Financial mathematics

Financial mathematics
Xavier Charvet
March 21, 2019

Contents
I. Fundamentals 5
1. Arbitrage pricing theory 7
2. Equivalent martingale measure and arbitrage 9
2.1. Radon-Nikodym theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2. Suﬃcient condition for no-arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3. Complete markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3. Derivative security pricing and complete markets 11
3.1. Change of numeraire technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2. Girsanov’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
II. From Black-Scholes to local-stochastic volatility models 13
4. The Black-Scholes Model 15
4.1. Derivation of the Black-Scholes equation . . . . . . . . . . . . . . . . . . . . . . . 15
4.1.1. Equity derivatives (without dividends) . . . . . . . . . . . . . . . . . . . . . 15
4.1.2. Equity derivatives (with dividends) and FX derivatives . . . . . . . . . . . . . 16
4.1.3. Terminal conditions and present value . . . . . . . . . . . . . . . . . . . . . 17
4.2. Feynman-Kac formula and risk-neutral expectation . . . . . . . . . . . . . . . . . . 18
4.3. Risk-neutrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.3.1. Equity derivatives (without dividends) . . . . . . . . . . . . . . . . . . . . . 19
4.3.2. FX derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.3.3. Foreign risk-neutral measure . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.3.4. Connection between domestic and foreign risk-neutral measures . . . . . . . . . 22
4.4. European Options Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.4.1. First method: risk-neutrality argument . . . . . . . . . . . . . . . . . . . . 22
4.4.2. Second method: solving the Black-Scholes PDE . . . . . . . . . . . . . . . . 24
5. Local volatility models 25
5.1. The Fokker-Planck equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.2. The Breeden-Litzenberger formula . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.3. Dupire’s local volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.4. The pricing PDE for local volatility models . . . . . . . . . . . . . . . . . . . . . . 29
6. Stochastic volatility models 31
6.1. Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.2. The pricing PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.3. Stochastic volatility models and risk-neutral measures . . . . . . . . . . . . . . . . . 34
6.4. The Heston model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Contents
III. Interest rates modeling 37
7. Diffusing the yield-curve 39
7.1. Arbitrage-free dynamics: the HJM framework . . . . . . . . . . . . . . . . . . . . . 39
7.2. Linear Gaussian models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
7.2.1. Linear Gaussian 1F models . . . . . . . . . . . . . . . . . . . . . . . . . . 39
7.2.2. Zero-coupon bond option pricing . . . . . . . . . . . . . . . . . . . . . . . 41
7.3. Libor market models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
7.3.1. Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
7.3.2. Parametrized functional forms of the covariance matrix . . . . . . . . . . . . . 42
7.3.3. Factor reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
7.3.4. Spot measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
8. Standard products 45
8.1. Libor in arrears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
8.2. Constant maturity swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
8.2.1. Method 1: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
8.2.2. Method 2: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
8.3. Quantoed CMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
8.3.1. Method 1: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
8.3.2. Method 2: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
9. Long-dated FX modeling 49
9.1. The 3 factors model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
9.1.1. The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
9.1.2. Pricing European FX options . . . . . . . . . . . . . . . . . . . . . . . . . 50
IV. Inflation modeling 53
10. Inflation derivatives 55
10.1. The Jarrow-Yildirim model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
10.2. Pricing zero-coupon inflation-indexed swaps . . . . . . . . . . . . . . . . . . . . . 55
10.3. Pricing year-on-year inflation-indexed swaps . . . . . . . . . . . . . . . . . . . . . 55
10.4. Pricing LPIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
V. The Greeks in the Black-Scholes model 57
11. Calls and puts 59
11.1. The premium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
11.2. The Delta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
11.3. The Gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
11.4. The Vega . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
11.5. The Theta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
11.6. The Rho . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
12. Binary options 63
12.1. The premium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
12.2. The Delta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
12.3. The Gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
12.4. The Vega . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
12.5. The Theta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
12.6. The Rho . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Xavier Charvet, Janvier 2019. Tous droits réservés.

Contents 5
13. Barrier options 67
13.1. Pricing and replicating barrier options . . . . . . . . . . . . . . . . . . . . . . . . 67
13.2. Down-and-in regular call . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
13.3. The premium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
13.3.1. Method 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
13.3.2. Method 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
13.3.3. Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
13.4. The Delta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
13.5. The Vega . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
VI. Numerical methods 71
14. Time series 73
14.1. Autoregressive models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
14.1.1. The AR models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
14.1.2. The ARMA models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
14.2. Autoregressive conditional heteroskedasticity models . . . . . . . . . . . . . . . . . 73
14.2.1. Original formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
14.2.2. Alternative formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
15. Tests 75
16. Monte-Carlo methods 77
16.1. Simulating a standard normal variable . . . . . . . . . . . . . . . . . . . . . . . . 77
16.2. Simulating a multinormal variable . . . . . . . . . . . . . . . . . . . . . . . . . . 77

1. Arbitrage pricing theory
To be improved later.

2. Equivalent martingale measure and
arbitrage
2.1. Radon-Nikodym theorem
Let P, ˆP be two equivalent measures. Then
• ˆP is associated to P with the density process ζt = EP
t
d ˆP
dP ∀t ∈ [0, T]
• ζt is a P-martingale, ζ0 = 1
• ∀YT which is FT -measurable, E
ˆP
t (YT ) = EP
t (YT
ζT
ζt
)
2.2. Sufficient condition for no-arbitrage
Let Dt be a deflator, i.e. a strictly positive Ito process. Given an asset process X, we
note XD the normalized asset process XD := X/D. QD is said to be an equivalent
martingale measure if XD is a QD-martingale (for any permissible asset process X).
If there exists a deflator D that allows for a QD-martingale measure, then there
is no arbitrage among all permissible strategies.
Theorem 2.1
If there exists such a deflator D, D is called the numeraire associated to the QD-
martingale measure.
2.3. Complete markets
A market is said to be complete if for all VT which is FT − measurable, VT is attainable
with a permissible strategy.

12 Equivalent martingale measure and arbitrage
A market is complete if and only if there exists a deﬂator D inducing a unique
martingale measure.
Theorem 2.2

3. Derivative security pricing and
complete markets
We suppose that the market is complete and D is a deflator inducing a unique equiva-
lent martingale measure QD. Then the price of a product whose payoff is VT at time T,
is given by Vt = DtEQD
t
VT
DT
.
3.1. Change of numeraire technique
Let Mt and Nt be two numeraires inducing equivalent martingale measures QM and QN .
If the market is complete, then
• The density process ζt := EQN
t
dQM
dQN satisfies the relation ζt = Mt/M0
Nt/N0
• For all VT which is FT measurable, EQM
t (VT ) = EQN
t VT
ζT
ζt
3.2. Girsanov’s theorem
Let P and Q be two equivalent measures, and ζt the density process ζ(t) := EP
t
dQ
dP .
One writes dζ(t)
ζ(t) = −θ(t) · dWt (martingale’s representation theorem, Wt being a n-
dimensional brownian motion under P). Then W∗(t) := W(t) +
t
0 θ(s)ds is a brownian
motion under Q.

Part II.
From Black-Scholes to
local-stochastic volatility models

4. The Black-Scholes Model
Black and Scholes (1973) consider a geometric Brownian motion
dSt = µStdt + σStdWt (4.1)
and make the following assumptions:
• The spot price St of a unit of the underlying asset follows a lognormal process (4.1),
driven by a constant drift µ and a constant volatility σ.
• Short selling is permissible.
• There is no transaction costs.
• There is no riskless arbitrage
• Trading is continuous
Remark: We will also use this chapter to introduce the main tools and concepts of quantitative ﬁnance,
which will be encountered regularly throughout this book. This includes:
• Backward pricing PDEs (Feynman-Kac)
• Risk-neutrality
• Change of measure / numeraire
4.1. Derivation of the Black-Scholes equation
In this section we derive the partial diﬀerential equation governing the price of an equity
derivative.
4.1.1. Equity derivatives (without dividends)
Let V (St, t) denote the value of a contingent claim at time t, conditional on the equity
price St. Applying Ito’s formula on V (St, t), we get
dV (St, t) =
∂V
∂t
dt +
∂V
∂S
dSt +
1
2
∂2V
∂S2
dS2
t (4.2)
Writing the quadratic variation of St as dS2
t = σ2S2
t dt and separating the deterministic
and stochastic parts, we get

18 The Black-Scholes Model
dV (S, t) =
∂V
∂t
+
1
2
σ2
S2
t
∂2V
∂S2
dt +
∂V
∂S
dSt (4.3)
We now consider a portfolio Πt which is long one unit of the contingent claim V (St, t)
and short ∆t units of the underlying asset. We will then chose ∆t such that Πt is riskless.
Πt = Vt − ∆tSt (4.4)
Diﬀerentiating Πt gives
dΠt = dVt − ∆tdSt (4.5)
which we rewrite as
dΠt =
∂V
∂t
+
1
2
σ2
S2
t
∂2V
∂S2
dt +
∂V
∂S
− ∆t dSt (4.6)
By taking ∆t = ∂V
∂S , we make the portfolio riskless. In that case, the growth of Πt
needs to be equal to the growth of a risk-free bond Bt, whose evolution is described by
(assuming that interest rates are constant):
dBt = rBtdt (4.7)
Therefore we have
dΠt =
∂V
∂t
+
1
2
σ2
S2
t
∂2V
∂S2
dt = rΠtdt (4.8)
Replacing Πt by Πt = Vt − ∂V
∂S St, this leads to the well-known Black-Scholes equation
∂V
∂t
+
1
2
σ2
S2 ∂2V
∂S2
+ rS
∂V
∂S
− rV = 0 (4.9)
As we can see, the real-world growth rate µ does not appear in the Black-Scholes
equation. This is because the dSt term has been completely removed from 4.6 by taking
∆t = ∂V
∂S .
4.1.2. Equity derivatives (with dividends) and FX derivatives
In FX, or when the stock has dividends, the situation is more complicated because St is
not a tradeable asset.
In case where there is dividends, the tradeable asset is ˜St = St exp(qt) where q is the
dividend yield. In the FX case, the tradeable asset is StBf
t where Bf
t is the underlying
foreign bond, St being the FX rate.
In this section, we derive the Black-Scholes formula for an FX derivative - the situation
in equity for a stock with dividends being similar.

