Local Volatility Models
Copyright ยฉ Changwei Xiong 2016
June 2016
last update: October 28, 2017
TABLE OF CONTENTS
Table of Contents .........................................................................................................................................1
1. Kolmogorov Forward and Backward Equations ..................................................................................2
1.1. Kolmogorov Forward Equation ....................................................................................................2
1.2. Kolmogorov Backward Equation ..................................................................................................5
2. Local Volatility .....................................................................................................................................6
2.1. Local Volatility by Vanilla Call .....................................................................................................6
2.2. Local Volatility by Forward Call ...................................................................................................8
2.2.1. Local Variance as a Conditional Expectation of Instantaneous Variance ......................... 9
2.2.2. Formula in Log-moneyness ............................................................................................. 10
2.3. Local Volatility by Implied Volatility .......................................................................................... 11
2.3.1. Formula in Log-strike ...................................................................................................... 12
2.3.2. Formula in Log-moneyness ............................................................................................. 13
2.3.3. Equivalency in Formulas ................................................................................................. 14
3. Local Volatility: PDE by Finite Difference Method ..........................................................................16
3.1. Date Conventions of Equity and Equity Option..........................................................................16
3.2. Deterministic Dividends ..............................................................................................................17
3.3. Forward PDE ...............................................................................................................................18
3.3.1. Treatment of Deterministic Dividends ............................................................................ 19
3.4. Backward PDE ............................................................................................................................21
3.4.1. PDE in Centered Log-spot .............................................................................................. 21
3.4.1.1. Treatment of Deterministic Dividends ............................................................................ 22
3.4.1.2. Vanilla Call ...................................................................................................................... 23
