3. Finance

Quantitative Finance: a brief overview
Part 3/4 - Finance
Dr. Matteo L. BEDINI
Central University of Finance and Economics, Beijing, PRC
27 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 III: Finance CUFE - 27 March 2015 2 / 35

Objective
The objective of this lecture is to show some of the mathematical
techniques used in derivatives pricing. In particular we shall consider:
1 The Change-of-Numéraire technique at work: from theory to
pseudo-implementation.
2 An optimal control problem that can be tackled using dierent
approaches and pricing methods.

Agenda - III
1 Quanto Plain-Vanilla products
Preliminaries
Application to Quanto Plain-Vanilla
2 Passport Options
Denition
Pricing techniques: an overview
3 Bibliography and Financial Brainteasers

Quanto Plain-Vanilla products Preliminaries
Agenda - III
Preliminaries
2 Passport Options
Denition

Change of measure: RadonNikodym density
On a measurable space (E, E) let µ be a measure, ξ be a positive, E-measurable
function.
A new measure: ν (A) :=
ˆ
A
ξdµ, ∀A ∈ E.
P, Q probability measures on (Ω, FT ) measurable space:
if Q P ⇒ ∃ r.v. ξ ≥ 0 s.t. Q(A) =
ˆ
A
ξdP, ∀A ∈ FT .
The above r.v. ξ1
is the RadonNikodym density of Q w.r.t. P (ξ = dQ
dP ):
EQ [X] = EP [ξX] .
If Q P and P Q then P ∼ Q and we can also proceed the
other-way-round (ξ−1 = dP
dQ):
EP [X] = EQ 1
ξ
X .
1
Measurable w.r.t. what? What's its P-expectation?

Change of measure: Bayes formula2
Theorem
1 On the ltered probability space (Ω, F, F, P) let Q ∼ P on FT . Then, ∃ a
strictly positive (P, F)-martingale ξ = (ξt, t ≤ T) (the the RadonNikodym
density of Q w.r.t. P) s.t. EQ [Xt] = EP [ξtXt] ∀Xt ∈ Ft.
2 Q ∼ P on FT with RadonNikodym density ξ = (ξt, t ≤ T), X ∈ FT
Q-integrable:
EQ [X|Ft] =
EP [XξT |Ft]
ξt
(Bayes' formula).
Interesting analogous when changing, not the measure, but the ltration.
Exercise
Compute EP [ξu|Ft], where u, t ∈ [0, T].
2
See, e.g., [JYC], Chapter 1, Propositions 1.7.1.1, 1.7.1.5.

Semimartingales for Financial Markets in 1 slide
Financial market: risky assets S(i) N
i=1
as (P, F)-semimartingales plus
non-risky asset dS
(0)
t = S
(0)
t rtdt + 0 · noise.
Strategy u = ut = u
(i)
t
N
i=0
: F-predictable process dening a portfolio
V t (u) :=
N
i=0 u
(i)
t S
(i)
t .
Self-nancing portfolios are such that dVt (u) = i u
(i)
t dS
(i)
t , or:
Vt (u)
S
(0)
t
= V0 (u) +
N
i=1
tˆ
0
u(i)
r d
S
(i)
r
S
(0)
r
.
Arbitrage: self-nancing u s.t. V0 (u) = 0, VT (u) ≥ 0, P(VT (u) 0) 0.
If Q ∼ P on FT is s.t. S(i)
/S(0) N
i=1
are Q-(local) martingales, then Q is
an equivalent martingale measure.
No Arbitrage ⇔ ∃Q ∼ P such that S(i)
/S(0) N
i=1
are Q-(local)
martingales.
Under any e.m.m. the value of any discounted self-nancing portfolio is a
(local) martingale.

Comments
Semimartingales (for asset prices) and predictable processes (for
investment strategies) allows to dene a stochastic integral that
eectively models the dynamics of the prot and loss of a portfolio.
Arbitrage is the possibility of making a prot without risk (more prot
than investing in the riskless asset). Mathematicians don't like it, but
practitioners do like it (when they gain).
For further discussion on the above-mentioned topics (and more on
complete markets, NFLVR conditions, etc.) we refer to the standard
textbook of Björk [B]. In the previous slides we have used material
extracted from the book of Brigo and Mercurio [BM], Chapter 2, and
[JYC], Chapters 1, 2 and the interested people will nd there the
references to plenty of interesting material. We suggest also the
introduction of Protter [P].
Equivalence of measures, arbitrage conditions, (semi)martingale
stability under ltration expansion are deeply connected topics.