Derivation of the Black-Scholes equation 19
We denote V (St, t) the price of a contingent claim. Again, we have:
dV (S, t) =
∂V
∂t
+
1
2
σ2
S2
t
∂2V
∂S2
dt +
∂V
∂S
dSt (4.10)
We now consider a portfolio Πt which is long one unit of the contingent claim V (St, t)
and short ∆t units of StBf
t . We will chose ∆t such that Πt is riskless.
Πt = Vt − ∆tStBf
t (4.11)
dΠt = dVt − ∆td(StBf
t )
= dVt − ∆tBf
t dSt − ∆tStdBf
t
= dVt − ∆tBf
t dSt − ∆tStrf
Bf
t dt
which we rewrite as
dΠt =
∂V
∂t
+
1
2
σ2
S2
t
∂2V
∂S2
− ∆tStrf
Bf
t dt +
∂V
∂S
− ∆tBf
t dSt (4.12)
By taking ∆t = 1
Bf
t
∂V
∂S , we make the portfolio riskless. In that case, the growth of
Πt needs to be equal to the growth of a risk-free domestic bond Bd
t , whose evolution is
described by (assuming that interest rates are constant):
dBd
t = rd
Bd
t dt (4.13)
Therefore we have
dΠt =
∂V
∂t
+
1
2
σ2
S2
t
∂2V
∂S2
− ∆tStrf
Bf
t dt = rd
Πtdt (4.14)
Replacing Πt by Πt = Vt − 1
Bf
t
∂V
∂S St, this leads to the Black-Scholes equation in the
FX context:
∂V
∂t
+
1
2
σ2
S2 ∂2V
∂S2
+ (rd
− rf
)S
∂V
∂S
− rd
V = 0 (4.15)
4.1.3. Terminal conditions and present value
The PDEs above are called parabolic equations and are solved backwards in time. Once
the terminal condition is known (V (ST , T) being the payoff of the contingent claim), the
present value of the option can be computed by solving backwards the PDE.
Remark: This PDE is also called convection-diffusion equation. In physics, it describes phenomena
where physical quantities such as particles or energy are transferred inside a physical system due to two
processes: diffusion and convection. The contribution in S2 ∂2
V
∂S2 describes diffusion. The contribution in
S ∂V
∂S
describes convection. The last contribution in V is called a forcing term.

4.2. Feynman-Kac formula and risk-neutral expectation
The Feynman-Kac formula makes a connection between the solution of a backward
parabolic partial differential equation and the expectation of the payoff of the deriva-
tive under an artificial measure - called the risk-neutral measure.
Let’s suppose that we have a backward parabolic equations of the form
∂V
∂t
+ a(x, t)
∂V
∂x
+
1
2
b2
(x, t)
∂2V
∂x2
= 0 (4.16)
with terminal condition V (x, T) = h(x), then the solution of 4.16 can be ex-
pressed as
V (x, 0) = Ed
[h(XT )|X0 = x] (4.17)
where Xt follows the diffusion
dXt = a(Xt, t)dt + b(Xt, t)dWd
t (4.18)
Theorem 4.1 (Feynman-Kac)
Proof Let’s consider the process Vt = V (Xt, t) and apply Ito’s forumla:
dVt =
∂V
∂t
dt +
∂V
∂x
dXt +
1
2
∂2
V
∂x2
dX2
t
=
∂V
∂t
+ a(Xt, t)
∂V
∂x
+
1
2
b2
(Xt, t)
∂V 2
∂x2
dt + b(Xt, t)
∂V
∂x
dWd
t
= b(Xt, t)
∂V
∂x
dWd
t
Therefore V (X0, 0) = Ed
(VT ) = Ed
(h(XT ))
In the presence of a forcing term f,
∂V
∂t
+ a(x, t)
∂V
∂x
+
1
2
b2
(x, t)
∂2V
∂x2
+ f(x, t)V = 0 (4.19)
we put ˆVt = exp
t
0 f(Xs, s)ds Vt and we find that
d ˆVt = exp
t
0
f(Xs, s)ds b(Xt, t)
∂V
∂x
dWd
t (4.20)
Therefore, we have in that case
V0 = Ed
exp
t
0
f(Xs, s) VT (4.21)

Risk-neutrality 21
In the FX context for instance, from 4.15, we have that the diffusion of St under the
risk-neutral measure is
dSt = (rd
− rf
)Stdt + σStdWd
t (4.22)
The present value of a contingent claim is the discounted risk-neutral expectation of
the payoff under the domestic risk-neutral measure
V0 = exp(−rd
T)Ed
[VT ] (4.23)
4.3. Risk-neutrality
In this section, we give an hands-on outline about risk-neutrality. We have outlined
in the previous section that the risk-neutral measure is the measure under which (all)
tradeable assets evolve with a drift equal to the interest rates r. We have also seen that
in the FX context, where there is a domestic and a foreign currency, the drift of the FX
rate under the (domestic) risk-neutral measure is equal to rd − rf . These considerations
wil be formalized in a theoretical way later on in this book - this section is merely an
hands-on, pragmatic approach.
4.3.1. Equity derivatives (without dividends)
Let’s consider a stock which evolves, in the real-world measure, as
dSt = µStdt + σStdWt (4.24)
At time t, the distribution of St is given by
St = S0 exp (µ −
σ2
2
)t + σWt (4.25)
To see this, we apply Ito’s formula on f(St) = log(St) to find that
d log(St) =
1
St
dSt −
1
2S2
t
dS2
t = (µ −
σ2
2
)dt + σdWt (4.26)
and 4.25 is then a straightforward consequence of 4.26.
A risk-neutral investor must expect St to have the same expected return as a domestic
bond Bd
t = erdt. In other words, the ratio between St and Bd
t must be a martingale under
the risk-neutral measure. Bd
t is the numeraire associated with the risk-neutral measure.

dZt = St/Bd
t
= S0 exp (µ −
σ2
2
)t + σWt exp(−rd
t)
= S0 exp −
σ2
2
t + σWt exp (µ − rd
)t
For Zt to be martingale, we then need to work under the measure Pd under which µ
is equal to rd:
dSt = µStdt + σStdWt
= rd
Stdt + σStdWd
t (4.27)
We see from 4.27 that the drift change to go from the real-world measure to the risk-
neutral measure is
Wd
t = Wt +
µ − rd
σ
t (4.28)
For each time t, the Radon-Nikodym density of Pd with respect to P, conditional to
Ft is given by Girsanov’s theorem:
dPd
dP
|
Ft
= exp −γWt −
1
2
γ2
t (4.29)
with γ = µ−rd
σ
4.3.2. FX derivatives
We have seen that in the FX context, the risky asset is StBf
t . Again, a risk-neutral
investor must expect StBf
t to have the same expected return as a domestic bond Bd
t .
Under the (domestic) risk-neutral measure, the ratio between these two assets must then
be a martingale.
dZt = StBf
t /Bd
t
= S0 exp (µ −
σ2
2
)t + σWt exp((rf
− rd
)t)
= S0 exp −
σ2
2
t + σWt exp (µ + rf
− rd
)t
For Zt to be martingale, we then need to work under the measure Pd under which µ
is equal to µd = rd − rf .

Risk-neutrality 23
= (rd
− rf
)Stdt + σStdWd
t (4.30)
We see from 4.30 that the drift change to go from the real-world measure to the risk-
neutral measure is
Wd
t = Wt +
µ − µd
σ
t (4.31)
For each time t, the Radon-Nikodym density of Pd with respect to P, conditional to
dPd
dP
|
Ft
= exp −γd
Wt −
1
2
(γd
)2
t (4.32)
with γd = µ−µd
σ
4.3.3. Foreign risk-neutral measure
Alternatively, in the FX context, we can work with Bd
t /St = Bd
t
ˆSt as being the risky asset,
where ˆSt = 1/St is the ﬂipped spot FX rate. In that case, a risk-neutral investor must
expect Bd
t /St to have the same expected return as a foreign bond Bf
t . Under the foreign
risk-neutral measure, the ratio between these two assets must then be a martingale. Bf
t
is the numeraire associated with the foreign risk-neutral measure.
dZt = ˆStBd
t /Bf
t
= ˆS0 exp (−µ +
σ2
2
)t − σWt exp((rd
− rf
)t)
= ˆS0 exp −
σ2
2
t − σWt exp (−µ + rd
− rf
+ σ2
)t
For Zt to be martingale, we then need to work under the measure Pf under which µ
is equal to µf = rd − rf + σ2.
= (rd
− rf
+ σ2
)Stdt + σStdWf
t (4.33)
We see from 4.33 that the drift change to go from the real-world measure to the foreign
risk-neutral measure is
Wf
t = Wt +
µ − µf
σ
t (4.34)

