight possible by humans (eg, flash crash) and increasedsystemic risk; HF trading algorithms do not use economicfundamentals(e.g. value and profitability of a firm.)

18.2 HFT Mishaps

Flash crash: Dow Jones IA plunged about 1000 points (recoveredin mintes) biggest one-day point decline.

Other mishaps: AP Twitter Feed/etc.

18.3 Limit Order Book

List of all waiting buy and sell orders.1. The prices are multiples of the tick size.2. For a given price, orders are arranged FIFO queue3. At each timet,

(a) Thebid priceBt is the price of the highest waiting buyorder.

(b) The askpriceAt is the price of the lowest waiting sellorder.

4. The state of the order book is modified by order book events:Limit orders, market orders, cancelations

5. Consolidated Order book: If the stock is traded in severalvenues, one aggregates over all visible trading venues. (Then

the question becomes, which exchange should we submit theorder to?)

6. Here, little or no discussion of pools.

18.4 Role of order book

The LOB is crucial in high frequency finance: it explains thetransaction costs. The liquidity providers post trading intentions:bids and offers. Liquidity takers execute certain orders: Adverseselection. We can construct a plot by showing the price in thelimit order on horizontal axis and the volume of desired shareson the vertical axis. Because of the FIFO nature we probablycan not get in at the highest bid or lowest ask.

18.4.1 DELL

Can we see a selling panic on the order book? Yes. Data is giant.Critical point is the bid/ask spread

18.5 Limit order

A limit order sits in the order book until it is either 1) executedagainst a matching market order or 2) it is canceled. A Limit order

1. May be executed very quickly if it corresponds to a price nearthe bid and the ask

2. It may take a long time if

(a) The market price moves away from the requested price.(b) The requested price is too far from the bid/ask.

3. Can be canceled at any time.

Typically, a limit order waits for a match. The transaction costisknown, the execution time is uncertain.

18.6 Market Order

The market order is an order to buy/sell a certain quantity ofthe asset at the best available price in the book. Agents canput a market order that, for a buy (sell) order,

1. The first share will be traded at the ask (bid) price.2. The remaining one(s) will be traded some ticks above(below).

in order to fill the order size. The ask (bid) price is then modifiedaccordingly.

18.7 Cancellation

We can also cancel orders.

18.8 LOB Dynamics summary

Agents can put a limit order aand wait for a match.Agents can put a market order that consumes the cheapest

limit orders in the book.Agents can put a cancellation.

18.9 Market impact of large fills

Current mid-price: average. Fill sizeN= 76015, eg buy. We filln1 shares available at best bid p1; etc. We fillnk shares at price

pk> pk1, such thatN=

ni. The transaction cost is

nipi.Then the effective price is 1N

nipi. The new mid-price is now

the new average of the order book(with the filled ones removed,32


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18.9. MARKET IMPACT OF LARGE FILLS 33

naturally.) Note that there is a widening of the bid-ask spread,as well as a change in the height of the bars (because there isless volume available now.)


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Chapter 19

High-Frequency Trading & Limit Order Book 2 - Carmona

19.1 Hidden Liquidity

Some exchanges allow agents to submit hidden orders. Theyare made visible to the broader market after being executed.This is a controversial issue: it is a barrier to the implementationof a full ytransparent market, and is an impediment to pricediscovery and information dissemination.

The results of the first empirical analyses:

1. Encourage fishing2. After it is revealed that a hidden order was executed: a rash

increase of order placement inside the bid-ask after.3. HF traders are divided into two groups: Traders trying to

take advantage of the remaining hidden liquidity, and traderstrying to steal execution priority from the fully hidden order.

Distinction between iceberg or fully hidden order.

19.1.1 Partially Hidden Orders: Iceberg Orders

Dark liquidity posted inside the LOB. Two components: Shownquantity and the hidden remainder. Order queued with the litpart of the LIB, only the shown quantity is visible. When theorder reaches the front of the queue, only the display quantityis filled. Then the grade (price and quantity filled) is revealed.The hidden part is put at the back of the queue. Sometimesan extra execution fee is charged by the exchange.

19.1.2 Fully Hidden Order

19.2 Dark Pools

Dark pools are an electronic engine that matches buy and sellorders without routing them to lit exchanges. The reason is tomove large amounts without impacting the price (no need foriceberg orders). These are run by private brokerages:

1. Ex: Liquidnet, Pipeline, ITG Posit, Goldman SIGMA X2. Participatnts submit lists of orders to matching engine3. Matched orders are executed at the midpoint of the bid-ask

spread.4. PROS: trade at midpoint can be better than lit market5. CONS: may have to wait a long time.

