2. Mathematics

Quantitative Finance: a brief overviewPart 2/4 - Mathematics

Dr. Matteo L. BEDINI

Central University of Finance and Economics, Beijing, PRC

25 March 2015

Disclaimer

The opinions expressed in these lectures are solely of the author and do notrepresent in any way those of the present/past employers.

Dr. Matteo L. BEDINI Quantitative Finance II: Mathematics CUFE - 25 March 2015 2 / 35

Objective

The objective of this lecture is to give a short review of the basicmathematical tools used in mathematical models for nance, with emphasison the practical/nancial meaning of some standard concepts.

Warning:

The material presented in this lecture cannot and does not cover alltheoretical tools that are needed in Mathematical Finance (it will probablycover less than 5%), but only a couple of mathematical concepts that mayhave relevant applications in pricing techniques or theoretical models ofnancial markets.


Agenda - II

1 Basic ToolsUp to conditional expectationPricing American options

2 Some Advanced ConceptsOptional and Predictable σ-algebras.High Frequency Trading, Default times and Insider Trading

3 Bibliography and Mathematical Brainteasers


Basic Tools Up to conditional expectation

Agenda - II






Probability space (Ω,F ,P)

Ω is a set, the sample space (space of possible outcomes)

F is a set of set, the σ-algebra on Ω, set of events (space of eventsthat shall be considered)

P is a function: P : F → [0, 1], called probability measure (how eventsare evaluated)



Random element X : (Ω,F)→ (E , E)

E is the state space (where we observe X (ω), the realization of X )

E is a σ-algebra on E (contains what we are actually interested inmeasuring)

X is a measurable function (X−1 (A) ∈ F , ∀A ∈ E )

Examples (E , E) =...

(R+,B (R+)), X is a random time (default time)

(R,B (R)), X is a random variable (asset value)

(C , C), X is a stochastic process (asset path)



The law PX := P X−1

A new probability space (E , E ,PX )

(E , E) as before

PX is a function: PX : E → [0, 1], called law of X

Indeed, the actual choice of Ω plays no role, and the interest focusesinstead on the various induced distributions. [K]

True indeed, even if in fact, as we shall see, some important theoreticaltools refers to the abstract probability space (Ω,F ,P).



Conditional Expectation E [X |G] 1

Modern probability theory can be said to begin with the notions ofconditioning and disintegration. [K]

Conditional expectation of an integrable random element X w.r.t. theσ-algebra G is a G-measurable random element ξ such thatˆ

G

ξdP =

ˆ

G

XdP, ∀G ∈ G.

For convenience, instead of using ξ, we use the symbol E [X |G].

Special case

If G = Ø,Ω, then we recover the expected value of X :

E [X | Ø,Ω] = E [X ] .

1see, e.g., [K, S]Dr. Matteo L. BEDINI Quantitative Finance II: Mathematics CUFE - 25 March 2015 9 / 35


Conditional Expectation: questions

1 On the probability space (Ω,F ,P), consider the σ-algebra G ⊆ F andlet X be a P-integrable and G-measurable random element. ComputeE [X |G].

2 Probability space ([0, 1] ,B ([0, 1]) , λ), G := σ[0, 13) (

13 ,

23

),[23 , 1],

X (ω) := ω. Draw the graph of the following functions:

1 ω 7→ E [X ]2 ω 7→ E [X |G].3 Let now H be the following σ-algebra: H = B

([2

3, 1]). Draw the map

ω 7→ E [X |G ∨ H] (G ∨ H := σ (G ∪ H)).

3 You are pricing an asset whose nal value is modeled with the r.v. X .You have the extraordinary possibility of choosing between:

1 G , i.e. E [X |G].2 G ∨ H, i.e. E [X |G ∨ H].

What do you choose? Why? Can you explain the nancial meaning ofyour choice? What if instead of X you had a derivative f (X )?


Basic Tools Pricing American options

Agenda - II






Conditional Expectation, Least Squares and Projection

Problem

You are given a Monte-Carlo engine and you have well-understoodconditional expectation. Design an algorithm for pricing American Options.

Warning:

Despite recent advances [...] the valuation and optimal exercise ofAmerican options remains one of the most challenging problems inderivatives nance [...]. This is primarily because nite dierence andbinomial techniques become impractical [...]. [LS].