3.4.2. PDE in Log-spot .............................................................................................................. 23
3.4.2.1. Treatment of Deterministic Dividends ............................................................................ 24
3.5. Local Volatility Surface ...............................................................................................................24
3.6. Barrier Option Pricing .................................................................................................................25
References ..................................................................................................................................................27
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The note is prepared for the purpose of summarizing local volatility models frequently
encountered in derivative pricing. It at first derives the Kolmogorov forward and backward equations,
which fundamentally govern the transition probability density of the diffusion process in derivative price
dynamics. Subsequently, it introduces the local volatility model in the context of Dupire formula and
then presents a PDE based local volatility model, in which the local volatility function is parametrized to
be piecewise linear in log-moneyness and piecewise constant in time.
1. KOLMOGOROV FORWARD AND BACKWARD EQUATIONS
The time evolution of the transition probability density function is governed by Kolmogorov
forward and backward equations, which will be introduced as follows, without loss of generality, in
multi-dimension.
1.1. Kolmogorov Forward Equation
Letโs consider the following ๐-dimensional stochastic spot process ๐๐ก โ โ๐ driven by an ๐-
dimensional Brownian motion ๐๐ก whose correlation matrix ๐ is given by ๐๐๐ก = ๐๐๐ก๐๐๐กโฒ
๐๐๐ก๐ร1
= ๐ด(๐ก, ๐๐ก)๐ร1
๐๐ก1ร1
+ ๐ต(๐ก, ๐๐ก)๐ร๐
๐๐๐ก๐ร1
(1)
We derive the dynamics of โ , where โ:โ๐ โถโ in this case is a scalar-valued Borel-measurable
function only on variable ๐๐ก
๐โ(๐๐ก)1ร1
= ๐ฝโ1ร๐
๐๐๐ก๐ร1
+1
2๐๐๐ก
โฒ
1ร๐๐ปโ๐ร๐
๐๐๐ก๐ร1
= ๐ฝโ๐ด๐๐ก + ๐ฝโ๐ต๐๐๐ก +1
2๐๐๐ก
โฒ๐ตโฒ๐ปโ๐ต๐๐๐ก (2)
where ๐ฝโ is the 1 ร ๐ Jacobian (i.e. the same as gradient if โ is a scalar-valued function) and ๐ปโ the
๐ ร๐ Hessian (with subscripts of ๐ now denoting the indices of vector components)
[๐ฝโ]๐ =๐โ
๐๐๐ and [๐ปโ]๐๐ =
๐2โ
๐๐๐๐๐๐ (3)
Expanding the expression in (2), we have
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๐โ =โ๐โ
๐๐๐๐ด๐๐๐ก
๐
๐=1
+โ๐โ
๐๐๐
๐
๐=1
โ๐ต๐๐๐๐๐
๐
๐=1
+1
2โ
๐2โ
๐๐๐๐๐๐โ๐ต๐๐๐๐๐๐ต๐๐๐๐ก
๐
๐=1
๐
๐,๐=1
= (โ๐โ
๐๐๐๐ด๐
๐
๐=1
+1
2โ
๐2โ
๐๐๐๐๐๐๐ด๐๐
๐
๐,๐=1
)๐๐ก +โ๐โ
๐๐๐
๐
๐=1
โ๐ต๐๐๐๐๐
๐
๐=1
(4)
where ๐ด = ๐ต๐๐ตโฒ is the ๐ ร๐ instantaneous variance-covariance matrix of ๐๐ . Integrating on both
sides of (4) from ๐ก to ๐, we have
โ(๐๐) โ โ(๐๐ก) = โซ (โ๐โ
๐๐๐๐ด๐
๐
๐=1
+1
2โ
๐2โ
๐๐๐๐๐๐๐ด๐๐
๐
๐,๐=1
)๐๐ข๐
๐ก
+โซ โ๐โ
๐๐๐
๐
๐=1
โ๐ต๐๐๐๐๐
๐
๐=1
๐
๐ก
(5)
Taking expectation on both sides of (5), we get (using notation ๐ผ๐ก[โ] = ๐ผ[โ|โฑ๐ก])
LHS = ๐ผ๐ก[โ(๐๐)] โ โ(๐๐ก) = โซโ๐ฆ๐๐,๐ฆ|๐ก,๐ฅ๐๐ฆ๐บ
โ โ๐ฅ
RHS = ๐ผ๐ก [โซ (โ๐โ
๐๐๐๐ด๐
๐
๐=1
+1
2โ
๐2โ
๐๐๐๐๐๐๐ด๐๐
๐
๐,๐=1
)๐๐ข๐
๐ก
] + ๐ผ๐ก [โซ โ๐โ
๐๐๐
๐
๐=1
โ๐ต๐๐๐๐๐
๐
๐=1
๐
๐ก
]โ
=0
= โซ โ๐ผ๐ก [๐โ
๐๐๐๐ด๐]
๐
๐=1
๐๐ข๐
๐ก
+1
2โซ โ ๐ผ๐ก [
๐2โ
๐๐๐๐๐๐๐ด๐๐]
