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Monte Carlo Methods and Black Scholes model
Christophe Chorro ([email protected])
MASTER MMMEF
22 Janvier 2008
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 1 / 87
Bibliography
The slides of this lecture (and other documents) are available on thefollowing website
http : //christophe.chorro.fr/enseignements.html
J. JACOD, P. PROTTER : Probability essentials, Springer, 2004.
Course of ANNIE MILLET in Paris 1: “ftp : //samos.univ −paris1.fr/pub/SAMOS/cours/millet/Master1/PolyM1_English.pdf ′′
http://christophe.chorro.fr/docs/CSangl.pdf
D. LAMBERTON, B. LAPEYRE : Introduction to stochastic calculus appliedto finance
MARK BROADIE AND PAUL GLASSERMAN, Estimating Security PriceDerivatives Using Simulation, Management Science, 1996, Vol. 42, No.2, 269–285.
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 2 / 87
Study Plan of the Lecture
Introduction to Monte Carlo MethodsPresentation of the problemAn historical exampleStrong law of large numbersControl of the error (the central limit theorem)Conclusion
Black scholes modelSimulations of Gaussian random variablesSimulation of the Brownian motionReminder on the Black Scholes modelThe greeksFinite difference method for GreeksIntegration by parts method for Greeks
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 3 / 87
Plan
1 Introduction to Monte Carlo MethodsPresentation of the problemAn historical exampleStrong law of large numbers (SLLN)Control of the error: the CLTConclusion
2 Black Scholes modelSimulations of Gaussian random variablesSimulation of Brownian motionReminder on the Black Scholes modelThe greeksFinite difference method for GreeksIntegration by parts method for GreeksNumerical resultsConcluding remarks
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 4 / 87
Presentation of the problem
The basic problem is to estimate a multi-dimensional integral∫Rd
f (x)dµ(x)
for which an analytic answer is not known.
More precisely we look for a stochastic algorithm that gives:
A numerical estimate of this integral,
An estimate of the error,
A good accuracy with an interesting computational cost.
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 5 / 87
Presentation of the problem
This kind of problem appears in a wide field of areas:
Von Neuman, Ulam: Neutron diffusion in fissionable material (Manhattanproject 1947)
Biology
Mathematical finance (pricing and hedging of contingents claims)
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 6 / 87
Plan
1 Introduction to Monte Carlo MethodsPresentation of the problemAn historical exampleStrong law of large numbers (SLLN)Control of the error: the CLTConclusion
2 Black Scholes modelSimulations of Gaussian random variablesSimulation of Brownian motionReminder on the Black Scholes modelThe greeksFinite difference method for GreeksIntegration by parts method for GreeksNumerical resultsConcluding remarks
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 7 / 87
The Buffon needle
We consider a floor with equally spaced lines, a distance δ apart and a needleof length 0 < l < δ dropped on it
Question: What is the probability that a needle dropped randomly intersectsone of the lines?
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 8 / 87
The Buffon needle
If θ = θ0 is fixed,
P(Intersection) =l sinθ0
δ.
If the needle is dropped randomly, θ is uniformly distributed on [0, π[ and
P(Intersection) =
∫ π
0
l sinθ0
δ
dθ0
π=
2lπδ
.
Now, if we drop N needles and denote by X the number of them crossing aline, one has
XN≈ 2l
πδ.
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 9 / 87
The Buffon needle
Laplace (1812) suggested to use this experiment to approximate π:
π ≈ 2lNXδ
.
Lazzarini (1901) made the experiment with l = 2.5cm, δ = 3cm andN = 3408. He obtained X = 1808 thus
π ≈ 355133
≈ 2.66917....
Problem: Time consuming and bad accuracy.......
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 10 / 87
The Buffon needleSee http://www.mste.uiuc.edu/reese/buffon/bufjava.html for computersimulation of this experiment.
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 11 / 87
The Buffon needle in modern language
We consider a random variable Z such that
Z = 1 if the randomly dropped needle crosses a line
Z = 0 otherwise.
We have
E[Z ] = P(Intersection) =2lπδ
and if we denote by Z1, ......ZN a N-sample of Z we previously use the
following approximation
Z1 + ... + ZN
N≈ E[Z ].
Aim: Prove the validity of this approximation.
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 12 / 87
Plan
1 Introduction to Monte Carlo MethodsPresentation of the problemAn historical exampleStrong law of large numbers (SLLN)Control of the error: the CLTConclusion
2 Black Scholes modelSimulations of Gaussian random variablesSimulation of Brownian motionReminder on the Black Scholes modelThe greeksFinite difference method for GreeksIntegration by parts method for GreeksNumerical resultsConcluding remarks
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 13 / 87
SLLN
We consider a probability space (Ω,A, P) and we denote by E the expectationunder P.