Simple Numerical Example 1/5 2

American put on no-dividend stock S = (St , t = 0, 1, 2, 3): StrikeK = 1.10, Risk-free rate r = 6%, Start date t = 0, Exercise datest = 1, 2, 3, # of paths in Monte-Carlo engine N = 8.

Path t = 0 t = 1 t = 2 t = 3

1 1.00 1.09 1.08 1.34

2 1.00 1.16 1.26 1.54

3 1.00 1.22 1.07 1.03

4 1.00 0.93 0.97 0.92

5 1.00 1.11 1.56 1.52

6 1.00 0.76 0.77 0.90

7 1.00 0.92 0.84 1.01

8 1.00 0.88 1.22 1.34

Table: Stock-price path under risk-neutral measure

2Next slides follow exactly [LS], Section 1.Dr. Matteo L. BEDINI Quantitative Finance II: Mathematics CUFE - 25 March 2015 13 / 35


Simple Numerical Example 2/5

Path t = 0 t = 1 t = 2 t = 3

1 - - - 0.00

2 - - - 0.00

3 - - - 0.07

4 - - - 0.18

5 - - - 0.00

6 - - - 0.20

7 - - - 0.09

8 - - - 0.00

Table: Cash-ows at maturity conditional on not early-exercise

If the put is in the money at time t = 2 (path # 1, 3, 4, 6, 7), the holdermust decide whether to exercise or to continue.



Simple Numerical Example 3/5Y discounted cash ow received at t = 3 if the put is not exercised att = 2.

Path Y St=2

1 0.00×0.94176 1.08

2 - -

3 0.07×0.94176 1.07

4 0.18×0.94176 0.97

5 - -

6 0.20×0.94176 0.77

7 0.09×0.94176 0.84

8 - -

Table: Use of in-the-money path for the sake of eciency of the algorithm.

Choice: regression of Y against constant, St=2, (St=2)2. Result:

E [Y |St=2] = −1.070 + 2.983× St=2 − 1.813× (St=2)2 .Dr. Matteo L. BEDINI Quantitative Finance II: Mathematics CUFE - 25 March 2015 15 / 35



Path Exercise Continuation

1 0.02 0.0369

2 - -

3 0.03 0.0461

4 0.13 0.1176

5 - -

6 0.33 0.1520

7 0.26 0.1565

8 - -

Table: Optimal Exercise Decision at t = 2.




Path t = 0 t = 1 t = 2 t = 3

1 - - 0.00 0.00

2 - - 0.00 0.00

3 - - 0.00 0.07

4 - - 0.13 0.00

5 - - 0.00 0.00

6 - - 0.33 0.00

7 - - 0.26 0.00

8 - - 0.00 0.00

Table: Cash-ows at t = 2 conditional on not early-exercise

Then, proceed recursively.



Price of the American put

The key insight [...] is that this conditional expectation can be estimated[...] by using least squares. Specically, we regress the ex-post realizedpayo from continuation on functions of the values of the state variables.The tted value from this regression provides a direct estimate of theconditional expectation function. [LS]

Once, for each path, the stopping decision is known, the associatedcash-ows (with the proper discount factor) are known as well and theiraverage will give 0.1144 as the fair-price of the American put.

Exercise:

Compute the price of the European put (EP) and compare it with the theprice of the American put (AP) above. Will the EP be cheaper than AP?Why?


Some Advanced Concepts Optional and Predictable σ-algebras.

Agenda - II






Filtration F = (Ft)t≥0

On a measurable space (Ω,F) a ltration F = (Ft)t≥0 is a collection ofσ-algebras such that

if s ≤ t, then Fs ⊆ Ft ,

for all t ≥ 0, Ft ⊆ F .

F = (Ft)t≥0 models the ow of information on a market.

Left-continuous: if σ(⋃

s<t Fs

)=: Ft− = Ft .

Right-continuous: if⋂

n∈NFt+ 1n

=: Ft+ = Ft .

(Ω,F ,P) probability space, NP collection of (F ,P)-null sets.

F0 =(F0t

)t≥0 raw ltration (no assumptions).

FP =(FPt)t≥0 completed ltration: FPt := F0

t ∨NP.

F = (Ft)t≥0 usual augmentation: Ft := F0t+ ∨NP.



Filtration: some questions.The collection (Ω,F ,P,F) is called ltered probability space: the basicset-up of a nancial market.

Questions

1 Why for right-continuous Ft+ :=⋂

n∈NFt+ 1n, while for

left-continuous Ft− := σ(⋃

s<t Fs

)?