For each time t, the Radon-Nikodym density of Pf with respect to P, conditional to
dPf
dP
|
Ft
= exp −γf
Wt −
1
2
(γf
)2
t (4.35)
with γf = µ−µf
σ
4.3.4. Connection between domestic and foreign risk-neutral measures
In this paragraph, we establish the connection between the domestic and the foreign
risk-neutral measures.
Let’s recall that
dPd
dP
|
Ft
= exp −γd
Wt −
1
2
(γd
)2
t , where γd
=
µ − µd
σ
(4.36)
and
dPf
dP
|
Ft
= exp −γf
Wt −
1
2
(γf
)2
t , where γf
=
µ − µf
σ
(4.37)
Since
Wf
t = Wd
t +
µd − µf
σ
t = Wd
t − σt
we then have that the density of Pf with respect to Pd is given by
dPf
dPd
|
Ft
= exp σWd
t −
1
2
σ2
t (4.38)
4.4. European Options Pricing
In this section, we derive the prices of calls and puts under the Black-Scholes model. Let
us then consider a call option, whose payoff is VT = (ST −K)+ at time T. We will assume
here that St is a stock with a divident yield equal to q. Two methods are exposed. The
first one is the most straightforward and uses the concept of risk-neutrality. The second
one solves the Black-Scholes PDE with the appropriate boundary conditions.
4.4.1. First method: risk-neutrality argument
We have seen that the present value of a contingent European claim is the discounted
expected value of the payoff under the risk-neutral measure:
V0 = e−rdT
Ed
(ST − K)+
(4.39)

European Options Pricing 25
V0 = e−rdT
Ed
(ST − K)+
= e−rdT
Ed
[(ST − K)1ST ≥K]
= e−rdT
Ed
[ST 1ST ≥K] − e−rdT
KEd
[1ST ≥K]
= e−rdT
Ed
[ST 1ST ≥K] − e−rdT
KPd
[ST ≥ K] (4.40)
Ed [ST 1ST ≥K] can be rewritten as
Ed
[ST 1ST ≥K] = Ed
S0 exp (r − q −
σ2
2
)t + σWd
t 1ST ≥K
= S0 exp ((r − q)t) Ed dPf
dPd
|
Ft
1ST ≥K
= S0 exp ((r − q)t) Ef
[1ST ≥K]
= S0 exp ((r − q)t) Pf
[ST ≥ K] (4.41)
where Pf denotes the measure whose density respectively to Pd is exp σWd
t − 1
2σ2t .
In the FX context, this is the foreign risk-neutral measure. In the situation of a stock
with dividends, this is the measure associated with the numeraire exp(qt).
Straightfoward calculations lead to
Pf
[ST ≥ K] = Φ(d1)
where
d1 =
ln(S0/K) + (r − q + σ2
2 )T
σ
√
T
and Φ denotes the cumulative distribution function (CDF) of a gaussian variable.
Similarly, we have
Pd
[ST ≥ K] = Φ(d2)
where
d2 =
ln(S0/K) + (r − q − σ2
2 )T
σ
√
T
The price of a call option is then given by
V C
0 = S0e−rf T
Φ(d1) − Ke−rdT
Φ(d2)
and the price of a put option is given by
V P
0 = Ke−rdT
Φ(−d2) − S0e−rf T
Φ(−d1)

4.4.2. Second method: solving the Black-Scholes PDE
We need to solve the Black-Scholes PDE
∂C
∂t
+
1
2
σ2
S2 ∂2C
∂S2
+ rS
∂C
∂S
− rC = 0 (4.42)
with the following boundary conditions
C(S, t) = 0 ∀t
C(S, t) → S asS → ∞
C(S, T) = max(S − K, 0) (4.43)
This PDE can be transformed into a diﬀusion equation with the following change of
variables:
τ = T − t
u = Cerτ
x = ln(
S
K
) + (r −
σ2
2
)τ (4.44)
Such that 4.42 becomes
∂u
∂τ
=
1
2
σ2 ∂2u
∂x2
(4.45)
with initial condition u(x, 0) = K(emax(x,0) − 1)
At time τ, u(x, τ) can be expressed as
u(x, τ) =
1
σ
√
2πτ
∞
−∞
u(y, 0) exp −
(x − y)2
2σ2τ
dy
which yields the following for u(x, τ)
u(x, τ) = Kex+1
2
σ2τ
Φ(d1) − KΦ(d2)
where d1 and d2 are well-known. Reverting u, x and τ to the opriginal set of variables
yields the solution of the Black-Scholes equation for a call option.

5. Local volatility models
The main limitation of the Black-Scholes model is that it is unable to reproduce the
term-structure of volatilities and tbe smile which is encountered on the options market.
While it is fairly straightforward to account for the term-structure of ATM volatilities
by imposing a time-dependent, deterministic volatility in the Black-Scholes model, it is
somehow more complicated to adapt the model to account for the smile on each maturity.
The following modiﬁcation to the Black-Scholes model enables to reproduce the forward
value and the ATM volatilities for each maturity correctly
dSt = µtStdt + σtStdWt
However for each maturity, the model implied lognormal smile is perfectly ﬂat, which
is not consistent with what is observed on the markets.
Dupire (1993) shows that it is possible to construct a state-dependent instantaneous
volatility σloc(St, t) that recovers the entire implied volatility surface. In this chapter we
show how to determine such a local volatility function, given an implied volatility surface
σimp(K, T).
dSt = µtStdt + σloc(St, t)StdWt
5.1. The Fokker-Planck equation
This equation is the dual of the Feynman-Kac equation. While Feynman-Kac equation
governs the evolution of the price of a contingent claim and is solved backwards in time,
Fokker-Planck equation governs the evolution of the probability density and is solved
forward in time.

28 Local volatility models
Let’s suppose that we have a process Xt of the form
dXt = a(Xt, t)dt + b(Xt, t)dWt (5.1)
with initial conditions (x0, t0). Then the probability density function p(x, t)
satisﬁes the following PDE (Fokker-Planck equation)
∂p(x, t)
∂t
= −
∂[a(x, t)p(x, t)]
∂x
+
1
2
∂2[b2(x, t)p(x, t)]
∂x2
(5.2)
Theorem 5.1 (Fokker-Planck)
Proof Suppose that g(x) is an arbitrary function such that limx→±∞ g(x) = 0. Applying
Ito’s formula on Gt = g(Xt), we have
dGt =
∂g
∂x
dXt +
1
2
∂2
g
∂x2
dX2
t
= a(x, t)
∂g
∂x
+
1
2
b2
(x, t)
∂2
g
∂x2
dt + b(x, t)
∂g
∂x
dWt (5.3)
such that
d
dt
E[Gt] = E a(x, t)
∂g
∂x
+
1
2
b2
(x, t)
∂2
g
∂x2
=
x
p(x, t) a(x, t)
∂g
∂x
+
1
2
b2
(x, t)
∂2
g
∂x2
dx (5.4)
Integrating by parts and using the property of g when x → ±∞ gives
d
dt
E[Gt] = −
x
∂[a(x, t)p(x, t)]
∂x
g(x)dx +
1
2 x
∂2
[b(x, t)p(x, t)]
∂x2
g(x)dx (5.5)
Diﬀerentiating
E[Gt] =
x
p(x, t)g(x)dx (5.6)
with respect to t gives
d
dt
E[Gt] =
x
∂p(x, t)
∂t
g(x)dx (5.7)
Finally, equating 5.5 and 5.7 gives, since g is arbitrary,
∂p(x, t)
∂t
= −
∂[a(x, t)p(x, t)]
∂x
+
1
2
∂2
[b2
(x, t)p(x, t)]
∂x2
(5.8)

The Breeden-Litzenberger formula 29
5.2. The Breeden-Litzenberger formula
Let’s suppose that the distribution of an asset S at a given expiry T is given by the risk-
neutral probability density fST
(s). We suppose that the interest rates are constant and
equal to rd. We also suppose that we have a continuum of call prices for this maturity,
given by the function K → C(K, T).
The Breeden-Lizenberger formula makes a connection between the risk-neutral prob-
ability density of the asset at time T and the prices of the call options on this asset for
this maturity.
fd
ST
(K) = erdT ∂2C(K, T)
∂K2
(5.9)
Theorem 5.2 (Breeden-Litzenberger formula)
Proof The price of a call option is given by the formula
C(K, T) = e−rd
T
Ed
[(ST − K)+
]
= e−rd
T
∞
0
(s − K)+
fd
ST
(s)ds
= e−rd
T
∞
K
(s − K)fd
ST
(s)ds (5.10)
We now use the fact that if
F(x) =
b(x)
a(x)
f(x, s)ds (5.11)
then its derivative with respect to x is given by
d
dx
F(x) = f(x, b(x))
db(x)
dx
− f(x, a(x))
da(x)
dx
+
b(x)
a(x)
∂
∂x
f(x, s)ds (5.12)
Applying this to C(K, T) with a(K) = K and b(K) = ∞ we get
∂C(K, T)
∂K
= −e−rd
T
∞
K
fd
ST
(s)ds (5.13)
Diﬀerentiating again gives
∂2
C(K, T)
∂K2
= e−rd
T
fd
ST
(s)ds (5.14)