The SEC regulates this in the US as Alternative Trading Systems.They have little to no public disclosure, and little transparency.

Supposedly, 32% of trades in 2012 were on dark pools.

19.3 Order book Modeling Objectives

Offer a framework to investigate order impact on execution prices.

1. Optimal mult-period liquidation strategies against a limitorder book

2. Detailed but tractable stochastic model of spread andtransaction costs

3. Benchmark tracking slippage4. Opportunity costs of delayed trading.

19.4 Order Book Models

Roughly speaking, LOB is a set of two histograms (Bids & Asks).The reduced form model sets it up as a Markov process (Ot)ton a large state space of order books O. The simplest model:

1. Smith-Farmer-Guillemot-Kirshnamurthy(SFGK) Model2. Market orders (buys and sells) arrive according to a Poisson

process with rate/2.3. Cancellation of existing limit orders: outstanding limit orders

die at a rate.

A little bit better one: Cont-Stoikov-Talreja:

1.P= {1, 2,..,n} is a price model2. LOB at timet is O(t).3. Admissible state space:

O = {O Zn;1 k n, Op< 0forp k, Op= 0forp(19.1)

4. Ask price at timet

PA(t) := (n + 1) inf{p; 1 p n, Op(t)>0} (19.2)5. Bid price at timet

PB(t) := 0 sup{p; 1 p n, Op(t) 0 (19.5)if it is true at time t= 0.

In summary:

1. This is a descriptive analysis2. Uses ideas from queuing theory: first passage times of

Birth-and -Death processes3. Laplace transform techniques4. We can compute/estimate probabilities of condition events5. But.... its not sufficient for optimal order book strategies.

Optimization problems: The goal of a LOB model is to

1. Understand the costs of transactions2. Develop efficient (or optimal) trading strategies

Typical challenge: Sellx0 units of an asset and maximize thesales revenues, using a limited number of market orders only!

sup1...n


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19.4. ORDER BOOK MODELS 35

where Uis a utility function and E is the expectation over amodel for the dynamics of the LOB Ot.

This is a prohibitively large-dimensional model.


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Chapter 20

High-Frequency Trading & Limit Order Book 3 - CarmonaPrice Impact Models & Optimal Execution

20.1 queries

We already saw that we should split and spread large orders, so:

1. How can we capture market price impact in a model?2. What are the desirable properties of a price impact model?3. How can we compute optimal execution trading strategies?4. What happens when several execution strategies interact?

20.2 Amlgren-Chriss Price Impact Model

Here we assume that the unaffected (fair) price is given by a semi-martingale. The mid-price is affected by trading, via two parts:

1. Permanent price impact given by a function g of the trading

speed:dPmidt =g(v(t))dt + dWt (20.1)

2. A temporary price impact given by a functionh of the tradingspeed:

Ptranst =Pmidt + h(v(t)) (20.2)

The problem is: we want find a deterministic continuoustransaction path to maximize the mean-variance reward

1. Closed form solution when permanent and instaneous priceimpact functionsg andh are linear.

2. Efficient frontier: the speed of trading and hence risk/returnis controlled by a risk aversion parameter.

This is widely used within the industry.

20.2.1 Criticisms

1. Mid pricePmidt is arithmetic brownian motion with drift...so we can see negative prices, reasonable only for short times,maybe that price never actually happens.

2. Are there issues with rate of trading in continuous time?3. Price impact is more complex than instantaneous and

permanent.4.5. Empirical evidence that it is stochastic.

20.2.2 Optimal Execution

An execution algorithm has three layers:

1. Highest: How to slice the order, when to trade, what size,how long?2. Mid: Given a slice, market or limit order? What price level?3. Low: Given an order, which venue? (we will ignore this!)

Set-up: bf goal: sellx0> 0 shares by time T >0.

1. X = (Xt)0tT execution strategy2. Xt: position (the number of shares held) at timet3. AssumeXt is absolute continuous (to get differentiability)

4. Take Pt the mid-price(unaffected price),Pt transaction price,andIt is the price impact; that is:

Pt= Pt+ It (20.3)

(for example, with the Linear Impact Amlgren-Chriss model:

It= [Xt X0] + Xt). (20.4)Our objective is to maximize some form of revnue at time

T, with revenue R(X) from the execution strategy X

R(X) = T0

(Xt)Pt)dt (20.5)

20.3 Challenges

The first generation considered price impact models: Risk

neutral framework, more complex portfolios (eg, with options), orrobustness and performance constraints(e.g. slippage or trackingmarket VWAP).