2 Would you prefer a right or left-continuous ltration for yourmarket-information? Why?

3 What is the dierence between FP and Ft where, if Nt is thecollection of (Ft ,P)-null sets, Ft =

(F tt := F0

t ∨Nt

)t≥0? If

Ω = [0, 1], F = B ([0, 1]), P = λ and F0 = Ø,Ω, what is thedierence between NP and N0?

In mathematical models for nance one always assume that F isright-continuous and completed (why?).



σ-algebras 3

Let F0 =(F0t

)t≥0 be a ltration:

A process X = (Xt , t ≥ 0) is progressive if (ω, t) 7→ Xt (ω) from(Ω× [0, t] ,F0

t × B ([0, t]))into (E , E) is measurable for all t. A set

A ⊆ Ω× R+ is progressive if IA (t, ω) is progressive. The set of allprogressive set is σ-algebra, called the progressive σ-algebra and it isdenoted byM

(F0).

The optional σ-algebra O(F0), is the σ-algebra dened on R+ × Ω

generated by all processes X = (Xt , t ≥ 0) with càdlàg paths that areadapted to F0.

The predictable σ-algebra P(F0), is the σ-algebra dened on R+ × Ω

generated by all processes X = (Xt , t ≥ 0) with left-continuous pathson (0,+∞) that are adapted to F0.

P(F0)⊆ O

(F0)⊆M

(F0).

3see, e.g., [N, DM]Dr. Matteo L. BEDINI Quantitative Finance II: Mathematics CUFE - 25 March 2015 22 / 35


Stopping timesLet τ be a random time and F0 =

(F0t

)t≥0 be a ltration.

τ is an F0-stopping time if

τ ≤ t ∈ F0t , ∀t ≥ 0.

τ is predictable if[τ,+∞) ∈ P

(F0).

τ is accessible if ∃ τnn∈N of predictable stopping times s.t.

P

(⋃n

ω : τn (ω) = τ (ω) < +∞

)= 1.

τ is a totally-inaccessible stopping time if

P (τ = T ) = 0, ∀T predictable.



Stopping times: questions 4

1 If τ is a stopping time is it predictable/accessible/totally inaccessible?

2 If τ is predictable/accessible/totally inaccessible is it a stopping time?

3 If τ is predictable is it accessible? Or totally inaccessible?

4 If τ is a predictable w.r.t. F0 and I change the measure from P to Q is itstill predictable?

5 If τ is predictable w.r.t. F0, is it predictable also w.r.t. FP? And w.r.t. F?And w.r.t. G bigger than F? And w.r.t. D smaller than F? If not, what isthe most likely change?

6 If τ is an accessible (resp. totally inaccessible) F-stopping time and I changethe measure from P to Q is it still accessible (resp. totally inaccessible)?

7 If τ is accessible (resp. totally inaccessible) w.r.t. F, is it accessible (resp.totally inaccessible) also w.r.t. G bigger than F? And w.r.t. D smaller thanF? If not, what is the most likely change?

(shall we give the denition of optional time and have fun with other questions?)

4see, e.g., [N, DM]Dr. Matteo L. BEDINI Quantitative Finance II: Mathematics CUFE - 25 March 2015 24 / 35

Some Advanced Concepts High Frequency Trading, Default times and Insider Trading

Agenda - II






Why bothering with all these tedious and nitty-grittyconcepts?

Answer

Because they can make the dierence between becoming rich or poor!



How so?

Predictable and Optional σ-algebras may be used, for example, to modeldierent types of traders, where those using optional-strategies gain moneyat expenses of those using predictable-strategies:

[...] we construct a model to show that high frequency tradersa [...] cancreate increased volatility and mispricings [...] that they exploit to theiradvantage.[...] high frequency traders [may create] trend in market prices that theyexploit to the disadvantage of ordinary traders. [...]Their speed advantage is captured by making the high frequency traders'strategies optional processes, instead of predictable processes. [JP12]

aBTW: which programming language would you use in HFT?



Another example from Credit Risk

Let τ be the (random) time at which the default of a company (or a state)occurs:

Structural models: predictable default time (default is announced).Used for modeling knowledge of the manager of the rm.

Reduced-form models: totally inaccessible default time (default occursby surprise). Used by the market (credit-spread is not 0 for shortmaturities).

See, for example, [G, JP04, JL07] among others.



