Dynamic Asset Pricing
Lecture 1 - NYU
Bruno Dupire
Part I
A quick review of stochastic calculus and applications
We survey the main concepts of stochastic calculus that are useful for financial
applications. As it has already been covered elsewhere, we focus on intuition over
formalism, for which a good reference is Karatzas and Shreve, Brownian Motion and
Stochastic Calculus.
1
Construction of the Brownian motion
The Brownian Motion remains the undisputed king of the realm of uncertainty modeling,
even though it is being increasingly challenged by jump processes. We present an
elementary introduction of the construction of Brownian motion.
1.1
Discrete time
Consider a set of Gaussian i.i.d. increments: X nt 
n
g
i 1
i
t , where gi  N (0,1) are
independent. In the case of independent random variables, the variance of the sum is
the sum of the variances:
n
var[ X nt ]   var[ gi ]t  nt
i 1
It is the probabilistic counterpart of the Pythagorean theorem, which states that the
square of the norm (variance) of the sum of orthogonal (independent) vectors (random
variables) is the sum of the squares of the norms. This geometric analogy is actually very
powerful, and it is always possible to represent n random variables as n vectors in
dimension n, from their covariance matrix.
The variance appears to be linear in time, actually equal to time. This is a feature we wish
to conserve in the continuous time.
1.2
Brownian bridge construction
1
To understand this subtle passage to continuous time, it is helpful to have a path-by-path
construction. We can build a Brownian trajectory by an iteration process that refines at
each step the previous trajectory without altering the already computed pivotal points.
This is the Brownian bridge construction. It consists in "filling the gaps" between the
already computed points, by using the law of the midpoint conditioned by the endpoints.
For those familiar with fractal constructions such as the Von Koch curve, Brownian
motion can be seen as a fractal with a random seed.
1.3
Harmonic decomposition
𝑡
We take any orthonormal basis {fi} of L2 [0, T ] and define 𝐹𝑖 (𝑡) = ∫0 𝑓 𝑖 (𝑠)𝑑𝑠.
Then Wt   gi Fi (t ) , where 𝑔𝑖 are independent normal samples as above, is a Brownian
motion.
As an exercise, check that in fact E[Wt 2 ]  t , under this representation.
For example, the graph below uses the Haar basis and leads to Brownian bridge
construction:
𝑓1
𝑓4
𝑓3
𝑓2
𝑇/2
𝑇
𝑇/2
𝑇
𝐹2
𝐹1
𝑇
𝑇/2
3𝑇/4
𝑇/4
𝐹3
𝐹4
𝑇/2 3𝑇/4 𝑇
𝑇/4 𝑇/2
𝑇
𝑇/2
𝑇
/4
4
𝑔1 𝐹1 + 𝑔2 𝐹2
𝑔1 𝐹1
𝑇
𝑔1 𝐹1 + 𝑔2 𝐹2
+ 𝑔3 𝐹3
𝑇
𝑇
2
෍ 𝑔𝑖 𝐹𝑖
𝑖=1
𝑇
The Brownian motion is obtained as the limit of smooth paths, like in a Fourier
decomposition. The basis can be chosen to optimize the L2convergence, as with Principal
Component Analysis (PCA). It is then called the Karhunen-Loeve decomposition.
1.4 Interpretations of the Wiener space
The three constructions all lead to Monte Carlo schemes. The discrete time approach is
essentially the Euler scheme, which builds the path step by step in time. The other two
approaches lead to schemes which refine a gross skeleton by adding sharper harmonics.
It corresponds in the first case to a binary tree or to the vertices of a high dimensional
hypercube, and in the second case, to a kind of "shoe box", by analogy with the PCA,
with dimension of decreasing importance, related to the eigenvalues of a variance
operator. Methods designed for low dimensions can be applied to the first dimensions
of this decomposition, leading to efficient schemes.
2
Quadratic Variation
The quadratic variation of a process X is defined as
𝑄𝑉 = lim ෍(𝑋𝑡 𝑖+1 − 𝑋𝑡 𝑖 )
2
∆𝑡→0
For Brownian motion, the Central Limit Theorem tells us that QVTW  T , as seen in the
first construction mentioned above. Notice that this gives a heuristic for a binomial tree
to approximate Brownian Motion: each of i.i.d. Gaussians with variance t has a standard
deviation of  t A natural approximation to the increment is a one-period binomial
model going up and down by t
This can be seen as a kind of converse of the CLT: adding many little binomials leads
to a Gaussian density, but Gaussian increments can be seen locally as binomial. This has
important consequences in terms of completeness: if a bond and the stock trade and
their relative price is driven by one Brownian motion, we are in the situation of two
assets and two states of the world, hence complete market.
3
Stochastic Integrals
Consider the following strategy (assuming 0 risk free interest):
1. at time t1 buy 20 shares at $100.
2. at time t2 buy 10 shares at $120.
3. at time t3 sell 20 shares at $110.
4. at time t4 sell 10 shares at $100.
Question: How does one calculate the P&L here? There are many ways to account for
3
the final wealth, for instance considering that the shares bought at time t1 have been sold
at time t 3 . However it is more convenient to compute everyday the number of shares in
the portfolio and to multiply it by the price change to obtain the P&L change. Repeating
this operation over each period and going to the limit in the number of periods leads us
to the definition of the stochastic integral as the sum:
a
ti
( X ti1  X ti )   at dWt
4 Stochastic Differential Equations
The Brownian motion allows formalizing uncertainty and enters modeling of security
prices in the following way: the local evolution is supposed to be the sum of a
deterministic component and of a random term. The motion over an infinitesimal time
interval is depicted as a stochastic differential equation (SDE)
dxt  a dt  b dWt
where W is a standard Brownian motion. The term a represents the drift of the
process and expresses the expected move. The term b is the diffusion coefficient or
volatility of the process x and expresses the uncertainty or amplitude of the noise. Both a
and b can take quite a general form; they can be processes themselves, provided they
satisfy some regularity and integrability conditions.
An important case is the one of Itô diffusions, where the coefficients are deterministic
functions of x and t :
dxt  a( xt , t ) dt  b( xt , t ) dWt
If a and b are Lipschitz in x and satisfy a slow growth condition (they are dominated in
absolute value by an affine function of x , then the SDE has a pathwise solution, which
means that the knowledge of a path of W provides a full knowledge of the associated path
for x .
5
Itô Formula and the Black-Scholes PDE
The above consideration gives the following multiplication table:
dt
4
dt dW
0
0
dW
0
dt
where we consider any power of dt of order greater than 1 negligible.
For a twice differentiable function f, applied to a stochastic process X, we can formally
approximate
1
f ( X  dX )  f ( X )  f ( X )dX  f ( X )dX 2
2
Assuming dX  adt  bdWt , the multiplication table above gives: dX 2  b 2 dt , and so,
adding time as an argument to our function,
1
df  f x dX  ft dt  b2 f xx dt
2
Writing the coefficients in a table:
dt
dW
b
a
dX
1
ft  af x  b 2 f xx
2
1
ft  b 2 f xx
2
df
df  f x dX
bf x
0
In the Black-Scholes model we take S = X to be a stock price, C (t , S )  f (t , X ) and
assume
dS
 rdt   dw
S
Applying Itô’s formula to C (t , S ) :

