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Transcription

Numerical methods in mathematical finance
Tobias Jahnke
Karlsruher Institut f¨
ur Technologie
Fakultät f¨
ur Mathematik
Institut f¨
ur Angewandte und Numerische Mathematik
[email protected]
© Tobias Jahnke, Karlsruhe 2014
Version: December 4, 2014
Preface
These notes are the basis of my lecture Numerical methods in mathematical finance given
at Karlsruhe Institute of Technology in the winter term 2014/15. The purpose of this
notes is to help students who have missed parts of the course to fill these gaps, and to
provide a service for those students who can concentrate better if they do not have to
copy what I write on the blackboard.
It is not the purpose of these notes, however, to replace the lecture itself, or to write a
text which could compete with the many excellent books about the subject. This is why
the style of presentation is rather sketchy. As a rule of thumb, one could say that these
notes only cover what I write during the lecture, but not everything I say.
There are still many typos and possibly also other mistakes. Of course, I will try to
correct any mistake I find as soon as possible, but please be aware of the fact that you
cannot rely on these notes.
Karlsruhe, winter term 2014/15,
Tobias Jahnke
i
ii
Contents
I
Mathematical models in option pricing
1 Options and arbitrage
1.1 European options . . . . . . . . . . .
1.2 More types of options . . . . . . . . .
1.3 Arbitrage and modelling assumptions
1.4 Arbitrage bounds . . . . . . . . . . .
1.5 A simple discrete model . . . . . . .
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2
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6
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2 Stochastic differential equations
2.1 The Wiener process . . . . . . . . . . . . . . . . . .
2.2 Stochastic differential equations and the Itô formula
2.3 Martingales . . . . . . . . . . . . . . . . . . . . . .
2.4 The Feynman-Kac formula . . . . . . . . . . . . . .
2.5 Extension to higher dimensions . . . . . . . . . . .
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9
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19
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27
31
3 The
3.1
3.2
3.3
3.4
3.5
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Black-Scholes equation
Geometric Brownian motion . . . . . . . . . . . . . . . . .
Derivation of the Black-Scholes equation . . . . . . . . . .
Black-Scholes formulas . . . . . . . . . . . . . . . . . . . .
Risk-neutral valuation and equivalent martingale measures
Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Numerical methods
32
4 Binomial methods
33
4.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3 Discrete Black-Scholes formula . . . . . . . . . . . . . . . . . . . . . . . . . 37
5 Numerical methods for stochastic differential
5.1 Motivation . . . . . . . . . . . . . . . . . . . .
5.2 Euler-Maruyama method . . . . . . . . . . . .
5.2.1 Derivation . . . . . . . . . . . . . . . .
iii
equations
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39
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41
41
5.3
5.4
5.2.2 Weak and strong convergence . . . . . . . . . . . .
5.2.3 Existence and uniqueness of solutions of SDEs . . .
5.2.4 Strong convergence of the Euler-Maruyama method
5.2.5 Weak convergence of the Euler-Maruyama method .
Higher-order methods . . . . . . . . . . . . . . . . . . . . .
Numerical methods for systems of SDEs . . . . . . . . . .
A Some definitions from probability theory
B The
B.1
B.2
B.3
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42
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47
50
56
57
Itˆ
o integral
59
The Wiener process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Construction of the Itô integral . . . . . . . . . . . . . . . . . . . . . . . . 61
Sketch of the proof of the Itô formula (Theorem 2.2.2). . . . . . . . . . . . 68
iv
Part I
Mathematical models in option
pricing
1
Chapter 1
Options and arbitrage
References: [BK04, Sey09]
1.1
European options
Financial markets trade investments into stocks of a company, commodities (e.g. oil, gold),
etc.
Stocks and commodities are risky assets, because their future value cannot be predicted.
Bonds are considered as riskless assets in this lecture. If B(t0 ) is invested at time t0 into
a bond with a risk-free interest rate r > 0, then the value of the bond at time t ≥ t0 is
simply
B(t) = er(t−t0 ) B(t0 ).
(1.1)
Simplifying assumption: continuous payment of interest
Spot contract: buy or sell an asset (e.g. a stock, a commodity etc.) with immediate delivery
Financial derivatives: contracts about future payments or deliveries with certain conditions
1. Forwards and futures: agreement between two parties to buy or sell an asset at a
certain time in the future for a certain delivery price
2. Swaps: contracts regulating an exchange of cash flows at different future times
(e.g. currency swap, interest rate swaps, credit default swaps)
3. Options
Definition 1.1.1 (European option)
• A European call option is a contract which gives the holder (=buyer) of the option
the right to buy an underlying risky asset at a future maturity date (expiration time)
T at a fixed exercise price (strike) K from the writer (=seller) of the option.
Typical assets: stocks, parcels of stocks, stock indices, currencies, commodities, ...
Difference to forwards and futures: At maturity the holder can choose if he wants
to buy the asset or not.
2
Tobias Jahnke
December 4, 2014 –
3
• European put option: Similar to call option, but vice versa, i.e. the holder can
sell the underlying to the writer.
Example: At time t = 0 Mr. J. buys 5 European call options. Each of these options
gives him the right to buy 10 shares of the company KIT at maturity T > 0 at the exercise
price of K = 120e per share.
• Case 1: At time t = T , the market price of KIT is 150e per share.
Mr. J. exercises his options, i.e. he buys 5 · 10 = 50 KIT shares at the price of
K = 120e per share and sells the shares on the market for 150e per share. Hence,
he wins 50 · 30 = 1500e.
• Case 2: At time t = T , the market price of KIT is 100e per share.
Hence, Mr. J. does not exercise his options.
What are options good for?
• Speculation
• Hedging (“insurance” against changing market values)
Since an option gives an advantage to the holder, the option has a certain value.
For given T and K the value V (t, S) of the option must depend on the time t and the
current price S of the underlying.
For an European option we know that the value at the maturity T is
(
(S − K)+ := max{S − K, 0} (European call)
V (T, S) =
(K − S)+ := max{K − S, 0} (European put).
The functions S 7→ (S − K)+ and S 7→ (K − S)+ are called the payoff functions of a
call or put, respectively.
The goal of this course is to answer the following question:
What is the fair price V (t, S) of an option for t < T ?
Why is this question important? In order to sell/buy an option, we need to know the fair
price.
Why is this question non-trivial? Because the value of the risky asset is random. In
particular, the price S(T ) at the future expiration time T is not yet known when we
buy/sell the option at time t = 0.
1.2
More types of options
Variations of the basic principle:
• European options can be exercised only at the maturity date.
4 –
December 4, 2014
• American options can be exercised at any time before and including the maturity
date.
• Bermuda options can be exercised at a set of times.
The names “European”, “American”, “Bermuda” etc. have no geographical meaning.
American options can be traded in Europe, European options can be traded in the USA,
etc.
• Vanilla options = standard options, i.e. European, American or Bermuda calls/puts
• Exotic options = non-standard options
Examples for exotic options:
• Path-dependent option: The payoff function does not only depend on the price
S(T ) of the underlying at time T , but on the entire path t 7→ S(t) for t ∈ [0, T ].
◦ Asian options: The payoff function depends on the average price, e.g.