TheoremLet (Xn)n∈N be i.i.d random variables with values in R such that E[|X1|] < ∞.
Then, denoting Sn = X1 + ... + Xn, one has
Sn
n→
a.s and L1E[X1].
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 14 / 87
SLLN
Numerical illustration (Buffon needle)
0 50 100 150 200 250 300 350 400 450 5000.0
0.1
0.2
0.3
0.4
0.5
0.6
Illustration of the SLLN when X1 → B( 12 ) and n = 500
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 15 / 87
SLLN
Remark 1: The hypothesis E[|X1|] < ∞ is necessary
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000!25
!20
!15
!10
!5
0
5
10
SLLN is not fulfilled when X1 → C(1) (here n = 10000)
Remark 2: Possible extension for random variables with values in Rd
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 16 / 87
SLLN: Sketch of the proof
We suppose that E[|X1|4] < ∞ and without loss of generality we assume thatE[X1] = 0.
Using that the random variables are independent and centered we obtain
E[(
Snn
)4]
= 1n4
(n∑
k=1E[X 4
k ] + 3∑i 6=j
E[X 2i ]E[X 2
j ]
)= 1
n4
(nE[X 4
1 ] + 3n(n − 1)E[X 21 ]2)
≤CS
3E[X 41 ]
n2 .
Thus
E
[(Sn
n
)4]→ 0 ⇒ E
[∣∣∣∣Sn
n
∣∣∣∣]→ 0.
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 17 / 87
SLLN: Sketch of the proof
Moreover from the preceding inequality and the monotone convergencetheorem:
E
[ ∞∑n=1
(Sn
n
)4]
=∞∑
n=1
E
[(Sn
n
)4]
< ∞
thus∞∑
n=1
(Sn
n
)4
< ∞ a.s.
This implies that
Sn
n→a.s
0.
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 18 / 87
Plan
1 Introduction to Monte Carlo MethodsPresentation of the problemAn historical exampleStrong law of large numbers (SLLN)Control of the error: the CLTConclusion
2 Black Scholes modelSimulations of Gaussian random variablesSimulation of Brownian motionReminder on the Black Scholes modelThe greeksFinite difference method for GreeksIntegration by parts method for GreeksNumerical resultsConcluding remarks
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 19 / 87
Control of the errorAn asymptotic bound via the Central Limit Theorem (CLT)
DefinitionWe say that a sequence of random variables (Xn)n>0 converges toward X indistribution ( Xn →
DX) if ∀f ∈ Cb(R, R),
E[f (Xn] →n→∞
E[f (X )].
This convergence extends when f is an indicator function.
TheoremLet (Xn)n∈N be i.i.d random variables with values in R such that E[|X1|2] < ∞.Then
Sn − nE[X1]√nσ
→DN (0, 1)
where σ2 = Var(X1).Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 20 / 87
Control of the error
Numerical illustration:
!! !" !# $ # " !$%$$
$%$&
$%#$
$%#&
$%"$
$%"&
$%!$
$%!&
$%'$
$%'&
Illustration of the CLT when X1 → U([0, 1]) and n = 500
We have an obvious extension of this result in any finite dimension
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 21 / 87
Control of the error
From the preceding theorem we obtain that
P(−aσ√
n≤ Sn
n− E[X1] ≤
aσ√n
)→
n→∞
∫ a
−a
1√2π
e−x22 dx .
In practice we know from tables (cf next slide that)
P(|N (0, 1)| ≤ 1.96) = 0.95
thus when n is large enough, with a confidence of 95%,
E[X1] ∈[
Sn
n− 1.96σ√
n,
Sn
n+
1.96σ√n
].
The magnitude of the error is given by 1.96σ√n : the size of σ is fundamental for
the speed of convergence.
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 22 / 87
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 23 / 87
Control of the error
When σ is unknown it may be estimated easily:
TheoremLet (Xn)n∈N be i.i.d random variables with values in R such that E[|X1|2] < ∞.
Then if we define
σn2 =
nn − 1
(1n
n∑i=1
X 2i − (
1n
n∑i=1
Xi)2
)one has
a)E[σn
2] = σ2
b)σn
2 →a.s
σ2.
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 24 / 87
Control of the error
We have the following result that allows the building of confidence intervalseven if σ is unknown.
