CHAPTER 9

The Volatility Smile

9.1. The Smile Problem

Remark 9.1. Assuming the LFM, we have

Cpl(0, T, S, N, K1) = NP (0, S)τ(T, S) Bl(K1, F (0;T, S), vS(T ))

and

Cpl(0, T, S, N, K2) = NP (0, S)τ(T, S) Bl(K2, F (0;T, S), vS(T )),

where the volatility parameter is given by

vS(T ) =

√∫ T

0

σ2(u; T, S)du.

However, for K1 �= K2, this is not realistic to hold with the same volatility param-

eter.

Definition 9.2 (The Volatility Smile). If caplet prices are given by

Cpl(0, T, S, N, K) = NP (0, S)τ(T, S) Bl(K, F (0;T, S), vS(T, K)),

then the curve

K �→ vS(T, K)√T

is called the volatility smile of the T -expiry caplet.

Remark 9.3. In the LFM, the volatility smile is “flat”. However, the volatility

smile is commonly seen to exhibit “smiley” or “skewed” shapes.

9.2. Shifted Lognormal Model

Definition 9.4 (Forward-rate dynamics in the shifted lognormal model). In

the shifted lognormal model, the simply-compounded forward interest rate for the

period [T, S] is assumed to satisfy the stochastic differential equation

dF (t; T, S) = σ(t; T, S) (F (t; T, S)− α) dWS(t),

53

54 9. THE VOLATILITY SMILE

where σ is deterministic, α �= 0, and WS is a Brownian motion under the S-forward

measure.

Theorem 9.5 (Pricing of caplets in the shifted lognormal model). In the shifted

lognormal model, the price of a caplet with notional value N , cap rate K, expiry

time T , and maturity time S is given by

Cpl(t, T, S, N, K)

= NP (t, S)τ(T, S) Bl

⎛⎝K − α, F (t; T, S)− α,

√∫ T

t

σ2(u; T, S)du

⎞⎠ .

Theorem 9.6 (The volatility smile in the shifted lognormal model). If α > 0,

then the volatility smile in the shifted lognormal model is increasing. If α < 0, then

the volatility smile in the shifted lognormal model is decreasing.

9.3. Brigo–Mercurio Local Volatility Model

Definition 9.7 (Forward-rate dynamics in the Brigo–Mercurio local volatil-

ity model). In the Brigo–Mercurio local volatility model, the simply-compounded

forward interest rate for the period [T, S] is assumed to satisfy the stochastic dif-

ferential equation

dF (t; T, S) = σ(t, F (t; T, S))F (t; T, S)dWS(t),

where WS is a Brownian motion under the S-forward measure QS ,

σ(t, y) =

√√√√√√√√√

n∑i=1

λiv2i (t, y)pi

t(y)

n∑i=1

λiy2pi

t(y)

,n∑

i=1

λi = 1,

and

λi > 0, pit(y) =

d(QS(Gi(t) ≤ y))dy

, 1 ≤ i ≤ n

such that

dGi(t) = vi(t, Gi(t))dWS(t), Gi(0) = F (0;T, S), 1 ≤ i ≤ n.

9.4. LOGNORMAL MIXTURE MODEL 55

Remark 9.8 (Fokker–Planck equation). Using the result that the pdf ft of the

solution of the stochastic differential equation

dX(t) = μ(t, X(t))dt + σ(t, X(t))dW (t)

satisfies the Fokker–Planck equation

∂

∂tft(y) = − ∂

∂y(μ(t, y)ft(y)) +

12

∂2

∂y2(σ2(t, y)ft(y)),

we may show that the pdf pt of F (t; T, S) in the Brigo–Mercurio local volatility

model is given by

pt(y) =n∑

i=1

λipit(y).

Theorem 9.9 (Pricing of caplets in the Brigo–Mercurio local volatility model).

In the Brigo–Mercurio local volatility model, the price of a caplet with notional value

N , cap rate K, expiry time T , and maturity time S is given by

Cpl(0, T, S, N, K) = NP (0, S)τ(T, S)n∑

i=1

λiES((Gi(T )−K)+).

9.4. Lognormal Mixture Model

Definition 9.10 (LM model). A lognormal mixture model, briefly LM model,

is a Brigo–Mercurio local volatility model in which

vi(t, y) = σi(t)y, 1 ≤ i ≤ n,

where σi are deterministic for all 1 ≤ i ≤ n.