+
ZT
1
S(t) dt − K 
T
0
(payoff of an average price call).
◦ Barrier options: The payoff depends on the question if the price of the
underlying has crossed a certain (upper or lower) barrier.
◦ Lookback options: The payoff depends on maxt∈[0,T ] S(t) or mint∈[0,T ] S(t).
• Options on several assets:
◦ Basket options: The payoff depends on the weighted sum of the prices Si of
several assets, e.g.
!+
d
X
ci Si − K
,
ci > 0
i=1
(payoff of a basket call)
◦ Rainbow options: The payoff depends on the relation between the assets,
e.g. max{S1 , . . . , Sd }.
• Binary options: The payoff function has only two possible values
• Compound options: Options on options
Remark: There are even more types of options.
1.3
Arbitrage and modelling assumptions
Example. Consider
• a stock with price S(t)
• a European call option with maturity T = 1, strike K = 100, and value V (t, S(t))
Tobias Jahnke
5
• a bond with price B(t)
Initial data: S(0) = 100, B(0) = 100, V (0) = 10.
Assumption: At time t = 1, we either have
“up”:
B(1) = 110, S(1) = 120
“down”: B(1) = 110, S(1) = 80
or
At t = 0, Mrs. C. buys 0.4 bonds, one call option and sells 0.5 stock (“short selling”).
Value of the portfolio at t = 0:
0.4 · B(0) + 1 · V (0) − 0.5 · S(0) = 0.4 · 100 + 1 · 10 − 0.5 · 100 = 0
Value of the portfolio at t = 1 is
0.4 · B(1) + 1 · V (1, S(1)) −0.5 · S(1)
| {z }
=(S(1)−K)+
Two cases:
“up”:
“down”:
0.4 · 110 + 1 · (120 − 100)+ − 0.5 · 120 = 44 + 20 − 60 = 4
0.4 · 110 + 1 · (80 − 100)+ − 0.5 · 80 = 44 + 0 − 40 = 4
In both cases, Mrs. C. wins 4e without any risk or investment!
Why is this possible? Because the price V (0) = 10 of the option is too low!
Definition 1.3.1 (Arbitrage) Arbitrage is the existence of a portfolio, which
• requires no initial investment, and
• which cannot cause any loss, but very likely a gain.
Remark. A bond will always yield a risk-less gain, but it requires an investment.
Assumptions for modelling an idealized market:
(A1) Arbitrage is impossible (no-arbitrage principle).
(A2) There is a risk-free interest rate r > 0 which applies for all credits. Continuous
payment of interest according to (1.1).
(A3) No transaction costs, taxes, etc. Trading is possible at any time. Any fraction of
an asset can be sold. Liquid market, i.e. selling an asset does not change its value
significantly.
(A4) A seller can sell assets he/she does not own yet (“short selling”, cf. Mrs. C. above)
(A5) No dividends on the underlying asset are paid.
6 –
December 4, 2014
Remark. Discrete payment of interest: obtain r · ∆t · B(0) after time ∆t. Value at
t = n∆t:
˜ = (1 + r · ∆t)n B(0) = (1 + rt/n)n B(0)
B(t)
For n −→ ∞ and ∆t −→ 0:
˜ = lim (1 + rt/n)n B(0) = ert B(0) = B(t)
lim B(t)
n→∞
n→∞
(continuous payment of interest)
1.4
Arbitrage bounds
Consider European options with strike K > 0 and maturity T on an underlying with price
S(t). Let VP (t, S) and VC (t, S) be the values of a put option and call option, respectively.
Lemma 1.4.1 (Put-call parity) Under the assumptions (A1)-(A5) we have
S(t) + VP (t, S(t)) − VC (t, S(t)) = e−r(T −t) K
for all t ∈ [0, T ].
Proof. Buy one stock, buy a put, write (sell) a call. Then, the value of this portfolio is
π(t) = S(t) + VP (t, S(t)) − VC (t, S(t))
and at maturity
π(T ) = S(T ) + VP (T, S(T )) − VC (T, S(T )) = S(T ) + (K − S(T ))+ − (S(T ) − K)+ = K.
Hence, the portfolio is risk-less. No arbitrage: The profit of the portfolio must be the
same as the profit for investing π(t) into a bond at time t:
!
π(T ) = K = er(T −t) π(t)
=⇒
e−r(T −t) K = π(t) = S(t) + VP (t, S(t)) − VC (t, S(t)).
Lemma 1.4.2 (Bounds for European calls and puts) Under the assumptions (A1)(A5), the following inequalities hold for all t ∈ [0, T ] and all S = S(t) ≥ 0:
S − e−r(T −t) K
+
≤ VC (t, S) ≤ S
(1.2)
e−r(T −t) K − S
+
≤ VP (t, S) ≤ e−r(T −t) K
(1.3)
Proof.
• It is obvious that VC (t, S) ≥ 0 and VP (t, S) ≥ 0 for all t ∈ [0, T ] and S ≥ 0.
Tobias Jahnke
7
• Assume that VC (t, S(t)) > S(t) for some S(t) ≥ 0.
Write (sell) a call, buy the stock and put the difference δ := VC (t, S(t)) − S(t) > 0
in your pocket.
At t = T , there are two scenarios:
If S(T ) > K: Must sell stock at the price K to the owner of the call.
Gain: K + δ > 0
If S(T ) ≤ K: Gain S(T ) + δ > 0
=⇒ Arbitrage! Contradiction!
• Put-call parity:
S − e−r(T −t) K = VC (t, S) − VP (t, S) ≤ VC (t, S)
| {z }
≥0
This proves (1.2). The proof of (1.3) is left as an exercise.
Remark. Similar inequalities can be shown for American options (exercise).
1.5
A simple discrete model
Consider
• a stock with price S(t)
• a European option with maturity T , strike K, and value V (t, S(t))
• a bond with price B(t) = ert B(0)
Suppose that the initial data S(0) = S0 and B(0) = 1 are known, and that (A1)-(A5)
hold. Goal: Find V (0, S0 ).
Simplifying assumption: At time t = T , there are only two scenarios
or
“up”:
S(T ) = u · S0
“down”: S(T ) = d · S0
with probability p
with probability 1 − p
Assumption: 0 < d ≤ erT ≤ u and p ∈ (0, 1)
In both cases, we have B(T ) = erT B(0) = erT .
Replication strategy: Construct portfolio with c1 bonds and c2 stocks such that
!
c1 B(t) + c2 S(t) = V (t, S(t))
for t ∈ {0, T }. For t = T , this means
!
case “up”: c1 erT + c2 uS0 = V (T, uS0 ) =: Vu
!
case “down”: c1 erT + c2 dS0 = V (T, dS0 ) =: Vd
8 –
December 4, 2014
Vu and Vd are known if u and d are known. The unique solution is (check!)
c1 =
uVd − dVu
(u − d)erT
c2 =
Vu − Vd
.
(u − d)S0
Hence, the fair price of the option is
uVd − dVu
Vu − Vd
V (0, S0 ) = c1 B(0) +c2 S0 =
+
| {z }
(u − d)erT
(u − d)
=1
which yields (check!)
V (0, S0 ) = e−rT qVu + (1 − q)Vd
with
q :=
erT − d
.
u−d
(1.4)
Remark: The value of the option does not depend on p.
Since 0 < d ≤ erT ≤ u by assumption, q ∈ [0, 1] can be seen as a probability. Now, define
a new probability distribution Pq by
Pq S(T ) = uS0 = q,
Pq S(T ) = dS0 = 1 − q
(q instead of p). Then, we have
Pq V (T, S(T )) = Vu = q,
Pq V (T, S(T )) = Vd = 1 − q
and hence
qVu + (1 − q)Vd = Eq V (T, S(T ))
can be regarded as the expectation of the payoff V (T, S(T )) with respect to Pq . In (1.4),
this expectation is multiplied by an discounting factor e−rT .
Interpretation: In order to have an amount of B(t) at time t, we have to invest B(0) =
e−rt B(t) into a bond at time t = 0.
The probability q has the property that
Eq (S(T )) = quS0 + (1 − q)dS0 =
erT − d
u − erT
uS0 +
dS0 = erT S0 .
u−d
u−d
Hence, the expected (with respect to Pq ) value of S(T ) is exactly the amount we obtain
when we invest S0 into a bond. Therefore, Pq is called the risk-neutral probability.
Moral of the story so far:
Under the risk-neutral probability, the price of a European
option is the discounted expectation of the payoff.
Chapter 2
Stochastic differential equations
Let (Ω, F, P) be a probability space1 : Ω 6= ∅ is a set, F is a σ-algebra (or σ-field) on Ω,
and P : F −→ [0, 1] is a probability measure.
A probability space is complete if F contains all subsets G of Ω with P-outer measure
zero, i.e. with
P∗ (G) := inf P(F ) : F ∈ F and G ⊂ F = 0.
Any probability space can be completed. Hence, we can assume that every probability
space in this lecture is complete.
2.1
The Wiener process
Definition 2.1.1 (Stochastic process) Let T be an ordered set (e.g. T = [0, ∞), T =
N). A stochastic process is a family X = {Xt : t ∈ T } of random variables
Xt : Ω −→ Rd .
Below, we will often simply write Xt instead of {Xt : t ∈ T }.
Equivalent notations: X(t, ω), X(t), Xt (ω), Xt , . . .
For a fixed ω ∈ Ω, the function t 7→ Xt (ω) is called a realization (or path or trajectory)
of X.
The path of a stochastic process is associated to some ω ∈ Ω. As time evolves, more
information about ω becomes available.
Example (cf. chapter 2 in [Shr04]). Toss a coin three times. Possible results are:
ω1
HHH
ω2
ω3
ω4
ω5
ω6
HHT HTH HTT THH THT
ω7
ω8
TTH TTT
(H = heads, T = tails).
1
See “2.2.2 What is (Ω, F, P) anyway?” in the book [CT04] for a nice discussion of this concept.
9
10 –
December 4, 2014
• Before the first toss, we only know that ω ∈ Ω.
• After the first toss, we know if the final result will belong to
{HHH, HHT, HT H, HT T } or to {T HH, T HT, T T H, T T T }.
These sets are “resolved by the information”. Hence, we know in which of the sets
{w1 , w2 , w3 , w4 }, {w5 , w6 , w7 , w8 }
ω is.
• After the second toss, the sets
{HHH, HHT }, {HT H, HT T }, {T HH, T HT }, {T T H, T T T }
are resolved, and we know in which of the sets
{w1 , w2 }, {ω3 , w4 }, {w5 , w6 }, {w7 , w8 }
ω is.
This motivates the following definition.
Definition 2.1.2 (Filtration)
• A filtration is a family {Ft : t ≥ 0} of sub-σ-algebras of F such that Fs ⊂ Ft for
all t ≥ s ≥ 0.
A filtration models the fact that more and more information about a process is known
as time evolves.
• If {Xt : t ≥ 0} is a family of random variables and Xt is Ft -measurable, then
{Xt : t ≥ 0} is adapted to (or nonanticipating with respect to) {Ft : t ≥ 0}.
Interpretation: At time t we know for each set S ∈ Ft if ω ∈ S or not. The value
of Xt is revealed at time t.
• For every s ∈ [0, t] let σ{Xs } be the σ-algebra generated by Xs , i.e. the smallest
σ-algebra on Ω containing the sets
Xs−1 (B) for all B ∈ B
where B denotes the Borel σ-algebra. By definition σ{Xs } is the smallest σ-algebra
where Xs is measurable.
A very important stochastic process is the Wiener process; cf. section B.1 in the appendix.
Filtration of the Wiener process. The natural filtration of the Wiener process on
[0, T ] is given by
{Ft : t ∈ [0, T ]},
Ft = σ{Ws , s ∈ [0, t]}
(cf. Definition 2.1.2). For technical reasons, however, it is more advantageous to use
an augmented filtration called the standard Brownian filtration. See pp. 50-51 in
[Ste01] for details.
Tobias Jahnke
2.2
11
Stochastic differential equations and the Itˆ
o formula
Definition 2.2.1 (SDE) A stochastic differential equation (SDE) is an equation
of the form
Zt
X(t) = X(0) +
f s, X(s) ds +
0
Zt
g s, X(s) dW (s).
(2.1)
0
The solution X(t) of (2.1) is called an Itˆ
o process.
The last term is an Itô integral, with W (t) denoting the Wiener process. An informal
construction of the Itô integral can be found in the appendix of these notes and will be
presented in the problem class. The functions f : R × R −→ R and g : R × R −→ R
are called drift and diffusion coefficients, respectively. These functions are typically given
while X(t) = X(t, ω) is unknown.
This equation is actually not a differential equation, but an integral equation! Often
people write
dXt = f (t, Xt )dt + g(t, Xt )dWt
as a shorthand notation for (2.1). Some people even “divide by dt” in order to make the
equation look like a differential equation, but this is more than audacious since “dWt /dt”
does not make sense.
Two special cases:
• If g t, X(t) ≡ 0, then (2.1) is reduced to
Zt
X(t) = X(0) +
f s, X(s) ds.
0
If X(t) is differentiable, this is equivalent to the initial value problem
dX(t)
= f t, X(t) ,
X(0) = X0 .
dt
• For f t, X(t) ≡ 0, g t, X(t) ≡ 1 and X(0) = 0, (2.1) turns into
Zt
X(t) = X(0) +
| {z }
=0
|0
f s, X(s) ds +
Zt
0
{z
=0
}
g s, X(s) dW (s) = W (t) − W (0) = W (t).
| {z }
=1
12 –
December 4, 2014
Computing Riemann integrals via the basic definition is usually very tedious. The fundamental theorem of calculus provides an alternative which is more convenient in most
cases. For Itô integrals, the situation is similar: The approximation via elementary functions which is used to define the Itô integral is rarely used to compute the integral. What
is the counterpart of the fundamental theorem of calculus for the Itô integral?
Theorem 2.2.2 (Itˆ
o formula) Let Xt be the solution of the SDE
dXt = f (t, Xt )dt + g(t, Xt )dWt
and let F (t, x) be a function with continuous partial derivatives
we have for Yt := F (t, Xt ) that
∂F ∂F
, ∂x ,
∂t
∂F
1 ∂ 2F 2
∂F
dt +
dXt +
g dt
∂x
2 ∂x2
∂t
∂F
1 ∂ 2F 2
∂F
∂F
+
f+
dt
+
gdWt .
g
=
∂t
∂x
2 ∂x2
∂x
and
∂2F
.
∂x2
Then,
dYt =
with f = f (t, Xt ), g = g(t, Xt ),
(2.2)
∂F
∂F (t, Xt )
=
, and so on.
∂x
∂x
Notation. From now on, the partial derivatives of some function u(t, x) will be denoted
by
∂u
∂u
∂ 2u
,
∂x u :=
,
∂x2 u :=
∂t
∂x
∂x2
and so on. Evaluations of the derivatives of F are to be understood in the sense of, e.g.,
∂x F (s, Xs ) := ∂x F (t, x)
∂t u :=
(t,x)=(s,Xs )
and so on.
The proof of Theorem 2.2.2 is sketched in the Appendix, cf. B.3.
Remarks:
1. If y(t) is a smooth deterministic
fuctions, then according to the chain rule the
derivative of t 7→ F t, y(t) is
dy(t)
d
F t, y(t) = ∂t F t, y(t) + ∂x F t, y(t) ·
dt
dt
and in shorthand notation
dF = ∂t F dt + ∂x F dy.
The Itô formula can be considered as a stochastic version of the chain rule, but the
term 21 ∂x2 F · g 2 dt is surprising since such a term does not appear in the deterministic
chain rule.
Tobias Jahnke
13
2. Let f (t, Xt ) = 0, g(t, Xt ) = 1, Xt = Wt and suppose that F (t, x) = F (x) does not
depend on t. Then, the Itô formula yields for Yt := F (Wt ) that
1
dYt = F 0 (Wt )dWt + F 00 (Wt )dt
2
which is the shorthand notation for
Zt
F (Wt ) = F (W0 ) +
1
F 0 (Ws )dWs +
2
0
Zt
F 00 (Ws )ds.
0
This can be seen as a counterpart of the fundamental theorem of calculus. Again,
the last term is surprising, because for a suitable deterministic function v(t) = vt
we obtain
Zt
F (vt ) = F (v0 ) +
F 0 (vs )dvs .
0
Example 1. Consider the integral
Z
t
Ws dWs .
0
Xt := Wt solves the SDE with f (t, Xt ) ≡ 0 and g(t, Xt ) ≡ 1. For
F (t, x) = x2 ,
Yt = F (t, Xt ) = Xt2 = Wt2
the Itô formula
dYt =
1 2
2
∂t F + ∂x F · f + ∂x F · g dt + ∂x F · g dWt
2
yields
1
· 2 · 12 dt + 2Wt · 1 dWt = dt + 2Wt dWt
2
1
2
Wt dWt =
d(Wt ) − dt
2
d(Wt2 ) = 0 + 0 +
=⇒
This means that
Zt
0
1
Ws dWs =
2
Zt
0
1
1 d(Ws2 ) −
2
Zt
0
1
1
1 ds = Wt2 − t.
2
2
14 –
December 4, 2014
Example 2. The solution of the SDE
dYt = µYt dt + σYt dWt
with constants µ, σ ∈ R and deterministic initial value Y0 ∈ R is given by
σ2 Yt = Y0 exp µ −
t + σWt .
2
This process is called a geometric Brownian motion and is often used in mathematical
finance to model stock prices (see below).
Proof. Let f (t, Xt ) ≡ 0, g(t, Xt ) ≡ 1, Xt = Wt as before, but now with
σ2 t + σx Y0
F (t, x) = exp
µ−
2
and derivatives
σ2 F (t, x),
∂t F (t, x) = µ −
2
∂xi F (t, x) = σ i F (t, x),
i ∈ {1, 2}.
Hence, the Itô formula applied to Yt = F (t, Xt ) = F (t, Wt ) yields
σ2 1 2
µ−
dYt =
Yt + 0 + σ Yt · 1 dt + σYt · 12 dWt
2
2
= µYt dt + σYt dWt .
2.3
Martingales
Recall the definition of the expectation of a random variable X:
Z
E(X) = X(ω) dP(ω)
Ω
Definition 2.3.1 (conditional expectation) Let X be an integrable random variable,
and let G be a sub-σ-algebra of F. Then, Y is a conditional expectation of X with
respect to G if Y is G-measurable and if
Z
⇔
A
E(X1A ) = E(Y 1A )
Z
X(ω) dP(ω) = Y (ω) dP(ω)
A
In this case, we write Y = E(X | G).
for all A ∈ G,
for all A ∈ G.
Tobias Jahnke
15
“This definition is not easy to love. Fortunately, love is not required.”
J.M. Steele in [Ste01], p. 45.
Interpretation. E(X | G) is a random variable on (Ω, G, P) and hence on (Ω, F, P), too.
Roughly speaking, E(X | G) is the best approximation of X detectable by the events in
G. The more G is refined, the better E(X | G) approximates X.
Examples.
1. If G = {Ω, ∅}, then E(X | G) = E(X).
2. If G = F, then E(X | G) = X.
3. If F ∈ F with P(F ) > 0 and
G = {∅, F, Ω \ F, Ω}
then it can be shown that
Z

1


XdP
if ω ∈ F


 P(F )
F
Z
E(X | G)(ω) =
1


XdP if ω ∈ Ω \ F.