๐
๐,๐=1
๐๐ข๐
๐ก
(6)
where ๐๐,๐ฆ|๐ก,๐ฅ is the transition probability density having ๐๐ = ๐ฆ at ๐ given ๐๐ก = ๐ฅ at ๐ก (i.e. if we solve
the equation (1) with the initial condition ๐๐ก = ๐ฅ โ โ๐ , then the random variable ๐๐ = ๐ฆ โ ๐บ has a
density ๐๐,๐ฆ|๐ก,๐ฅ in the ๐ฆ variable at time ๐). Differentiating (6) with respect to ๐ on both sides, we have
โซ โ๐ฆ๐๐๐,๐ฆ|๐ก,๐ฅ๐๐
๐๐ฆ๐บ
=โ๐ผ๐ก [๐โ
๐๐๐๐ด๐]
๐
๐=1
+1
2โ ๐ผ๐ก [
๐2โ
๐๐๐๐๐๐๐ด๐๐]
๐
๐,๐=1
=โโซ๐โ๐ฆ๐๐ฆ๐
๐ด๐๐๐,๐ฆ|๐ก,๐ฅ๐๐ฆ๐บ
๐
๐=1
+1
2โ โซ
๐2โ๐ฆ๐๐ฆ๐๐๐ฆ๐
๐ด๐๐๐๐,๐ฆ|๐ก,๐ฅ๐๐ฆ๐บ
๐
๐,๐=1
(7)
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If we assume ๐บ โก โ๐ and also assume the probability density ๐ and its first derivatives ๐๐/๐๐ฆ๐ vanish
at a higher order of rate than โ and ๐โ/๐๐ฆ๐ as ๐ฆ๐ โ ยฑโ โ ๐ = 1,โฏ ,๐, then we can integrate by parts
for the right hand side of (7), once for the first integral and twice for the second
โซ๐โ๐ฆ๐๐ฆ๐
๐ด๐๐๐๐ฆ๐บ
= โซ โ๐ฆ๐ด๐๐|๐ฆ๐ =โโ+โ
โ =0
๐๐ฆโ๐๐บโ๐
โโซ โ๐ฆ๐(๐ด๐๐)
๐๐ฆ๐๐๐ฆ
๐บ
and
โซ๐2โ๐ฆ๐๐ฆ๐๐๐ฆ๐
๐ด๐๐๐๐๐ฆ๐บ
= โซ๐โ๐ฆ๐๐ฆ๐
๐ด๐๐๐|๐ฆ๐=โโ
+โ
โ =0
๐๏ฟฝฬ ๏ฟฝ๐๏ฟฝฬ ๏ฟฝ๐
โโซ๐โ๐ฆ๐๐ฆ๐
๐(๐ด๐๐๐)
๐๐ฆ๐๐๐ฆ
๐บ
= โโซ โ๐ฆ๐(๐ด๐๐๐)
๐๐ฆ๐|๐ฆ๐=โโ
+โ
โ =0
๐๏ฟฝฬ ๏ฟฝ๐๏ฟฝฬ ๏ฟฝ๐
+โซ โ๐ฆ๐2(๐ด๐๐๐)
๐๐ฆ๐๐๐ฆ๐๐๐ฆ
๐บ
where โซ (โ)๐๏ฟฝฬ ๏ฟฝ๐๏ฟฝฬ ๏ฟฝ๐
= โซ โฏโซ โซ โฏโซ(โ)๐๐ฆ1โ
โฏ๐๐ฆ๐โ1โ
๐๐ฆ๐+1โ
โฏ๐๐ฆ๐โ
(8)
Plugging the results of (8) into (7), we have
โซ โ๐ฆ๐๐
๐๐๐๐ฆ
๐บ
= โโโซ โ๐ฆ๐(๐ด๐๐)
๐๐ฆ๐๐๐ฆ
๐บ
๐
๐=1
+1
2โ โซ โ๐ฆ
๐2(๐ด๐๐๐)
๐๐ฆ๐๐๐ฆ๐๐๐ฆ
๐บ
๐
๐,๐=1
โนโซ โ๐ฆ (๐๐
๐๐+โ
๐(๐ด๐๐)
๐๐ฆ๐
๐
๐=1
โ1
2โ
๐2(๐ด๐๐๐)
๐๐ฆ๐๐๐ฆ๐
๐
๐,๐=1
)๐๐ฆ๐บ
= 0
(9)
By the arbitrariness of โ, we conclude that for any ๐ฆ โ ๐บ
๐๐
๐๐+โ
๐(๐ด๐๐)
๐๐ฆ๐
๐
๐=1
โ1
2โ
๐2(๐ด๐๐๐)
๐๐ฆ๐๐๐ฆ๐
๐
๐,๐=1
= 0, ๐ด = ๐ต๐๐ตโฒ (10)
This is the Multi-dimensional Fokker-Planck Equation (a.k.a. Kolmogorov Forward Equation) [1]. In
this equation, the ๐ก and ๐ฅ are held constant, while the ๐ and ๐ฆ are variables (called โforward variablesโ).
In the one-dimensional case, it reduces to
๐๐
๐๐+๐(๐ด๐)
๐๐ฆโ1
2
๐2(๐ต2๐)
๐๐ฆ2= 0 (11)
where ๐ด = ๐ด(๐, ๐ฆ) and ๐ต = ๐ต(๐, ๐ฆ) are then scalar functions.
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1.2. Kolmogorov Backward Equation
Letโs express conditional expectation of โ(๐๐ก) by ๐(๐ก, ๐๐ก) = ๐ผ๐ก[โ(๐๐)]. Because for ๐ โค ๐ก โค ๐
we have
๐(๐, ๐๐) = ๐ผ๐[โ(๐๐)] = ๐ผ๐[๐ผ๐ก[โ(๐๐)]] = ๐ผ๐[๐(๐ก, ๐๐ก)] (12)
the ๐(๐ก, ๐๐ก) is a martingale by the tower rule (i.e. If โ holds less information than ๐ข , then
๐ผ[๐ผ[๐|๐ข]|โ] = ๐ผ[๐|โ]). The dynamics of the ๐(๐ก, ๐๐ก) is given by
๐๐ =๐๐
๐๐ก๐๐ก + ๐ฝ๐
1ร๐
๐๐๐ก๐ร1
+1
2๐๐๐ก
โฒ
1ร๐๐ป๐๐ร๐
๐๐๐ก๐ร1
=๐๐
๐๐ก๐๐ก + ๐ฝ๐๐ด๐๐ก + ๐ฝ๐๐ต๐๐๐ก +
1
2๐๐๐ก
โฒ๐ตโฒ๐ป๐๐ต๐๐๐ก
(13)
where ๐ฝ๐ is the Jacobian (i.e. the same as gradient if ๐ is a scalar-valued function) and ๐ป๐ the Hessian of
๐ with respect to ๐ (with subscripts denoting the indices of vector components)
๐ฝ๐ = (๐๐
๐๐1โฏ
๐๐
๐๐๐) , ๐ป๐ =
(
๐2๐
๐๐12 โฏ
๐2๐
๐๐1๐๐๐โฎ โฑ โฎ๐2๐
๐๐๐๐๐1โฏ
๐2๐
๐๐๐2 )
(14)
Expanding (13), we have
๐๐ = (๐๐
๐๐ก+โ
๐๐
๐๐๐๐ด๐
๐
๐=1
+1
2โ
๐2๐
๐๐๐๐๐๐๐ด๐๐
๐
๐,๐=1
)๐๐ก +โ๐๐
๐๐๐
๐
๐=1
โ๐ต๐๐๐๐๐
๐
๐=1
(15)
Since ๐(๐ก, ๐๐ก) is a martingale, the ๐๐ก-term must vanish, which gives
๐๐
๐๐ก+โ
๐๐
๐๐๐๐ด๐
๐
๐=1
+1
2โ
๐2๐
๐๐๐๐๐๐๐ด๐๐
๐
๐,๐=1
= 0 (16)
This is the multi-dimensional Feynman-Kac formula1.