TheoremLet (Xn)n∈N be i.i.d random variables with values in R such that E[|X1|2] < ∞,then
Sn − nE[X1]√nσn
→DN (0, 1)
Proof: It is just a consequence of the classical CLT and of the Slutsky lemma:
LemmaIf Xn →
DX and Yn →
Da (a being a constant) then
XnYn →D
Xa.
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 25 / 87
Plan
1 Introduction to Monte Carlo MethodsPresentation of the problemAn historical exampleStrong law of large numbers (SLLN)Control of the error: the CLTConclusion
2 Black Scholes modelSimulations of Gaussian random variablesSimulation of Brownian motionReminder on the Black Scholes modelThe greeksFinite difference method for GreeksIntegration by parts method for GreeksNumerical resultsConcluding remarks
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 26 / 87
Reminder of the method
To basically compute E[f (X )] by Monte Carlo Methods we have to
“Generate” a n-sample of the distribution of X
Compute 1n
n∑k=1
f (Xk ) for large n
Precise the confidence interval coming from the CLT
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 27 / 87
Advantages
Easy to implement on any software
No regularity on f
The control of the error ( σ√n ) is independent of the dimension of the
problem
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 28 / 87
Limits
The error is a random variable (we only have confidence intervals)
This method may be slow if we don’t use extra-techniques
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 29 / 87
Limits
Consider that we want to approximate by Monte Carlo simulations
I = E[eβN (0,1)]
(even if the exact value is given by eβ22 ).
In this way we generate (see part) a n-sample (G1, ...., Gn) of a N (0, 1) anduse
In =eβG1 + ... + eβGn
n≈ E[eβN (0,1)].
By the CLT the order of magnitude of the relative error is given by
In − II
≈ σ√(n)I
=
√eβ2 − 1√
n.
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 30 / 87
Limits
When β = 1, we need n ≥ 1000 to obtain a relative error ≤ 10%
When β = 5, we need n ≥ 7 ∗ 1010 to obtain a relative error ≤ 100%(THIS IS NUMERICALLY IMPOSSIBLE) Moreover, in this case
Exact value= 268357
Approximated value (for n=100000)= 107709
Confidence interval (level 95%)= ]20188, 195229[
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 31 / 87
Plan
1 Introduction to Monte Carlo MethodsPresentation of the problemAn historical exampleStrong law of large numbers (SLLN)Control of the error: the CLTConclusion
2 Black Scholes modelSimulations of Gaussian random variablesSimulation of Brownian motionReminder on the Black Scholes modelThe greeksFinite difference method for GreeksIntegration by parts method for GreeksNumerical resultsConcluding remarks
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 32 / 87
Simulations of Gaussian random variables
Starting from i.i.d uniform random variables on [0, 1] (Un)n∈N
( See: http://christophe.chorro.fr/docs/MC1.pdf)
we want to generate random samples of a Gaussian distribution.
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 33 / 87
Gaussian random variables from Inversion Method
Let X be a real random variable, the distribution function of X is defined as
FX (x) = P(X ≤ x); x ∈ R.
Properties: non-decreasing, right C0, limx→+∞
F (x) = 1 and limx→−∞
F (x) = 0.
DefinitionWe define the generalized inverse of FX denoted by F−X where ∀u ∈]0, 1[,
F−X (u) = infx | FX (x) ≥ u.
Remark: When FX is strictly increasing and continuous, F−X = F−1X .
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 34 / 87
Gaussian random variables from Inversion Method
PropositionIf U → U([0, 1]) then F−X (U) has the same distribution then X.
Proof: We just have to remark that ∀u ∈]0, 1[, ∀x ∈ R,
F−X (u) ≤ x ⇔ u ≤ FX (x).
Figure: Illustration of the definition of F−X
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 35 / 87
Gaussian random variables from Inversion Method
Example 1: Particular continuous distributions
If X → E(λ), λ > 0 then FX (x) = (1− e−λx)1x≥0 and
−1λ
Log(1− U) → E(λ).
If X → C(a), a > 0, then FX (x) = 1π [Arctan( x
a ) + π2 ] and
a tan(
π(U − 12
)
)→ C(a).
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 36 / 87
Gaussian random variables from Inversion Method
Example 2: When X → N (0, 1), FX is unknown but we have the followingapproximation:
If u > 0; 5, let t =√−2log(1− u)
F−X (u) ≈ t − c0 + t(c1 + tc2)
1 + t(d1 + t(d2 + td3)).
If u ≤ 0; 5, let t =√−2log(u)
F−X (u) ≈ c0 + t(c1 + tc2)
1 + t(d1 + t(d2 + td3))− t .