 P(Ω \ F )
Ω\F
Lemma 2.3.2 (Properties of the conditional expectation) For all integrable random variables X and Y and all sub-σ-algebras G ⊂ F, the conditional expectation has the
following properties:
• Linearity: E(X + Y | G) = E(X | G) + E(Y | G)
• Positivity: If X ≥ 0, then E(X | G) ≥ 0.
• Tower property: If H ⊂ G ⊂ F are sub-σ-algebras, then
E E(X | G) | H = E(X | H)
• E E(X | G) = E(X)
• Factorization property: If Y is G-measurable and |XY | and |Y | are integrable, then
E(XY | G) = Y E(X | G)
Proof: Exercise.
Definition 2.3.3 (martingale) Let Xt be a stochastic process which is adapted to a
filtration {Ft : t ≥ 0} of F. If
1. E(|Xt |) < ∞ for all 0 ≤ t < ∞, and
2. E(Xt |Fs ) = Xs for all 0 ≤ s ≤ t < ∞,
then Xt is called a martingale. A martingale Xt is called continuous if there is a set
Ω0 ⊂ Ω with P(Ω0 ) = 1 such that the path t 7→ Xt (ω) is continuous for all ω ∈ Ω0 .
16 –
December 4, 2014
Interpretation: A martingale models a fair game. Observing the game up to time s
does not give any advantage for future times.
Examples. It can be shown that each of the following processes is a continuous martingale
with respect to the standard Brownian filtration:
α2
2
exp αWt − t
Wt ,
Wt − t,
2
Proof: Exercise.
Theorem 2.3.4 (The Itˆ
o integral as a martingale) The Itô integral
Zt
X(t, ω) =
u(s, ω) dW (s, ω).
0
of a function u ∈ H2 [0, T ] is a continuous martingale with respect to the standard Brownian filtration.
Proof: Theorem 6.2, p. 83 in [Ste01].
Remark: The space H2 [0, T ] is defined in Definition B.2.1 in the appendix. If u ∈ L2loc
(see step 4 in section B.2 in the appendix), then the Itô integral is only a local martingale;
cf. Proposition 7.7 in [Ste01].
2.4
The Feynman-Kac formula
Let Xt be the solution of the SDE
dXt = f (t, Xt )dt + g(t, Xt )dWt ,
t ∈ [t0 , T ],
Xt0 = ξ
with suitable functions f and g. Let u(t, x) be the solution of the (deterministic) partial
differential equation (PDE)
1
∂t u(t, x) + f (t, x)∂x u(t, x) + g 2 (t, x)∂x2 u(t, x) = 0,
2
t ∈ [t0 , T ],
x∈R
with terminal condition
u(T, x) = ψ(x)
for some ψ : R −→ R. Apply the Itô formula (Theorem 2.2.2) to u(t, Xt ):
1 2
2
du(t, Xt ) = ∂t u(t, Xt ) + f (t, Xt )∂x u(t, Xt ) + g (t, Xt )∂x u(t, Xt ) dt + g(t, Xt )∂x u(t, Xt ) dWt
2
|
{z
}
=0
Tobias Jahnke
17
Equivalent:
ZT
u(T, XT ) = u(t0 , Xt0 ) +
|{z}
| {z }
ξ
ψ(XT )
g(t, Xt )∂x u(t, Xt ) dWt
t0
Taking the expectation and applying Lemma B.2.7 yields the Feynman-Kac formula
(Richard Feynman, Mark Kac)
E ψ(XT ) = u(t0 , ξ).
Remark: This derivation is informal, because we have tacitly assumed that all terms
exist. See, e.g., Chapter 15 in [Ste01] for a correct proof.
2.5
Extension to higher dimensions
In order to model options on several underlying assets (e.g. basket options), we have to
consider vector-valued Itô integrals and SDEs. A d-dimensional SDE takes the form
Zt
Xj (t) = Xj (0) +
fj (s, X(s)) ds +
m Z
X
t
gjk (s, X(s)) dWk (s)
k=1 0
0
(2.3)
(j = 1, . . . , d)
for d, m ∈ N and suitable functions
fj : R × Rd −→ R,
gjk : R × Rd −→ R.
W1 (s), . . . , Wm (s) are one-dimensional scalar Wiener processes which are pairwise independent. (2.3) is equivalent to
Zt
Zt
f (s, X(s)) ds +
X(t) = X(0) +
0
g(s, X(s)) dW (s)
0
with vectors
T
W (t) = W1 (t), . . . , Wm (t) ∈ Rm
T
f (t, x) = f1 (t, x), . . . , fd (t, x) ∈ Rd
and a matrix

g11 (t, x) · · ·

..
g(t, x) = 
.
gd1 (t, x) · · ·

g1m (t, x)

..
d×m
∈R
.
gdm (t, x)
(2.4)
18 –
December 4, 2014
Theorem 2.5.1 (Multi-dimensional Itˆ
o formula ) Let Xt be the solution of the SDE
(2.4) and let F : [0, ∞) × Rd −→ Rn be a function with continuous partial derivatives
∂t F , ∂xj F , and ∂xj ∂xk F . Then, the process Y (t) := F (t, Xt ) satisfies
dY` (t) = ∂t F` (t, Xt ) dt
+
d
X
∂xi F` (t, Xt ) · fi (t, Xt ) dt
i=1
d
d
1 XX
+
∂x ∂x F` (t, Xt ) ·
2 i=1 j=1 i j
+
d
X
i=1
∂xi F` (t, Xt ) ·
m
X
m
X
!
gik (t, Xt )gjk (t, Xt )
dt
k=1
gik (t, Xt ) dWk
k=1
or equivalently
1 T 2
T
dt + (∇F` )T g dW (t)
dY` = ∂t F` + f ∇F` + tr g (∇ F` )g
2
where ∇F` is the gradient and ∇2 F` is the Hessian of F` , and where tr(A) =
the trace of a matrix A = (aij )i,j ∈ Rm×m .
Pm
j=1
ajj is
Proof: Similar to the case d = m = 1.
Final remarks.
1. Existence and uniqueness of solutions. Ordinary differential equations can
have multiple solutions with the same initial value, and solutions do not necessarily
exist for all times. Hence, we cannot expect that every SDE has a unique solution.
As in the ODE case, however, existence and uniqueness can be shown under certain
assumptions concerning the coefficients f and g. See 4.5 in [KP99] for details.
2. Itˆ
o vs. Stratonovich. The Itô integral is not the only stochastic integral, and
the Stratonovich integral is a famous alternative. The Stratonovich integral has
the advantage that the ordinary chain rule remains valid, i.e. the additional term
in the Itô formula does not appear when the Stratonovich integral is used. Their
disadvantage is the fact that Stratonovich integrals are not martingales, whereas Itô
integrals are. Stratonovich integrals can be transformed into Itô integrals and vice
versa. See 3.1, 3.3 in [Øks03] and 3.5, 4.9 in [KP99].
Actually, the SDEs (2.1) or (2.3) should be called “Itô SDE” or “SDE of the Itô
type”. Since only Itô SDEs and no Stratonovich SDEs will appear in this lecture,
however, we simply use the term “SDE” for “Itô SDE”.
Chapter 3
The Black-Scholes equation
References: [BK04, GJ10, Sey09]
Goal: Find equations to determine the value of an option on a single underlying asset.
Throughout this chapter, we make the assumptions (A1)-(A5) from 1.3 unless otherwise
stated.
3.1
Geometric Brownian motion
First step: Model the price of the underlying by a suitable process St .
For the value of a bond with interest rate r > 0, we have Bt = B0 ert . Try to “stochastify”
this equation with a Wiener process in order to model the underlying.
First attempt: St = S0 eat + σWt for some a, σ ∈ R. Problem: St can have negative values.
Not good.
Second attempt: For the bond we have
ln Bt = ln B0 + rt.
which motivates the ansatz
ln St = ln S0 + at + σWt
for some a, σ ∈ R to model the underlying. The parameter σ is called the volatility.
Applying exp(. . .) gives
St = S0 exp(at + σWt )
and hence St ≥ 0 if S0 ≥ 0. In fact, St is the geometric Brownian motion from 2.2 and
solves the SDE
dSt = µSt dt + σSt dWt
19
20 –
December 4, 2014
with µ = a + σ 2 /2. Interpretation:
dSt
=
µdt
+
σdWt
St
relative change = deterministic trend + random fluctuations
Lemma 3.1.1 (moments of GBM) The geometric Brownian motion
St = S0 exp(at + σWt ),
a = µ − σ 2 /2
with µ ∈ R, σ ∈ R and fixed (deterministic) initial value X0 has the following properties:
1. E(St ) = S0 eµt
2
2. E(St2 ) = S02 e(2µ+σ )t
2
3. V(St ) = S02 e2µt eσ t − 1
Proof: Exercise.
Definition 3.1.2 (log-normal distribution) A vector-valued random variable X(ω) ∈
Rd is log-normal (=log-normally distributed) if ln X = (ln X1 , . . . , ln Xd )T ∈ Rd is normally distributed, i.e. ln X ∼ N (ξ, Σ) for some ξ ∈ Rd and a symmetric, positive definite
matrix Σ ∈ Rd×d . The expectation and the covariance matrix have the entries
Ei (X) = eξi +Σii /2 ,
1
Vij (X) = E Ei (X) − Xi Ej (X) − Xj = eξi +ξj + 2 (Σii +Σjj ) eΣij − 1 .
For d = 1 and Σ = σ 2 the corresponding density is

(ln x − ξ)2
√ 1
exp −
2σ 2
φ(x) = φ(x, ξ, σ) =
2πσx

0
Proof: Exercise.
Example. The (one-dimensional) geometric Brownian motion
St = S0 exp(at + σWt )
is log-normal, because
ln St = ln S0 + at + σWt ∼ N (ln S0 + at, σ 2 t)
if x > 0
else.
Tobias Jahnke
3.2
21
Derivation of the Black-Scholes equation
Situation: St value of an underlying, Bt value of a bond.
Goal: Determine the fair price Vt of an option.
Replication strategy: Consider a portfolio containing at ∈ R underlyings and bt ∈ R
bonds such that
Vt = at St + bt Bt
(cf. section 1.5). Assume that the portfolio is self-financing: no cash inflow or outflow,
i.e. buying an item must be financed by selling another one. Consequence:
Vt+δ − Vt = (at+δ St+δ − at St ) + (bt+δ Bt+δ − bt Bt )
≈ at (St+δ − St ) + bt (Bt+δ − Bt )
for all t ≥ 0 and small δ > 0. For δ −→ 0 we obtain (in an integral sense)
dVt = at dSt + bt dBt .
Now suppose that
dSt = µSt dt + σSt dWt ,
dBt = rBt dt
(3.1)
with µ, σ, r ∈ R. This yields
dVt = at µSt dt + σSt dWt + bt rBt dt
= at µSt + bt rBt dt + at σSt dWt .
(3.2)
Now assume that the value of the option is a function of t and St , i.e. Vt = V (t, St ).
Apply the Itô formula:
1 2
2 2
dV (t, St ) = ∂t V (t, St ) + ∂S V (t, St ) · µSt + ∂S V (t, St ) · σ St dt
2
(3.3)
+ ∂S V (t, St ) · σSt dWt
Equating the dWt -terms in (3.2) and (3.3) yields
at = ∂S V (t, St ),
while equating the dt-terms yields
=⇒
=⇒
1
at µSt + bt rBt = ∂t V (t, St ) + ∂S V (t, St ) · µSt + ∂S2 V (t, St ) · σ 2 St2
2
1 2
bt rBt = ∂t V (t, St ) + ∂S V (t, St ) · σ 2 St2
2
1
1 2
2 2
bt =
∂t V (t, St ) + ∂S V (t, St ) · σ St
Bt r
2
22 –
December 4, 2014
if we assume that Bt r 6= 0. The formulas for at and bt yield
1
1 2
2 2
V (t, St ) = at St + bt Bt = ∂S V (t, St ) · St +
∂t V (t, St ) + ∂S V (t, St ) · σ St Bt .
Bt r
2
Since this is true for every value of St , we can consider S = St as a parameter. Multiplying
with r yields the Black-Scholes equation
∂t V (t, S) +
σ2 2 2
S ∂S V (t, S) + rS∂S V (t, S) − rV (t, S) = 0.
2
Fischer Black and Myron Scholes 1973, Robert Merton 1973
Nobel Prize in Economics 1997
The Black-Scholes equation is a partial differential equation (PDE): It involves partial
derivatives with respect to t and S. This PDE must be solved backwards in time:
instead of an initial condition, we have the terminal condition
V (T, S) = ψ(S)
where T is the expiration time and ψ(S) is the payoff function, i.e. ψ(S) = (S − K)+ for
a call and ψ(S) = (K − S)+ for a put.
The Black-Scholes equation must be solved for S ∈ R+ := [0, ∞) because only nonnegative prices make sense. At the boundary S = 0, no boundary condition is required,
because
σ2 2 2
S ∂S V (t, S) = 0,
S→0 2
lim
lim rS∂S V (t, S) = 0
S→0
if V is sufficiently smooth. For S = 0, we obtain
0 = ∂t V (t, 0) − rV (t, 0)
=⇒
V (t, 0) = e−r(T −t) V (T, 0).
(3.4)
This yields V (t, 0) = 0 for calls and V (t, 0) = e−r(T −t) K for puts.
Remark. Surprisingly, the parameter µ from (3.1) does not appear in the Black-Scholes
equation. A similar observation has been made for the simple discrete model from 1.5.
3.3
Black-Scholes formulas
First goal: Solve the Black-Scholes equation for an European call, i.e.
σ2 2 2
S ∂S V (t, S) + rS∂S V (t, S) − rV (t, S) = 0
2
V (T, S) = (S − K)+
∂t V (t, S) +
with parameters r, σ, K, T > 0.
t ∈ [0, T ], S > 0
Tobias Jahnke
23
Step 1: Transformation to the heat equation
Define new variables:
x(S) = ln(S/K)
σ2
τ (t) = (T − t)
2
V (t, S)
w(τ, x) =
K
Derivatives in new variables:
∂t V (t, S) = K∂t w(τ, x) = K∂τ w(τ, x)
x : (0, ∞) −→ (−∞, ∞)
τ : [0, T ] −→ [0, σ 2 T /2]
w : [0, σ 2 T /2] × (−∞, ∞) −→ R
dτ
σ2
= −K ∂τ w(τ, x)
dt
2
dx
K
∂S V (t, S) = K∂x w(τ, x)
= ∂x w(τ, x)
dS
S
K
∂S2 V (t, S) = . . . = 2 ∂x2 w(τ, x) − ∂x w(τ, x)
S
Insert into the Black-Scholes equation:
dx
1
1
1
because
=
·
=
dS
S/K K
S
σ2 2 2
0 = ∂t V (t, S) + S ∂S V (t, S) + rS∂S V (t, S) − rV (t, S)
2
σ2
σ2 K
K
= −K ∂τ w(τ, x) + S 2 2 ∂x2 w(τ, x) − ∂x w(τ, x) + rS ∂x w(τ, x) − rKw(τ, x)
2
2
S
S
2
Divide by K σ2 :
∂τ w(τ, x) = ∂x2 w(τ, x) − ∂x w(τ, x) + c∂x w(τ, x) − cw(τ, x)
with c := 2r/σ 2 . Next, we eliminate the last three terms. Ansatz:
u(τ, x) = e−αx−βτ w(τ, x),
α, β ∈ R
Substitute:
∂τ u(τ, x) = −βu(τ, x) + e−αx−βτ ∂τ w(τ, x)
= −βu(τ, x) + e−αx−βτ ∂x2 w(τ, x) + (c − 1)∂x w(τ, x) − cw(τ, x)
Since
∂x w(τ, x) = ∂x eαx+βτ u(τ, x) = αeαx+βτ u(τ, x) + eαx+βτ ∂x u(τ, x)
∂x2 w(τ, x) = α2 eαx+βτ u(τ, x) + 2αeαx+βτ ∂x u(τ, x) + eαx+βτ ∂x2 u(τ, x)
it follows that
∂τ u(τ, x) = − βu(τ, x) + α2 u(τ, x) + 2α∂x u(τ, x) + ∂x2 u(τ, x)
+ (c − 1) αu(τ, x) + ∂x u(τ, x) − cu(τ, x)
2
2
=∂x u(τ, x) + 2α + (c − 1) ∂x u(τ, x) + − β + α + (c − 1)α − c u(τ, x)
24 –
December 4, 2014
Hence, the terms including u(τ, x) and ∂x u(τ, x) vanish if
−β + α2 + (c − 1)α − c = 0
2α + (c − 1) = 0.
and
The solution is
1
α = − (c − 1),
2
1
β = − (c + 1)2 = −(1 − α)2 .
4
With these parameters, u(τ, x) solves the heat equation
∂τ u(τ, x) = ∂x2 u(τ, x),
x ∈ R, τ ∈ [0, σ 2 T /2]
with initial condition
u(0, x) = e−αx w(0, x) = e−αx
V (T, S)
(S − K)+
= e−αx
= e−αx (ex − 1)+
K
K
since S = Kex .
Step 2: Solving the heat equation
Lemma 3.3.1 (solution of the heat equation) Let u0 : R −→ R be a continuous
function which satisifes the growth condition
2
|u0 (x)| ≤ M eγx .
with constants M > 0 and γ ∈ R. Then, the function
1
u(τ, x) = √
4πτ
Z∞
(x − ξ)2
exp −
u0 (ξ) dξ
4τ
−∞
is the unique solution of the heat equation
∂τ u(τ, x) = ∂x2 u(τ, x),
x ∈ R, τ > 0
and for all x ∈ R we have
lim u(τ, x) = u0 (x).
τ →0
Proof. The fact that u solves the PDE can be checked by substituting and computing
the partial derivatives
(exercise). The last assertion can be verified via the transformation
√
η = (ξ − x)/ 4τ (exercise). Uniqueness follows from the maximum principle.
Tobias Jahnke
By a tedious1 calculation, it can be shown that
u(τ, x) = exp (1 − α)x + (1 − α)2 τ Φ(d1 ) − exp −αx + α2 τ Φ(d2 )
Zx
1
2
Φ(x) = √
e−s /2 ds
2π
−∞
σ2
S
ln K + r ± 2 (T − t)
√
d1/2 =
σ T −t
25
(3.5)
(3.6)
Remark: Φ(x) is the cumulative distribution function of the standard normal distribution.
Step 3: Inverse transform
Since β = −(1 − α)2 and β + α2 = 2α − 1 = −c it follows that
V (t, S) = Kw(τ, x) = K exp(αx + βτ )u(τ, x)
= K exp(αx + βτ ) exp (1 − α)x + (1 − α)2 τ Φ(d1 )
− K exp(αx + βτ ) exp −αx + α2 τ Φ(d2 )
2
= K exp(x) Φ(d1 ) − K exp (β + α ) τ Φ(d2 )
| {z }
| {z }
−c
=S
= SΦ(d1 ) − K exp(−r(T − t))Φ(d2 )
Check boundary: Since limS→0 Φ d1/2 (S) = 0 we obtain
h
i
lim V (t, S) = lim SΦ d1 (S) − KΦ d2 (S) = 0 ⇐⇒ (3.4) X
S&0
S&0
Check terminal condition:
V (T, S) = SΦ(d1 ) − KΦ(d2 )
By definition of d1/2 = d1/2 (t)
ln
lim d1/2 (t) = lim
t→T
t→T
S
K