Using the transition probability density ๐๐,๐ฆ|๐ก,๐ฅ, we can write the expectation as
1 https://en.wikipedia.org/wiki/Feynman-Kac_formula
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๐๐ก,๐ฅ = ๐ผ๐ก[โ(๐๐)] = โซโ๐ฆ๐๐,๐ฆ|๐ก,๐ฅ๐๐ฆ๐บ
(17)
The formula (16) defines that
๐
๐๐กโซโ๐ฆ๐๐๐ฆ๐บ
+โ๐ด๐๐
๐๐ฅ๐โซโ๐ฆ๐๐๐ฆ๐บ
๐
๐=1
+1
2โ ๐ด๐๐
๐2
๐๐ฅ๐๐๐ฅ๐โซโ๐ฆ๐๐๐ฆ๐บ
๐
๐,๐=1
= 0
โนโซ โ๐ฆ (๐๐
๐๐ก+โ๐ด๐
๐๐
๐๐ฅ๐
๐
๐=1
+1
2โ ๐ด๐๐
๐2๐
๐๐ฅ๐๐๐ฅ๐
๐
๐,๐=1
)๐๐ฆ๐บ
= 0
(18)
By the arbitrariness of โ, we have
๐๐
๐๐ก+โ๐ด๐
๐๐
๐๐ฅ๐
๐
๐=1
+1
2โ ๐ด๐๐
๐2๐
๐๐ฅ๐๐๐ฅ๐
๐
๐,๐=1
= 0, ๐ด = ๐ต๐๐ตโฒ (19)
This is the multi-dimensional Kolmogorov Backward Equation. In this equation, the ๐ and ๐ฆ are held
constant, while the ๐ก and ๐ฅ are variables (called โbackward variablesโ). In the 1-D case, it reduces to
๐๐
๐๐ก+ ๐ด
๐๐
๐๐ฅ+1
2๐ต2๐2๐
๐๐ฅ2= 0 (20)
where ๐ด = ๐ด(๐ก, ๐ฅ) and ๐ต = ๐ต(๐ก, ๐ฅ) are then scalar functions.
2. LOCAL VOLATILITY
In local volatility models, the volatility process is assumed to be a function of both the spot level
and the time. It is one step generalization of the well-known Black-Scholes model. Under risk neutral
measure, the spot process (e.g. an equity or an FX rate) is assumed to follow a geometric Brownian
motion
๐๐๐ก๐๐ก= ๐๐ก๐๐ก + ๐(๐ก, ๐๐ก)๐๏ฟฝฬ๏ฟฝ๐ก , ๐๐ก = ๐๐ก โ ๐๐ก (21)
with cash rate ๐๐ก and dividend rate ๐๐ก (or foreign cash rate for FX).
2.1. Local Volatility by Vanilla Call
Under the assumption of deterministic ๐๐ก , the European (vanilla) call option price can be
expressed as a function of maturity time ๐ and strike ๐พ
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๐๐,๐พ|๐ก,๐ฅ = ๏ฟฝฬ๏ฟฝ๐ก[๐ท๐ก,๐(๐๐ โ ๐พ)+] = ๐ท๐ก,๐โซ (๐ฆ โ ๐พ)๐๐,๐ฆ|๐ก,๐ฅ๐๐ฆ
โ
๐พ
(22)
where ๐ท๐ก,๐ = exp (โโซ ๐๐ข๐๐ข๐
๐ก) is the deterministic discount factor and ๐๐,๐ฆ|๐ก,๐ฅ is the transition
probability density having spot ๐๐ = ๐ฆ at ๐ given initial condition ๐๐ก = ๐ฅ at ๐ก. Differentiating (22) with
respect to ๐พ, we have the first order and second order partial derivative
๐๐๐,๐พ|๐ก,๐ฅ๐๐พ
= โ๐ท๐ก,๐โซ ๐๐,๐ฆ|๐ก,๐ฅ๐๐ฆโ
๐พ
,๐2๐๐,๐พ|๐ก,๐ฅ๐๐พ2
= ๐ท๐ก,๐๐๐,๐พ|๐ก,๐ฅ (23)
which gives the transition probability density function by
๐๐,๐พ|๐ก,๐ฅ =1
๐ท๐ก,๐
๐2๐๐,๐พ|๐ก,๐ฅ๐๐พ2
(24)
The (24) is also known as Breeden-Litzenberger formula.