Where
c0 = 2.515517, c1 = 0.802853, c3 = 0.010328,
d1 = 1.432788, d2 = 0.189269, d3 = 0.001308.
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 37 / 87
Gaussian random variables from Inversion Method
−4 −2 0 2 4
0.0
0.1
0.2
0.3
0.4
Figure: Empirical density of 10000 standard normal distribution obtained by Inversionmethod
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 38 / 87
Gaussian random variables from Transformationmethod
Here we try to express X as a function of another random variable Y easy togenerate.
One of the main tool is the following result
PropositionLet D and ∆ be two open sets of Rd and Φ = (Φ1, ...Φd ) : D → ∆ aC1-diffeomorphism. If g : ∆ → R is measurable and bounded then∫
∆
g(v)dv =
∫D
g(Φ(u)) | JΦ(u)) | du
where JΦ(u) = det[(
∂Φi
∂uj(u))
1≤i,j≤d
].
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 39 / 87
Gaussian random variables from TransformationmethodExample 1: Box-Muller method
PropositionIf U1 and U2 are two independent uniform random variables on [0, 1] then
G1 =√−2log(U1)cos(2πU2) and G2 =
√−2log(U1)sin(2πU2)
are two independent N (0, 1).
Proof: Let us define the following C1-diffeomorphism
Ψ : (x , y) ∈]0, 1[2→ (u =√−2log(x)cos(2πy), v =
√−2log(x)sin(2πy))
fulfilling |JΨ(x , y)| = 2πx . Since u2 + v2 = −2log(x), according to the change
of variables theorem (Φ = Ψ−1), one has for F ∈ Cb(R2, R),∫]0,1[2
F (Ψ(x , y))dxdy =
∫R2−(R+×0)
F (u, v)1
2πe−
u2+v22 dudv .
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 40 / 87
Gaussian random variables fromTransformationmethod
−4 −2 0 2 4
−4
−2
02
4
Figure: Simulation of 5000 pairs of independent N (0, 1) by Box-Muller method
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 41 / 87
Plan
1 Introduction to Monte Carlo MethodsPresentation of the problemAn historical exampleStrong law of large numbers (SLLN)Control of the error: the CLTConclusion
2 Black Scholes modelSimulations of Gaussian random variablesSimulation of Brownian motionReminder on the Black Scholes modelThe greeksFinite difference method for GreeksIntegration by parts method for GreeksNumerical resultsConcluding remarks
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 42 / 87
Simulation of Brownian motion
We consider a probability space (Ω,A, P).
DefinitionStandard Brownian motion (B.M) is a stochastic process (Bt)t∈[0,T ] fulfilling :
a) B0 = 0 P-a.s.
b) B is continuous i.e t → Bt(w) is continuous for P almost all w.
c) B has independent increments: For Si t > s, Bt − Bs is independent ofFBs = σ(Bu, u ≤ s).
d) the increments of B are stationary and gaussian: For t ≥ s, Bt − Bs followsa N (0, t − s).
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 43 / 87
Simulation of Brownian motion
We consider a subdivision 0 = t0 < ... < tn = T of [0, T ]. We want to simulate
(Bt0 , ...Btn).
Idea:Btk = Btk−1 + Btk − Btk−1︸ ︷︷ ︸
N (0,tk−tk−1)⊥Btk−1 ,...,B0
.
PropositionIf (G1, ...Gn) are i.i.d N (0, 1), we define
X0 = 0, Xi =i∑
j=1
√tj − tj−1Gj i > 0.
Then(X0, ..., Xn) =
D(Bt0 , ...Btn).
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 44 / 87
Brownian motion
0.0 0.4 0.8−
0.8
−0.
20.
40.0 0.4 0.8
−1.
5−
0.5
0.0 0.4 0.8
−0.
60.
00.
4
0.0 0.4 0.8−
0.5
0.5
Figure: 4 paths of the Brownian motion on [0, 1] generated using the precedingmethod with the regular subdivision of step 0.001.
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 45 / 87
Brownian motion
If we want to add points to the preceding simulation, the following result isuseful.
PropositionFor t > s,
D(
B t+s2| (Bt , Bs)
)= N
(Bt + Bs
2,
t − s4
).
Idea of the proof
B t+s2
=Bt + Bs
2+ Z
where
• Z is independent of σ(Bu | u ≥ t) and σ(Bu | u ≤ s).
• Z → N(0, t−s
4
).