Change of Numéraire
Denition
A Numéraire is any strictly positive price process Z = (Zt, t ≥ 0).
Since it is a price process, its discounted value Zt/S
(0)
t , t ≥ 0 is a
strictly positive Q-martingale.
A new measure: QZ
(A) :=
ˆ
A
Zt
S
(0)
t
dQ, ∀A ∈ Ft,
and for any price process X = (Xt, t ≥ 0), the process (Xt/Zt , t ≥ 0) is a
QZ -martingale, and we are happy because we like martingales (no drift).

Quanto Plain-Vanilla products Application to Quanto Plain-Vanilla
Agenda - III
Preliminaries
2 Passport Options
Denition

Denition
Quanto products commonly refer to nancial derivative involving a
domestic equity market D, a foreign equity market F and a market for the
exchange rate between them (see, e.g. [B], Chapter 17). We are interested
in computing
PD
(0, T)ED
Φ SF
T
Stock-price process in the foreign market SF = SF
t , t ≥ 0
Pricing measures in the domestic (resp. foreign) market QD (resp.
QF).
Discount factors in the domestic (resp. foreign) market P (0, T)D
(resp. P (0, T)F
)
χF→1D = χF→1D
t , t ≥ 0 denotes the exchange rate from currency F
to currency D, where χF→1D
t represents the amount of F-money for
having, at time t, 1 unit of D-money.
Deterministic interest rates (for simplicity): ZCB = Discount factors.

No-Arbitrage considerations
The key-question is: how to change the pricing measure from QF to QD (or
vice-versa)?
In the foreign market:
V F
0
:= PF
(0, T) EF
Φ SF
T
In terms of domestic currency:
V D
0
:= χF→1D
0
V F
0
= χF→1D
0
PF
(0, T) EF
Φ SF
T
However, it shouldn't matter to change now or at expiry:
V D
0
= PD
(0, T)ED
Φ SF
T χF→1D
T
⇒ PD
(0, T)ED
Φ SF
T χF→1D
T = χF→1D
0
PF
(0, T) EF
Φ SF
T

Changing the Numéraire3
Theorem
The Radon-Nikodym derivative dQF
dQD dening the change of measure from the
currency-F risk-neutral probability measure QF to the currency-D risk-neutral
probability measure QD is given by
dQF
dQD
=
χF→1D
0
χF→1D
T
PD (0, T)
PF (0, T)
.
Theorem
Changing the measure from QF to QD is equivalent to changing the Numéraire
from PF (·, T) to PD (·, T) χF→1D.
3
See [BM], Section 2.9, Theorems 2.9.1, 2.9.2.

In practice
From the above theorems we have that
PD
(0, T) ED
Φ SF
T = PF
(0, T) EF χF→1D
T
χF→1D
0
Φ SF
T
=
PF (0, T)
χF→1D
0
EF
χF→1D
T Φ SF
T .
Question
Why we are happy?

Answer
If we were in a pure Black-Scholes world the above wouldn't be necessary:
closed-form formulas are available (see, e.g. [B], Chapter 17).
Since no specic model has been assumed, the previous formula allows the
standard use of Monte-Carlo methods in more general context (local
volatility, stochastic volatility, etc):
1 Write the dynamics of the stock-price process SF = (SF
t , t ≥ 0)
and of the exchange rate χF→1D = χF→1D
t , t ≥ 0 under the
foreign pricing measure QF.
2 Calibrate the necessary parameters with the quotes listed in
the F-market.
3 Run your Monte-Carlo engine accordingly to simulate the
values χF→1D
T and Φ (SF
T ).
4 Compute the average (sample mean will indeed be the estimator
of the QF-expectation).
5 Multiply the obtained number with the F-ZCB PF (0, T) and
divide by the spot FX-rate χF→1D
0 . The QD-price is served.

Concluding questions/remarks
The key-point is that the Change-of-Numéraire technique allows to
perform all necessary computation in the F-world, to solve a problem
that was mixing the F-dynamics of the underlying with the D-pricing
measure.
Exercise: what if stock-price and the exchange rate were
independent4
?
How can we test that the previous pseudo-algorithm is correct?
What to do if interest rates were stochastic? Need to use the
forward-measure; again: Change-of-Numéraire!
4
Independent w.r.t. what?