30 Local volatility models
5.3. Dupire’s local volatility
We assume for now that interest rates are equal to 0.
The risk-neutral probability density function satisfies Breeden-Litzenberger formula
p(K, T) =
∂2C(K, T)
∂K2
(5.15)
and also satisfies the Fokker-Planck equation
∂p(K, T)
∂T
=
1
2
∂2[b2(K, T)p(K, T)]
∂K2
(5.16)
since in this case, we assume there is no interest rates. Differentiating 5.15 with respect
to T gives
∂p(K, T)
∂T
=
∂2
∂K2
∂C(K, T)
∂T
(5.17)
and equating and integrating twice 5.16 and 5.17 with respect to K gives
1
2
b2
(K, T)p(K, T) =
∂C
∂T
+ α(T)K + β(T) (5.18)
When K goes to +∞, p(K, T) and C(K, T) both tend to (smoothly) 0 for all T, so
α(T) = 0 and β(T) = 0.
Therefore the diffusion term is equal to
b(K, T) =
2∂C/∂T
∂2C/∂K2
(5.19)
Since b(K, T) = σloc(K, T)K we have the Dupire’s formula
In the absence of interest rates,
σloc(K, T) =
2∂C/∂T
K2∂2C/∂K2
(5.20)
Theorem 5.3 (Dupire’s formula)

The pricing PDE for local volatility models 31
5.4. The pricing PDE for local volatility models
Let’s work in the general FX context. The diﬀusion of the spot FX interest rate is given
by
dSt = (rd
t − rf
t )Stdt + σloc(St, t)StdWt (5.21)
The price of a contingent claim is governed by the following PDE
∂V
∂t
+
1
2
σ2
loc(St, t)S2 ∂2V
∂S2
+ (rd
− rf
)S
∂V
∂S
− rd
V = 0 (5.22)

6. Stochastic volatility models
We have seen that a local volatility model has the ability to fit the market smile that
is observed for European vanilla options. However this kind of model still suffers from
a lack of realism. In fact, empirical analysis show that the volatility of a stock or more
generally of any non-riskless asset is itself stochastic. A pure local-volatility model tends
to underestimate the forward skew and forward smile that is observed on the forward
starting options market. Stochastic volatility models are built in order to remedy this
problem.
6.1. Generalities
Let’s consider a generic stochastic volatility model of the form
dSt = rtStdt +
√
vtStdW1
t
dvt = a(vt)dt + b(vt)dW2
t
under a risk-neutral measure Pd, with < dW1
t , dW2
t >= ρdt
The following formula connects the local volatility σ2
loc(K, T) implied by this model,
with the risk-neutral expectation of the instantaneous variance vT at time T conditional
on the asset price ST being equal to K.
σ2
loc(K, T) = Ed
[vT |ST = K] (6.1)
Theorem 6.1 (Gatheral’s formula)
Proof Let’s assume for simplicity that interest rates are equal to zero and consider the value
of a European option of strike K:
C(ST , K) = Ed
(ST − K)+
Differentiating with respect to K twice gives
∂C
∂K
= −Ed
[H(ST − K)]
∂2
C
∂K2
= Ed
[δ(ST − K)] (6.2)

34 Stochastic volatility models
where H and δ are respectively the Heaviside and the Dirac delta function. Using Ito’s formula
on f(St) = (St − K)1St≥K, we get:
d(St − K)+
= df(St)
= f (St)dSt +
1
2
f (St)dS2
t
= H(St − K)dSt +
1
2
δ(St − K)vtS2
t dt (6.3)
The stochastic diﬀential of the price of a call option C being
dCt = dEd
(St − K)+
= Ed
H(St − K)dSt +
1
2
δ(St − K)vtS2
t dt
=
1
2
Ed
δ(St − K)vtS2
t dt
=
1
2
Ed
δ(St − K)vtS2
t dt
=
1
2
Ed
[vt|St = K] Ed
[δ(St − K)] K2
=
1
2
Ed
[vt|St = K] p(St, K)K2
(6.4)
Therefore
∂C
∂T
=
1
2
Ed
[vT |ST = K] p(ST , K)K2
(6.5)
which can be rewritten as
σ2
loc(K, T) = Ed
[vT |ST = K] (6.6)
6.2. The pricing PDE
Suppose that the stock price and its variance satisfy the following SDEs
dSt = µtStdt +
√
vtStdW1
t
dvt = α(St, vt, t)dt + β(St, vt, t)
√
vtdW2
t
under the real-world measure, with < dW1
t , dW2
t >= ρdt.
Contrary to the Black-Scholes model, there is here two sources of randomness. There-
fore we also seek to hedge changes in volatility in order to form a riskless portfolio.

The pricing PDE 35
Consider a portfolio Πt which is long one unit of a contingent claim and short ∆ units
of the stock and ∆1 units of another asset whose value V1 depends on volatility.
Πt = Vt(St, vt, t) − ∆tSt − ∆1V1(St, vt, t) (6.7)
dΠt =
∂V
∂t
+
1
2
vS2 ∂2V
∂S2
+ ρvβS
∂2V
∂v∂S
+
1
2
vβ2 ∂2V
∂v2
dt
− ∆1
∂V1
∂t
+
1
2
vS2 ∂2V1
∂S2
+ ρvβS
∂2V1
∂v∂S
+
1
2
vβ2 ∂2V1
∂v2
dt
+
∂V
∂S
− ∆1
∂V1
∂S
− ∆ dSt
+
∂V
∂v
− ∆1
∂V1
∂v
− ∆ dvt (6.8)
To make the portfolio riskless, we choose ∆ and ∆1 such that
∂V
∂S
− ∆1
∂V1
∂S
− ∆ = 0
∂V
∂v
− ∆1
∂V1
∂v
− ∆ = 0 (6.9)
such that both dS and dv terms vanish in 6.8.
Therefore
dΠt =
∂V
∂t
+
1
2
vS2 ∂2V
∂S2
+ ρvβS
∂2V
∂v∂S
+
1
2
vβ2 ∂2V
∂v2
dt
− ∆1
∂V1
∂t
+
1
2
vS2 ∂2V1
∂S2
+ ρvβS
∂2V1
∂v∂S
+
1
2
vβ2 ∂2V1
∂v2
dt
= rΠtdt
= r(V − ∆S − ∆1V1)dt (6.10)
Separating V and V1 terms leaves us with
∂V
∂t
+
1
2
vS2 ∂2V
∂S2
+ ρvβS
∂2V
∂v∂S
+
1
2
vβ2 ∂2V
∂v2
+ rS
∂V
∂S
− rV /
∂V
∂v
=
∂V1
∂t
+
1
2
vS2 ∂2V1
∂S2
+ ρvβS
∂2V1
∂v∂S
+
1
2
vβ2 ∂2V1
∂v2
+ rS
∂V1
∂S
− rV1 /
∂V1
∂v
(6.11)
Equation 6.11 is valid for any pair (V, V1) of contingent claims which depend on both
St and vt. Therefore there must exist a function f(St, vt, t) independent of the contingent
claim V such that

36 Stochastic volatility models
∂V
∂t
+
1
2
vS2 ∂2V
∂S2
+ ρvβS
∂2V
∂v∂S
+
1
2
vβ2 ∂2V
∂v2
+ rS
∂V
∂S
− rV = f(St, vt, t)
∂V
∂v
(6.12)
Without loss of generality, we can write
f(St, vt, t) = −α(St, vt, t) + φ(St, vt, t)β(St, vt, t)
√
vt (6.13)
φ(St, vt, t) is called the market price of volatility risk.
Similarly to the Black-Scholes model, we can see that the drift term µt does not appear
in the pricing PDE. However, there is still one unknown, due to the additional source of
randomness: the market price of volatility risk, which needs to be estimated.
6.3. Stochastic volatility models and risk-neutral measures
The reason φ is called the market price of volatility risk is that it represents the extra
return per unit of volatility risk dW2
t for a delta hedeged - but non-vega hedged, portfolio.
To see this, consider the portfolio Π1 which is long one unit of a contingent claim and
short ∆ = ∂V
∂S units of the stock.
Π1 = V − ∆S (6.14)
Applying Ito’s formula on Π1 we have:
dΠ1 =
∂V
∂t
+
1
2
vS2 ∂2V
∂S2
+ ρvβS
∂2V
∂v∂S
+
1
2
vβ2 ∂2V
∂v2
dt
+
∂V
∂S
− ∆ dSt +
∂V
∂v
dv (6.15)
Therefore
dΠ1 − rΠ1dt =
∂V
∂t
+
1
2
vS2 ∂2V
∂S2
+ ρvβS
∂2V
∂v∂S
+
1
2
vβ2 ∂2V
∂v2
+ rS
∂V
∂S
− rV dt
+
∂V
∂v
dv
= (−α + φβ
√
vt)
∂V
∂v
dt +
∂V
∂v
(αdt + β
√
vtdW2
t )
= β
√
v
∂V
∂v
φ(St, vt, t)dt + dW2
t (6.16)

The Heston model 37
By setting d ˜W2
t = φ(St, vt, t)dt + dW2
t , we define the risk-neutral measure associated
with the market price of volatility risk φ. It is important to notice that for a stochas-
tic volatility model, there is an infinity of risk-neutral measures (one for each φ function).
Under the risk-neutral measure associated with φ, we then have
dSt = rtStdt +
√
vtStd ˜W1
t
dvt = [α(St, vt, t) − β(St, vt, t)
√
vtφ(St, vt, t)] dt + β(St, vt, t)
√
vtd ˜W2
t (6.17)
Careful: < dW1
t , dW2
t >=< d ˜W1
t , d ˜W2
t >
6.4. The Heston model
The Heston model (1993) assumes the following diffusion for a stock or a FX rate
dSt = µtStdt +
√
vtStdW1
t
dvt = κ(m − vt)dt + α
√
vtdW2
t (6.18)
under the real-world measure, with < dW1
t , dW2
t >= ρdt.

