1. The document discusses using Malliavin calculus to compute Greeks, which measure the sensitivity of derivative prices to changes in variables like the underlying asset price. Malliavin calculus can improve the convergence speed for computing Greeks compared to finite difference methods, especially for options with discontinuous payoffs. 2. Traditional approaches to computing Greeks include using binomial trees or the Black-Scholes PDE. Finite difference methods introduce errors from both the Monte Carlo simulation and the derivative approximation. Malliavin calculus avoids differentiating the discontinuous payoff function. 3. The document provides an overview of Malliavin calculus, including the Malliavin derivative operator and integration by parts formula. It then compares

1. 1
Derivatives Pricing and computation of the Greeks
using Malliavin Calculus
DARMAN Guillaume - DELATTE Hugo — Tutor: Mr.Xavier BAY
Abstract—The use of derivatives has increased within the last
30 years. The annual exchange-traded futures and options are
worth 25 000 billions of dollars on all markets in 2015[1]. Deriva-
tives are mainly used to hedge, arbitrate and also to speculate.
The Black-Scholes model allows to price a derivative which means
computing the premium. When detaining a derivative an investor
must keep an eye on some parameters named Greeks to manage
its risks. Malliavin calculus is an alternative to improve the
convergence speed for the computation of Greeks particularly for
discontinuous payoffs compared to finite difference approach.
I. INTRODUCTION
THIS paper is to explain where and how Malliavin cal-
culus can be applied to financial engineering. We will
first remind briefly the mathematical model that underlies
the variation of the stock price of an asset. Then we will
introduce the use of binomial trees for pricing derivatives
in discrete-time then state the Black-Scholes formula for the
continuous-time evaluation of derivatives. We will finally give
the definition of Greeks parameters. We will focus on the ∆
and its corresponding interpretation for hedgers. We will then
compute ∆ with finite difference method on the one hand and
with Malliavin calculus on the other hand. We will compare
these two methods regarding convergence speed.
II. BROWNIAN MOTION, BLACK-SCHOLES AND GREEKS
PARAMETERS
A. Classical approach of stock price variation
The common mathematical model to simulate the stock
price, S, of an asset is based on a Brownian motion. The
variation of the stock price ∆S has a deterministic dimension
and also a random dimension thanks to the Wiener process z.
One example of such a process is shown below.
dS = µSdt + σSdz
Notations:
S: underlying asset’s price
µ: the expected annual return
σ: the annual volatility of the underlying asset
z: a standard Wiener process
In the following sections we will state that µ is equal to the
risk-free interest rate r as we assume being in a risk-neutral
world.
B. Binomial trees for derivatives pricing
A common method for pricing a derivative is based on
constructing a binomial tree to consider all the possible paths
that a share might follow until a given maturity T. To do so we
simulate n steps. Three parameters are necessary to construct
a binomial tree. These are p, u and d. For each step there is a
probability p for an up move and (1-p) for a down move. The
increasing coefficient of the stock price for each step is noted
u for an up move and d is the coefficient for a down move.
For each terminal node of the binomial tree the method is to
compute the payoff of the derivative. Then to obtain the initial
value of the derivative, also called the premium, we have to
discount the weighted payoffs at the risk-free rate according to
the risk-neutral world hypothesis. The following formulas sum
up all the important equations involved in the pricing of an
option according to binomial trees model. These formulas are
given for a n-step binomial tree. Let us consider j up moves
and (n-j) down moves until the maturity T.
• Terminal node probability: n!
j!(n−j)! pj
(1 − p)n−j
• Terminal stock price value: S0uj
dn−j
• Up move rate: u = eσ
√
T
• Down move rate: d = e−σ
√
T
Fig. 1: Two-step binomial tree