 2 S 2 C SS
dC   C t 
2


 dt  C S dS

Hedged position C  C S S is locally riskless and earns Ct 
(Ct 
 2 S 2 C SS
2
 2 S 2 C SS
2
, therefore
) dt  (C  C S S ) rdt
where r is the risk-free rate.
which gives us, under the no-arbitrage condition, the Black-Scholes equation
Ct  
 2 S 2 C SS
2
5
 r (C  C S S )
6 Martingale Representation Theorems
The Martingale Representation Theorem (MRT1) tells us that if X is a continuous
martingale, with appropriate integrability and measurability conditions, it can be written
as an integral of Brownian motion:
t
X t  X 0    t dw
0
here X t can be viewed as a claim price, X 0 is the initial price and  is the hedge.
A very important use of MRT is the following:
Under the no-arbitrage condition, there exists an equivalent martingale measure for the
asset S and its derivatives, e.g. the option Vt  Et [ H T ] . Putting this together with MRT1
we get
t
Vt  V0   as dWs .
0
The stock price, itself an asset, can be written
t
St  S0    s dWs .
0
giving
T
VT  V0  
0
aS
s
dS s
This tells us that the claim can be replicated from an initial premium (an expectation)
by conducting a hedging strategy (a stochastic integral). The hedge converts the initial
premium into the target claim; it transforms an expectation into a pathwise replication.
An alternative formulation (MRT2) states that X can be represented as
X t  X 0  W X t
here  X  t is the quadratic variation of X. We can think of this representation as
stochastic volatility, or business time delta-hedge.
7
Feynman-Kac representation
The Feynman-Kac representation establishes some equivalence between the two
classical approaches to option valuation: pricing by finite difference discretization of a
PDE and by martingale methods by computing an expectation by Monte Carlo
simulation for instance. It says that under some conditions,
a)
the solution of a PDE can be computed by expectation
b)
an expectation seen as a function of the initial conditions satisfies a PDE
It relies essentially on Itô’s lemma and the simple example gives a flavor of it.
6
We take the following SDE:
dxt  a dt  b dWt
and a smooth function f ( x, t ) with f ( x, T )  f ( x) . Applying Itô’s lemma to f gives
f
df  Lf dt  b dW
x
where
f
f b 2  2 f
Lf 
a 
t
x 2 x 2
a) yt  f ( xt , t ) is a martingale if and only if Lf  0 , in which case
f ( xt , t )  yt  Et [ yT ]  Et [ f ( xT )]
and f can be computed as an expectation.
b) Conversely, if we define f by f ( x, t )  E x ,t [ f ( xT )] then yt  f ( xt , t ) is a
martingale and its drift is nil, which means that f satisfies
Lf  0
f ( x, T )  f ( x )
Part II
Change of Measure
dP
, and
dQ
 dP 
dQ
E Q [ X ]   X ( )dQ( )   X ( )
( )dP( ) E P  X