Passport Options Denition
Agenda - III
Preliminaries
2 Passport Options
Denition

Denition 1/2
In its simplest form, a Passport Option (PO) is a 0-strike call option on the
wealth generated by trading on a given underlying S. PO were initially
considered by Bankers Trust (see [HLP]) and rstly appeared in second half
of the 90s in the FX and futures markets.
The payo at expiry is equal to
(V u
T )+
:= max {V u
T , 0} ,
where V u
T is the value of the trading account at the expiry date T.

Denition 2/2
The trading account V u
= (V u
t , 0 ≤ t ≤ T) is a stochastic process depending on
three variables:
1 the initial value x of the trading account (x ∈ R),
2 the underlying price process S = (St, t ≥ 0) on which the trading strategy is
executed (a semimartingale),
3 the strategy u = (ut, 0 ≤ t ≤ T), chosen by the buyer (ut ∈ [−L, L] for all
t, where L 0 is the strategy bound).
Example
Two examples covering the most popular situations. Left: no dividends, no
interest rate. Right: no dividends, constant interest rate.
V u
0 = x
dV u
t = utdSt
V u
0 = x
dV u
t = r (V u
t − utSt) dt + utdSt

Cash-ows and price 1/2
Fix a strategy u: the value V u
t of a PO, at time t ∈ [0, T] is dened by
V u
t := P (t, T) E (V u
T )+
|Ft .
The price of a PO, is always computed under the assumption that the
option holder will follow the optimal strategy. Hence, according to this
framework, the price Vt at time t of the PO is equal to
Vt := sup
u
V u
t = sup
u
P (t, T) E (V u
T )+
|Ft .

Cash-ows and price 2/2
For the option holder −V0 + P (0, T) (V u
T )
+
:
⇒ −V0 + P (0, T) sup
u
E (V u
T )
+
. (1)
For the option issuer V0 − P (0, T) (V u
T )
−
(why?):
⇒ V0 + P (0, T) inf
u
E − (V u
T )
−
= V0 − P (0, T) sup
u
E (V u
T )
−
. (2)
Imposing equality between (1) and (2)
V0 =
1
2
P (0, T) sup
u
E (V u
T )
+
+ sup
u
E (V u
T )
−
= P (0, T) sup
u
E (V u
T )
+
because the problem is symmetric: V u
T = −V −u
T ⇒ (V u
T )
+
= V −u
T
−
and
u ∈ [−L, L] ⇒ sup
u∈[−L,L]
E (V u
T )
+
= sup
u∈[−L,L]
E (V u
T )
−
.

The optimal strategy
In some situations it can be shown that the strategy u = (ut)t≥0
dened by
ut :=
−L, if V u
t 0
L, if V u
t ≤ 0
= −Lsign (V u
t )
is optimal, i.e
V q
t ≤ V u
t , ∀q = u.
Roughly speaking, the optimal strategy for the option holder is to invest up
to the position limit on the underlying, buying when the account is
negative and selling when the account is positive. See, e.g.
[SV, HH00, HH01, DY, APW, AAB].

Passport Options Pricing techniques: an overview
Agenda - III
Preliminaries
2 Passport Options
Denition

Pricing techniques
1 Monte Carlo.
2 Closed-form formulas in a Black-Scholes-world ([HH00, AAB] only in
two special cases, [SV] more general).
3 PDE [AAB, APW, HLP].
4 Probabilistic Approach [DY, HH00], and [SV].
5 Discrete-time binomial model.

Monte-Carlo
1 Use of Least Square à la Longsta and Schwartz ([LS]) for the
conditional expectations. Trading times 0 = t0 t1 ... tN = T :
1 At expiry: VN (x) = max {x, 0} , x ∈ R.
2 For each i = N − 1, ..., 0
Vi (x) = max
u∈[−L,L]
E[Vi+1 (x + u (Si+1 − Si )) |Fti
] , x ∈ R.
In practice:
1 MC forward simulation of (St, t ≥ 0).
2 Backward projection for the computation of Vi .
2 Just simulate the asset path and use the optimal strategy when this
can be used (e.g. if the underlying is a Geometric Brownian motion -
see [DY], Theorem 6.1).

Closed-form formulas in a Black-Scholes-world
Asset and account are martingales (dSt = σStdWt, dV u
t = utdSt) ([AAB]).
Vt = e−rτ
St
Vt
St
+
+ N (d+) − 1 +
Vt
St
N (d−) + Ω (t, T) ,
where τ := T − t, N (x) := 1
√
2π
exp −x2
2
and
d± :=
− log 1 + Vt
St
± 1
2
σ2
τ
σ
√
τ
,
Ω (t, T) :=
1
2
d+σ
√
τ − 1 N (d+) + 1 +
Vt
St
N (d−) + σ
√
τN (d+) .
Analogous formula in [HH00] when discounted asset and discounted
account are martingales.