if S > K
∞
S
+ r±
(T − t)
ln K
√
= lim √
= 0
if S = K
t→T σ T − t

σ T −t

−∞ if S < K
σ2
2
and hence


if S > K
1
lim Φ(d1/2 (t)) = 1/2 if S = K
t→T


0
if S < K
1
=⇒


S − K
lim V (t, S) = 0
t→T


0
if S > K
if S = K
if S < K
... so tedious that we do not even dare to ask the reader to prove this as an exercise.
X
26 –
December 4, 2014
All in all, we have shown the following
Theorem 3.3.2 (Black-Scholes formula for calls) If r, σ, K, T > 0, then the BlackScholes formula
V (t, S) = SΦ(d1 ) − K exp(−r(T − t))Φ(d2 )
with Φ and d1/2 from (3.5) and (3.6), respectively, is the (unique) solution of the BlackScholes equation for European calls, i.e.
σ2 2 2
∂t V (t, S) + S ∂S V (t, S) + rS∂S V (t, S) − rV (t, S) = 0
2
V (T, S) = (S − K)+ .
t ∈ [0, T ], S > 0
Corollary 3.3.3 (Black-Scholes formula for puts) The Black-Scholes equation for a
European put
σ2 2 2
∂t V (t, S) + S ∂S V (t, S) + rS∂S V (t, S) − rV (t, S) = 0
2
V (T, S) = (K − S)+
t ∈ [0, T ], S > 0
with r, σ, K, T > 0 has the unique solution
V (t, S) = K exp(−r(T − t))Φ(−d2 ) − SΦ(−d1 )
with Φ and d1/2 from (3.5) and (3.6), respectively.
Proof. Let VC (t, S) be the value of a call with the same T and K. The put-call-parity
(Lemma 1.4.1) and Theorem 3.3.2 imply
V (t, S) = e−r(T −t) K + VC (t, S) − S
= e−r(T −t) K + SΦ(d1 ) − K exp(−r(T − t))Φ(d2 ) − S
= e−r(T −t) K 1 − Φ(d2 ) + S Φ(d1 ) − 1
= e−r(T −t) KΦ(−d2 ) − SΦ(−d1 )
because Φ(x) + Φ(−x) = 1.
European Call (K = 100, T = 1)
European Put (K = 100, T = 1)
100
80
100
t=0
t = 0.25
t = 0.5
t = 0.75
t=1
t=0
t = 0.25
t = 0.5
t = 0.75
t=1
80
V(t,S)
60
V(t,S)
60
40
40
20
20
0
0
50
100
S
150
200
0
0
50
100
S
150
200
Tobias Jahnke
27
Definition 3.3.4 (Greeks) For a European option with value V (t, S) we define “the
greeks”
delta:
gamma:
vega/kappa:
∆ = ∂S V
Γ = ∂S2 V
κ = ∂σ V
theta:
rho:
θ
ρ
= ∂t V
= ∂r V
These partial derivatives can be considered as “condition numbers” which measure the
sensitivity of V (t, S) with respect to the corresponding parameters. This information is
important for stock broker.
Remark: Explicit formulas for the greeks can be derived from the Black-Scholes formulas
(execise).
3.4
Risk-neutral valuation and equivalent martingale
measures
In 1.5 we have seen that in the simplified two-scenario model the value of an option can be
priced by replication. The same strategy was applied to the refined model in the previous
section. In the simple situation considered in 1.5, the value of an option turned out to
be the discounted expectation of the payoff under the risk-neutral probability. In this
subsection, we will see that this is also true for the refined model from 3.2.
Theorem 3.4.1 (Option price as discounted expectation) If V (t, S) is the solution
of the Black-Scholes equation
σ2 2 2
S ∂S V (t, S) + rS∂S V (t, S) − rV (t, S) = 0
2
V (T, S) = ψ(S)
∂t V (t, S) +
t ∈ [0, T ], S > 0
with payoff function ψ(S), then
V (t? , S? ) = e−r(T −t? )
Z∞
ψ(x)φ(x, ξ, β) dx
(3.7)
0
for all t? ∈ [0, T ] and S? > 0. The function φ is the density of the log-normal distribution
(cf. Definition 3.1.2) with parameters
p
σ2
(T − t? ),
β = σ T − t? .
ξ = ln S? + r −
(3.8)
2
The assertion can be shown by showing that the above representation coincides with the
Black-Scholes formulas for puts and calls. Such a proof, however, involves several changes
28 –
December 4, 2014
of variables in the integral representations and rather tedious calculations. We give a
shorter and more elegant proof:
Proof. Step 1: In our derivation of the Black-Scholes model, we have assumed that
dSt = µSt dt + σSt dWt ,
i.e. that the price of the underlying is a geometric Brownian motion with drift µSt ;
cf. (3.1). It turned out, however, that the parameter µ does not appear in the BlackScholes equation. Hence, we can choose µ = r and consider the SDE
dSbt = rSbt dt + σ Sbt dWt ,
t ∈ [t? , T ]
Sbt? = S?
as a model for the stock price.
Step 2: The function u(t, S) := er(T −t) V (t, S) solves the PDE
∂t u(t, S) +
σ2 2 2
S ∂S u(t, S) + rS∂S u(t, S) = 0,
2
t ∈ [0, T ]
because
σ2 2 2
S ∂S u(t, S) + rS∂S u(t, S)
2
σ2
= − rer(T −t) V (t, S) + er(T −t) ∂t V (t, S) + S 2 er(T −t) ∂S2 V (t, S) + rSer(T −t) ∂S V (t, S)
2
2
σ 2 2
r(T −t)
=e
−rV (t, S) + ∂t V (t, S) + S ∂S V (t, S) + rS∂S V (t, S) = 0.
2
|
{z
}
∂t u(t, S) +
=0
(Black-Scholes equation)
Moreover, u satisfies the terminal condition
u(T, S) = V (T, S) = ψ(S).
Step 3: Applying the Feynman-Kac formula (cf. 2.4) with f (t, S) = rS and g(t, S) = σS
yields
E ψ(SbT ) = u(t? , S? ) = er(T −t? ) V (t? , S? )
and thus
V (t? , S? ) = e−r(T −t? ) E ψ(SbT ) .
We know that SbT is log-normal, i.e.
Z∞
E ψ(SbT ) =
0
where φ(x, ξ, β) is the density of the log-normal distribution with parameters (3.8); cf. the
example after Definition 3.1.2.
Tobias Jahnke
29
Interpretation. We know from Definition 3.1.2 that
Z∞
E SbT = xφ(x, ξ, β) dx
0
β2
= exp ξ +
2
2 1 p
σ2
(T − t? ) +
σ T − t?
= exp ln S? + r −
2
2
= exp (ln S? + r(T − t? ))
= S? exp (r(T − t? )) .
This means that for µ = r the expected value of the stock is exactly the money obtained
by investing S? into a bond at time t? and waiting until T − t? . Hence, the log-normal
distribution with parameters (3.8) defines the risk-neutral probability; cf. 1.5. The
integral in (3.7) is precisely the expected payoff under the risk-neutral probability, and
(3.7) states that the price of the option is obtained by discounting the expected payoff.
A different perspective. Consider now the geometric Brownian motion
dSt = µSt dt + σSt dWt
with µ 6= r. Since E(St ) = S0 eµt , an investor expects µ > r as a compensation for the
risk, because otherwise he might prefer to invest into the riskless bond Bt = B0 ert . The
term
γ=
µ−r
σ
is called market price of risk, and we have
dSt = rSt dt + σSt (γ dt + dWt ) = rSt dt + σSt dWtγ ,
with Wtγ = γ t + Wt .
Problem: Wtγ is not a Wiener process under the probability measure P, because E(Wtγ ) =
γ t + E(Wt ) = γ t 6= 0 for t > 0 and µ 6= r.
Question: Is there another probability measure Q such that dWtγ is a Wiener process
under Q?
Definition 3.4.2 (equivalent martingale measure) Let (Ω, F, P) be a probability space
with filtration {Ft : t ≥ 0}. A probability measure Q is called an equivalent martingale
measure or risk-neutral probability if there is a random variable Y > 0 such that
R
• Q(A) = E(1A · Y ) = A Y (ω)dP(ω) for all events A ∈ F, and
• e−rt St is a martingale under Q with respect to the filtration {Ft : t ≥ 0}.
30 –
December 4, 2014
Remark. The first property implies that P(A) > 0 ⇐⇒ Q(A) > 0 (“equivalent”).
Now let
γ2
YT := exp −γWT − T
2
and
Q(A) = E(1A · YT ).
Then, Girsanov’s theorem states that Wtγ = γ t + Wt is a Wiener process under Q (see
e.g. 4.4 in [Ben04], 8.6 in [Øks03]). Moreover, the Itô formula yields
d(e−rt St ) = −re−rt St dt + e−rt rSt dt + σSt dWtγ = σe−rt St dWtγ .
Hence, e−rt St is a martingale under Q, and Q is an equivalent martingale measure. All
in all, we have exchanged
µ −→ r,
P −→ Q,
Wt −→ Wtγ .
If µ = r, then P = Q and Wt = Wtγ . Now we are back in the situation of Theorem 3.4.1,
and it follows that
V (t? , S? ) = e−r(T −t? ) EQ (ψ(ST ))
(3.9)
where St is the solution of
dSt = rSt dt + σSt dWtγ ,
St? = S? .
t ∈ [t? , T ]
General pricing formula. Up to now, we have only considered European options, i.e.
options with a payoff that depends only on the value of the underlying at maturity. For
Asian or barrier options, the pricing formula (3.9) can be generalized to
Vt = e−r(T −t? ) EQ (VT |Ft )
(cf. 5.2.4 in [Shr04]).
Fundamental theorems of option pricing:
• If a market model has at least one equivalent martingale measure, then there is no
arbitrage possibility (cf. Theorem 5.4.7 in [Shr04]).
• Consider a market model with at least one equivalent martingale measure. Then,
the equivalent martingale measure is unique if and only if the model is complete, i.e.
if every derivative (options, forwards, futures, swaps, ...) can be replicated (hedged)
(cf. Theorem 5.4.9 in [Shr04]).
Tobias Jahnke
3.5
31
Extensions
The “standard” Black-Scholes model can be generalized in several ways:
• Options with d > 1 underlyings (e.g. basket options) are modeled by the d-dimensional
Black-Scholes equation
∂t V +
d
d
X
1X
ρij σi σj Si Sj ∂Si ∂Sj V + r
Si ∂Si V − rV = 0
2 i,j=1
i=1
where V = V (t, S1 , . . . , Sd ), r, σi > 0 and ρij ∈ [−1, 1] are the correlation coefficients.
• Non-constant interest rate and volatility: r = r(t, S), σ = σ(t, S)