dP
 dQ 


If W is Brownian Motion under P, under Q there will be a drift of, say, a, or
If P  Q, we have the Radon-Nikodym derivative
EQ[dW]=adt. Girsanov Theorem then tells us that
a
dQ
 adW  2
e
dP
In the other direction - introducing a drift into the paths when the two measures are
given - the formula is simpler:
dS
  dt   dW P
S
and
dS
  dt   dW Q
S
7
can be solved:
   
E Q [W P ]  
t
  
and
WQ W P 
1
 
t.

Change of Numeraire
Using the change of measure, we can write
𝐸 𝑄 [𝑆 𝑋] = 𝑆0 𝐸 𝑄𝑆 [𝑋]
where the measure 𝑄𝑆 is defined by
𝑑𝑄𝑆
𝑑𝑄
𝑆
= 𝑆 . For example, the price of a call option can
0
be written:
𝐶 = 𝐸 𝑄 [(𝑆 − 𝐾)+ ] = 𝐸 𝑄 [(𝑆 − 𝐾)𝑋] = 𝑆0 𝐸𝑄𝑆 [𝑋] − 𝐾𝐸 𝑄 [𝑋] = 𝑆0 𝑄𝑆 [𝑆 > 𝐾] − 𝐾𝑄[𝑆 > 𝐾]
where 𝑋 = 1𝑆>𝐾
The first probability in the last expression is N (d1 ) the second is N (d 2 ) .
Under the martingale measure,
dS
  dW and changing measure using Girsanov
S
Theorem we get under QS :
dS
  2 dt   dW P
S
and
EQS [dW Q ]   dt
Now suppose we have an asset A, with associated risk neutral measure QA , so that
𝑄
𝑋𝐴 (𝑡) = 𝐸𝑡 𝐴 [𝑋𝐴 (𝑇)] . The subscript A means we take the value in units of A, the
numeraire, and 𝑋𝐴 (𝑡)denotes the price of X in terms of A at time t. If we wish to change
to $ as a numeraire, we get (absence of subscript denotes price in $).
𝑋(𝑡)
𝐴(𝑡)
𝑄
= 𝐸𝑡 𝐴 [
𝑋(𝑇)
𝐴(𝑇)
]
This gives
𝑄
𝑋(𝑇)
]
𝐴(𝑇)
𝑋(𝑡) = 𝐴(𝑡)𝐸𝑡 𝐴 [
Perhaps the most common choice of numeraire would be the money market account
t
 (t )  e
 0 rs ds
, which gives:
8
𝑇
𝑄
𝑋(0) = 𝐸0 𝛽 [𝑋(𝑇)𝑒 − ∫0
𝑟𝑠 𝑑𝑠
]
There are many other choices of numeraire that are useful for various purposes.
For example, if we choose the zero coupon bond Z T as a numeraire, we have
X ( 0)
 E QT
Z T (0)
 X (T ) 