2. 2
For instance, the price of a call option according to the
binomial trees pricing method would be given by the following
formula:
c = e−rT
n
j=0
n!
j!(n − j)!
pj
(1 − p)n−j
max(S0uj
dn−j
− K; 0)
The risk-neutral world hypothesis states that the expected
return for any asset is the risk-free interest rate r over any
given period of time. This means that pu+(1−p)d = erT
. The
parameters u and d depend on the volatility σ and determine
the return for an up move (u−1) and the down move (d−1).
The valuation of derivatives using binomial trees is a
discrete-time model. In the limit as the time steps become ever
more smaller we obtain a continuous-time valuation known as
Black-Scholes formula.
C. Itˆo’s lemma and Black-Scholes PDE
As the stock price S follows an Itˆo’s process, the Itˆo’s
lemma states that a function f depending on S and t follows
the process:
df =
∂f
∂t
+
∂f
∂S
µS +
1
2
∂2
f
∂S2
σ2
S2
dt +
∂f
∂S
σSdz
Combining Itˆo’s lemma and the general Wiener process we
can demonstrate that within a risk-neutral world we obtain the
partial differential equation of Black-Scholes that follows. This
is the partial differential equation version of Black-Scholes.
In order to demonstrate this formula we have to use the no-
arbitrage argument. This argument means that any portfolio
must earn the risk-free interest rate over any given period of
time. Otherwise an investor could lock in a profit buying or
selling the portfolio depending on the situation.
∂f
∂t
+ rS
∂f
∂S
+
1
2
σ2
S2 ∂2
f
∂S2
= rf
We also have another version which uses the probability
density function. The Black-Scholes-Merton formula that al-
lows to compute the value of the premium of an European call
noted c and put noted p are given below.
c = SN(d1) − K e(−rτ)
N(d2)
p = K e(−rτ)
N(−d2) − SN(−d1)
With: 


d1 =
ln( S
K ) + (r + σ2
2 )T
σ
√
T
d2 = d1 − σ
√
T
Once we have priced a derivative an investor may want to
keep an eye on its investment and in particular the evolution
of its risks exposure. This is the role of the Greeks.
D. The Greek letters
Each Greek letter measures a different dimension to the
risk in an option position and the aim of a trader is to manage
the Greeks so that all risks are acceptable [2]. The Greeks
measures the variation of a variable of an option, for example
the option price noted f, compared to the variation of another
variable that can be the price of the underlying asset S or
the volatility σ or the risk free rate r, if we take into account
a risk-free valuation and also the time derivative of option’s
price. The five fundamentals Greeks are summarized in the
following tab.
TABLE I: Greeks letters
Delta ∆ = ∂f
∂S
Gamma Γ = ∂2
f
∂S2
Theta Θ = ∂f
∂t
Rho ρ = ∂f
∂r
Vega ν = ∂f
∂σ
E. Interpretation of the Greek ∆
The delta of an option is the rate of change of its price with
respect to the price of the underlying asset. Delta hedging,
also referred to as a delta-neutral position, implies to buy
or sell ∆ shares for each unit of derivative sold or bought.
The buy-position or sell-position is determined regarding the
sign of ∆. If ∆ is positive, a delta-neutral position implies
to buy ∆ shares. Otherwise if ∆ is negative we must sell ∆
shares whatever the derivatives.The benefits (losses) on the
shares will be offset by the losses (benefits) on the derivative.
It is very important to emphasize that ∆ is computed for
infinitesimal time-steps and as a consequence the value of ∆
varies along time. As ∆ is the number of share to buy or sell
to cover the risk, an investor has to modify at every time the
number of shares in its portfolio.
III. MALLIAVIN CALCULUS
The Malliavin calculus is the stochastic calculus of
variations. It is an infinite dimensional differential calculus
on the Wiener space [3].
First approach of Malliavin derivation
Let {Wt}t∈[0,T ] be a standard one-dimensional Brownian
motion defined on a complete probability space (Ω, F, P).
With Ω the space for continuous functions over [0, T].We
assume that F = {Ft}t∈[0,T ] is generated by W. F is a
filtration and represents all the occurrences of the Brownian
motion at time T. P is a measure of probability so that
(Wt)0 t T is a Brownian motion where:
Wt :
Ω → R
ω → ωt(ω) = ω(t)
Let F :
Ω → R
ω → F(ω)
be an Ft measurable functional.We can
define the directional derivative of F as follows:

