2. Mathematics

Quantitative Finance: a brief overview
Part 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 not
represent 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 basic
mathematical tools used in mathematical models for nance, with emphasis
on the practical/nancial meaning of some standard concepts.
Warning:
The material presented in this lecture cannot and does not cover all
theoretical tools that are needed in Mathematical Finance (it will probably
cover less than 5%), but only a couple of mathematical concepts that may
have relevant applications in pricing techniques or theoretical models of
nancial markets.

Agenda - II
1 Basic Tools
Up to conditional expectation
Pricing American options
2 Some Advanced Concepts
Optional 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
1 Basic Tools

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 events
that shall be considered)
P is a function: P : F → [0, 1], called probability measure (how events
are 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 in
measuring)
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 focuses
instead on the various induced distributions. [K]
True indeed, even if in fact, as we shall see, some important theoretical
tools 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 of
conditioning and disintegration. [K]
Conditional expectation of an integrable random element X w.r.t. the
σ-algebra G is a G-measurable random element ξ such that
ˆ
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] .
1
see, e.g., [K, S]

Conditional Expectation: questions
1 On the probability space (Ω, F, P), consider the σ-algebra G ⊆ F and
let X be a P-integrable and G-measurable random element. Compute
E [X|G].
2 Probability space ([0, 1] , B ([0, 1]) , λ), G := σ 0, 1
3
1
3
, 2
3
, 2
3
, 1 ,
X (ω) := ω. Draw the graph of the following functions:
1 ω → E[X]
2 ω → E[X|G].
3 Let now H be the following σ-algebra: H = B 2
3 , 1 . Draw the map
ω → 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 of
your choice? What if instead of X you had a derivative f (X)?

Basic Tools Pricing American options
Agenda - II
1 Basic Tools

Conditional Expectation, Least Squares and Projection
Problem
You are given a Monte-Carlo engine and you have well-understood
conditional expectation. Design an algorithm for pricing American Options.
Warning:
Despite recent advances [...] the valuation and optimal exercise of
American options remains one of the most challenging problems in
derivatives nance [...]. This is primarily because nite dierence and
binomial techniques become impractical [...]. [LS].

Simple Numerical Example 1/5 2
American put on no-dividend stock S = (St, t = 0, 1, 2, 3): Strike
K = 1.10, Risk-free rate r = 6%, Start date t = 0, Exercise dates
t = 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
2
Next slides follow exactly [LS], Section 1.

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 holder
must decide whether to exercise or to continue.

Y discounted cash ow received at t = 3 if the put is not exercised at
t = 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
.

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 realized
payo from continuation on functions of the values of the state variables.
The tted value from this regression provides a direct estimate of the
conditional expectation function. [LS]
Once, for each path, the stopping decision is known, the associated
cash-ows (with the proper discount factor) are known as well and their
average 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 the
price of the American put (AP) above. Will the EP be cheaper than AP?
Why?

Some Advanced Concepts Optional and Predictable σ-algebras.
Agenda - II
1 Basic Tools

Filtration F = (Ft)t≥0
On a measurable space (Ω, F) a ltration F = (Ft)t≥0
is a collection of
σ-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 st Fs =: Ft− = Ft.
Right-continuous: if n∈N Ft+1
n
=: Ft+ = Ft.
(Ω, F, P) probability space, NP collection of (F, P)-null sets.
F0
= F0
t t≥0
raw ltration (no assumptions).
FP = FP
t t≥0
completed ltration: FP
t := F0
t ∨ NP.
F = (Ft)t≥0
usual augmentation: Ft := F0
t+ ∨ NP.

Filtration: some questions.
The collection (Ω, F, P, F) is called ltered probability space: the basic
set-up of a nancial market.
Questions
1 Why for right-continuous Ft+ := n∈N Ft+1
n
, while for
left-continuous Ft− := σ st Fs ?
2 Would you prefer a right or left-continuous ltration for your
market-information? Why?
3 What is the dierence between FP and Ft where, if Nt is the
collection of (Ft, P)-null sets, Ft = Ft
t := F0
t ∨ Nt t≥0
? If
Ω = [0, 1], F = B ([0, 1]), P = λ and F0 = {Ø, Ω}, what is the
dierence between NP and N0?
In mathematical models for nance one always assume that F is
right-continuous and completed (why?).