 Z T (T ) 
Z T (T )  1, therefore
X (0)  Z T (0) E QT [ X (T )]  DF E QT [ X (T )]
Where DF is the discount factor between 0 and T.
Please note that it is a discounted expectation as opposed to the previous
expression which was the expectation of the discounted value (under another
measure).
2 Application: Swaptions
A swap contract exchanges a set of fixed coupon payments between times T1 and T2
for a set of payments paying the market rate (LIBOR) . Swap rate (or par swap rate) is
the fixed coupon that makes the price of this contract zero.
Assume that at time T0 a swap contract was originated with swap rate SR0 . Initially
the contract has zero value. At time t , market swap rate is SRt while our swap contract
pays fixed coupon SR0 . The price of this contract is
swap(t )  (SR0  SRt ) LVLT1,T2 (t )
where LVLT1,T2 (t ) is the annuity (sum of discount factors for the dates of coupon
payment)
We can rewrite it as
swap (t )
 SR0  SRt
LVLT1 ,T2 (t )
On the left we have a ration of prices of two tradable securities. Under Q LVL it is a
martingale. Therefore SRt is also a martingale under Q LVL .
Therefore under Q LVL the swap rate has no drift and we can write
dSR(t )
  t dw .
SR(t )
A swaption contract is an option to enter a swap contract paying a fixed coupon K .
A swaption with maturity T1 has payoff
swaption(T1 )  ( K  SRT1 )  LVLT1 ,T2 (T1 )
9
Under Q LVL measure
swaption(t )
is a martingale:
LVLT1 ,T2 (t )
 swaption(T1 ) 
swaption(t )
Q

 E QLVL 
  E LVL ( K  SRT1 )
LVLT1 ,T2 (t )
 LVLT1 ,T2 (T1 ) 