The second generation uses simplified LOB models, forexample a simple liquidation problem or performance constraintsand using both market and limit orders.

20.4 Optimal Execution

First we can expand on the definition

R(X) = T0

(Xt)Pt)dt (20.6)

= T

0

XtPtdt T

0

XtItdt (20.7)

=x0P0+

T0

XtdPt C(x) (20.8)

whereC(x) = (20.9)

Can try to maximize expected revenue, but get a boring answer.A better way is instead to maximize:

E [R(X)] var[R(X)] (20.10)Here, is a risk aversion parameterlate trades carry somevolatility risk.

For a DETERMINISTIC trading strategy X, we can find theexpectation.

Instead, we might include risk aversion via a utility function:

max E [U(R(XT))] (20.11)All of these models have some shortcomings:

1. they are deterministic2. do not react to price changes3. are time-inconsistent4. counter-intuitive

Furthermore, the computations require

1. solving nonlinear PDEs2. singular terminal conditions.

36


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20.5. RECENT DEVELOPMENTS 37

20.5 Recent Developments

1. Gatheral/Schied(2011)2. Schied(2012)3. Almgren-Li (2012): Hedging a large option position. Explicit

solution in some gases)

Modeling the LOB by a shape function:

1. Obizhaeva-Wang (2006)2. Alfonsi-Fruth-Schied(2010)

3. Alfonsi-Schied-Schulz(2011)4. Predoiu-Shaikhet-Shreve(2011)


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Chapter 21

High-Frequency Trading & Limit Order Book 4 - CarmonaPredatory Trading

Large traders facing forced liquidation. Especially if the needto liquidate is known to other traders. These can be hedge fundswith a nearing margin call, or traders who use portfolio insurance,stop loss orders, etc. Some institutions/funds can not hold onto downgrade instruments. Finally, index replication funds atre-balancing dates(for example, the Russell 3000.)

Forced liquidation can be very costly because of price impact.Collapses: LTCM vs AmaranthReference: Cramer 2002.Goals:

1. Understand predation2. Illustrate benefit of stealth trading

3. Illustrate benefit of sunshine tradingThe two extremes are:

1. Elastic: temporary impact dominates2. Plastic: Permanent impact dominates

Optimization problem needs a model of the dynamics of theorder books. We will model it as poisson.

38


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Chapter 22

High Frequency Trading Lecture 5Heterogeneous Beliefs and HF Market Making (Carmona)

Wed like to build a model of agents and have the limit orderbook model happen automatically.

Agents:

1. Market Maker: agent that places competitive orders on bothsides of the order book in exchange for privileges. LiquidityProvider Strategy: adapt pricing and volumes by readingclient flows.

2. Clients: Liquidity Takers, agents who trade with themarket maker. Clients place market orders. Each client hashis/her own information and acts accordingly.

Well assume an ordering in time.

22.1 Theoretical Literature1. Early approaches2. Inventory models3. Informed traders4. Zero-Intelligence models5. Price impact models.

22.2 Objective

Wed like to propose a stochastic agent-based model in whichexistence and tractable and realistic properties of the limit orderbook appear as a result of the analysis. The client model shouldcapture the dependence between trades and price dynamics. Themarket maker assumes clients are rational and optimizes theirorder book choice.

See Carmona, Webster (2012). Mathematics:1. (, F,F = (Ft)t0,P) withW a P-DM that generates F.2. Fk Fgenerated by a P-BMWk.3. Fk such that Pk|Fkt P|Fkt .4. Pt is an Ito process adapted to all (Fk)k=0,..,n.Note that

1. Each agent has his/her own filtration and probability measure.2. The filtrations (information structures) are potentially different3. THe price process is adapted to all of them (i.e., each client

sees the price.)

Trades:

1. MidpricePt is announced by the market at time t.

2. Market maker proposes an order book aroundPt3. The market maker cannot differentiate clients pre-trade4. Client triggers a trade of volumet5. CLient obtains volume tand pays cash flow Ptt+ct(t) (with

ct() representing the transaction cost function at time t.)6. The market maker learns the identity of the client post-trade

(assumption depends upon market, true for FX)

The market maker controls transactions cost function ct(),and the clienti controls trading volumes/speedsit.