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 46 / 87
Plan
1 Introduction to Monte Carlo MethodsPresentation of the problemAn historical exampleStrong law of large numbers (SLLN)Control of the error: the CLTConclusion
2 Black Scholes modelSimulations of Gaussian random variablesSimulation of Brownian motionReminder on the Black Scholes modelThe greeksFinite difference method for GreeksIntegration by parts method for GreeksNumerical resultsConcluding remarks
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 47 / 87
Black Scholes model
We consider the time interval [0, T ] and r the risk free rate (supposed to beconstant) during this period.
Non-risky asset: Its dynamic is given by
S00 = 0, S0
t = ert .
Risky asset: Under the historical probability P its dynamic is given by thefollowing SDE:
dSt = µStdt + σStdBt (1)
with initial condition S0 = x0 > 0 and where B is a standard BM under P.
Itô formula ⇒ St = x0e(µ− 12 σ2)t+σBt .
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 48 / 87
Black Scholes model
0.0 0.4 0.81
35
sigma=1, mu=1
0.0 0.4 0.8
0.4
0.7
1.0
sigma=1, mu=−1
0.0 0.4 0.8
1.0
2.5
4.0
sigma=0.5, mu=1
0.0 0.4 0.80.
40.
71.
0
sigma=0.5, mu=−1
Figure: Simulation of a path of the risky asset in the B&S model for differentparameters
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 49 / 87
Black Scholes model
What is, in this model, the price at t of a contingent claim with payoff ΦT at T?
PropositionIn the B&S model there exists a unique probablity Q ∼ P such that the priceat t of a contingent claim with payoff ΦT at T is given by
Pt = E∗[e−r(T−t)ΦT | Ft ].
Moreover the dynamic of the risky asset under Q is given by
dSt = rStdt + σStdWt (µ ⇔ r) (2)
where W is a standard BM under Q.
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 50 / 87
Black Scholes model
Examples: The Black Scholes Formulas
For Call options (ΦT = (ST − K )+) one has
E∗[(ST − K)+ | Ft] = StN(d1(t, St))− Ke−r(T−t)N(d2(t, St))
where
d1(t , x) =log( x
K ) + (r + σ2
2 )(T − t)σ√
T − tet d2(t , x) =
log( xK ) + (r − σ2
2 )(T − t)σ√
T − t
and where N is the distribution function of a N (0, 1).
For Put options (ΦT = (K − ST )+) one has
E∗[(K− ST)+ | Ft] = −StN(−d1(t, St)) + Ke−r(T−t)N(−d2(t, St)).
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 51 / 87
Black Scholes modelThis formulas are fundamental because
They are easy to compute in practiceThey are a Benchmark to test numerical methodsIn the case where we don’t have closed form formulas we may use MonteCarlo Methods
price = e−rT E∗[ΦT ] ≈ e−rT 1N
N∑i=1
ΦiT
where the ΦiT are independent realizations of ΦT .
Morever we have a control of the error (CLT): with a probability of 95%
price ∈
[e−rT 1
N
N∑i=1
ΦiT −
1.96e−2rT Σ√N
, e−rT 1N
N∑i=1
ΦiT +
1.96e−2rT Σ√N
]
Σ being the (empirical) variance of ΦT .
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 52 / 87
Black Scholes model
Examples: We take x = 100, K = 100, σ = 0, 15, r = 0, 05, T = 1, t = 0.
Price of a call on mean: e−r E[
(S 1
2+S1
2 − K)
+
]
Estimated value (N=10000) = 6.05, confidence interval at 95% :[5.89; 6.20]Estimated value (N=100000) = 6.04, confidence interval at 95% : [5.99; 6.09]
Price of a call on max: e−r E[(Max(S 12, S1)− K )+] with σ = 0.5 and
K = 1.Estimated value (N=10000) = 8.68, confidence interval at 95% : [8.49; 8.87]Estimated value (N=10000) = 8.88, confidence interval at 95% : [8.82; 8.95]
For prices, the error only comes from Monte-Carlo approximations
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 53 / 87
Plan
1 Introduction to Monte Carlo MethodsPresentation of the problemAn historical exampleStrong law of large numbers (SLLN)Control of the error: the CLTConclusion
2 Black Scholes modelSimulations of Gaussian random variablesSimulation of Brownian motionReminder on the Black Scholes modelThe greeksFinite difference method for GreeksIntegration by parts method for GreeksNumerical resultsConcluding remarks
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 54 / 87
The greeks
We suppose that the payoff of a contingent claim is given by ΦT = f (ST ), thus,
Pt = E∗[e−r(T−t)ΦT | Ft ] = F (t , St) (Markov process)
where
F (t , x) = e−r(T−t)∫ +∞
−∞f (xe(r− 1
2 σ2)(T−t)+σy√
T−t)1√2π
e−y2
2 dy .