Remark 9.11 (Probability density functions in the LM model). In the LM

model, the probability density functions pit are given by

pit(y) =

1yVi(t)

√2π

exp

{− 1

2V 2i (t)

(ln

(y

F (0;T, S)

)+

12V 2

i (t))2

},

where

Vi(t) =

√∫ t

0

σ2i (u)du, 1 ≤ i ≤ n.

The forward-rate dynamics in the LM model is then given by

dF (t; T, S) = σ(t, F (t; T, S))F (t; T, S)dWS(t),

56 9. THE VOLATILITY SMILE

where WS is a Brownian motion under the S-forward measure QS and

σ(t, y) =

√√√√√√√√√

n∑i=1

λiσ2i (t)pi

t(y)

n∑i=1

λipit(y)

.

Note also that

σ2(t, y) =n∑

i=1

Λi(t, y)σ2i (t) with Λi > 0 such that

n∑i=1

Λi = 1.

Theorem 9.12 (Pricing of caplets in the LM model). In the LM model, the

price of a caplet with notional value N , cap rate K, expiry time T , and maturity

time S is given by

Cpl(0, T, S, N, K) = NP (0, S)τ(T, S)n∑

i=1

λi Bl(K, F (0;T, S), Vi(T )),

where Vi are as in Remark 9.11.

9.5. Lognormal Mixture Model with Different Means

Definition 9.13 (Forward-rate dynamics in the LMDM model). In the logormal

mixture model with different means, briefly LMDM model, the simply-compounded

forward interest rate for the period [T, S] is assumed to satisfy the stochastic dif-

ferential equation

dF (t; T, S) = σ(t, F (t; T, S))F (t; T, S)dWS(t),

where WS is a Brownian motion under the S-forward measure QS ,

σ(t, y) =

√√√√√√√√√

n∑i=1

λiσ2i (t)pi

t(y)

n∑i=1

λipit(y)

+

2n∑

i=1

λiμi(t)∫ ∞

y

xpit(x)dx

n∑i=1

λiy2pi

t(y)

,n∑

i=1

λi = 1,

μi and σi are deterministic for all 1 ≤ i ≤ n such that σ(t, y) is well defined and

such thatn∑

i=1

λi exp{∫ t

0

μi(u)du

}= 1,

and

λi > 0, pit(y) =

d(QS(Gi(t) ≤ y))dy

, 1 ≤ i ≤ n

9.6. SECOND BRIGO–MERCURIO LOCAL VOLATILITY MODEL 57

such that

dGi(t) = μi(t)Gi(t)dt + σi(t)Gi(t)dWS(t), Gi(0) = F (0;T, S), 1 ≤ i ≤ n.

Remark 9.14 (Probability density functions in the LMDM model). In the

LMDM model, the probability density functions pit are given by

pit(y) =

1yVi(t)

√2π

exp

{− 1

2V 2i (t)

(ln

(y

F (0;T, S)

)−Mi(t) +

12V 2

i (t))2

},

where

Mi(t) =∫ t

0

μi(u)du, Vi(t) =

√∫ t

0

σ2i (u)du, 1 ≤ i ≤ n.

Also, the pdf pt of F (t; T, S) in the LMDM model is given by

pt(y) =n∑

i=1

λipit(y).

Theorem 9.15 (Pricing of caplets in the LMDM model). In the LMDM model,

the price of a caplet with notional value N , cap rate K, expiry time T , and maturity

time S is given by

Cpl(0, T, S, N, K) = NP (0, S)τ(T, S)n∑

i=1

λieMi(T ) Bl(Ke−Mi(T ), F (0;T, S), Vi(T )),

where Mi and Vi are as in Remark 9.14.

9.6. Second Brigo–Mercurio Local Volatility Model

Definition 9.16 (Second Brigo–Mercurio local volatility model). In the second

Brigo–Mercurio local volatility model, the simply-compounded forward interest rate

for the period [T, S] is assumed to be given in the form

F (t) = F (t; T, S) = h(t, WS(t)),

where WS is a Brownian motion under the S-forward measure and h is a positive

function in two variables which is continuously differentiable in the first variable

and twice continuously differentiable in the second variable, strictly increasing in

the second variable satisfying

limx→∞h−1(T, x) =∞,

where we write h−1(t, x) = y if h(t, y) = x, and such that F is a martingale under

WS .