Taking the first derivative of ๐๐,๐พ|๐ก,๐ฅ in (22) with respect to ๐, we find
๐๐๐,๐พ|๐ก,๐ฅ๐๐
= โ๐๐๐๐,๐พ|๐ก,๐ฅ + ๐ท๐ก,๐โซ (๐ฆ โ ๐พ)๐๐๐,๐ฆ|๐ก,๐ฅ๐๐
๐๐ฆโ
๐พ
= โ๐๐๐๐,๐พ|๐ก,๐ฅ + ๐ท๐ก,๐โซ (๐ฆ โ ๐พ)(1
2
๐2(๐๐,๐ฆ2 ๐ฆ2๐๐,๐ฆ|๐ก,๐ฅ)
๐๐ฆ2โ๐(๐๐๐ฆ๐๐,๐ฆ|๐ก,๐ฅ)
๐๐ฆ)๐๐ฆ
โ
๐พ
(25)
using the Kolmogorov Forward Equation (11)
๐๐๐,๐ฆ|๐ก,๐ฅ๐๐
=1
2
๐2(๐๐,๐ฆ2 ๐ฆ2๐๐,๐ฆ|๐ก,๐ฅ)
๐๐ฆ2โ๐(๐๐๐ฆ๐๐,๐ฆ|๐ก,๐ฅ)
๐๐ฆ (26)
Applying integration by parts to the integrals on the right hand side of (25) yields
โซ (๐ฆ โ ๐พ)๐(๐๐๐ฆ๐๐,๐ฆ|๐ก,๐ฅ)
๐๐ฆ๐๐ฆ
โ
๐พ
= (๐ฆ โ ๐พ)๐๐๐ฆ๐๐,๐ฆ|๐ก,๐ฅ|๐ฆ=๐พโ
โ =0
โโซ ๐๐๐ฆ๐๐,๐ฆ|๐ก,๐ฅ๐๐ฆโ
๐พ
= โ๐๐ (โซ (๐ฆ โ ๐พ)๐๐,๐ฆ|๐ก,๐ฅ๐๐ฆโ
๐พ
+ ๐พโซ ๐๐,๐ฆ|๐ก,๐ฅ๐๐ฆโ
๐พ
) =๐๐๐พ
๐ท๐ก,๐
๐๐๐,๐พ|๐ก,๐ฅ๐๐พ
โ๐๐๐๐,๐พ|๐ก,๐ฅ๐ท๐ก,๐
and
(27)
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8
โซ (๐ฆ โ ๐พ)๐2(๐๐,๐ฆ
2 ๐ฆ2๐๐,๐ฆ|๐ก,๐ฅ)
๐๐ฆ2๐๐ฆ
โ
๐พ
= (๐ฆ โ ๐พ)๐(๐๐,๐ฆ
2 ๐ฆ2๐๐,๐ฆ|๐ก,๐ฅ)
๐๐ฆ|๐ฆ=๐พ
โ
โ =0
โโซ๐(๐๐,๐ฆ
2 ๐ฆ2๐๐,๐ฆ|๐ก,๐ฅ)
๐๐ฆ๐๐ฆ
โ
๐พ
= ๐๐,๐ฆ2 ๐ฆ2๐๐,๐ฆ|๐ก,๐ฅ|๐ฆ=๐พ
โ= ๐๐,๐พ
2 ๐พ2๐๐,๐พ|๐ก,๐ฅ =๐๐,๐พ2 ๐พ2
๐ท๐ก,๐
๐2๐๐,๐พ|๐ก,๐ฅ๐๐พ2
where we have ๐๐,โ|๐ก,๐ฅ = 0 and ๐๐๐,โ|๐ก,๐ฅ/๐๐ฆ = 0 assuming the probability density ๐๐,๐ฆ|๐ก,๐ฅ and its first
derivative vanish at a higher order of rate as ๐ฆ โ โ. Plugging (27) into (25), we find
๐๐๐,๐พ|๐ก,๐ฅ๐๐
= โ๐๐๐๐,๐พ|๐ก,๐ฅ +๐๐,๐พ2 ๐พ2
2
๐2๐๐,๐พ|๐ก,๐ฅ๐๐พ2
โ ๐๐๐พ๐๐๐,๐พ|๐ก,๐ฅ๐๐พ
+ ๐๐๐๐,๐พ|๐ก,๐ฅ
=๐๐,๐พ2 ๐พ2
2
๐2๐๐,๐พ|๐ก,๐ฅ๐๐พ2
โ ๐๐๐พ๐๐๐,๐พ|๐ก,๐ฅ๐๐พ
โ ๐๐๐๐,๐พ|๐ก,๐ฅ
(28)
and eventually the Dupire formula for the local volatility ๐๐,๐พ expressed in terms of vanilla call price
๐๐,๐พ2
2=
๐๐๐,๐พ|๐ก,๐ฅ๐๐
+ ๐๐๐พ๐๐๐,๐พ|๐ก,๐ฅ๐๐พ
+ ๐๐๐๐,๐พ|๐ก,๐ฅ
๐พ2๐2๐๐,๐พ|๐ก,๐ฅ๐๐พ2
(29)
2.2. Local Volatility by Forward Call
Sometimes, it is more convenient to express the Dupire formula in terms of a forward (i.e.