Part III.
Interest rates modeling

7. Diﬀusing the yield-curve
7.1. Arbitrage-free dynamics: the HJM framework
Modeling interest rates dynamics represents a conceptual gap with respect to the equi-
ties world, since one now needs to study the dynamics of a whole yield curve along time,
instead of the dynamics of a single asset.
For the dynamics of the zero-coupon bonds to be arbitrage-free, the continuum of zero-
coupon bonds need to satisfy the following dynamics under the risk-neutral measure:
dP(t, T)
P(t, T)
= rtdt + σP (t, T)dWT
t (7.1)
where {WT
t ; 0 ≤ T < ∞} are a continuum of Brownian motions under the risk-neutral
measure (i.e. associated to the numeraire Bt) and σ(t, T) is a Ft-adapted process.
In the one-factor case (i.e. when all zero-coupons are driven by a same Brownian
motion Wt), the relation P(t, T) = exp −
T
t F(t, s)ds leads to the following SDE for
the instantaneous forward rates:
dF(t, T) = σP (t, T)
∂σP (t, T)
∂T
dt +
∂σP (t, T)
∂T
dWt (7.2)
The HJM framework is the most general formulation of interest rates dynamics. How-
ever this class of models is in general non-Markovian (in which case they are computa-
tionally intractable, therefore ineﬃcient in practice).
When σP (t, T) is deterministic, the model is called Gaussian HJM.
7.2. Linear Gaussian models
7.2.1. Linear Gaussian 1F models
This class of models stands for the case where instantenous forward rates are driven by
a unique Brownian motion and σP (t, T) is deterministic for all T.
The Hull-White model
The dynamics of the short-rate under the risk-neutral measure in the Hull-White model
are as follows:

42 Diﬀusing the yield-curve
drt = κ(θt − rt)dt + σdWt (7.3)
where θt is deterministic (calibrated to the initial yield curve), which leads to:
r(t) = e−κt
r(0) +
t
0
eκ(s−t)
θsds + σe−κt
t
0
eκu
dWu (7.4)
This model can be straightforwardly expressed in the HJM framework as (WT
t being
this time a brownian motion under the T-forward measure):
dF(t, T) = σe−κ(T−t)
dWT
t (7.5)
which leads to:
P(t, T) = exp(A(t, T) − B(t, T)rt) (7.6)
where
B(t, T) = 1−exp(−κ(T−t))
κ
A(t, T) = (θt − σ2
2κ2 )(B(t, T) − (T − t)) − σ2B(t,T)2
4κ
The correlation between r(t) and r(s) for t < s is given by:
ρ(rt, rs) =
cov(rt, rs)
stddev(rt)stddev(rs)
=
e2κt − 1
e2κs − 1
Remark: the term κ is the speed to which the process reverts to its mean value. In-
creasing κ reduces the volatility of the forward rates and the correlation between them.
The extended Vasicek model
The Hull-White model is a particular case of the extended Vasicek model, where the
dynamics of the short-rate under the risk-neutral measure are as follows:
drt = κt(θt − rt)dt + σtdWt (7.7)
where κt, θt, σt are deterministic. In the HJM framework, the speciﬁcation is
dF(t, T) = σt exp −
T
t
κudu dWT
t (7.8)
which leads to:
P(t, T) =
P(0, T)
P(0, t)
exp(−x(t)G(t, T) −
1
2
y(t)G(t, T)2
) (7.9)
where



G(t, T) =
T
t exp(−
u
t κsds)du
x(t) = r(t) − F(0, t)
y(t) =
t
0 exp(−2
t
u κsds)σ2
udu

Libor market models 43
7.2.2. Zero-coupon bond option pricing
We consider the contingent claim whose payoff at T is VT = (P(T, T∗) − K)+ where
T∗ ≥ T and K is the strike. Its value at time t < T is given by:
Vt = P(t, T)ET
t [(P(T, T∗
) − K)+
] (7.10)
Since P(T, T∗) is log-normally distributed under the T-forward measure, this leads to
Vt = P(t, T∗
)Φ(d+) − P(t, T)Φ(d−)K (7.11)
where d± =
ln
P (t,T ∗)
KP (t,T )
±v/2
√
v
and v =
T
t |σP (u, T∗) − σP (u, T)|2
du
7.3. Libor market models
In this framework, one diffuses directly the 3M or 6M forward Libor rates (they are
martingales under their respective forward measures).
7.3.1. Variants
The Brace-Gatarek-Musiela model
Under the Ti+1-forward measure, the dynamics of the forward Libor rate Li(t, Ti, Ti+1)
is given by:
dLi(t)
Li(t)
= σi(t)dWi(t)
where σi is assumed to be deterministic (lognormal distribution), and dWi, dWj =
ρi,jdt.
Local-volatility models
To take into account the skew observed on the market, one usually uses local-volatility
models, such as the one below (this is a displaced diffusion, the original lognormal BGM
is a particular case, with q = 1):
dLi(t) = qLi(t) + (1 − q)Li(0) σi(t)dWi(t)
Remark: alternative popular local-volatility Libor market models use a CEV process
instead of the displaced diffusion (but the CEV does not allow for negative rates).

Stochastic volatility models
Using stochastic volatility gives some curvature to the smile (kurtosis). One of the most
popular Libor market model in the financial industry is as follows:
dLi(t) = qLi(t) + (1 − q)Li(0) σi(t) z(t)dWi(t)
dz(t) = κ(θ − z)dt + ν
√
zd ˜W(t)
with θ := z0 := 1 and d ˜W, dWi = 0.
The process of the stochastic volatility is called a CIR process. When q = 1, this is a
multi-Heston model.
7.3.2. Parametrized functional forms of the covariance matrix
A popular choice for the paramerisation of the volatilities and correlations is as follows:
σi(t) = (a + b(Ti − t)) exp(−c(Ti − t)) + d
ρij(t) = exp(−α |i − j|)
7.3.3. Factor reduction
One usually uses much less Brownian motions than there are Libor rates involved in the
diffusion. Typically, 3 to 5 factors are enough to take into account the main variations
of the yield curve.
Which means that one can use ˜Wk, k = 1...5 independent Brownian motions and one
then writes:
dWi(t) =
k
aikd ˜Wk(t)
where dWi, dWj = k aikajk dt = ρijdt.
7.3.4. Spot measure
In practice, one usually diffuses the forward Libor rates under the spot measure, i.e. the
measure associated to the numeraire N(t) :=
m(t)
j=0 (1 + Lj(Tj)τj)P(t, Tm(t)+1), where
Tm(t) ≤ t < Tm(t)+1.

Libor market models 45
Calculating the drifts under the spot measure
Let’s write ¯Li(t) = 1 + τiLi(t). Since
¯Li(t)Pi+1(t)
N(t) and Pi(t)
N(t) are martingales under the spot
measure, one has:
EN
t d
¯Li(t)Pi+1(t)
N(t) = 0 and EN
t d Pi(t)
N(t) = 0
Consequently, if one poses Xi(t) = Pi+1(t)/N(t) one has:
EN
t d ¯Li(t)Xi(t) = 0 = EN
t d ¯LiXi+dXi
¯Li+d ¯LidXi = EN
t d ¯LiXi +EN
t d ¯LidXi
(the term EN
t dXi
¯Li disappears since Xi(t) is drift-free under the spot-measure).
Therefore, EN
t
d ¯LiXi
¯LiXi
+ d ¯LidXi
¯LiXi
= 0.
Let’s pose, by deﬁnition: X, Y := E dXdY
XY . Then one has:
¯Li, Xi = ¯Li, Pi+1 − ¯Li, N
= ¯Li,
i
k=0
(1 + Lkτk)−1
− ¯Li,
m(t)
k=0
(1 + Lkτk)−1
=
i
k=0
¯Li, (1 + Lkτk)−1
−
m(t)
k=0
¯Li, (1 + Lkτk)−1
=
i
k=m(t)+1
¯Li, (1 + Lkτk)−1
= −
i
k=m(t)+1
¯Li, (1 + Lkτk)
Therefore,
EN
t
−τidLi
1 + τiLi
+
i
m(t)+1
τidLi
1 + τiLi
τkdLk
1 + τkLk
= 0 (7.12)
Deriving the drifts in the no-skew case
Under the spot measure: dLi
Li
= µidt + σidWi. Replacing in 7.12 gives:
−τi
1 + τiLi
(µiLidt) +
k
τi
1 + τiLi
τk
1 + τkLk
dLidLk = 0
which gives:
µi =
i
m(t)+1
τkLk
1 + τkLk
σiσkρikdt

Deriving the drifts in the skew case
If one poses ˜Li = qLi +(1−q)Li(0), one has under the spot measure: d ˜Li
˜Li
= ˜µidt+qσidWi
The equation 7.12 can then be rewritten as EN
t
−τid ˜Li
1+τiLi
+ i
m(t)+1
τid ˜Li
1+τiLi
τkd ˜Lk
1+τkLk
1
q = 0
which gives:
˜µi =
i
m(t)+1
τk
˜Lk
1 + τkLk
σiσkρikdt