58 9. THE VOLATILITY SMILE

Theorem 9.17 (Forward-rate dynamics in the second Brigo–Mercurio local

volatility model). In the second Brigo–Mercurio local volatility model, the simply-

compounded forward interest rate for the period [T, S] satisfies the stochastic differ-

ential equation

dF (t) =∂

∂wh(t, h−1(t, F (t)))dWS(t),

where WS is a Brownian motion under the S-forward measure.

Lemma 9.18 (Transition density in the second Brigo–Mercurio local volatility

model). In the second Brigo–Mercurio local volatility model, we have

QS(F (T ) ≤ x|F (t) = y) = Φ(

h−1(T, x)− h−1(t, y)√T − t

),

and the density of F (T ) conditional on F (t) = y is given by

p(t, y; T, x) =1√

2π(T − t)exp

(− (h−1(T, x)− h−1(t, y))2

2(T − t)

)ddx

h−1(T, x).

Theorem 9.19 (Pricing of caplets in the second Brigo–Mercurio local volatility

model). In the second Brigo–Mercurio local volatility model, the price of a caplet

with notional value N , cap rate K, expiry time T , and maturity time S is given by

Cpl(t, T, S, N, K)

= NP (t, S)τ(T, S)

⎧⎪⎪⎨⎪⎪⎩

∫ ∞

h−1(T,K)−h−1(t,F (t))

h(T, h−1(t, F (t)) + w)e−w2

2(T−t) dw√2π(T − t)

−KΦ(

h−1(t, F (t))− h−1(T, K)√T − t

)}.

The price of the caplet at time 0 is given by

Cpl(0, T, S, N, K)

= NP (0, S)τ(T, S)

{1√2πT

∫ ∞

h−1(T,K)

h(T, w)e−w22T dw −KΦ

(−h−1(T, K)√

T

)}.

Theorem 9.20 (Volatility smile in the second Brigo–Mercurio local volatility

model). The volatility smile in the second Brigo–Mercurio local volatility model

satisfies the equation

ddK

(Bl(K, F (0), v(K))) = −Φ(−h−1(T, K)√

T

).

9.7. GEOMETRIC BROWNIAN MOTION MIXTURE MODEL 59

9.7. Geometric Brownian Motion Mixture Model

Definition 9.21 (GBM mixture model). The geometric Brownian motion mix-

ture model, briefly GBM mixture model, is a second Brigo–Mercurio local volatility

model in which the function h is given by

h(t, w) =n∑

i=1

αie− 1

2 β2i t+βiw,

where αi, βi > 0 for all 1 ≤ i ≤ n.

Remark 9.22 (GBM mixture model). Indeed the given function h satisfies all

necessary requirements. Moreover, F satisfies

dF (t) = σ(t, F (t))F (t)dWS(t),

where

σ(t, y) =

n∑i=1

αiβie− 1

2 β2i t+βih

−1(t,y)

n∑i=1

αie− 1

2 β2i t+βih

−1(t,y)

.

Note also that

σ(t, y) =n∑

i=1

Λi(t, y)βi with Λi > 0 such thatn∑

i=1

Λi = 1

so that the local volatility may be viewed as a stochastic average of the basic

volatilities βi.

Theorem 9.23 (Pricing of caplets in the GBM mixture model). In the GBM

mixture model, the price of a caplet with notional value N , cap rate K, expiry time

T , and maturity time S is given by

Cpl(t, T, S, N, K) = NP (t, S)τ(T, S)×

×{

n∑i=1

αie− 1

2 β2i t+βih

−1(t,F (t))Φ(

βi(T − t)− h−1(T, K) + h−1(t, F (t))√T − t

)

−KΦ(

h−1(t, F (t))− h−1(T, K)√T − t

)}.

60 9. THE VOLATILITY SMILE

The price of the caplet at time 0 is given by

Cpl(0, T, S, N, K)

= NP (0, S)τ(T, S)

{n∑

i=1

αiΦ(

βiT − h−1(T, K)√T

)−KΦ

(−h−1(T, K)√

T

)}.