undiscounted) call, which is defined as
๐ถ๐,๐พ|๐ก,๐ฅ = ๏ฟฝฬ๏ฟฝ๐ก[(๐๐ โ ๐พ)+] = โซ (๐ฆ โ ๐พ)๐๐,๐ฆ|๐ก,๐ฅ๐๐ฆ
โ
๐พ
=๐๐,๐พ|๐ก,๐ฅ๐ท๐ก,๐
(30)
with ๐๐,๐พ|๐ก,๐ฅ given in (22) (note that (30) is true only if the interest rate is deterministic). Following a
similar derivation, we find that
๐๐ถ๐,๐พ|๐ก,๐ฅ๐๐พ
= โโซ ๐๐,๐ฆ|๐ก,๐ฅ๐๐ฆโ
๐พ
,๐2๐ถ๐,๐พ|๐ก,๐ฅ๐๐พ2
= ๐๐,๐พ|๐ก,๐ฅ and
๐๐ถ๐,๐พ|๐ก,๐ฅ๐๐
= โซ (๐ฆ โ ๐พ)๐๐๐,๐ฆ|๐ก,๐ฅ๐๐
๐๐ฆโ
๐พ
=1
2๐๐,๐พ2 ๐พ2
๐2๐ถ๐,๐พ|๐ก,๐ฅ๐๐พ2
+ ๐๐(๐ถ๐,๐พ|๐ก,๐ฅ โ ๐พ๐๐ถ๐,๐พ|๐ก,๐ฅ๐๐พ
)
(31)
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
9
Therefore the Dupire formula for ๐๐,๐พ expressed in terms of forward call price reads
๐๐,๐พ2
2=
๐๐ถ๐,๐พ|๐ก,๐ฅ๐๐
+ ๐๐๐พ๐๐ถ๐,๐พ|๐ก,๐ฅ๐๐พ
โ ๐๐๐ถ๐,๐พ|๐ก,๐ฅ
๐พ2๐2๐ถ๐,๐พ|๐ก,๐ฅ๐๐พ2
(32)
2.2.1. Local Variance as a Conditional Expectation of Instantaneous Variance
The forward call (30) can be expressed as
๐ถ๐,๐พ|๐ก,๐ฅ = ๏ฟฝฬ๏ฟฝ๐ก[(๐๐ โ ๐พ)+] = ๏ฟฝฬ๏ฟฝ๐ก[๐ฝ(๐๐ โ ๐พ)(๐๐ โ ๐พ)] = ๏ฟฝฬ๏ฟฝ๐ก[๐ฝ(๐๐ โ ๐พ)๐๐] โ ๐พ๏ฟฝฬ๏ฟฝ๐ก[๐ฝ(๐๐ โ ๐พ)] (33)
where ๐ฝ is the Heaviside step function1. Differentiating once with respect to ๐พ, we get
๐๐ถ๐,๐พ|๐ก,๐ฅ๐๐พ
= โโซ ๐๐,๐ฆ|๐ก,๐ฅ๐๐ฆโ
๐พ
= โ๏ฟฝฬ๏ฟฝ๐ก[๐ฝ(๐๐ โ ๐พ)] โน ๏ฟฝฬ๏ฟฝ๐ก[๐ฝ(๐๐ โ ๐พ)๐๐] = ๐ถ๐,๐พ|๐ก,๐ฅ โ ๐พ๐๐ถ๐,๐พ|๐ก,๐ฅ๐๐พ
(34)
Differentiating again with respect to ๐พ, we have
๐2๐ถ๐,๐พ|๐ก,๐ฅ๐๐พ2
= ๐๐,๐พ|๐ก,๐ฅ = ๏ฟฝฬ๏ฟฝ๐ก[๐ฟ(๐๐ โ ๐พ)] (35)
where ๐ฟ is the Dirac delta function2. Applying Itoโs Lemma to the terminal payoff of the option gives