Thus, F (t , x) = E∗[e−r(T−t)f (SxT−t)].
PropositionUnder mild hypotheses on f , F∈ C1,2([0, T [×R, R).
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 55 / 87
The greeks
The greeks measure the sensitivity of the price with respect to a givenparameter.
• ∆ measures the sensitivity of the price with respect to the underlying
∆t(St) =∂F∂x
(t , St)
It is also the quantity of risky asset in the hedging portfolio!!!!• Γ measures the sensitivity of the delta with respect to the underlying
Γt(St) =∂2F∂x2 (t , St)
It is also a measure of the frequence a position must be re-hedgedin order to maintain a delta neutral position
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 56 / 87
The greeks
• Θ measures the sensitivity of the price with respect to the underlying
Θt(St) =∂F∂t
(t , St)
• ρ measures the sensitivity of the price with respect to interest rate
ρt(St) =∂F∂r
(t , St)
• vega (which is not a greel letter!!!) measures the sensitivity of the pricewith respect to the volatility
vegat(St) =∂F∂σ
(t , St)
precautions to take for the estimation of σ!
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 57 / 87
The greeksFor call and put options ( strike K maturity T ) the greeks at t = 0 are given by:
Call Put
∆0(x) N(d1) > 0 −N(−d1) < 0
Γ0(x) 1xσ√
TN ′(d1) > 0 1
xσ√
TN ′(d1) > 0
Θ0(x) − xσ
2√
TN ′(d1)− Kre−rT N(d2) < 0 xσ
2√
TN ′(d1) + Kre−rT (N(d2)− 1) ??
ρ0(x) TKe−rT N(d2) > 0 TKe−rT (N(d2)− 1) < 0
vega0(x) x√
TN ′(d1) > 0 x√
TN ′(d1) > 0
where
d1(x) =log( x
K ) + (r + σ2
2 )T
σ√
Tet d2(x) =
log( xK ) + (r − σ2
2 )T
σ√
T(3)
and where N is the distribution function of a N (0, 1).Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 58 / 87
The greeks
• In the preceding table, to obtain the values of the greeks at time t we justhave to change T into T − t
• They are easy to compute in practice
• They are a Benchmark to test numerical methods
We will restrict ourselves to ∆ and Γ.
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 59 / 87
Plan
1 Introduction to Monte Carlo MethodsPresentation of the problemAn historical exampleStrong law of large numbers (SLLN)Control of the error: the CLTConclusion
2 Black Scholes modelSimulations of Gaussian random variablesSimulation of Brownian motionReminder on the Black Scholes modelThe greeksFinite difference method for GreeksIntegration by parts method for GreeksNumerical resultsConcluding remarks
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 60 / 87
Finite difference method for Greeks
A classical method is to use a finite difference scheme
∆t(x) ≈ F (t , x + h)− F (t , x − h)
2h
Γt(x) ≈ F (t , x + h) + F (t , x − h)− 2F (t , x)
h2
where h is sufficiently small.
F (t , x + h), F (t , x − h) et F (t , x) are computed by Monte Carlo methods.
Reminder: F(t, x) = E∗[e−r(T−t)f(SxT−t)].
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 61 / 87
Finite difference method for Greeks
• Contrary to prices, for the greeks there are two factors of approximationtwo factors of approximation
Error from finite difference︸ ︷︷ ︸E1
+ Error from Monte Carlo︸ ︷︷ ︸E2
• The choice of h may be difficult (cf: Broadie-Glasserman):
• When h is too big, E1 may strongly increase.
• When h is too small, the variance of the Monte Carlo estimator may explode.
• When we use ∆t(x) ≈ F (t,x+h)−F (t,x−h)2h since
Var(F (t, x + h)− F (t, x − h)) = Var(F (t, x + h)) + Var(F (t, x + h))− 2Cov(F (t, x + h), F (t, x − h))
it is (often) better to use the same random sample for the two MonteCarlo simulations!
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 62 / 87
Finite difference method for Greeks
A priori (it will be confirmed by numerical results) this methods should bemore efficient when the prices are regular in x :
• When f (y) = 1y≥M (Binary options)
E∗[| f(Sx+hT )− f(Sx
T) |2] = P∗(
Mx + h
< e(r− 12 σ2)T+σBT <
Mx
)= O(h).