• Stochastic volatility: Either σ = σ(ω) is a random variable with known distribution
or σ = σt (ω) is a stochastic process.
• Dividends: When a dividend δ · St with δ ≥ 0 is paid at time t, the price of the
underlying drops by the same amount due to the no-arbitrage assumption. Hence,
a continuous flow of dividends can be modeled by
dSt = (µ − δ)St dt + σSt dWt ,
which yields the Black-Scholes equation
∂t V (t, S) +
σ2 2 2
S ∂S V (t, S) + (r − δ)S∂S V (t, S) − rV (t, S) = 0.
2
Black-Scholes formulas with dividends: 4.5.1 in [GJ10].
• Nonzero transaction costs =⇒ nonlinear Black-Scholes equation
• Discontinuous underlyings: Jump-diffusion models, Black-Scholes PDE with additional integral term
Remark: Some of these extensions will be considered in the lecture Numerical methods
in mathematical finance II (summer term).
Part II
Numerical methods
32
Chapter 4
Binomial methods
Situation: Let S(t) be the value of an underlying, and let V (t, S) be the value of an
option with maturity T > 0.
Assumptions: Assume (A1)-(A5) from Section 1.3.
Goal: Approximate V (t, S) by a numerical method.
Idea: Refine the simple discrete model from 1.5 such that it approximates the continuoustime Black-Scholes model.
Remark: For European calls/puts, such an approximation is not necessary, because
V (t, S) can be computed via the Black-Scholes formula. Nevertheless, such options will
serve as a model problem. The numerical method can be extended to other types of
options.
4.1
Derivation
Discretize the time-interval [0, T ]: Choose N ∈ N, let τ = T /N and tn = n · τ .
Let Sn be the price of the underlying at time tn . Bond: B(t) = B(0)ert
Additional assumptions:
1. For a given price Sn , the price at tn+1 = tn + τ is
(
u · Sn with probability p
Sn+1 =
d · Sn with probability 1 − p
with (unknown) u > 1, 0 < d < 1, p ∈ [0, 1].
2. The expected profit from investing into the underlying is the same as for the bond:
E Sn+1 |Sn = erτ Sn .
2
2
3. E Sn+1
|Sn = e(2r+σ )τ Sn2 with given volatility σ ∈ R.
Remark: In the continuous-time model where S(t) is modeled by a geometric Brownian
motion, the last two conditions hold for Sn = S(tn ) if µ = r (risk-neutral pricing).
33
34 –
December 4, 2014
Compute u, d, p:
erτ − d
1. erτ Sn = E Sn+1 |Sn = uSn p + dSn (1 − p) ⇐⇒ p =
u−d
Since p ∈ [0, 1], we have d ≤ erτ ≤ u.
2
2
2
2
2. e(2r+σ )τ Sn2 = E Sn+1
|Sn = uSn p + dSn (1 − p)
Only two conditions for three unknowns u, d, p. Choose third condition:
!
3. u · d = 1
Solution of 1.-3.:
p
β2 − 1
p
1
d = = β − β2 − 1
u
u=β+
1 −rτ
2
e
+ e(r+σ )τ
2
erτ − d
p=
.
u−d
β :=
Replication strategy: Consider a portfolio containing a ∈ R underlyings and b ∈ R bonds
such that
!
aSn + bB(tn ) = V tn , Sn =: Vn
It follows that
E Vn+1 | Vn = aE(Sn+1 | Sn ) + berτ B(tn )
= erτ aSn + bB(tn )
= erτ Vn
4.2
(4.1)
Algorithm
Cox, Ross & Rubinstein 1979
1. Forward phase: initialization of the tree. For all n = 0, . . . , N and j = 0, . . . , n
let
Sjn = uj dn−j S(0) = (approximate) price of the underlying at time tn
after j “ups” and n − j “downs”.
The condition d · u = 1 implies that
S(0) = S00 = S12 = S24 = . . .
S11 = S23 = S35 = . . .
S01 = S13 = S25 = . . .
At tn there are only n + 1 possible values S0n , . . . , Snn of the underlying.
Tobias Jahnke
35
S00 = S(0)
for n = 0, 1, 2, . . . , N − 1
S0,n+1 = dS0,n
for j = 0, . . . , n
Sj+1,n+1 = uSj,n
end
end
2. Backward phase: compute the option values. Let Vjn be the value of the option
after j “ups” and n − j “downs” of the underlying. At maturity, we have
VjN = ψ(SjN ),
(
(SjN − K)+
ψ(SjN ) =
(K − SjN )+
(call)
(put)
Use (4.1):
=⇒
erτ Vjn = E V (tn+1 ) | Vjn = pVj+1,n+1 + (1 − p)Vj,n+1
−rτ
Vjn = e
pVj+1,n+1 + (1 − p)Vj,n+1
(a) European options:
for j = 0, . . . , N
VjN = ψ(SjN )
end
for n = N − 1, N − 2, . . . , 0
for j = 0, . . . ,n
Vjn = e−rτ pVj+1,n+1 + (1 − p)Vj,n+1
end
end
Result:
V00
(b) American options: Check in each step if early exercise is advantageous. At time
tn the value of the option must not be less than ψ(Sjn ) for all j.
36 –
December 4, 2014
15
S
jn
10
5
0
0
0.2
0.4
0.6
0.8
1
time
Figure 4.1: Illustration of the binomial method (S(0) = 5, T = 1, N = 10).
for j = 0, . . . , N
VjN = ψ(SjN )
end
for n = N − 1, N − 2, . . . , 0
for j = 0, . . . ,n
−rτ
˜
Vjn = e
pVj+1,n+1 + (1 − p)Vj,n+1
Vjn = max{V˜jn , ψ(Sjn )}
end
end
Result:
V00
Remarks.
1. The result V00 is only an approximation for the true value V (0, S(0)) of the option,
because the price process has been approximated.
2. The result V00 ≈ V (0, S(0)) depends on the initial value S(0). For a different value
of S(0), the entire computation must be repeated.
3. An efficient implementation of the binomial method requires only O(N ) operations;
see [Hig02].
Examples: see slides
Tobias Jahnke
4.3
37
Discrete Black-Scholes formula
Lemma 4.3.1 Let V (t, S) be the value of a European option with payoff function ψ(S)
and maturity T > 0. Then, the binomial method yields the approximation
V00 = e
−rT
N
X
B(j, N, p)ψ(SjN )
j=1
where
N j
B(j, N, p) =
p (1 − p)N −j
j
is the binomial distribution.
Proof: exercise.
Remarks:
1. This result explains the name “binomial method”.
2. Interpretation: V00 is the discounted expected payoff under a suitable probability;
cf. 1.5 and 3.4.
Proposition 4.3.2 (Discrete Black-Scholes formula) Under the conditions of Lemma 4.3.1,
an equivalent formula for the option price is
V00 = S(0)Ψ(m, N, q) − K exp(−rT )Ψ(m, N, p)
where
q = upe−rτ
m = min{0 ≤ j ≤ N : (SjN − K) ≥ 0}.
Ψ(m, N, p) =
N
X
B(j, N, p)
j=m
Proof: exercise.
Question: What happens if we let N −→ ∞ and τ = T /N −→ 0?
Proposition 4.3.3 (Convergence of the discrete Black-Scholes formula) Let σ
ˆ=
√
√
√
(N )
σ
ˆ τ
−ˆ
σ τ
and d = 1/u = e
. If V00 = V00 is the approximation given
(ln u)/ τ , i.e. u = e
(N )
by the binomial method with τ = T /N , then V00 converges to the value given by the
38 –
December 4, 2014
(continuous) Black-Scholes equation:
(N )
lim V00
N →∞
= S(0)Φ(d1 ) − K exp(−rT )Φ(d2 )
Zx
1
2
Φ(x) = √
e−s /2 ds
2π
−∞
σ
ˆ2
+
r
±
T
ln S(0)
K
2
√
d1/2 =
σ
ˆ T
Proof: See 3.3 in [GJ10] (use central limit theorem).
Chapter 5
Numerical methods for stochastic
differential equations
5.1
Motivation
According to 3.4 the value of a European option is the discounted expected payoff under
the risk-neutral probability:
V (0, S0 ) = e−rT EQ ψ S(T )
For the standard Black-Scholes model:
−rT
Z∞
V (0, S0 ) = e
0
with log-normal density φ and parameters
σ2
T,
ξ = ln S0 + r −
2
√
β = σ T.
Two ways to price the option:
1. Quadrature formula. Let w(x) := ψ(x)φ(x, ξ, β).
Choose 0 ≤ xmin < xmax such that w(x) ≈ 0 for x 6∈ [xmin , xmax ].
xmin = K and xmax sufficiently large for calls, xmin = 0 and xmax = K for puts.
Choose large N ∈ N, let h = (xmax − xmin )/N and xk = xmin + kh. Approximate
Z∞
x
Zmax
w(x) dx ≈
0
w(x) dx =
xmin
N
−1
X
x
Zk+1
w(x) dx ≈
k=0 x
k
with suitable nodes cj ∈ [0, 1] and weights bj .
39
N
−1
X
k=0
h
s
X
j=1
bj w(xk + cj h)
40 –
December 4, 2014
2. Monte-Carlo method. In the Black-Scholes model, S(t) is defined by the SDE
t ∈ [0, T ],
dS(t) = rS(t)dt + σS(t)dW (t),
S0 given
(risk-neutral, µ = r)
Solution: Geometric Brownian motion
σ2
S(t) = S0 exp
r−
t + σW (t) .
2
This is the process which corresponds to φ(x, ξ, β), because S(T ) is log-normal with
the same parameters. Estimate the expectated payoff as follows:
• Generate many realizations S(T, ω1 ), . . . , S(T, ωm ), m ∈ N “large”.
• Approximate
m
V (0, S0 ) ≈ e
−rT
1 X
ψ S(T, ωj )
m j=1
Consider now a more complicated price process:
dS(t) = rS(t)dt + σS(t)dW 1 (t)
p
2
2
1
2
2
dσ (t) = κ θ − σ (t) dt + ν ρdW (t) + 1 − ρ dW (t)
(5.1a)
(5.1b)
Heston model with parameters r, κ, θ, ν > 0, initial values S0 , σ0 , independent scalar
Wiener processes W 1 (t), W 2 (t) , correlation ρ ∈ [−1, 1]
Steven L. Heston 1993
Now the volatility is not a parameter, but a stochastic process defined by a second SDE.
We do not have an explicit formula for S(t) and σ(t), but the Monte-Carlo approach is
still feasible:
• Choose N ∈ N, define step-size τ = T /N and tn = nτ . For each ω1 , . . . , ωm compute
approximations
Xn(1) (ωj ) ≈ S(tn , ωj ),
Xn(2) (ωj ) ≈ σ 2 (tn , ωj ),
n = 0, . . . , N
by solving the SDEs (5.1a), (5.1b) numerically.
• Approximate
m
V (0, S0 ) ≈ e
−rT
1 X
(1)
ψ XN (ωj )
| {z }
m j=1
≈S(T,ωj )
The Monte-Carlo approach even works for other types of options. As an example, consider
an Asian option with payoff
+