Hypotheses:

1. Marginal costs are defined (ie, ct() is differentiable in)2. Clients may choose not to trade

3. The midprice is well defined4. Marginal costs increase with volume (ie, ct is convex)5. ct has compact domain.

We can do the Legrendre Transform:

t() := supsupp(ct)

( ct()) (22.1)

According to Duality:

ctconvex with compact domain tis a positive finite meas(22.2)

The distribution t represents the order book formed by the

orders of the market maker. Ift has a density f(x), it is theshape function we used earlier.

More about the client model: We are NOT trying to implementan optimal trading strategy. We assume the client is only tryingto predict!

22.3 Client Optimization Problem

Exogeneous state variables

1. Pt is a nonnegative Ito process2. ct is a random adapted convex function in a fixed domain.

Endogeneous state variables:

dLit=

itdt

dXit=Li

tdPt

ct(

i

t)dt

(22.3)

whereit is the rate at which the clients trade (control variable),Lit is the volume or total position of the client, and X

it is the

wealth, marked to mid price.Then the objective function is

Ji = EPiUi(Xii , Pi)

(22.4)

whereUi is the utility function and i is the stopping time. Thatis, each client maximizes the utility according to his OWN beliefsabout the probability of things happening.

Theorem 1 Under suitable integrability assumptions onUi andi, the optimal strategy is

it:=ct(it) = EQiPi Pt|Fit (22.5)

withdQi

dPi =

xUi(Xii, Pi)

EPixUi(Xii , Pi)

(22.6)Note that these are NOT the from the CAPM

modelUsing it = c

t(

it) or

it = [c

t]1(it) =

t(

it), we can

transform to dual variables:dLt= 1n

i

t(

it)dt

dXt= LtdPt+ 1n

i

it

t(

it) t(it)

dt

(22.7)

39


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40CHAPTER 22. HIGH FREQUENCY TRADING LECTURE 5HETEROGENEOUS BELIEFS AND HF MARKET MAKING

Assume the market maker is risk-neutral.This is super complicated, so well do the natural thing: let

ntend to infinity,So critically, well need to model the it; well have two choices:

1. Microscopic model:

dit= itdt + dBit+ dBt (22.8)2. Macroscopic model: Stochastic Partial Differential Equation

dt() =

1

2(2 + 2) + t() + (t())

dtt()dBt

(22.9)

What can all of this tell us about Pt? We do not want tomake an explicit model for the price process. Instead, we wouldlike to infer the price from the client trades.

We can do this via an entropic feedback.Finally we have a stochastic control problem:

J =

0

etE [] dt (22.10)

under the constraint 0

et logt

t, t

dt

C

We can use the PontryaginCan define

m() = ( )

(22.11)

if >0, and then we have

H() (22.12)


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Chapter 23

HFT Day 6

23.1 Market impact model

These describe the quantitative feedback of a trade executionstrategy on asset prices.

A revenue is given by

RT(X) = T0

SXt dXt (23.1)

and the liquidation costs are

CT(X) = X0S00 RT(X). (23.2)(We may need to add correction terms to these formulas when

Xis not continuous.)We can define regularity in the following ways

1. The model must admit optimal trade execution strategiesfor reasonable risk criteria.

2. Optimal strategies ought to be well-behaved.3. Regularity conditions should be independent of investor

preferences4. One should distinguish the effects of price impact from

profitable investment strategies that can arise via trendfollowing. Therefore, we will assume from now on that:

S0 is a martingale (23.3)

Definition 2 (Price Manipulation) A round trip is a trade

execution strategyX withX0= XT= 0. A price manipulationstrategy is a round tripXwith strictly positive expected revenues,

E [RT(X)]>0 (23.4)

41


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Part V

Guest Lectures

42


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Chapter 24

Risk Measures - Rudloff

Overview:

1. Risk measures: Primal representation, acceptance sets, dualrepresentation, examples

2. Generalizations: Multivariate risks

Basics: We define a probability space (, F, P) (the samplespace,-algebra, and probability measure, respectively). Randomvariables map the probability space to real numbers.

Risk measures are mathematical models to quantify uncertainy.It is a functional on the spaceLp(, F, P) withp [0, ] (orsubspaces of random variables):

:Lp R {+} (24.1)

The values will be real numbers, in USD or Euros, etc.Interpretation: The higher(X), the higher the risk.There are many different risk measures. There is not one

absolutely objective risk measure. Key questions:

1. Which properties a function should have to be a reasonablerisk measure?

2. Which random variables are acceptable for a rm?3.

Some features:

1. R0:Normalization: (0) = 0.2. R1:Monotonicity X1, X2 Lp: X1 X2 =

(X1) (X2).3. R2:Translation properties (Cash invariance)X

Lp,c

R,

= ,(X+ c) = (X) c.Extra features:

1. R3: Convexity in2. R0-R3: Convex Risk Measure3. R4: Positive Homogeneity (scaling property)4. R0-R4: Coherent Risk Measure

Value-At-Risk is a typical Risk Measure, but it is not convex!