• When f (y) = (y − K )+ (Call)
E∗[| f(Sx+hT )− f(Sx
T) |2] ≤ E∗[(Sx+hT − Sx
T)2] = h2E∗[e2(r− 12 σ2)T+2σBT ] = O(h2).
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 63 / 87
Plan
1 Introduction to Monte Carlo MethodsPresentation of the problemAn historical exampleStrong law of large numbers (SLLN)Control of the error: the CLTConclusion
2 Black Scholes modelSimulations of Gaussian random variablesSimulation of Brownian motionReminder on the Black Scholes modelThe greeksFinite difference method for GreeksIntegration by parts method for GreeksNumerical resultsConcluding remarks
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 64 / 87
Integration by parts method for Greeks
The idea is simple: Write the greeks on the following form
Greeks = E ∗ [PAYOFF× weight]
with
• A weight independent of the Payoff.
• A weight such that the variance of PAYOFF× weight is minimal.
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 65 / 87
Integration by parts method for Greeks
Here we suppose that ΦT = f (ST ). The following result is independent of f :
PropositionOne has
∆t(x) = e−r(T−t)E∗[
BT−t
xσ(T − t)f (Sx
T−t)
]and
Γt(x) = e−r(T−t)E∗[(
−BT−t
x2σ(T − t)+
B2T−t − (T − t)(σ(T − t)x)2
)f (Sx
T−t)
].
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 66 / 87
Integration by parts method for GreeksProof in the case of delta: Let f ∈ C1
K (R, R) (use approximation otherwise).
According to the Lebesgue theorem of differentiation under the integral sign,
∆t(x) = e−r(T−t)∫ +∞
−∞
∂
∂xf (xe(r− 1
2 σ2)(T−t)+σy√
T−t)︸ ︷︷ ︸g(x,y)
1√2π
e−y2
2 dy .
But
∂g∂x
(x , y) =1
xσ√
T − t∂g∂y
(x , y).
Thus, using Integration by parts,
∆t(x) =e−r(T−t)
xσ√
T − t
∫ +∞
−∞f (xe(r− 1
2 σ2)(T−t)+σy√
T−t)y√2π
e−y2
2 dy
and
∆t(x) = e−r(T−t)E∗[
BT−t
xσ(T − t)f (Sx
T−t)
].
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 67 / 87
Integration by parts method for Greeks
Advantages:
• One factor of approximation (Monte Carlo)
• This method doesn’t depend on the Payoff (the weight is independent off ).
Question:• Is there a criteria (in terms of variance) to choose among all the possible
weights?
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 68 / 87
Integration by parts method for Greeks
PropositionLet Π being a square integrable weight such that for all Payoffs of the formf (ST )
Greek = E∗[f (ST )× Π].
Thus, the weight minimzing the variance of f (ST )× Π is given by
Π0 = E∗[Π | FT ].
Rk: The weights in the preceding proposion are optimal.
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 69 / 87
Integration by parts method for Greeks
Proof: Let Π fulfilling greek = E∗[f (ST )Π], we want to minimize Var(f (ST )Π).
One has
Var(f (ST )Π) = E∗[(
f (ST )Π− greek)2]
= E∗[(
f (ST )(Π− Π0) + f (ST )Π0 − greek)2]
= E∗[(
f (ST )(Π− Π0))2]
+ Var(f (ST )Π0)
+ 2E∗[(
f (ST )(Π− Π0)(Π0f (ST )− greek))]
.
The last line being equal to zero, the result follows.
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 70 / 87
Plan
1 Introduction to Monte Carlo MethodsPresentation of the problemAn historical exampleStrong law of large numbers (SLLN)Control of the error: the CLTConclusion
2 Black Scholes modelSimulations of Gaussian random variablesSimulation of Brownian motionReminder on the Black Scholes modelThe greeksFinite difference method for GreeksIntegration by parts method for GreeksNumerical resultsConcluding remarks
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 71 / 87
Numerical results Call-PutWe take x = 100, K = 100, σ = 0, 15, r = 0, 05, T = 1, t = 0.