8. Standard products
The products below can be priced directly using any interest rates model (or dual cur-
rency model if the trade is quantoed) by using Monte-Carlo simulations. Their price will
obviously (slightly) depend on the model used (LG1F, LG2F, multi-factor LMM etc).
However it is worth knowing a few shortcuts for pricing these standard products, which
we present in the following paragraphs.
8.1. Libor in arrears
In a Libor in arrears contract, the Libor rate Li(Ti, Ti+1) is paid at time Ti (in a regular
forward rate agreement, this would be at time Ti+1).
For conveniency, we note Li(t) := Li(t, Ti, Ti+1). From the relation P(t,Ti+1)
P(t,Ti) = 1
1+τiLi(t) ,
one gets Li(t) = P(t,Ti)−P(t,Ti+1)
τiP(t,Ti+1) therefore Li(t) is a martingale under the Ti+1-forward
measure.
The value at time t of such a payment is:
Vt = P(t, Ti)ETi
t [Li(Ti)] = P(t, Ti)E
Ti+1
t [Li(Ti)] = P(t, Ti)E
Ti+1
t Li(Ti)
dPi/dPi+1(Ti)
dPi/dPi+1(t)
= P(t, Ti)E
Ti+1
t Li(Ti)
P(Ti, Ti)/P(t, Ti)
P(Ti, Ti+1)/P(t, Ti+1)
= P(t, Ti+1)E
Ti+1
t Li(Ti)(1 + τiLi(Ti))
= P(t, Ti+1)Li(0) + τiP(t, Ti+1)E
Ti+1
t Li(Ti)2
(which is consistent with the fact that a payment of Li(Ti, T+1) at time Ti is equivalent
to a payment of Li(Ti, Ti+1)(1 + τiLi(Ti, Ti+1)) at time Ti+1).
The price of a Libor in arrears in then model-dependent. In a log-normal Libor market
model, one has ETi+1 Li(Ti)2 = Li(0)2eσ2
i Ti (more generally, the distribution of Li(Ti)
may be extracted from a continuum of caplets prices).
This is called a convexity adjustment.

48 Standard products
8.2. Constant maturity swaps
A CMS payment consists in paying at time T a swap rate which is observed at time
Ts ≤ T, whose underlying maturity is Te (in general Te > T).
From the relation S(t, Ts, Te) = P(t,Ts)−P(t,Te)
τiP(t,Ti) one gets that S(t) is a martingale un-
der its swap measure (i.e. the measure associated to the numeraire annuity A(t, Te) :=
τiP(t, Ti)).
The value at time t of such a payment is
Vt = P(t, T)ET
t S(Ts)
= P(t, T)EA
t S(Ts)
dPT /dPA(Ts)
dPT /dPA(t)
= P(t, T)EA
t S(Ts)
P(Ts, T)/A(Ts, Te)
P(t, T)/A(t)
The price of a CMS payment is then model-dependent.
8.2.1. Method 1:
One assumes that S(Ts) and the Radon-Nikodym derivative R(Ts) := P(Ts,T)/A(Ts,Te)
P(t,T)/A(t)
are lognormally distributed and correlated with a correlation ρ. In that case, one has
ET S(Ts) = S(0, Ts, Te)eρσSσRT .
8.2.2. Method 2:
A usual assumption is to work under a Linear Swap Rate model, where the zero-coupon
bonds discounted by the annuity can be expressed as follows:
ˆP(t, T) := P(t, T)/A(t) = A+BT S(t), where the parameters A and B are ﬁtted using
• the relation τi
ˆP(t, Ti) = 1, which holds for all values of S(t), and therefore gives
A = 1/ τi
• and the martingale property ˆP(0, T) = EA[ ˆP(t, T)] = A + BT S(0) which gives
BT = ( ˆP(0, T) − A)/S(0)
Using this LSM assumption, one then has Vt = A(t)EA
t S(TS)(A + BT S(TS))
Since S(t) is a martingale under the swap measure, Vt only depends on EA
t S(TS)2 ,
which is easily tractable if one works under a log-normal Swap rate Market model (more
generally the distribution of S(Ts) can be extracted from a continuum of swaption prices).

Quantoed CMS 49
8.3. Quantoed CMS
A quantoed CMS payment consists in paying at time T, in a domestic currency, a foreign
swap rate which is observed at time Ts ≤ T, whose underlying maturity is Te (in general
Te > T).
The value at time t of such a payment is
Vt = Pd
(t, T)ET,d
t Sf
(Ts) = Pd
(t, T)EA,f
t Sf
(Ts)
dPT,d/dPA,f (Ts)
dPT,d/dPA,f (t)
= Pd
(t, T)EA,f
t Sf
(Ts)
Pd(Ts, T)/(Af (Ts)X(Ts)d/f )
Pd(t, T)/(Af (t)X(t)d/f )
(we use the fact that from the domestic economy perspective, X(t)d/f Af (t) is the do-
mestic numeraire associated to the PA,f measure).
One then needs to evaluate:
EA,f
t Sf
(Ts)
Pd(Ts, T)
(Af (Ts)X(Ts)d/f )
= EA,f
t Sf
(Ts)X(Ts)f/d Pd(Ts, T)
Pf (Ts, T)
Pf (Ts, T)
Af (Ts)
= EA,f
t Sf
(Ts)X(Ts, T)f/d Pf (Ts, T)
Af (Ts)
where X(t, T)f/d is the forward FX rate maturing at T, seen from time t.
8.3.1. Method 1:
One assumes that Sf (Ts) and the Radon-Nikodym derivative R(Ts) are lognormally
distributed and correlated with a correlation ρ. In that case, one has ET Sf (Ts) =
Sf (0, Ts, Te)eρσSσRT .
8.3.2. Method 2:
As seen previously, one assumes that one can write Pf (Ts,T)
Af (Ts)
= Af + Bf
T Sf (Ts)
The price of the quantoed CMS then depends on the marginal distributions of Sf (Ts)
and X(Ts, T)f/d under the PA,f measure, and the dependence between them. If one
assumes that both are lognormally distributed and are correlated with a correlation
factor ρX,S, this leads to straightforward calculations. It then remains to evaluate
EP,f (X(Ts, T)f/d): this is achieved using the fact that the Radon-Nykodym derivative is
a martingale under PA,f (and using the Linear Swap Rate assumption).

9. Long-dated FX modeling
One of the most popular (and simplest) model one encounters in the long-dated FX fi-
nancial industry is the following 3 factors model: one diffuses the domestic and foreign
currencies using a Hull-White model and one uses a lognormal (or local volatility) dif-
fusion for the spot FX rate. More advanced models involve a stochastic volatility for
the FX rate and a multifactor diffusion for the domestic and foreign yield curves (dual
currency Libor Market Model).
9.1. The 3 factors model
9.1.1. The model
The model is as follows:
dSt
St
= (rd
t − rf
t )dt + σS
(St, t) · dWd
t
drd
t = κd
t (θd
t − rd
t )dt + σd
t · dWd
t
drf
t = κf
t (θf
t − rf
t )dt + σf
t · dWf
t
where · stands for the inner product and Wd
t , Wf
t are 3-dimensional brownian motions,
respectively under the domestic and foreign risk-neutral measures (therefore σS, σd, σf
are 3-dimensional vectors).
We have seen in 4.38 that Wf
t = Wd
t −σS(St, t)t (that was for a 1-dimensional brownian
motion but it is also true in the multidimensional case). Therefore the 3-F model can be
rewritten as:
dSt
St
= (rd
t − rf
t )dt + σS
(St, t) · dWd
t
drd = κd
t (θd
t − rd
t )dt + σd
t · dWd
t
drf = κf
t (θf
t − rf
t ) − σS
t (St, t) · σf
t dt + σf
t · dWd
t
where Wd
t is a 3-d brownian motion under the domestic risk-neutral measure. An
equivalent usual formulation is as follows:

52 Long-dated FX modeling
dSt
St
= (rd
t − rf
t )dt + σS
(St, t) dW1,d
t
drd = κd
t (θd
t − rd
t )dt + σd
t dW2,d
t
drf = κf
t (θf
t − rf
t ) − ρS,f
t σS
t (St, t) σf
t dt + σf
t dW3,d
t
where || stands for the 3-d norm, and each Wi,d
t , i = 1...3 is a 1-d BM under the
domestic risk-neutral measure. The correlations between these 1-d bm are then given by:
dW1,d
t , dW2,d
t = ρS,d
t dt =
σS(St, t) · σd
t
|σS(St, t)| σd
t
dt
dW1,d
t , dW3,d
t = ρS,f
t dt =
σS(St, t) · σf
t
|σS(St, t)| σf
t
dt
dW2,d
t , dW3,d
t = ρd,f
t dt =
σd
t · σf
t
σd
t σf
t
dt
Finally, the classical formulation is as follows (that was a bit long but at least we have
properly shown where the new drift comes from, which is rarely done in books):
dSt
St
= (rd
t − rf
t )dt + σS
(St, t)dW1,d
t
drd = κd
t (θd
t − rd
t )dt + σd
t dW2,d
t
drf = κf
t (θf
t − rf
t ) − ρS,f
t σS
t (St, t)σf
t dt + σf
t dW3,d
t
(we have removed the || so now the volatilities are scalars).
9.1.2. Pricing European FX options
Deterministic FX volatility case
The payoﬀ at time T being VT = (FT − K)+, the value at time t of such an option is
then Vt = Pd(t, T)ET
t FT − K
+
where the forward FX rate F(t, T) = StPd(t,T)
Pf (t,T)
is a martingale under the domestic
T-forward measure.
When the volatility of the FX rate is deterministic, FT is then lognormally distributed
under the foreign T-forward measure, consequently one has:

The 3 factors model 53
Vt = Pd
(t, T)Ef,T
t FT − K
+
= Pd
(t, T) Ft,T N(d+) − KN(d−)
= StPf
(t, T)N(d+) − KPd
(t, T)N(d−)
where:
Ft,T =
StPf (t, T)
Pd(t, T)
d± =
ln(Ft,T /K) ± 1
2
T
t σ(s)2ds
T
t σ2(s)ds
σ(t)2
= σS
(t)2
+ σd,T
(t)2
+ σf,T
(t)2
− 2ρS,d
σS
(t)σd,T
(t) + 2ρS,f
σS
(t)σf,T
(t) − 2ρd,f
σd,T
(t)σf,T
(t)
σd,T (t) and σf,T (t) being the volatilities of Pd(t, T) and Pf (t, T).