3. 3
DhF(ω) = lim
→0
1
F(ω +
.
0
h(s)ds) − F(ω)
where h ∈ L2
[0, T] and
.
0
h(s)ds the function that asso-
ciates
t
0
h(s)ds to t ∈ [0, T]. We can demonstrate under some
hypothesis that the Malliavin derivative operator can be written
as a scalar product on the space L2
[0, T] as follows:
DhF(ω) = h, DF(ω) L2[0,T ]
where
DF :
Ω → L2
[0, T]
ω → DF(ω)
Derivative Operator of Malliavin
The Malliavin derivative operator can be summarized as
follows:
D :
L2
(Ω) → L2
(Ω, L2
[0, T])
F → DF
Ω → L2
[0, T]
ω → DF(ω)
Integration by parts formula
The adjoint operator of the Malliavin calculus is named
the Skorohod integral and is the very basis of the integration
by parts formula used in the Malliavin calculus. This adjoint
operator is defined as follows:
δ : Lq
(Ω, L2
[0, T]) → Lq
(Ω)
The adjoint operator δ is a stochastic integral if u(·, t) is Ft
measurable for any t. With the definition of the adjoint operator
we now have the integration by parts formula that follows:
E
T
0
DtF · u(·, t)dt = E (F · δ(u))
IV. ESTIMATION OF GREEKS
Monte Carlo with finite difference method may face com-
putational challenge for the Greeks of discontinuous payoffs
options because of their poor convergence. Malliavin calculus
enables to smoothen the payoff function. When using Monte
Carlo and simulation with finite difference approximation to
estimate the Greeks we make two errors: one on the numerical
computation via the Monte Carlo and another one on the
approximation of the derivative function of its finite difference.
First, we will introduce the finite different method and estimate
its error, then we will applied Malliavin calculus to estimate
Greeks.
A. Finite Difference approach
We consider a model that depends on a parameter θ and we
suppose that for each value of θ we can generate a random
variable Y (θ), representing the output. Let
f(θ) = E[Y (θ)]
In the application to option pricing, Y (θ) represent the
payoff of an option, f(θ) is its price, and θ could be any of the
model parameters that influence the price. For example, When
θ is the initial price of an underlying asset, then f(θ) represent
the options delta. An obvious approach to estimate the delta
proceeds as follows. We simulate independent replications
Y1(θ − h),...,Yn(θ − h) of the model at parameter θ − h and n
additional replications Y1(θ−h),...,Yn(θ−h) at θ+h, for some
h > 0. We average each set of replications to get Y n(θ − h)
and Y n(θ + h) and form the central-difference estimator
ˆ∆(n, h) =
Y n(θ + h) − Y n(θ − h)
2h
We have:
E[ ˆ∆] =
f(θ + h) − f(θ − h)
2h
and
Bias( ˆ∆) = E[ ˆ∆] − ∆
If f is twice differentiable in a neighborhood of θ, then
f(θ + h) = f(θ) + f (θ)h +
1
2
f (θ)h2
+ o(h2
)
f(θ − h) = f(θ) − f (θ)h +
1
2
f (θ)h2
+ o(h2
)
Then
Bias( ˆ∆) =
f(θ + h) − f(θ − h)
2h
− f (θ)
If f is three times differentiable at θ, we have
Bias( ˆ∆) =
1
6
f (θ)h2
+ o(h2
)
To improve accuracy of the central-difference estimators we
would like to chose small values of h. But the positive effect
of a small h on bias must be weighed against its negative effect
on variance. The variance of the central-difference estimator
is
V ar[ ˆ∆(n, h)] =
1
4h2
V ar[Y n(θ + h) − Y n(θ − h)]
If we suppose that the pairs (Yi(θ −h), Yi(θ +h)), i = 1, 2,...,
are i.i.d we have
V ar[Y n(θ + h) − Y n(θ − h)] =
1
n
V ar[Y (θ + h) − Y (θ − h)]
Then, we can rewrite the variance of the central-difference
estimator as
V ar[ ˆ∆(n, h)] =
1
4h2n
V ar[Y (θ + h) − Y (θ − h)]
If we simulate Y (θ −h) and Y (θ +h) independently we have
V ar[ ˆ∆(n, h)] =
1
4h2n
(V ar[Y (θ + h)] + V ar[Y (θ − h)])
≈
a
4h2n

4. 4
We saw that decreasing h can decrease bias while increasing
variance. Therefore, the use of finite different method to
estimate Greeks requires balancing these two considerations.
To find the optimal h, we want to minimized the mean square
error (MSE). In our case
MSE( ˆ∆) ≈
1
36
f 2
(θ)h4
+
a
4h2n
The optimal h is
h∗
≈
9a
2f 2(θ)n
1/6
And the convergence is in O(n1/3
)
But for discontinuous payoff like for binary option, the
finite difference method is very inefficient. To overcome this
inefficiency, we will use the Malliavin calculus to avoiding
the differentiation of the payoff function. We will be able to
compute the Greeks as the expectation of the original payoff
times a weigh.
B. Malliavin calculus for Greeks estimation
We suppose that there are no transactions costs or taxes,
all securities are perfectly divisible, there are no dividends
during the life of the derivative, there are no riskless arbitrage
opportunities and we can define a risk neutral probability
measure Q , the risk-free rate r is constant and security trading
is continuous with a finite horizon t ∈ [0, T]. We assume
that the market is complete. The uncertainty is modeled by
the complete probability space (Ω, F, Q) and the information
evolves according to the filtration {Ft, t ∈ [0, T]} generated
by the standard Wiener process (Wt). The evolution of the
underlying price (Xt)t∈[0,T ] in the risk neutral universe is
described by the standard one dimensional Black-Scholes
SDE:
dXt = rXtdt + σXtdWt
X0 = x
⇔ Xt = x e(r−σ2
/2)t+σWt
Where X0 denote the initial underlying asset price.
We denote the Ft measurable functional by F:
F(ω) = φ x e(r−σ2
/2)T +σWT (ω)
= f (WT (ω))
With φ the discounted payoff and