ZT
1
S(t) dt
(average strike call).
ψ t 7→ S(t) = S(T ) −
T
0
Tobias Jahnke
41
Now the payoff depends on the entire path t 7→ S(t). We approximate
S(tn , ωj ) ≈
1
T
Xn(1) (ωj ),
ZT
0
N
1 X (1)
S(t, ωj ) dt ≈
X (ωj )
N + 1 n=0 n
and hence
!+
m
N
X
X
1
1
(1)
V (0, S0 ) ≈ e−rT
XN (ωj ) −
X (1) (ωj )
m j=1
N + 1 n=0 n
Remark: In the original paper, Heston derives an explicit Black-Scholes-type formula
for European options by means of characteristic functions. Hence, European options in
the Heston model can also be priced by quadrature formulas, but for Asian options this
is impossible.
Goal: Construct and analyze numerical methods for SDEs.
5.2
Euler-Maruyama method
5.2.1
Derivation
Consider the one-dimensional SDE
dX(t) = f t, X(t) dt + g t, X(t) dW (t),
t ∈ [0, T ],
X(0) = X0
with suitable functions f and g and a given initial value X0 . Choose N ∈ N, define
step-size τ = T /N and tn = nτ .
tZn+1
X(tn+1 ) = X(tn ) +
tZn+1
g s, X(s) dW (s)
f s, X(s) ds +
tn
tn
≈ X(tn ) + (tn+1 − tn ) f tn , X(tn ) + g tn , X(tn ) W (tn+1 ) − W (tn )
| {z }
|
{z
}
=τ
=:∆Wn
Replacing X(tn ) −→ Xn and “≈” −→ “=” yields the
Euler-Maruyama method (Gisiro Maruyama 1955, Leonhard Euler 1768-70):
For n = 0, . . . , N − 1 let ∆Wn = W (tn+1 ) − W (tn ) and
Xn+1 = Xn + τ f tn , Xn + g tn , Xn ∆Wn .
Hope that Xn ≈ X(tn ):
SDE
X(t)
Xn
exact
recursion
=⇒
?
approx. ⇐=
approx.
exact
42 –
December 4, 2014
The exact solution X(tn ) and the numerical approximation Xn are random variables. For
every path t 7→ W (t, ω) of the Wiener process, a different result is obtained. X(t) is
called strong solution if t 7→ W (t, ω) is given, and weak solution if t 7→ W (t, ω) can
be chosen. Approximations of weak solutions: For each n, generate a random number
Zn ∼ N (0, 1) and let
√
∆Wn = τ Zn .
Question: Does Xn really approximate X(tn )? In which sense? How accurately?
5.2.2
Weak and strong convergence
Definition 5.2.1 (strong and weak convergence) Let T > 0, N ∈ N, τ = T /N and
tn = nτ . An approximation Xn (ω) ≈ X(tn , ω) converges
• strongly with order γ > 0, if there is a constant C > 0 independent of τ such that
max E |X(tn ) − Xn | ≤ Cτ γ
n=0,...,N
for all sufficiently small τ , and
• weakly with order γ > 0 with respect to a function F : R −→ R, if there is a
constant C > 0 independent of τ such that
max E F X(tn ) − E F (Xn ) ≤ Cτ γ
n=0,...,N
for all sufficiently small τ .
Remarks:
• Strong convergence =⇒ path-wise convergence
Weak convergence =⇒ convergence of moments (if F (x) = xk ) or probabilities (if
F (x) = 1[a,b] ).
• Strong convergence ←→ Asian options
Weak convergence ←→ European options
• Strong convergence of order γ implies weak convergence of order γ with respect to
F (x) = x (exercise).
5.2.3
Existence and uniqueness of solutions of SDEs
Theorem 5.2.2 (existence and uniqueness)
Let f : R+ × R −→ R and g : R+ × R −→ R be functions with the following properties:
• Lipschitz condition: There is a constant L ≥ 0 such that
|f (t, x) − f (t, y)| ≤ L|x − y|,
for all x, y ∈ R and t ≥ 0.
|g(t, x) − g(t, y)| ≤ L|x − y|
(5.2)
Tobias Jahnke
• Linear growth condition: There is a constant K ≥ 0 such that
|f (t, x)|2 ≤ K 1 + |x|2 ,
|g(t, x)|2 ≤ K 1 + |x|2
43
(5.3)
for all x ∈ R and t ≥ 0.
Then, the SDE
dX(t) = f t, X(t) dt + g t, X(t) dW (t),
t ∈ [0, T ]
with deterministic initial value X(0) = X0 has a continuous adapted solution and
sup E X 2 (t) < ∞.
t∈[0,T ]
e
If both X(t) and X(t)
are such solutions, then
e
P X(t) = X(t)
for all t ∈ [0, T ] = 1.
Proof: Theorem 9.1 in [Ste01] or Theorem 4.5.3 in [KP99].
Remark: The assumptions can be weakened.
5.2.4
Strong convergence of the Euler-Maruyama method
For simplicity, we only consider the autonomous SDE
dX(t) = f X(t) dt + g X(t) dW (t),
t ∈ [0, T ]
and the Euler-Maruyama approximation
Xn+1 = Xn + τ f Xn + g Xn ∆Wn .
with X(0) = X0 , T > 0, N ∈ N, τ = T /N , tn = nτ .
We assume that f = f (x) and g = g(x) satisfy the Lipschitz condition (5.2). In the
autonomous case, this implies the linear growth condition (5.3) (exercise).
Theorem 5.2.3 (strong error of the Euler-Maruyama method) Under these conditions, there is a constant Cˆ such that
ˆ 1/2
max E |X(tn ) − Xn | ≤ Cτ
n=0,...,N
for all sufficiently small τ . Cˆ does not depend on τ .
For the proof we need the following
44 –
December 4, 2014
Lemma 5.2.4 (Gronwall) Let α : [0, T ] −→ R+ be a positive integrable function. If
there are constants a > 0 and b > 0 such that
Zt
0 ≤ α(t) ≤ a + b
α(s) ds
0
for all t ∈ [0, T ], then α(t) ≤ aebt .
Proof: exercise.
Proof of Theorem 5.2.3.
Strategy:
• Define the step function
Y (t) =
N
−1
X
1[tn ,tn+1 ) (t)Xn
for t ∈ [0, T ),
Y (T ) := XN .
n=0
For n = 0, . . . , N − 1 this means that
⇐⇒ t ∈ [tn , tn+1 ).
• Define α(s) := sup E |Y (r) − X(r)|2 and prove the Gronwall inequality
Y (t) = Xn
r∈[0,s]
Zt
0 ≤ α(t) ≤ Cτ + b
α(s) ds.
(5.4)
0
• Apply Gronwall’s lemma. This yields α(t) ≤ τ Cˆ 2 with Cˆ 2 = CebT for all t ∈ [0, T ]
p
• Since1 E(Z) ≤ E(Z 2 ) for random variables Z, it follows that
max E |Xn − X(tn )| ≤ sup E |Y (t) − X(t)|
n=0,...,N
t∈[0,T ]
≤ sup
q
E |Y (t) − X(t)|2
t∈[0,T ]
p
√
= α(T ) ≤ τ Cˆ
Main challenge: Prove Gronwall inequality (5.4). Choose fixed t ∈ [0, T ] and let n be the
index with t ∈ [tn , tn+1 ).
Elementary calculation: 0 ≤ V(Z) = E (Z − E(Z))2 = E Z 2 − 2ZE(Z) + E(Z)2 = E Z 2 −
2
2
E(Z) and hence E(Z) ≤ E Z 2 .
1
Tobias Jahnke
45
Derive integral representation of the error:
Y (t) = Xn = X0 +
= X0 +
n−1
X
(Xk+1 − Xk )
k=0
n−1
X
τ f (Xk ) + g(Xk )∆Wk
k=0
t
t
n−1 Zk+1
n−1 Zk+1
X
X
f (Xk ) ds +
g(Xk ) dW (s)
= X0 +
k=0 t
k
k=0 t
k
Ztn
= X0 +
Ztn
f (Y (s)) ds +
0
g (Y (s)) dW (s)
0
This is not an SDE, because we have
solution
Zt
X(t) = X(0) +
R tn
0
. . . instead of
f X(s) ds +
0
Zt
Rt
0
. . . . Comparing with the exact
g X(s) dW (s)
0
yields the error representation
Y (t) − X(t) =
Ztn h
f (Y (s)) − f X(s)
|0
i
ds +
{z
}
=:T1
Zt
−
f X(s) ds −
tn
|
Zt
Ztn h
i
g (Y (s)) − g X(s) dW (s)
|0
{z
=:T2
}
g X(s) dW (s)
tn
{z
=:T3
}
{z
|
=:T4
}
= T 1 + T2 − T 3 − T 4 .
The Cauchy-Schwarz inequality gives
(T1 + T2 − T3 − T4 )2 = (1, 1, −1, −1)T
2
≤ 4kT k22 = 4 · (T12 + T22 + T32 + T42 )
and hence
E |Y (t) − X(t)|2 ≤ 4 · E T12 + T22 + T32 + T42 .
46 –
December 4, 2014
First term: For functions u ∈ L2 ([0, tn ]) the Cauchy-Schwarz inequality yields
t
2
Zn
Ztn
Ztn
2
 u(s) · 1 ds ≤ |u(s)| ds · 12 ds .
0
0
(5.5)
|0 {z }
=tn
Using the Lipschitz bound (5.2), we obtain
2 
 t
Zn h
i
f (Y (s)) − f X(s) ds 
E T12 = E 
0 t

Zn 2
≤ tn E  f (Y (s)) − f X(s) ds
0
Ztn 2 2
≤ TL
E Y (s) − X(s) ds
0
≤ T L2
Zt
α(s) ds
(t instead of tn )
0
because t ≥ tn by assumption.
Second term: It follows from the Itô isometry (Theorem B.2.5) and the Lipschitz bound
(5.2) that
2 
 t
Zn h
i
g (Y (s)) − g X(s) dW (s) 
E T22 = E 

0
t
n
Z
= E

2
g (Y (s)) − g X(s) ds
0
Ztn 2 2
≤L
E Y (s) − X(s) ds
0
≤ L2
Zt
α(s) ds
0
because t ≥ tn by assumption.
Tobias Jahnke
47
Third term: Equation (5.5) and the linear growth bound (5.3) yield
2 
 t
Z
E T32 = E  f X(s) ds 
tn

Zt
≤ (t − tn )E 

2
f X(s) ds
tn

Zt
≤ τK · E 

1 + |X(s)|2 ds ≤ cτ 2
tn
because Theorem 5.2.2 states that E (1 + |X(s)|2 ) remains bounded on [tn , t].
Last term: Using the Itô isometry and the linear growth bound (5.3) it follows that
 t
2 
Z
E T42 = E  g X(s) dW (s) 
tn

Zt
= E

g X(s) 2 ds
tn

Zt
≤ K · E

1 + |X(s)|2 ds ≤ cτ
tn
These bounds yield the Gronwall inequality (5.4) with b = 4(T + 1)L2 and with C depending on K and sups∈[0,T ] E (1 + |X(s)|2 ).
5.2.5
Weak convergence of the Euler-Maruyama method
Theorem 5.2.5 (weak error of the Euler-Maruyama method) Under the conditions
of 5.2.4, there is a constant Cˆ such that
ˆ
max E F X(tn ) − E F (Xn ) ≤ Cτ
n=0,...,N
for all sufficiently small τ and all smooth functions F . Cˆ does not depend on τ .
Proof. Define piecewise linear interpolation: In addition to the piecewise constant
Y (t), we define the piecewise linear interpolation
Z(t) = Xn + (t − tn )f Xn + g Xn W (t) − W (tn )
for t ∈ [tn , tn+1 ).
48 –
December 4, 2014
Since Y (t) = Xn for t ∈ [tn , tn+1 ), this is equivalent to
Zt
Z(t) = X(0) +
f Y (s) ds +
0
or
Zt
g Y (s) dW (s)
0
dZ = f (Y )dt + g(Y )dW (t).
Properties:
• Z(tn ) = Xn = Y (tn ) for all n = 0, . . . , N .
• For every δ ∈ [0, τ ], Z(tn + δ) is the Euler-Maruyama approximation after one step
with step-size δ and initial value Z(tn ) = Yn .
• t 7→ Z(t, ω) is continuous with probability 1.
Choose n ∈ {1, . . . , N } and consider the error at time tn .
Apply the Feynman-Kac formula: Let u(t, x) be the solution of the PDE
1
∂t u(t, x) + f (x)∂x u(t, x) + g 2 (x)∂x2 u(t, x) = 0,
2
t ∈ [0, tn ],
x∈R
with terminal condition u(tn , x) = F (x). Apply the Itô formula to u(t, Z(t)):
du(t, Z) =
1 2
2
∂t u(t, Z) +f (Y )∂x u(t, Z) + g (Y )∂x u(t, Z) dt
| {z }
2
= ... (PDE)
+ g(Y )∂x u(t, Z) dW (t)
2
1 2
2
= f (Y ) − f (Z) ∂x u(t, Z) + g (Y ) − g (Z) ∂x u(t, Z) dt
2
+ g(Y )∂x u(t, Z) dW (t)
Equivalent:
Ztn
u(tn , Z(tn )) = u(0, Z(0)) +
f (Y ) − f (Z) ∂x u(t, Z) dt
0
1
+
2
Ztn
g 2 (Y ) − g 2 (Z) ∂x2 u(t, Z) dt
0
Ztn
+
g(Y )∂x u(t, Z) dW (t)
0
By construction: u(tn , Z(tn )) = u(tn , Xn ) = F (Xn ) Feynman-Kac (see 2.4): u(0, Z(0)) = u(0, X(0)) = E F (X(tn ))
This yields
Tobias Jahnke
E[F (Xn )] − E F (X(tn )) ≤
49
Ztn E f (Y ) − f (Z) ∂x u(t, Z) dt
|0
1
+
2
|
{z
}
=:T1
Ztn 2
2
2
E g (Y ) − g (Z) ∂x u(t, Z) dt
0
{z
}
=:T2
Bounds for T1 and T2 : Define
G(t, x) = f (Y (t)) − f (x) ∂x u(t, x)
and apply the Itô formula to G(t, Z):
dG(t, Z) =
1 2
2
∂t G(t, Z) + ∂x G(t, Z) · f (Y ) + ∂x G(t, Z) · g (Y ) dt
2
+∂x G(t, Z) · g(Y )dW (t)
Equivalent:
Zt
G(t, Z(t)) = G(tn−1 , Z(tn−1 )) +
{z
}
|
Zt
∂t G(s, Z) ds +
tn−1
=0 (Def.)
Zt
1
+
2
∂x G(s, Z) · f (Y ) ds
tn−1
∂x2 G(t, Z)
2
Zt
· g (Y ) ds +
tn−1
∂x G(s, Z) · g(Y ) dW (s)
tn−1
where Z = Z(s) and Y = Y (s). Consider E(. . .):
E G(t, Z(t)) =
Zt
Zt E ∂t G(s, Z) ds +
E ∂x G(s, Z) · f (Y ) ds
tn−1
1
+
2
tn−1
Zt
2
2
E ∂x G(t, Z) · g (Y ) ds + 0
tn−1
It can be shown that all three integrands remain bounded. It follows that
E G(t, Z(t)) ≤ C · (t − tn−1 ) ≤ Cτ.
50 –
December 4, 2014
Consequence:
Ztn Ztn
T1 = E f (Y ) − f (Z) ∂x u(t, Z) dt = E G(t, Z(t)) dt ≤ Ctn τ ≤ CT τ.
0
0
In a similar way, it can be shown that T2 ≤ Ctn τ ≤ CT τ . This proves that
ˆ
E F X(tn ) − E F (Xn ) ≤ Cτ.
5.3
Higher-order methods
Consider again the one-dimensional SDE
dX(t) = f X(t) dt + g X(t) dW (t),
t ∈ [0, T ],
X(0) = X0
with suitable functions f and g and a given initial value X0 .
Goal: Construct numerical methods with higher order.
Caution! Numerical methods for ordinary differential equations can in general not be
extended to stochastic differential equations!
Example: Heun’s method.
Heun’s method for the ODE y(t)
˙ = f (y) with initial value y(0) = y0 takes the form
yen+1 = yn + τ f (yn )
τ
yn+1 )) .
yn+1 = yn + (f (yn ) + f (e
2
Similar to trapezoidal rule, but explicit. The natural modification of this method for
SDEs is
en+1 = Xn + τ f (Xn ) + g(Xn )∆Wn
X
1
τ
e
e
Xn+1 = Xn +
f (Xn ) + f (Xn+1 ) +
g(Xn ) + g(Xn+1 ) ∆Wn .
2
2
Consider the special case f (x) ≡ 0, g(x) = x. For the exact solution X(t), it follows that
 t