10. Inflation derivatives
10.1. The Jarrow-Yildirim model
The methodology is similar to the approach used in long-dated FX: the nominal rates
can be seen as the domestic interest rates, the real rates as the foreign interest rates, and
the inflation index as the FX rate. The JY model is then nothing else than the usual 3F
model encountered in the long-dated FX industry. The model is then as follows, under
the nominal risk-neutral measure:
dIt
It
= (nt − rt)dt + σIdWI(t)
dnt = κn
t (θd
t − nt)dt + σndWn(t)
drt = κr
t (θr
t − rt) − ρI,rσIσr dt + σrdWr(t)
10.2. Pricing zero-coupon inflation-indexed swaps
The fixed leg pays (1 + K)n − 1 at time Tm, the floating leg pays (I(Tm)
I0
− 1) at time Tm
where m is the number of years.
Receiving I(Tm) at time Tm is the same as receiving I(t)Pr(t, Tm) at t (just like in the
FX world) therefore the value of the floating leg at time t is
ZCIIS(t, Tm) =
It
I0
Pr(t, Tm) − Pn(t, Tm)
At time t = 0, this reduces to ZCIIS(0, Tm) = Pr(t, Tm) − Pn(t, Tm) and one can then
extract the values of the real zero-coupons from the prices of the zc inflation indexed
swaps (and it’s model-independent).
10.3. Pricing year-on-year inflation-indexed swaps
The fixed leg pays τiK and the floating leg pays τi
I(Ti)
I(Ti−1) − 1 , both at time Ti.
The value at time t of the payment at time Ti of floating-leg is then given by:

58 Inflation derivatives
Y Y IIS(t, Ti−1, Ti) = τiEn
t exp −
Ti
t
nudu
I(Ti)
I(Ti−1)
− 1
= τiEn
t exp −
Ti−1
t
nudu exp −
Ti
Ti−1
nudu
I(Ti)
I(Ti−1)
− 1
= τiEn
t exp −
Ti−1
t
nudu exp −
Ti
Ti−1
nudu
I(Ti)
I(Ti−1)
− 1
= τiEn
t exp −
Ti−1
t
nudu En
Ti−1
exp −
Ti
Ti−1
nudu
I(Ti)
I(Ti−1)
− 1
= τiEn
t exp −
Ti−1
t
nudu Pr(Ti−1, Ti) − Pn(Ti−1, Ti)
= τiPn(t, Ti−1)E
n,Ti−1
t Pr(Ti−1, Ti) − τiPn(t, Ti)
Therefore the price of a year-on-year inflation indexed swap is model-dependent.
10.4. Pricing LPIs
The LPI index of maturity TN capped at C and floored at F is defined as
LPIN =
N
i=1
1 + [Yi]F
C
where N is the number of years and Yi = I(Ti)
I(Ti−1) − 1 is the year-on-year inflation rate.
Method: extract the distributions of the Yi under the forward Ti-measure from the
YoY options, then apply a shift using the JY model to get the distributions under the
TN -forward terminal measure (first order of the change of measure), and use a (gaus-
sian) copula to specify the dependences between the YoY rates. Then use Monte-Carlo
simulations (Sobol to make it faster).

Part V.
The Greeks in the Black-Scholes
model

11. Calls and puts
11.1. The premium
The fair value of a call option in the Black-Scholes model is:
V = e−qτ SΦ(d1) − Ke−rτ Φ(d2).
Figure 11.1.: Premium call; Premium put: K = 100, σ = 0.2, r = 0.03, T = 3m; 1y
11.2. The Delta
The Delta is the sensitivity of the value of an option with respect to the underlying’s
price: ∆ = ∂V
∂S .
For a call:
∂C
∂S = ∂
∂S e−qτ SΦ(d1)−Ke−rτ Φ(d2) = e−qτ Φ(d1)+e−qτ S ∂d1
∂S Φ (d1)−Ke−rτ ∂d2
∂S Φ (d2)
Using the relations ∂d1
∂S = ∂d2
∂S = 1
σS
√
τ
and e−qτ SΦ (d1) = Ke−rτ Φ (d2), one then gets:
∆call = e−qτ Φ(d1).
Similarly, for a put, one calculates ∆put = −e−qτ Φ(−d1) = ∆call − 1.

62 Calls and puts
Figure 11.2.: Delta call; Delta put: K = 100, σ = 0.2, r = 0.03, T = 3m; 1y
11.3. The Gamma
The Gamma is the second derivative of the value of an option with respect to the under-
lying’s price: Γ = ∂2V
∂S2 .
One has Γcall = Γput = ∂2V
∂S2 = e−qτ φ(d1)
Sσ
√
τ
. As one can see on the graphs, the Gamma
reaches its maximum slightly out of the money.
Figure 11.3.: Gamma: K = 100, σ = 0.2, r = 0.03, T = 3m; 1y

The Vega 63
11.4. The Vega
The Vega is the sensitivity of the value of an option with respect to the volatility: V = ∂V
∂σ .
One has Vcall = Vput = e−qτ S
√
τφ(d1)
Figure 11.4.: Vega: K = 100, σ = 0.2, r = 0.03, T = 3m; 1y
11.5. The Theta
The Theta is the sensitivity of the value of an option with respect to the time to ma-
turity: Θ = −∂V
∂τ . One has Θcall = −e−qτ Sφ(d1)σ
2
√
τ
− rKe−rT Φ(d2) + qSe−qτ Φ(d1) and
Θput = −e−qτ Sφ(d1)σ
2
√
τ
+ rKe−rT Φ(−d2) − qSe−qτ Φ(−d1)
Figure 11.5.: Theta: K = 100, σ = 0.2, r = 0.03, r = 0.03, T = 3m; 1y

64 Calls and puts
11.6. The Rho
The Rho is the sensitivity of an option w.r.t. the risk-free interest rates: ρ = ∂V
∂r . One
has ρcall = Kτe−rτ Φ(d2) and ρput = −Kτe−rτ Φ(−d2).
Figure 11.6.: Theta: K = 100, σ = 0.2, r = 0.03, T = 3m; 1y

12. Binary options
12.1. The premium
The fair value of a digital call option is D = − ∂C
∂K where C is the value of the associated
European call option. In the Black-Scholes model, that gives Cbinary = e−rτ Φ(d2) and
Pbinary = e−rτ Φ(−d2). In the following, all the graphs are rescaled by a factor K.
Figure 12.1.: Premium call; Premium put: K = 100, σ = 0.2, r = 0.03, T = 3m; 1y

66 Binary options
12.2. The Delta
One has ∆b-call = −∆b-put = e−rτ 1
Sσ
√
τ
φ(d2).
Figure 12.2.: Delta binary: K = 100, σ = 0.2, r = 0.03, T = 3m; 1y
12.3. The Gamma
One has Γb-call = −Γb-put = −e−rτ φ(d2)
S2σ
√
τ
1 + d2
σ
√
τ
.
Figure 12.3.: Gamma binary: K = 100, σ = 0.2, r = 0.03, T = 3m; 1y

The Vega 67
12.4. The Vega
One has Vb-call = −Vb-put = −e−rτ φ(d2)d1
σ .
Figure 12.4.: Vega binary: K = 100, σ = 0.2, r = 0.03, T = 3m; 1y
12.5. The Theta
One has Θb-call = −Θb-put = re−rτ Φ(d2) − e−rτ φ(d2)2(r−q)τ−d1σ
√
τ
2τσ
√
τ
.
Figure 12.5.: Theta binary: K = 100, σ = 0.2, r = 0.03, T = 3m; 1y

68 Binary options
12.6. The Rho
One has ρb-call = −ρb-put = −τe−rτ Φ(d2) + e−rτ φ(d2)
√
τ
σ .
Figure 12.6.: Theta binary: K = 100, σ = 0.2, r = 0.03, T = 3m; 1y

13. Barrier options
Knock-in options: the option activates only if the underlying goes beyond the barrier
before maturity T. Knock-out options: the option ceases to exist if the underlying goes
beyond the barrier before maturity T.
Options regular: the barrier is out of the money. When the barrier is in the money, it
is called a reverse option.
The relations below show that one only needs to study up-and-in regular options.
1. KIt + KOt = Eurt.
2. KIt = KIreg
t + KIrev
t (by spliting the payoﬀ in two parts: one above the barrier
H, and one below)
13.1. Pricing and replicating barrier options
One then assumes that at time 0, the barrier H has not been reached yet.
1. Once the barrier is reached (if that happens), the value of the option at time τH is
then the value of the European option paying f(ST ) at time T,
i.e. VτH = EurτH (H, f(ST ), T).
2. Using the relation 13.1, one has EurτH (H, f(ST ), T) = EurτH H, ST
H
γ
f H2
ST
, T ,
whose value at time t is Vt = Eurt St, ST
H
γ
f H2
ST
, T .
3. Applying the formula 13.1 again, one obtains Vt = St
H
γ
Eurt St, f H2
S2
t
ST , T