f(z) =φ x e(r−σ2
/2)T +σz
f (z) =φ x e(r−σ2
/2)T +σz
σx e(r−σ2
/2)T +σz
The Malliavin derivative is
DtF(ω) = f (WT (ω))1[0,T ](t)
= φ (XT (ω))
∂XT (ω)
∂x
σx
= φ (XT (ω))
∂XT (ω)
∂x
∂Xt
∂x
−1
σXt
With
XT = x e(r−σ2
/2)T +σWT
We denote the price of the derivative by v:
v(x) = E [φ(XT )|X0 = x]
The Greek Delta is defined as
∆ =
∂v(x)
∂x
= E
∂φ(XT )
∂x
We rewrite Delta depending on the Malliavin derivative
∆ = E
∂φ(XT )
∂x
= E
∂φ
∂x
(XT )
∂XT
∂x
∂Xt
∂x
−1
σXt ×
∂Xt
∂x
(σXt)
−1
= E Dtφ(XT ) ×
∂Xt
∂x
(σXt)−1
We integrate from 0 to T and use the Malliavin integration
by parts formula
T × ∆ = E
T
0
Dtφ(XT )
∂Xt
∂x
(σXt)−1
= E φ(XT )
T
0
u(ω, t)dWt(ω)
With u(., t) = ∂Xt
∂x (σXt)−1
We have
∆ =
1
T
E φ(XT )
T
0
∂Xt
∂x
(σXt)−1
dWt
With
∂Xt
∂x
= e(r−σ2
/2)t+σWt
=
Xt
x
Finally
∆ = E φ(XT )
WT
xσT
WT
xσT is called the optimal Malliavin weight.
We compute the optimal Malliavin weight for the other
Greeks of an European option in a Black & Scholes model:
Greek Malliavin weight
Delta WT
xσT
Gamma 1
T 2σx
W 2
T
σT − 1
σ − WT
Vega
W 2
T
σT − 1
σ − WT
Rho WT
σ − T

5. 5
C. Analysis of the results
We implemented a pricing tool in Python to analyze, visual-
ize the convergence and compare the finite difference approach
and Malliavin calculus.
In the following example, we estimate the Delta of an
European Call and the Gamma of a Digital Call and we
compare the different approaches. We use the following input:
TABLE II: Greeks letters
Stock Price S = 100
Strike Price K = 100
Maturity T = 6 months
Expected return r = 0.01
Volatility 0.20
Number of Simulations 100, 000
Finite difference step h = 0.5
Delta estimation of an European Call
Fig. 2: Comparison between Finite difference and Malliavin
approach on the Delta of a continuous payoff
Eu Call’s Delta B&S Finite Diff Malliavin
Delta 0.54 0.55 0.54
Bias 0 αh2
0
Standard deviation 0 0.042 0.004
Gamma estimation of a Digital Call
Fig. 3: Comparison between Finite difference and Malliavin
approach on the Gamma of a discontinuous payoff
Digital’s Gamma B&S Finite Diff Malliavin
Gamma -2.1e-4 -7.6e-3 -2.1e-4
Bias 0 β + γh2
0
Standard deviation 0 0.015 1.5e-5
In both cases, Malliavin calculus approach has better con-
vergence proprieties. The time convergence is even faster com-
pared to finite difference method with discontinuous payoffs.
V. CONCLUSION
Given the results shown above we can notice that Malliavin
calculus improves the convergence speed for the computation
of Greeks. Not only this method is faster but is also a non-
biased estimation of these parameters. Furthermore, the very
application of Malliavin calculus and the very convenient
aspect is that this method applies for discontinuous payoffs
and can treat the singularities of all derivatives.
We would like to warmly thank our professor Xavier BAY
for his great help on the understanding of Malliavin calculus
and all the time that he devoted to our research project.
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