Zt
Z
E(X(t)) = E(X0 ) + E f X(s) ds + E  g s, X(s) dW (s),
| {z }
0
=0
| 0
{z
}
=0
Tobias Jahnke
51
i.e. that E(X(t)) is constant. The method simplifies to
en+1 = Xn + Xn ∆Wn
X
1
e
Xn + Xn+1 ∆Wn .
Xn+1 = Xn +
2
or equivalently
1
Xn+1 = Xn + Xn ∆Wn + Xn (∆Wn )2 .
2
This yields
τ
1
E(Xn+1 ) = E(Xn ) + E(Xn ∆Wn ) + E Xn (∆Wn )2 = E(Xn ) + E(Xn )
|
{z
} 2
2
=0
and thus for N −→ ∞ and τ = T /N −→ 0
N
T
lim E(XN ) = lim (1 + τ /2) X0 = lim 1 +
X0 = eT /2 X0 .
τ →0
τ →0
N →∞
2N
N
Hence, the method is not consistent! No convergence!
Stochastic Taylor expansions
Important tool for the construction of higher-order methods.
For smooth F : R −→ R, the Itô formula yields
1 00
2
0
dF (X) = F (X) · f (X) + F (X) · g (X) dt + F 0 (X) · g(X) dW (t)
|
{z
}
2
|
{z
}
=:L1 F (X)
=:L0 F (X)
(no time derivative, because F = F (x) does not depend on t) or equivalently
Zs
F X(s) = F X(tn ) +
tn
L0 F X(r) dr +
Zs
tn
L1 F X(r) dW (r)
(5.6)
Appendix A
Some definitions from probability
theory
The triple (Ω, F, P) is called a probability
Definition A.0.1 (Probability space)
space, if the following holds:
1. Ω 6= ∅ is a set, and F is a σ-algebra (or σ-field) on Ω, i.e. a family of subsets of
Ω with the following properties:
• ∅∈F
• If F ∈ F, then Ω \ F ∈ F
∞
[
• If Fi ∈ F for all i ∈ N, then
Fi ∈ F
i=1
The pair (Ω, F) is called a measurable space.
2. P : F −→ [0, 1] is a probability measure, i.e.
• P(∅) = 0 and P(Ω) = 1
• If Fi ∈ F for all i ∈ N are pairwise disjoint (i.e. Fi ∩ Fj = ∅ for i 6= j), then
P
∞
[
!
Fi
i=1
=
∞
X
P(Fi ).
i=1
Definition A.0.2 (Borel σ-algebra) If U is a family of subsets of Ω, then the σalgebra generated by U is
FU =
\
{F : F is a σ-algebra of Ω and U ⊂ F}.
If U is the collection of all open subsets of a topological space Ω (e.g. Ω = Rd ), then
B = FU is called the Borel σ-algebra on Ω. The elements B ∈ B are called Borel sets.
For the rest of this section (Ω, F, P) is a probability space.
52
Tobias Jahnke
53
Definition A.0.3 (Measurable functions, random variables)
• A function X : Ω −→ Rd is called F -measurable if
X −1 (B) := {ω ∈ Ω : X(ω) ∈ B} ∈ F
for all Borel sets B ∈ B. If (Ω, F, P) is a probability space, then every F-measurable
functions is called a random variable.
• Random variables X1 , . . . , Xn are called independent if
!
n
n
\
Y
−1
P
Xi (Ai ) =
P Xi−1 (Ai )
i=1
i=1
for all A1 , . . . , An ∈ B.
• If X : Ω −→ Rd is any function, then the σ-algebra generated by X is the
smallest σ-algebra on Ω containing all the sets
X −1 (B) for all B ∈ B.
Notation: F X = σ{X}
F X is the smallest σ-algebra where X is measurable.
Appendix B
The Itˆ
o integral
Let (Ω, F, P) be a complete probability space.
B.1
The Wiener process
Robert Brown 1827, Louis Bachelier 1900, Albert Einstein 1905, Norbert Wiener 1923
Definition B.1.1 (Normal distribution) A random variable X : Ω −→ Rd with d ∈ N
is normal if it has a multivariate normal (Gaussian) distribution with mean µ ∈ Rd
and a symmetric, positive definite covariance matrix Σ ∈ Rd×d , i.e.
Z
1
1
T −1
p
P(X ∈ B) =
exp − (x − µ) Σ (x − µ) dx
2
(2π)d det(Σ)
B
for all Borel sets B ⊂ Rd . Notation: X ∼ N (µ, Σ)
Remarks:
1. If X ∼ N (µ, Σ), then E(X) = µ and Σ = (σij ) with σij = E[(Xi − µi )(Xj − µj )].
2. Standard normal distribution ⇔ µ = 0, Σ = I (identity matrix).
3. If X ∼ N (µ, Σ) and Y = v + T X for some v ∈ Rd and a regular matrix T ∈ Rd×d ,
then
Y ∼ N (v + T µ, T ΣT T ).
(B.1)
Definition B.1.2 (Wiener process, Brownian motion)
(a) A continuous-time stochastic process {Wt : t ∈ [0, T )} is called a standard Brownian motion or standard Wiener process if it has the following properties:
1. W0 = 0 (with probability one)
54
Tobias Jahnke
55
2. Independent increments: For all 0 ≤ t1 < t2 < . . . < tn < T the random variables
Wt2 − Wt1 ,
Wt3 − Wt2 ,
...
,
Wtn − Wtn−1
are independent.
3. Wt − Ws ∼ N (0, t − s) for any 0 ≤ s < t < T .
˜ ⊂ Ω with P(Ω)
˜ = 1 such that t 7→ Wt (ω) is continuous for all ω ∈ Ω.
˜
4. There is a Ω
(1)
(d)
(b)
If Wt , . . . ,Wt are independent one-dimensional Wiener processes, then Wt =
(1)
(d)
Wt , . . . , W t
is called a d-dimensional Wiener process, and
Wt − Ws ∼ N (0, (t − s)I).
Remark: This process will serve as the “source of randomness” in our model of the
financial market.
Notation: Wt = Wt (ω) = W (t, ω) = W (t)
Numerical simulation of a Wiener process (d=1). Choose step-size τ > 0, put
˜ 0 = 0.
tn = n · τ and W
For n = 0, 1, 2, 3, . . .
Generate random number Zn ∼ N (0, 1)
˜ n+1 = W
˜ n + √τ Zn
W
˜ 0, W
˜ 1, W
˜ 2 , . . . approximates a path of the Wiener
For τ −→ 0 the interpolation of W
process.
How smooth is a path of a Wiener process? Consider only d = 1.
H¨
older continuity and non-differentiability
A function f : (a, b) −→ R is H¨
older continuous of order α for some α ∈ [0, 1] if there
is a constant C such that
|f (t) − f (s)| ≤ C|t − s|α
for all s, t ∈ (a, b).
If α = 1, then f is Lipschitz continuous.
If α > 0, then f is uniformly continuous.
If α = 0, then f is bounded.
A path of the Wiener process on a bounded interval is Hölder continuous of order α ∈ [0, 21 )
with probability one.
For α ≥ 12 , however, the path is not Hölder continuous of order α with probability one.
In particular, a path of the Wiener process is nowhere differentiable with probability one.
Proofs: [Ste01], chapter 5
56 –
December 4, 2014
Unbounded total variation
Let [a, b] be an interval and let
PN = (tn )N
n=0 ,
a = t0 < t1 < . . . < tN = b
be a partition of [a, b] with |PN | = maxn |tn − tn−1 |.
Example: equidistant partition, τ = (b − a)/N , tn = a + n · τ .
The total variation of a function f : (a, b) −→ R is
T Va,b (f ) =
lim
N
X
N −→ ∞
n=1
|PN | −→ 0
|f (tn ) − f (tn−1 )|.
If f is differentiable and f 0 is integrable, then it can be shown that
Zb
T Va,b (f ) =
|f 0 (t)| dt
a
Conversely: If a function f has bounded total variation, then its derivative exists for
almost all x ∈ [a, b].
Consequence: A path of the Wiener process has unbounded total variation with probability one.
B.2
Construction of the Itˆ
o integral
References: [KP99, Øks03, Shr04, Ste01]
The model considered in 1.5 is clearly
possible prices of S(T ).
Goal: Construct a more realistic model
Na¨ıve Ansatz:
dX
= f (t, X) +
|dt {z
}
too simple: only two discrete times, only two
for the dynamics of S(t).
g(t, X)Z(t),
|
{z
}
Z(t) = ?
random noise
ordinary differential
equation
Apply explicit Euler method: Choose t ≥ 0 and N ∈ N, let τ = t/N , tn = n · τ and define
approximations Xn ≈ X(tn ) by
Xn+1 = Xn + τ f (tn , Xn ) + τ g(tn , Xn )Z(tn )
(n = 0, 1, 2, . . .).
In the special case f (t, X) = 0 and g(t, X) = 1, we want that Xn = W (tn ) is the Wiener
process, i.e. we postulate that
!
W (tn+1 ) = W (tn ) + τ Z(tn ).
Tobias Jahnke
57
This yields
Xn+1
= Xn + τ f (tn , Xn ) + g(tn , Xn ) W (tn+1 ) − W (tn )
and after N steps
XN = X0 + τ
N
−1
X
f (tn , Xn ) +
n=0
N
−1
X
g(tn , Xn ) W (tn+1 ) − W (tn ) .
(B.2)
n=0
Keep t fixed, let N −→ ∞, τ = t/N −→ 0. Then, (B.2) should somehow converge to
Zt
X(t) = X(0) +
Zt
g(s, X(s))dW (s) .
f (s, X(s)) ds +
0
|0
{z
(?)
(B.3)
}
Problem: We cannot define (?) as a pathwise Riemann-Stieltjes integral! When N −→
∞, the sum
N
−1
X
g(tn , Xn (ω)) W (tn+1 , ω) − W (tn , ω)
n=0
diverges with probability one, because a path of the Wiener process has unbounded total
variation with probability one.
New goal: Define the integral
Zt
It [u](ω) =
u(s, ω) dWs (ω)
0
in a “reasonable” way for the following class of functions.
Definition B.2.1 Let (Ω, F, P) be a probability space, and let {Ft : t ∈ [0, T ]} be the
standard Brownian filtration. Then, we define H2 [0, T ] to be the class of functions
u = u(t, ω),
u : [0, T ] × Ω −→ R
with the following properties:
• (t, ω) 7→ u(t, ω) is (B × F)-measurable.
• u is adapted to {Ft : t ∈ [0, T ]}, i.e. u(t, ·) is Ft -measurable.
 T

Z
• E  u2 (t, ω) dt < ∞
0
58 –
December 4, 2014
Step 1: Itˆ
o integral for elementary functions
Definition B.2.2 (Elementary functions) A function φ ∈ H2 [0, T ] is called elementary if it is a stochastic step function of the form
φ(t, ω) = a0 (ω)1[0,0] (t) +
N
−1
X
= a0 (ω)1[0,t1 ] (t) +
an (ω)1(tn ,tn+1 ] (t)
n=0
N
−1
X
an (ω)1(tn ,tn+1 ] (t)
n=1
with a partition 0 = t0 < t1 < . . . < tN −1 < tN = T . The random variables an must be
Ftn -measurable with E(a2n ) < ∞. Here and below,
(
1 if t ∈ [c, d]
1[c,d] (t) =
(B.4)
0 else
is the indicator function of an interval [c, d].
Skizze
For 0 ≤ c < d ≤ T , the only reasonable way to define the Itô integral of an indicator
function 1(c,d] is
ZT
IT [1(c,d] ](ω) =
Zd
dW (s, ω) = W (d, ω) − W (c, ω).
1(c,d] (s) dW (s, ω) =
0
c
Hence, by linearity, we define the Itô integral of an elementary function by
IT [φ](ω) =
N
−1
X
an (ω) W (tn+1 , ω) − W (tn , ω) .
n=0
Lemma B.2.3 (Itˆ
o isometry for elementary functions) For all elementary functions
we have
 T

Z
E IT [φ]2 = E  φ2 (t, ω) dt
0
or equivalently
kIT [φ]kL2 (dP) = kφkL2 (dt×dP)
with

Z ZT
kφkL2 (dt×dP) = 
Ω
0
 21
  T
 12
Z
φ2 (t, ω) dt dP = E  φ2 (t, ω) dt .
0
Tobias Jahnke
59
Proof. Since
2
a20 (ω)1[0,0] (t)
φ (t, ω) =
+
N
−1
X
a2n (ω)1(tn ,tn+1 ] (t)
n=0
we obtain