Delta Call
0,64
0,645
0,65
0,655
0,66
0,665
0,67
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97
Nbre de Simulations (10E4)
del
ta
Mall
DF
Valeur Theo:0,6584855
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 72 / 87
Numerical results Call-Put
Gamma Call
0,0225
0,023
0,0235
0,024
0,0245
0,025
0,0255
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97
Nbre simulations(10E4)
Gam
ma Mall
DF
Valeur theo 0,0244688
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 73 / 87
Numerical results Call-Put
Delta Put
-0,349
-0,346
-0,343
-0,34
-0,337
-0,334
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97
Nbre simulations (10E4)
del
ta
Mall
DF
Valeur Theo -0,3415145
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 74 / 87
Numerical results Call-Put
Gamma Put
0,0225
0,023
0,0235
0,024
0,0245
0,025
0,0255
0,026
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97
Nbre simulations (10E4)
gam
ma Mall
DF
Valeur Theo 0,0244688
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 75 / 87
Numerical results Call-Put
Call Ratio of Variances (FD/ IBP)
∆ 0.134
Γ 0.132
Put Ratio of Variances (FD/ IBP)
∆ 0.30
Γ 1.32
• The payoff being regular, the finite difference performs quite well.• Integration by parts gives better results for Put than for Call (Explosion of
weight!!!)
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 76 / 87
Numerical results Digital
A Digital option is characterized by a payoff 1SxT >Kmin irregular in Kmin.
We can show easily that
F (0, x) = e−rT KN(d),
∆0(x) =e−rT
xσ√
Tn(d)
and
Γ0(x) =e−rT
x2σ2Tn(d)
(d + σ
√T)
where n is the density of a N (0, 1) and where d =log( x
Kmin)+(r−σ2
2 )(T )
σ√
T.
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 77 / 87
Numerical results DigitalWe take x = 100, K = 100, σ = 0, 15, r = 0, 05, T = 1, Kmin = 95.
Delta Digitale
0,02
0,0203
0,0206
0,0209
0,0212
0,0215
0,0218
0,0221
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97
Nbre simulations (10E4)
Del
ta
Mall
DF
Valeur Theo 0,0211279
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 78 / 87
Numerical results Digital
Gamma Digitale
-0,0015
-0,0014
-0,0013
-0,0012
-0,0011
-0,001
-0,0009
-0,0008
-0,0007
-0,0006
-0,0005
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97
Nbre Simulations (10E4)
Gam
ma Mall
DF
Valeur Theo 0,001058
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 79 / 87
Numerical results Digital
Digitale Ratio of Variances (FD/ IBP)
∆ 5.31
Γ 2354
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 80 / 87
Numerical results CorridorA corridor option is characterized by a payoff 1Kmax >Sx
T >Kmin (difference of twodigitals).
We take x = 100, K = 100, σ = 0, 15, r = 0, 05, T = 1, Kmin = 95,Kmax = 105.
Delta Corridor
-0,0052
-0,0047
-0,0042
-0,0037
-0,0032
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97
Nbre de simulation (10E4)
Del
ta
Mall
DF
Valeur Théo: -0,0041146
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 81 / 87
Numerical results Corridor
Gamma Corridor
-0,002
-0,0015
-0,001
-0,0005
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97
Nbre de simulations (10E4)
Gam
ma Mall
DF
Valeur Theo: -0,0009170
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 82 / 87
Numerical results Corridor
Corridor Ratio of Variances (FD/ IBP)
∆ 134
Γ 5785
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 83 / 87
Numerical results
• The more irregular is the payoff, the more efficient is the integration byparts method.
• The performance of the integration by parts method increases with theorder of derivation.
• In practice, the weights we found here are polynomials of WT . Thus theintegration by parts method will be more efficient for small maturities(small weights).
• The integration by parts method performs better for a put than for a call .
• The integration by part method is in fact a variance reduction technique..
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 84 / 87
Plan
1 Introduction to Monte Carlo MethodsPresentation of the problemAn historical exampleStrong law of large numbers (SLLN)Control of the error: the CLTConclusion
2 Black Scholes modelSimulations of Gaussian random variablesSimulation of Brownian motionReminder on the Black Scholes modelThe greeksFinite difference method for GreeksIntegration by parts method for GreeksNumerical resultsConcluding remarks
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 85 / 87
Conclusions
For a payoff of the form f (St0 , ..., Stn) one has:
Proposition
∆0(x) = e−r(T−t)E∗[
f (St0 , ..., Stn)n∑1
λi(Bti − Bti−1)
]with λ1 = 1
xσt1et ∀1 ≤ i < n − 1
λi+1 =
1x −
i∑j=1
(tj − tj−1)λjσ
(ti+1 − ti)σ.
So we may use (with optimality) the integration by parts method in the BlackScholes model for discrete Lookback or asian options.
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 86 / 87
Conclusions
• What happens for other payoffs in the Black Scholes model???
• What happens for other models???
Christophe Chorro ([email protected]) (MASTER MMMEF)Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier 2008 87 / 87