PDE
In [HLP] the price of the Passport option is computed as a solution of the
Hamilton-Jacobi-Bellman equation corresponding to the optimal control
problem (see also [APW] and [AAB]). Still, a Black-Scholes framework is
assumed.
Using Bellman's principle and after some computation one must solve a
PDE of the form
−
∂v
∂t
= −fv + fz
∂v
∂z
+
σ2
2
∂2
v
∂z2
max
|u|≤1
(z − u)2
,
which is maximized by u = −sign (z).

Probabilistic Approach 1/2
The local time L0
t t≥0
of V u at level 0 is a measure on how much
time the process V u spends near 0. If V u is a martingale
E (V u
T )+
= E (V u
0
)+
+
1
2
L0
T .
The more X crosses 0, the greater the value of L0
T . The strategy
maximizing the above expectation should be the one that try to force
the account to cross level 0 as much as possible.
This gives an heuristic explanation why the strategy given by
ut = −Lsign (V u
t )
(being long when the account is negative and short otherwise) is the
optimal one (see [HH00]). Note that what is a martingale and what is
not depends on the specications of the contract.

Probabilistic Approach 2/2
Figure: Local time of a stochastic process as
´ t
0 δx (Ws) ds [wiki].

Concluding remarks
Another interesting technique is described in the paper of S. Shreve
and J. Vecer [SV].
In the above-mentioned material those who are interested may nd
further references where more exotic products are considered, as well
as numerical methods for the implementation of the presented
solutions.
This example shows that the same problem may be tackled in many
dierent ways: it is up to the quant analyst to decide which method
should be implemented keeping in mind that, most of the times, The
Solution may not exists but, instead, a solution should be found with
the available instruments and the given constraints (time, computing
power, etc.).

Bibliography and Financial Brainteasers
Agenda - III
Preliminaries
2 Passport Options
Denition

Financial Brainteaser 1/3
Consider two European options: a call and a put with the same maturity,
underlying, strike. Is their implied volatility the same?
(source [X1])

Consider a deterministic asset, whose initial value (day 0) is 10.
On day 1, 3, 5, 7, 9, ... it grows of 1%. On days 2, 4, 6, 8, ... it grows of
-1%.
Money invested on this asset will be paid back after a great amount of time.
Would you invest in that asset?

Write down the Black-Scholes formula for the price of a European Call
option. Provide the shortest and most rigorous computation of the of
the option.
Hint: use the Euler's Theorem for computing the derivative of
homogeneous functions.
(source [X3])

Ahn H., Penaud A., Willmot P. Various passport options and their
valuation. Applied Mathematical Finance, 6:275-292 (1999).
Andersen L., Andreasen J., Brotherton Ratclie R. The passport
option. Journal of Computational Finance, Vol. 1, No. 3, 15-36 (1998).
T. Björk. Arbitrage Theory in Continuous Time. Oxford University
Press, Second Edition, 2004.
D. Brigo, F. Mercurio. Interest Rate Models - Theory and Practice.
Springer, Second Edition, 2006.
Delbaen F., Yor M. Passport options. Mathematical Finance
12(4):299-328 (2002).
Henderson V., Hobsong D.G. Local Time, coupling and the Passport
option. Finance and Stochastics, Vol. 4, 1, 69-80 (2000).
Henderson V., Hobsong D.G. Passport options with stochastic
volatility. Applied Mathematical Finance, 8:97-118 (2001).

Hyer T., Lipton-Lifschitz A., Pugachevsky D. Passport to success.
Risk, 10, pp. 127-131 (1997).
M. Jeanblanc, M. Yor and M. Chesney. Mathematical Methods for
Financial Markets. Springer, First edition, 2009.
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.
P. Protter. A partial introduction to nancial asset pricing theory.
Stochastic Processes and their Appl., 91:169204, 2001.
Shreve S., Vecer J. Options on a Traded Account: Vacation Calls,
Vacation Puts and Passport Options. Finance and Stochastics,
4:255-274 (2000).
http://en.wikipedia.org/wiki/Local_time_%28mathematics%
29#/media/File:Local_times_surface.png

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