70 Barrier options
In the Black-Scholes model, one has
Eurt(St, f(ST ), T) = Eurt St,
ST
St
γ
f
S2
t
ST
, T
where γ = 1 − 2r
σ2 .
Proposition 13.1
Proof Since ST
S0
γ
= eσγWT − 1
2 (σγ)2
T
is of expectation 1, it deﬁnes a probability mea-
sure ˜Q whose density w.r.t. the risk-neutral measure Q is d ˜Q
dQ |FT
:= ST
S0
γ
. One has
S2
0
ST
=
S0e(r−σ2
/2)T −σ ˜WT
, ˜Wt := Wt − σγt being a Brownian motion under ˜Q. Therefore, the law of
S2
0
ST
under ˜Q is the same as the law of ST under Q and one then has EQ
(f(ST )) = E
˜Q
f
S2
0
ST
=
EQ ST
S0
γ
f
S2
0
ST
13.2. Down-and-in regular call
We note L the barrier and we assume that the option is regular, i.e. L ≤ K.
13.3. The premium
13.3.1. Method 1
We use the formula from the previous paragraph. For S0 below the barrier, the value is
the value of the European option. For S0 above the barrier, the price of the down-and-in
call is given by:
DICreg
0 =
S0
L
γ
Eurt St, f
L2
S2
0
ST , T
=
S0
L
γ
Eurt St,
L2
S2
0
ST − K
+
, T
= S0e−qT L
S0
2λ
Φ(x) − Ke−rT L
S0
2λ−2
Φ(x − σ
√
T)
where λ = 1 − γ/2 and x = ln(L2/S0K)+(r−q+σ2/2)T
σ
√
T

The premium 71
13.3.2. Method 2
For S0 below the barrier, the value is the value of the European option. For S0 above
the barrier, the price of the down-and-in call is given by:
DICreg
0 = e−rT
E (ST − K)+
1τL≤T
= e−rT
P ST 1ST ≥K1τL≤T − e−rT
P K1ST ≥K1τL≤T
= e−qT
S0Pf
1ST ≥K1τL≤T − e−rT
KP 1ST ≥K1τL≤T
= ...use Girsanov and the reﬂexion principle a few times...
= S0e−qT L
S0
2λ
Φ(x) − Ke−rT L
S0
2λ−2
Φ(x − σ
√
T)
where λ = (r − q + σ2/2)/σ2 and x = ln(L2/S0K)+(r−q+σ2/2)T
σ
√
T
13.3.3. Graph
Figure 13.1.: Premium down-and-in call: K = 100, H = 90, σ = 0.2, r = 0.03, T =
3m; 1y

72 Barrier options
13.4. The Delta
Figure 13.2.: Delta down-and-in call: K = 100, H = 90, σ = 0.2, r = 0.03, T = 3m; 1y
13.5. The Vega
Figure 13.3.: Vega down-and-in call: K = 100, H = 90, σ = 0.2, r = 0.03, T = 3m; 1y

14. Time series
14.1. Autoregressive models
14.1.1. The AR models
The AR(p) model is deﬁned as Xt = a0 + p
i=1 aiXt−i + t where (ai)i are the parameters
of the model and t is a white noise. 1
Estimation of the parameters: least-squares regression or method of moments (Yule-
Walker equations).
14.1.2. The ARMA models
The ARMA(p, q) model is deﬁned as Xt = a0 + p
i=1 aiXt−i + t + q
j=1 θj t−j where
(ai)i, (θj)j are the parameters of the model and t is a white noise.
14.2. Autoregressive conditional heteroskedasticity models
One considers a process Xt that one can write as Xt = σt t. A (G)ARCH model is a
model for the variance σ2
t .
14.2.1. Original formulation
An ARCH(p) model assumes that the variance σ2
t at time t depends on the realizations
of (Xt−i)1≤i≤p according to
Var(Xt|Xt−1, · · · , Xt−p) = σ2
t = a0 +
p
i=1
aiX2
t−i
A GARCH(p, q) model assumes that the variance σ2
t at time t depends on the real-
izations of (Xt−i, σt−i)1≤i≤p according to
Var(Xt|Xt−1, · · · , Xt−p) = σ2
t = a0 +
p
i=1
aiX2
t−i +
q
j=1
bjσ2
t−j
The parameters (ai)i, (bj)j are all assumed to be positive.
1
White noise:= the discrete version of the increments of a Brownian motion.

76 Time series
14.2.2. Alternative formulation
Rather than dealing with two diﬀerent kind of quantities (σt on the left-hand side, Xt
on the right-hand side of the previous formulaes), we present this alternative formulation:
One writes X2
t = σ2
t + zt, it is straightforward to show that zt is a martingale and
therefore can be seen as a white noise. In a GARCH(p, q) model, one has:
X2
t = σ2
t + zt = a0 +
max(p,q)
i=1
(ai + bi)X2
t−i −
q
i=1
bjzt−j + zt
where we deﬁne ai = 0 for p < i ≤ max(p, q) and bj = 0 for q < j ≤ max(p, q).
Therefore X2
t is an ARMA process.

15. Tests
Suppose that (Xi)1≤i≤n are n i.i.d. multinomial random variables, i.e. Xi ∼ X
where X : Ω → {x1, · · · , xm}, P(X = xj) = pj, m
j=1 pj = 1. Then
Tn :=
m
j=1
n
i=1(1Xi=xj − pj)
2
npj
∼
n→∞
χ2
m−1
Theorem 15.1 (Pearson’s chi-squared test)
Therefore, if one wants to know whether a random variable X follows a probability
law P, or in other words, if one wants to know whether the null hypothesis
H0 := ’there is no diﬀerence between the observation and the theory’
is satisﬁed, one then computes the statistic Tn (for n large enough), which has to be
below F−1
χ2
m−1
(1 − α) with α = 0.05 to consider that the hypothesis H0 is true.
Exemple. One wants to know whether a die is rigged or not. In order to know, one rolls
it say 600 times and one calculates T. If T is below F−1
χ2
5
(0.95) = 11.07 one accepts that
the die is not rigged.
Suppose that (Xi)1≤i≤n are n i.i.d. random variables, with E(X) = µ and
V ar(X) = σ2. Let’s write Sn :=
n
i=1 Xi
n . Then
√
n(Sn − µ) ∼
n→∞
N(0, σ)
Theorem 15.2 (Central-limit theorem)
It may be easier to remember it on this form: n
i=1 Xi ∼
n→∞
N(nµ, σ
√
n)
Exemple. One tosses a fair coin 100 times. What is the probaility to observe at least
60 heads.

78 Tests
Answer: each Xi is a Bernoulli variable, of mean p = 0.5 and standard variation
σ = p(1 − p) = 0.5. Therefore the standard deviation of S100 is 0.5
√
100 = 5. We
then aim to estimate P(S100 ≥ 60). 60 is then two standard deviations away from the
mean, and therefore P(S100 ≥ 60) is approximately equal to P(X ≥ µ + 2σ) where X is
a normal variable of mean µ and standard deviation σ.
Tables show that P(X ≥ µ+2σ) (1−0.9545)/2 = 2.27%. The exact value is actually
P(B(100, 0.5) ≥ 60) = 2.84%.

16. Monte-Carlo methods
16.1. Simulating a standard normal variable
One assumes that one can draw U ∼ U(0, 1).
Methods to avoid:
1. Draw U and compute Φ−1(U)
2. Compute U1, · · · , U12 ∼ U(0, 1) then 12
i=1 Uk − 6 ∼ N(0, 1)
Recommended method: Box-Muller
It is based on the fact that 1√
2π
e−x2+y2
2 dxdy = (1
2e−s/2ds)( 1
2π dθ) where x = r cos θ,
y = r sin θ and s = x2 + y2.
1. Basic form: let U, V ∼ U(0, 1) i.i.d.. Then we draw S =
√
−2 ln U which follows
an exponential law of parameter 1/2.
Therefore X =
√
−2 ln U cos(2πV ), Y =
√
−2 ln U sin(2πV ) ∼ N(0, 1) i.i.d.
2. Polar form: let U, V ∼ U(−1, 1)i.i.d.. One computes S = U2 + V 2 and one rejects
S if S > 1. If S ≤ 1 then compute X = U −2 ln(S)
S and Y = V −2 ln(S)
S .
Therefore X and Y are N(0, 1) i.i.d.
16.2. Simulating a multinormal variable
Let µ ∈ Rn and Σ ∈ S+
n (R) a variance-covariance matrix (Si,j = Cov(Xi, Xj)).
We note N(µ, Σ) the law whose density probability is given by
f(x) :=
1
(2π)ndet(Σ)
1/2
exp −
1
2
(x − µ)T
Σ−1
(x − µ)
Let Yi ∼ N(0, 1) i.i.d. and consider X = µ + MY where M ∈ Mn(R), MMT = Σ.
Then X ∼ N(µ, Σ).

80 Monte-Carlo methods
Methods to ﬁnd M:
1. One can use a Cholesky factorisation Σ = LLT where L is a lower triangular matrix.
Then take M = L.
2. Or one can write Σ = QDQT where Q is orthogonal and D is diagonal. Then take
M = QD1/2.

Index
Black-Scholes
diﬀusion, 15
PDE, 16
Breeden-Litzenberger, 27
Feynman-Kac, 18
Fokker-Planck, 26

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