24.1 Acceptance Sets

We call A :={X Lp : (X)0 the acceptance set of therisk measure.

Lemma 1 Consider a function: Lp R {+}. It holds:1. monotone = A+ Lp+ subset ofAp2. convex impliesA convex3. positively homogeneous impliesA is a cone.4. closed impliesA is closed.

Reminder: A functionF is called closed (or lower semicon-tinuous) if

epif:= {(v, r) Lp R :f(v) r} Lp R (24.2)is closed inLp R.

Can we construct a risk measure from a given set of acceptablepositions?

Lemma 2 Consider a setA Lp. DefineA(X) := inf{t R :X+ t A} (24.3)

It holds

1. inf{t R :t A} = 0 = A(0) = 02. A + Lp+ A = A monotone.3. A satisfies the translation property.4. A convex = A convex5. A a cone = A is positively homogeneous6. A closed = A closed.Lemma 3 There is a one-to-one relationship between lower

semicontinuous risk measures and closed acceptance sets via1. A= {X Lp :(X) 0} and2. (X) = inf{t R :X+ t A}.Theorem 2 A function : Lp R {+inf} is l.s.c. convexrisk measure a represetnation of the form

(X) =QQ

{EQ[X] (Q)} (24.4)

where Q := {prob measuresQ : dQP Lp} and (Q) =XA

EQ[X] = (dQdP) is called the penalty function.

Aside: Iff: X

R, then the conjugate function

f(x) = supxX

{x(x) f(x)} (24.5)

Iffis l.s.c. convex, proper, then

f(x) = supxX

{x(x) f(x)] =f(x) (24.6)

(ie, it is the conjugate of the conjugate.)We just need to prove that Q andQ are the same; these fall

out from our original definitions. Start with

(0) = 0 (24.7)

which implies infyLp

(y) = 0 (24.8)

which implies that(y) 0 (24.9)

Next, since is monotone, we have

dom L (24.10)which implies that

(x) = supyL+

{E[Xy] (y)} (24.11)43


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44 CHAPTER 24. RISK MEASURES - RUDLOFF

Finally, from translative, we have that

dom = {y Lp : E[y] = 1 (24.12)and

supxA

{E[xy] } (24.13)

For a l.s.c. coherent risk measure we have that

(X) = supQQ

EQ [(something)] (24.14)

A few counterexamples:

1. Variance2(X) is not monotone or translative!2. Value at riskV aR(X) at level (0, 1) is not convex!

V aR(X) := inf{x R :P{X+ x 0} } = x(24.15)

Example: Consider two defaultable corporate bonds with facevalue $500,000. PayoffX1, X2 (r= 10%, default prob 0.8%, inde-pendent). CalculateV aR(X1),V aR(X2),V aR(

12

(X1+ X2)for = 1%.

X1=

50, 000 99.2%500, 000 0.8% (24.16)

y=1

2(X1+ X2) =

50, 000 98.4%

225, 000 1.58500, 000 ??0.006??

(24.17)

This example also shows VaR is not subadditive!A better one: Conditional Value at Risk:

CV aR(X) := 1

E[X1{Xxa}] + x( P[X x])

(24.18)

or

= 1

0

V aR(X)d (24.19)

Also called Average Value at Risk or Expected shortfall at level (0, 1).

Other ones: Expected loss, worst-case risk measure.(X) :=E[X],max

Consider the exponential utility function u(x) = 1 ex andthe set of acceptable positions whenever the expected utility isnonnegative.

24.2 Outlook

24.2.1 Multivariate risks

One extension is to now have many random variables instead

of just one that is, X Lp

d.Why would this be important?One application: Important fordealing with many banks which are interlinked and want capitalrequirements for each bank taking interconnected into account.

Another: Markets with transaction costsyou cant sum overindividual stocks because of transaction costs.

For such porfolios, we can look at

R(X) = {u Rd :X+ u A} (24.20)for some acceptance setA Lpd.

A similar approach (of defining these from their properties)can be followed here, with similar result.


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Chapter 25

Cumulant Moments & Queuing models

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Bibliography

princeton_financial_math.pdf

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