ZT
2
φ (t, ω) dt =
E
N
−1
X
E a2n (tn+1 − tn )
(B.5)
n=0
0
for the right-hand side. If we let ∆Wn = W (tn+1 ) − W (tn ), then
IT [φ]2 =
N
−1
X
!2
an ∆Wn
=
n=0
N
−1 N
−1
X
X
an am ∆Wn ∆Wm .
(B.6)
n=0 m=0
By definition, the Wiener process has independent increments with E(∆Wn ) = 0 and
E(∆Wn2 ) = V(∆Wn ) = tn+1 − tn . If n > m, then am an ∆Wm is Ftn -measurable, and since
∆Wn is independent of Ftn , it follows that
(
E (an am ∆Wm ) E (∆Wn ) = 0 if n 6= m
E (an am ∆Wn ∆Wm ) =
E (a2n ) (tn+1 − tn )
if n = m.
Hence, taking the expectation of (B.6) gives
E IT [φ]
2
=
N
−1
X
E a2n (tn+1 − tn ).
(B.7)
n=0
Comparing (B.5) and (B.7) yields the assertion.
Step 2: Itˆ
o integral on H2 [0, T ]
Lemma B.2.4 For any u ∈ H2 [0, T ] there is a sequence (φk )k∈N of elementary functions
φk ∈ H2 [0, T ] such that
lim ku − φk kL2 (dt×dP) = 0
k→∞
Proof: Section 6.6 in [Ste01].
Let u ∈ H2 [0, T ] and let (φk )k∈N be elementary functions such that
u = lim φk
k→∞
in L2 (dt × dP)
60 –
December 4, 2014
as in Lemma B.2.4. The linearity of IT [·] and Lemma B.2.3 yield
kIT [φj ] − IT [φk ]kL2 (dP) = kIT [φj − φk ]kL2 (dP) = kφj − φk kL2 (dt×dP) −→ 0
for j, k −→ ∞. Hence, (IT [φk ])k is a Cauchy sequence in the Hilbert space L2 (dP). Thus,
(IT [φk ])k converges in L2 (dP), and we can define
IT [u] = lim IT [φk ].
k→∞
The choice of the sequence does not matter: If (ψk )k∈N are elementary functions with
u = limk→∞ ψk in L2 (dt × dP), then by Lemma B.2.3 we obtain for k −→ ∞
kIT [φk ] − IT [ψk ]kL2 (dP) = kIT [φk − ψk ]kL2 (dP)
= kφk − ψk kL2 (dt×dP)
≤ kφk − ukL2 (dt×dP) + ku − ψk kL2 (dt×dP) −→ 0.
Theorem B.2.5 (Itˆ
o isometry) For all u ∈ H2 [0, T ] we have
kIT [u]kL2 (dP) = kukL2 (dt×dP) .
Proof: Exercise.
Step 3: The Itˆ
o integral as a process
So far we have defined the Itô integral IT [u](ω) over the interval [0, T ] for fixed T . For
applications in mathematical finance, however, we want to consider It [u](ω) : t ∈ [0, T ]
as a stochastic process.
If u(s, ω) ∈ H2 [0, T ], then 1[0,t] (s)u(s, ω) ∈ H2 [0, T ]. Can we define It [u](ω) by
IT [1[0,t] u](ω)?
Theorem B.2.6 For any u ∈ H2 [0, T ] there is a process {Xt : t ∈ [0, T ]} that is a
continuous martingale with repsect to the standard Brownian filtration Ft such that the
event
ω ∈ Ω : Xt (ω) = IT [1[0,t] u](ω)
has probability one for each t ∈ [0, T ].
A proof can be found in [Ste01], Theorem 6.2, pages 83-84.
Tobias Jahnke
61
Step 4: The Itˆ
o integral on L2loc [0, T ]
So far we have defined the Itô integral for functions u ∈ H2 [0, T ]; cf. Definition B.2.1.
Such functions must satisfy
 T

Z
E  u2 (t, ω) dt < ∞,
(B.8)
0
and this condition is sometimes too restrictive. With some more work, the Itô integral
can be extended to all functions
u : [0, T ] × Ω −→ R
u = u(t, ω),
with the following properties:
• (t, ω) 7→ u(t, ω) is (B × F)-measurable.
• u is adapted to {Ft : t ∈ [0, T ]}.
 T

Z
• P  u2 (t, ω) dt < ∞ = 1
0
This class is called L2loc [0, T ]. The first two conditions are the same as for H2 [0, T ], but
the third condition is weaker than (B.8). If y : R −→ R is continuous, then u(t, ω) =
y W (t, ω) ∈ L2loc [0, T ], because ω 7→ y W (t, ω) is continuous with probability one and
hence bounded on [0, T ].
Details: Chapter 7 in [Ste01].
Notation
The process X constructed above is called the Itˆ
o integral (Itô Kiyoshi 1944) of u ∈
2
Lloc [0, T ] and is denoted by
Zt
X(t, ω) =
u(s, ω) dW (s, ω).
0
The Itô integral over an arbitrary interval [a, b] ⊂ [0, T ] is defined by
Zb
Zb
u(s, ω) dW (s, ω) −
u(s, ω) dW (s, ω) =
a
Za
0
u(s, ω) dW (s, ω).
0
Alternative notations:
Zb
Zb
u(s, ω) dW (s, ω) =
a
Zb
u(s, ω) dWs (ω) =
a
Zb
us (ω) dWs (ω) =
a
us dWs
a
62 –
December 4, 2014
Properties of the Itˆ
o integral
Lemma B.2.7 Let c ∈ R and u, v ∈ L2loc [0, T ]. The Itô integral on [a, b] ⊂ [0, T ] has the
following properties:
1. Linearity:
Zb Zb
cu(s, ω) + v(s, ω) dWs (ω) = c
a
Zb
u(s, ω) dWs (ω) +
a
v(s, ω) dWs (ω)
a
with probability one.
 b

Z
2. E  u(s, ω) dWs (ω) = 0
a
Zt
u(s, ω) dWs (ω) is Ft -measurable for t ≥ a.
3.
a
4. Itô isometry on [a, b]:

Zb
E

 b

2
Z
u(s, ω) dWs (ω)  = E  u2 (s, ω) ds
a
a
(cf. Theorem B.2.5).
5. Martingale property: The Itô integral
Zt
X(t, ω) =
u(s, ω) dW (s, ω).
0
of a function u ∈ H2 [0, T ] is a continuous martingale with respect to the standard
Brownian filtration; cf. Theorem 2.3.4. If u ∈ L2loc [0, T ], then the Itô integral is only
a local martingale; cf. Proposition 7.7 in [Ste01].
The first four properties can be shown by considering elementary functions and passing
to the limit.
Tobias Jahnke
B.3
63
Sketch of the proof of the Itˆ
o formula (Theorem
2.2.2).
• Equation (2.2) is the shorthand notation for
Zt Yt = Y0 +
1 2
2
∂t F (s, Xs ) + ∂x F (s, Xs ) · f (s, Xs ) + ∂x F (s, Xs ) · g (s, Xs ) ds
2
0
Zt
∂x F (s, Xs ) · g(s, Xs )dWs
+
0
Assume that F is twice continuously differentiable with bounded partial derivatives.
(Otherwise F can be approximated by such functions with uniform convergence on
compact subsets of [0, ∞) × R.)
Assume that (t, ω) 7→ f (t, Xt (ω)) and (t, ω) 7→ g(t, Xt (ω)) are elementary functions.
(Otherwise approximate by elementary functions.) Hence, there is a partition 0 =
t0 < t1 < . . . < tN = t such that
f (t, Xt (ω)) = f (0, X0 (ω))1[0,t1 ] (t) +
N
−1
X
f (tn , Xtn (ω))1(tn ,tn+1 ] (t)
n=1
and the same equation with f replaced by g.
• Notation: For the rest of the proof, we define
f (n) := f (tn , Xtn ),
F (n) := F (tn , Xtn ),
g (n) := g(tn , Xtn ),
∂t F (n) := ∂t F (tn , Xtn )
and so on, and
∆tn = tn+1 − tn ,
∆Xn = Xtn+1 − Xtn ,
∆Wn = Wtn+1 − Wtn .
Since f and g are elementary functions, we have
Ztn
Xtn = X0 +
Xtn = X0 +
Ztn
f (s, Xs ) ds +
0
n−1
X
k=0
g(s, Xs ) dWs
0
f (tk , Xtk ) ∆tk +
| {z }
f (k)
n−1
X
k=0
g(tk , Xtk ) ∆Wk .
| {z }
g (k)
and hence
∆Xn = Xtn+1 − Xtn = f (n) ∆tn + g (n) ∆Wn .
64 –
December 4, 2014
• Telescoping sum:
Yt = YtN = Y0 +
N
−1
X
N
−1
X
n=0
n=0
(Ytn+1 − Ytn ) = Y0 +
F (n+1) − F (n)
Apply Taylor’s theorem:
F (n+1) − F (n)
1
= ∂t F (n) · ∆tn + ∂x F (n) · ∆Xn + ∂t2 F (n) · (∆tn )2 + ∂t ∂x F (n) · ∆tn ∆Xn
2
1 2 (n)
+ ∂x F · (∆Xn )2 + Rn (∆tn , ∆Xn )
2
with a remainder term Rn . Insert this into the telescoping sum.
• Consider the limit N −→ ∞, ∆tn −→ 0 with respect to k · kL2 (dP) . For the first two
terms, this yields
lim
N →∞
N
−1
X
∂t F
(n)
· ∆tn = lim
N
−1
X
N →∞
n=0
Zt
∂t F (tn , Xtn ) · ∆tn
=
n=0
∂t F (s, Xs ) ds
0
and
lim
N →∞
= lim
N →∞
N
−1
X
n=0
N
−1
X
∂x F (n) · ∆Xn
∂x F
(n)
·f
(n)
∆tn + lim
N →∞
n=0
Zt
∂x F (n) · g (n) ∆Wn
n=0
Zt
∂x F (s, Xs ) · f (s, Xs ) ds +
=
N
−1
X
0
∂x F (s, Xs ) · g(s, Xs ) dWs .
0
• Next, we investigate the “∂x2 F (n) term”. Since
2
(∆Xn )2 = f (n) ∆tn + g (n) ∆Wn
we have
N −1
N −1
2
1 X 2 (n)
1 X 2 (n)
2
∂x F · (∆Xn ) =
∂x F · f (n) (∆tn )2
2 n=0
2 n=0
+
+
N
−1
X
∂x2 F (n) · f (n) g (n) ∆tn ∆Wn
n=0
N
−1
X
1
2
n=0
∂x2 F (n) · g (n)
2
(∆Wn )2 .
(B.9)
(B.10)
(B.11)
Tobias Jahnke
65
For the right-hand side of (B.9), we obtain

2
!2 
N
−1
−1
N
X
X
2
2
=E 
∂x2 F (n) · f (n) (∆tn )2  −→ 0.
∂x2 F (n) · f (n) (∆tn )2 2
n=0
n=0
L (dP)
With the abbreviation α(n) := ∂x2 F (n) · f (n) g (n) we obtain for the right-hand side of
(B.10) that

2
N −1
!2 
N
−1
X
X
α(n) ∆tn ∆Wn = E
α(n) ∆tn ∆Wn 
2
n=0
n=0
L (dP)
=
N
−1 N
−1
X
X
E α(n) α(m) ∆Wn ∆Wm ∆tn ∆tm .
n=0 m=0
Since
E α(n) α(m) ∆Wn ∆Wm = E α(n) α(m) ∆Wn E (∆Wm ) = 0
| {z }
=0
for n < m and similar for m < n, only the terms with n = m have to be considered,
which yields
2
−1
N
−1
N
X
X
α(n) ∆tn ∆Wn =
E (α(n) )2 (∆tn )2 E (∆Wn )2 −→ 0.
n=0
2
{z
}
|
n=0
L (dP)
=∆tn
The third term (B.11), however, has a non-zero limit: We show that
t
Z
N −1
2
1 X 2 (n)
1
(n) 2
2
lim
∂x F · g
∂x2 F (s, Xs ) · g(s, Xs ) ds
(∆Wn ) =
N →∞ 2
2
n=0
0
which yields the strange
2 additional term in the Itô formula. With the abbreviation
β (n) = 12 ∂x2 F (n) · g (n) we have
2
−1
N
X
(n)
2
β
(∆W
)
−
∆t
n
n 2
n=0
L (dP)

!2 
N
−1
X

= E
β (n) (∆Wn )2 − ∆tn
n=0
=E
"N −1 N −1
XX
n=0 m=0
#
β (n) β (m) (∆Wn )2 − ∆tn
(∆Wm )2 − ∆tm
.
66 –
December 4, 2014
For n < m we have
E β (n) β (m) (∆Wn )2 − ∆tn (∆Wm )2 − ∆tm
= E β (n) β (m) (∆Wn )2 − ∆tn E (∆Wm )2 − ∆tm = 0
|
{z
}
=0
and vice versa for n > m. Hence, only the terms with n = m have to be considered,
and we obtain
2
#
"N −1
−1
N
X
X
2
2
(∆Wn )2 − ∆tn
β (n) (∆Wn )2 − ∆tn =E
β (n)
2
n=0
n=0
L (dP)
=
N
−1
X
n=0
h
E
β (n)
2 i h
2 i
E (∆Wn )2 − ∆tn
|
{z
}
−→0
according to Exercise 5.
• With essentially the same arguments, it can be shown that
N −1
1 X 2 (n)
∂t F · (∆tn )2 = 0
lim
N →∞ 2
n=0
lim
N →∞
N
−1
X
∂t ∂x F (n) · ∆tn ∆Xn = 0
n=0
and that the remainder term from the Taylor expansion can be neglected when the
limit is taken.
Bibliography
[Ben04] Fred Espen Benth. Option theory with stochastic analysis : an introduction to
mathematical finance. Springer, 2004.
[BK04] N. H. Bingham and R¨
udiger Kiesel. Risk-neutral valuation. Pricing and hedging
of financial derivatives. Springer Finance. Springer, London, 4nd ed. edition,
2004.
[CT04] Rama Cont and Peter Tankov. Financial modelling with jump processes. CRC
Financial Mathematics Series. Chapman & Hall, Boca Raton, FL, 2004.
[GJ10] Michael G¨
unther and Ansgar J¨
ungel. Finanzderivate mit MATLAB. Mathematische Modellierung und numerische Simulation. Vieweg, Wiesbaden, 2nd ed.
edition, 2010.
[Hig02] Desmond J. Higham. Nine ways to implement the binomial method for option
valuation in MATLAB. SIAM Rev., 44(4):661–677, 2002.
[KP99] Peter E. Kloeden and Eckhard Platen. Numerical solution of stochastic differential equations. Number 23 in Applications of Mathematics. Springer, Berlin,
corr. 3rd printing edition, 1999.
[Øks03] Bernt Øksendal. Stochastic differential equations. An introduction with applications. Universitext. Springer, Berlin, 6th ed. edition, 2003.
[Sey09] R¨
udiger U. Seydel. Tools for computational finance. 4th revised and extended ed.
Universitext. Springer, Berlin, 4th revised and extended ed. edition, 2009.
[Shr04] Steven E. Shreve. Stochastic calculus for finance. II: Continuous-time models.
Springer Finance. Springer, 2004.
[Ste01] J. Michael Steele. Stochastic calculus and financial applications. Number 45 in
Applications of Mathematics. Springer, New York, NY, 2001.
67

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