σ-algebras 3
Let F0
= F0
t t≥0
be a ltration:
A process X = (Xt, t ≥ 0) is progressive if (ω, t) → 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 all
progressive set is σ-algebra, called the progressive σ-algebra and it is
denoted by M 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 are
adapted 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 paths
on (0, +∞) that are adapted to F0
.
P F0
⊆ O F0
⊆ M F0
.
3
see, e.g., [N, DM]

Stopping times
Let τ be a random time and F0
= F0
t t≥0
be a ltration.
τ is an F0
-stopping time if
{τ ≤ t} ∈ F0
t , ∀t ≥ 0.
τ is predictable if
[τ, +∞) ∈ P F0
.
τ is accessible if ∃ {τn}n∈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 it
still 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 is
the most likely change?
6 If τ is an accessible (resp. totally inaccessible) F-stopping time and I change
the 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 than
F? If not, what is the most likely change?
(shall we give the denition of optional time and have fun with other questions?)
4
see, e.g., [N, DM]

Some Advanced Concepts High Frequency Trading, Default times and Insider Trading
Agenda - II
1 Basic Tools

Why bothering with all these tedious and nitty-gritty
concepts?
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 model
dierent types of traders, where those using optional-strategies gain money
at expenses of those using predictable-strategies:
[...] we construct a model to show that high frequency tradersa [...] can
create increased volatility and mispricings [...] that they exploit to their
advantage.
[...] high frequency traders [may create] trend in market prices that they
exploit 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]
a
BTW: 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 occurs
by surprise). Used by the market (credit-spread is not 0 for short
maturities).
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 I{τ≤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-predictable
accessible time.
See, for example, [DM], Chapter 4, Section 3, Theorem 107. The above
ltration is of crucial importance in credit risk and in the classical
progressive 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 :=
ut
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 to
understand 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)) occurs
before maturity T (r = 0, LGD = 1):
ExpectedLoss = E e−rτ
I{τT} LGD = 1 − e−λT
.
However, due to ill-bonus policy of the company, his actual time horizon is not T
but αT (α ∈ [0, 1]). So, for him, in fact:
ExpectedLossshort-termist
= 1 − e−λαT
.
Other market player (ignoring short-termist's bad-behavior), will misunderstand
the above as a model disagreement on the default intensity:
ExpectedLossshort-termist
= 1 − e−λαT
.
The (private) information of the short-termist player, the model-misunderstanding
of other market agents, may lead to nancial bubble!
5
Following example is taken by [Mo].

Bibliography and Mathematical Brainteasers
Agenda - II
1 Basic Tools

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 and
nothing behind the other two. You will receive whatever is behind the door
of your choice. However, before you choose, the game show host promises
that rather than immediately opening the door of your choice to reveal its
contents, he will open one of the two doors to reveal that it is empty. He
will 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 prize
lies behind either Door 3 or Door 1. Should you switch your choice to Door
1?
(source [X1])

C. Dellacherie and P.-A. Meyer. Probabilities and Potential.
North-Holland, 1978.
K. Giesecke. Default and information. Journal of Economic Dynamics
and Control, 30:2281-2303, 2006.
R. Jarrow and P. Protter. Structural versus Reduced Form Models: A
New Information Based Perspective. Journal of Investment
Management, 2004.
R. Jarrow and P. Protter. A Dysfunctional Role of High Frequency
Trading in Electronic Markets. International J. of Theoretical and
Applied 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 for
Financial 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 by
Simulation: A Simple Least-Squares Approach. The Review of
Financial Studies, Spring 2001 Vol. 14. No. 1, pp. 113-147.
R. Mansuy and M. Yor. Random Times and Enlargements of
Filtrations in Brownian Setting. Volume 1873 Lecture Notes in
Mathematics, 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.

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