That gives for the price of the swaption

swaption(t )  LVLT1 ,T2 (t ) E QLVL ( K  SRT1 ) 
3


Application: Quanto Options
Quanto is a claim for which the underlying is denominated in one currency while
the claim itself is settled in a different currency. An example is a claim paying in USD
the value of Nikkei index (i.e. if Nikkei is 7000 yen, the payyoff is 7000 dollars).
If S  Nikkei¥ (Nikkei index denominated in yen) then in yen risk-neutral measure
Q
¥
dS
 r¥ dt   S dw ¥
S
Our claim is denominated in US dollars. If the correlation between exchange rate
Q
and Nikkei index is  and volatility of exchange rate is  x , then E $ [dw ¥ ]   x . This
gives
dS
 r¥ dt   S dw ¥  (r¥   x S )dt   S dw $
S
Nikkei
ρ
$
¥
4 Application of Girsanov’s Theorem: The HJM Model
The instantaneous forward rate is defined as
10
f t ,T  lim
Bt ,T  Bt ,T  T
T Bt ,T
T 0
Thus it is a martingale under the EMM associate to the Zero Coupon of maturity T as a
numeraire. Let us assume
𝑑𝑓𝑡,𝑇 = 𝑎 𝑑𝑡 + 𝜎𝑡,𝑇 𝑑𝑊𝑡
Changing measure from the martingale measure QT to QB , we get:
EQB [dW QT ]   B dt
where
T
 B    t , s ds
t
so that
T
dft ,T   t ,T   t , s dsdt   t ,T dW
t
Q
.
Part III
FTAP
FTAP (Fundamental Theorem of Asset Pricing) refers to a collection of results linking
absence of arbitrage and completeness to existence and uniqueness of equivalent
martingale measures (EMM). In a nutshell, it states that
1) Absence of arbitrage is equivalent to the existence of an EMM
2) Absence of arbitrage and completeness is equivalent to existence and uniqueness
of an EMM
This compact formulation is as elegant (as maths should be) as it is misleading (as
maths should not be). This is due to the following complications:
- the notion of absence of arbitrage in continuous time is not well defined, or rather
can be defined in various ways (strictly positive probability of a gain, limit of strong
arbitrages...)
- the class of admissible trading strategies can be defined or constrained in various
ways (L2, bounded losses...)
- the set of contingent claims can be defined in various ways
We will set aside these intricacies arising from the topologies infinite dimensional
spaces can be endowed with and focus on the main principles and interpretations.
11
If there is an EMM, then all claims are priced by expectation and a positive pay-off
has a positive expectation, which precludes arbitrage.
If on top of it the EMM is unique, all claims have a unique possible price and it is not
surprising that this absence of lee-way implies they are attainable and that the market is
complete. If the market is complete, all claims are attainable and there is a unique
possible price which is the initial wealth of the self financing strategy, which implies a
unique pricing measure.
The reciprocal (absence of arbitrage  existence of an EMM) might be less natural,
and we then need the help of the Hahn-Banach theorem.
1 Hahn-Banach theorem
The Hahn-Banach theorem is one of the cornerstones of functional analysis and may look
fearsome, but it is actually of childlike simplicity. It states that two disjoint convex sets
can be separated by a hyperplane.
To illustrate the theorem, let us consider that apples are convex but that bananas are
not.
We can always separate disjoint apples:
We cannot always separate disjoint bananas:
nor even one apple from one banana:
12
This presentation may be seen insulting to Herren Hahn und Banach, the readership,
the author, convex sets, apples and bananas, but it encapsulates the principle that leads
to potent applications, including the one of interest here, the link between absence of
arbitrage and existence of an EMM.
The precise statement of the Hahn-Banach theorem is somewhat intricate, as there
are several formulations, either analytical or geometrical, with proofs relying on arcane
facts about topology in infinite dimensional spaces, and Zorn’s lemma, which you may
accept or not according to your liking of the axiom of choice. The interpretation of the
hyperplane can be revealing, the same way Lagrange multipliers in constrained
optimisation problems can be.
1.1 Back to FTAP
What is the link between the Hahn-Banach theorem and the FTAP? Let us define A the
set of claims that can be attained with a 0 cost and K the set of positive claims. Both A
and K are convex set and any claim belonging to both would be an arbitrage (positive
pay-off attainable for nothing). So absence of arbitrage is equivalent to void intersection
and the Hahn-Banach theorem can be applied: There is (at least) a hyperplane that
separates A and K.
The separating hyperplane contains A and can be seen as the kernel of a linear
functional, which is positive on K and 0 on A, and which we normalize by imposing the
value 1 on the claim that gives a constant pay-off equal to 1 (Zero coupon). This linear
functional is a pricing functional and defines in turn an EMM Q by the following
relationship:
13
Q[ B]  E Q [1B ]  (1B )
for any B measurable set of claims.
1.2 An Important Extension
Let X be a claim such that for each EMM Q, EQ[X]>0. Then, X can be decomposed as the
sum of a positive claim and of an attainable claim of price 0. If it were not the case, we
could separate X from the set A'=A+K which is convex as the sum of two convex sets
with an affine hyperplane.
We can associate to this hyperplane an EMM Q such that EQ[X]<0, which contradicts
the assumption, so there exists a in A and k in K such that X=a+k. By way of consequence,
we have the following result:
Any claim X can be decomposed in the following way
X  inf E Q [ X ]  a  k
Q EMM
with a in A and k in K. It suffices to realize that EQ[X-inf]>0. This means that X can be
subreplicated by inf+a
Similarly,
X  sup E Q [ X ]  a '  k '
Q EMM
with a' in A and k' in K. This means that X can be superreplicated by sup+a'.
Let us stress that we have only talked of attainable claims. This may contain for
instance dynamic hedging with the underlying as well as static hedging with vanilla
options, for instance. In this case, we have the following decomposition of the exotic
option X:
14
X  inf    t dS t    i Ci  pi   k
where C i is a vanilla option of price p i .
Part IV
Option Basics
1
Static relations between option prices
Let K1 , K 2 , K 3 be three strikes, K 3  K 2  K , K1  K 2  K
Payoffs at maturity:
C(K1)
K1
Taking K    0 , then
C(K1) - C(K2)
K1
C(K1) - 2C(K2)+C(K3)
K2
K1
C ( K   )  2C ( K )  C ( K   )
2
and the payoff is an approximation to a delta-function
K2
K3
 C ( K )
K-ε K K+ε
2 Risk-Neutral Density
At some time t before maturity the price of an option is given by
C ( K , t )  DF   ( S )( S  K )  ds
Take second derivative with respect to S of both sides of the equation and use our
result about the second derivative of the payoff to get
C ( K , t )  DF ( K )
15
Thus, if option prices are known for all strikes, the implied risk-neutral probability
density can be recovered via
 (K ) 
C ( K )
DF
3 Payoff decomposition
A general payoff f (S ) can be expressed as
f ( S )  f ( S 0 )  f ( S )( S  S 0 ) 
S0

f ( K ) ( K  S )  dK 
0
S0
 f ( K ) (S  K )

dK ,
0
therefore, the price of an option with a general payoff f (S ) can also be expressed as a
linear combination of prices of call and put options.
16