the identity
๐(๐๐ โ ๐พ)+ = ๐ฝ(๐๐ โ ๐พ)๐๐๐ +
1
2๐ฟ(๐๐ โ ๐พ)๐๐
2๐๐2๐๐
= ๐ฝ(๐๐ โ ๐พ)๐๐๐๐๐๐ +1
2๐ฟ(๐๐ โ ๐พ)๐๐
2๐๐2๐๐ + ๐ฝ(๐๐ โ ๐พ)๐๐๐๐๐๏ฟฝฬ๏ฟฝ๐
(36)
Taking conditional expectations on both sides gives
๐๐ถ๐,๐พ|๐ก,๐ฅ = ๐๏ฟฝฬ๏ฟฝ๐ก[(๐๐ โ ๐พ)+] = ๐๐๏ฟฝฬ๏ฟฝ๐ก[๐ฝ(๐๐ โ ๐พ)๐๐]๐๐ +
1
2๏ฟฝฬ๏ฟฝ๐ก[๐ฟ(๐๐ โ ๐พ)๐๐
2๐๐2]๐๐
โน๐๐ถ๐,๐พ|๐ก,๐ฅ๐๐
= ๐๐๏ฟฝฬ๏ฟฝ๐ก[๐ฝ(๐๐ โ ๐พ)๐๐] +1
2๏ฟฝฬ๏ฟฝ๐ก[๐ฟ(๐๐ โ ๐พ)๐๐
2๐๐2]
(37)
Notice that
1 Heaviside step function: ๐ฝ(๐ฅ) = {
0, ๐ฅ < 01, ๐ฅ โฅ 0
2 Dirac delta function can be viewed as the derivative of the Heaviside step function: ๐ฟ(๐ฅ) =๐๐ฝ(๐ฅ)
๐๐ฅ= {โ, ๐ฅ = 00, ๐ฅ โ 0
,
which is also constrained to satisfy the identity: โซ ๐ฟ(๐ฅ)๐๐ฅโ
= 1.
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
10
๏ฟฝฬ๏ฟฝ๐ก[๐ฟ(๐๐ โ ๐พ)๐๐2๐๐2] = ๐พ2๏ฟฝฬ๏ฟฝ๐ก[๐ฟ(๐๐ โ ๐พ)๐๐
2] (38)
and from Bayesโ rule
๏ฟฝฬ๏ฟฝ๐ก[๐ฟ(๐๐ โ ๐พ)๐๐2] = ๏ฟฝฬ๏ฟฝ๐ก[๐๐
2|๐๐ = ๐พ]๏ฟฝฬ๏ฟฝ๐ก[๐ฟ(๐๐ โ ๐พ)] (39)
we can derive from (37)
๐๐ถ๐,๐พ|๐ก,๐ฅ๐๐
= ๐๐๐ถ๐,๐พ|๐ก,๐ฅ โ ๐๐๐พ๐๐ถ๐,๐พ|๐ก,๐ฅ๐๐พ
+1
2๐พ2๐2๐ถ๐,๐พ|๐ก,๐ฅ๐๐พ2
๏ฟฝฬ๏ฟฝ๐ก[๐๐2|๐๐ = ๐พ]
โน๏ฟฝฬ๏ฟฝ๐ก[๐๐
2|๐๐ = ๐พ]
2=
๐๐ถ๐,๐พ|๐ก,๐ฅ๐๐
+ ๐๐๐พ๐๐ถ๐,๐พ|๐ก,๐ฅ๐๐พ
โ ๐๐๐ถ๐,๐พ|๐ก,๐ฅ
๐พ2๐2๐ถ๐,๐พ|๐ก,๐ฅ๐๐พ2
(40)
This is identical to (32). It means that the conditional expectation of the stochastic variance must equal
the Dupire local variance [2]. That is, local variance is the risk-neutral expectation of the instantaneous
variance conditional on the final stock price ๐๐ being equal to the strike price ๐พ [3].