Minimal ltration making a random time a stopping time

Let τ be the (random) time and

H =(Ht := σ

(Iτ≤s, 0 ≤ s ≤ t

)∨NP

).

Theorem

If the law Pτ is diuse τ is a totally inaccessible stopping time.If the law P is purely atomic and non-degenerate, τ is a non-predictableaccessible time.

See, for example, [DM], Chapter 4, Section 3, Theorem 107. The aboveltration is of crucial importance in credit risk and in the classicalprogressive enlargement of ltration.



Insider Trading

(Ω,F ,P) probability space, S = (St , t ≥ 0) stock-price process(semi-martingale), T ∈ (0,+∞) and

F = (Ft)t≥0 ltration generated by S (with usual conditions).

G = (Gt)t≥0 enlarged ltration dened by

Gt :=⋂u>t

Fu ∨ σ (ST ) .

Two agents on the market:

Normal agent: his ow of information is modeled by F.Insider trader: his ow of information is modeled by G.

The theoretical framework of enlargement of ltration (H-hypothesis,H′-hypothesis, etc. - see, e.g. [JYC, P, MY06]) is then very important tounderstand the role of information on nancial markets.



Example on Model Risk5

Protection seller in some credit derivative has a loss if default τ (∼ E (λ)) occursbefore maturity T (r = 0, LGD = 1):

ExpectedLoss = E[e−rτ Iτ<T

]LGD =

(1− e−λT

).

However, due to ill-bonus policy of the company, his actual time horizon is not Tbut αT (α ∈ [0, 1]). So, for him, in fact:

ExpectedLossshort-termist =(1− e−λαT

).

Other market player (ignoring short-termist's bad-behavior), will misunderstandthe above as a model disagreement on the default intensity :

ExpectedLossshort-termist =(1− e−λαT

).

The (private) information of the short-termist player, the model-misunderstandingof other market agents, may lead to nancial bubble!

5Following example is taken by [Mo].Dr. Matteo L. BEDINI Quantitative Finance II: Mathematics CUFE - 25 March 2015 31 / 35

Bibliography and Mathematical Brainteasers

Agenda - II






Mathematical Brainteaser 1/3

(You should answer in less than 5 seconds) How much is 19× 21?




Compute (quickly) ˆ

R

exp−ax2 + bx

dx

(a, b positive constant).




There are three doors. You know there is a prize behind one of them andnothing behind the other two. You will receive whatever is behind the doorof your choice. However, before you choose, the game show host promisesthat rather than immediately opening the door of your choice to reveal itscontents, he will open one of the two doors to reveal that it is empty. Hewill then give you the option of switching your choice. You choose Door 3,he opens Door 2 and reveals that it is empty. You now know that the prizelies behind either Door 3 or Door 1. Should you switch your choice to Door1?

(source [X1])



C. Dellacherie and P.-A. Meyer. Probabilities and Potential.North-Holland, 1978.

K. Giesecke. Default and information. Journal of Economic Dynamicsand Control, 30:2281-2303, 2006.

R. Jarrow and P. Protter. Structural versus Reduced Form Models: ANew Information Based Perspective. Journal of InvestmentManagement, 2004.

R. Jarrow and P. Protter. A Dysfunctional Role of High FrequencyTrading in Electronic Markets. International J. of Theoretical andApplied Finance, 15, No. 3, 2012.

M. Jeanblanc and Y. Le Cam. Reduced form modelling for credit risk.Preprint 2007.

M. Jeanblanc, M. Yor and M. Chesney. Mathematical Methods forFinancial Markets. Springer, First edition, 2009.



O. Kallenberg. Foundations of Modern Probability. Springer-Verlag,Second Edition, 2002.

F. A. Longsta, E. S. Schwartz. Valuing American Options bySimulation: A Simple Least-Squares Approach. The Review ofFinancial Studies, Spring 2001 Vol. 14. No. 1, pp. 113-147.

R. Mansuy and M. Yor. Random Times and Enlargements ofFiltrations in Brownian Setting. Volume 1873 Lecture Notes inMathematics, Springer, 2006.

M. Morini. Understanding and Managing Model Risk. Wiley Finance,2011.

A. Nikeghbali. An essay on the general theory of stochastic processes.Probability Surveys, 3:345-412, 2006.

P. Protter. Stochastic Integration and Dierential Equations. Springer,Berlin, Second edition, 2005.

A. Shiryaev. Probability. Springer-Verlag, Second Edition, 1991.


Science

2. Mathematics