2.2.2. Formula in Log-moneyness
In real applications, numerical methods are often in favor of log-moneyness to be the spatial
variable, which can be regarded as a centered log-strike
๐ = ln๐พ
๐น๐ก,๐ where
๐๐
๐๐พ=1
๐พ,
๐๐
๐๐= โ๐๐ (41)
We want to express the Dupire formula in the (๐, ๐)-plane using call option price ๐ถ๐,๐ (short for ๐ถ๐,๐|๐ก,๐ง
for ๐ง = ln๐ฅ
๐น๐ก,๐) equivalent to the forward call ๐ถ๐,๐พ (short for ๐ถ๐,๐พ|๐ก,๐ฅ). Note that although the ๐ถ๐,๐ and
๐ถ๐,๐พ are equivalent, they are two different functions. The conversion from (๐, ๐พ)-plane to (๐, ๐)-plane
is achieved by using the following partial derivatives derived by chain rule
๐๐ถ๐,๐พ๐๐
=๐๐ถ๐,๐๐๐
+๐๐ถ๐,๐๐๐
๐๐
๐๐=๐๐ถ๐,๐๐๐
โ ๐๐๐๐ถ๐,๐๐๐
๐๐ถ๐,๐พ๐๐พ
=๐๐ถ๐,๐๐๐
๐๐
๐๐พ+๐๐ถ๐,๐๐๐
๐๐
๐๐พ=1
๐พ
๐๐ถ๐,๐๐๐
๐2๐ถ๐,๐พ๐๐พ2
=๐
๐๐(๐๐ถ๐,๐๐๐
)๐๐
๐๐พ+๐
๐๐(๐๐ถ๐,๐๐๐
)๐๐
๐๐พ=1
๐พ
๐
๐๐(1
๐พ
๐๐ถ๐,๐๐๐
) =1
๐พ2(๐2๐ถ๐,๐๐๐2
โ๐๐ถ๐,๐๐๐
)
(42)
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
11
Plugging these partial derivatives into (32), we have the Dupire formula expressed in ๐
๐๐,๐2
2=
๐๐ถ๐,๐๐๐
โ ๐๐๐ถ๐,๐
๐2๐ถ๐,๐๐๐2
โ๐๐ถ๐,๐๐๐
(43)
where ๐๐,๐ is the local volatility in (๐, ๐)-plane equivalent to ๐๐,๐พ.
2.3. Local Volatility by Implied Volatility
It is a market standard to quote option prices as Black-Scholes implied volatilities. Hence, it is
more straightforward to express the local volatility in terms of the implied volatilities rather than option
prices. Taking ๐ก as of today, we can define the forward price as
๐น๐ก,๐ = ๐๐ก exp(โซ ๐๐ข๐๐ข๐
๐ก
) (44)
The forward call price in Black-Scholes model is then given by
๐๐,๐พ,๐ = ๐น๐ก,๐ฮฆ(๐+) โ ๐พฮฆ(๐โ) with ๐ยฑ =ln๐น๐ก,๐๐พ
๐๐,๐พโ๐ยฑ๐๐,๐พโ๐
2, ๐ = ๐ โ ๐ก (45)
where ๐๐,๐พ is the Black-Scholes implied volatility derived from market quotes of vanilla options and ฮฆ
the standard normal cumulative density function. Its partial derivatives can be derived as
๐๐๐,๐พ,๐
๐๐= ๐๐๐น๐ก,๐ฮฆ(๐+) + ๐น๐ก,๐๐(๐+)
๐๐+๐๐
โ ๐พ๐(๐โ)๐๐โ๐๐
= ๐๐๐น๐ก,๐ฮฆ(๐+) +๐พ๐(๐โ)๐
2โ๐
๐๐๐,๐พ,๐
๐๐พ= ๐น๐ก,๐๐(๐+)
๐๐+๐๐พ
โ ๐พ๐(๐โ)๐๐โ๐๐พ
โ ฮฆ(๐โ) = โฮฆ(๐โ),๐2๐๐,๐พ,๐
๐๐พ2=๐(๐โ)
๐พ๐โ๐
๐๐๐,๐พ,๐
๐๐= ๐น๐ก,๐๐(๐+)
๐๐+๐๐
โ ๐พ๐(๐โ)๐๐โ๐๐
= ๐พ๐(๐โ)โ๐,๐2๐๐,๐พ,๐
๐๐2=๐+๐โ๐พ๐(๐โ)โ๐
๐
๐2๐๐,๐พ,๐
๐๐๐๐พ=๐(๐โ)๐+
๐
(46)
where we have used
๐๐ยฑ๐๐
=๐๐
๐โ๐โ๐โ2๐,
๐๐ยฑ๐๐พ
= โ1
๐พ๐โ๐,
๐๐ยฑ๐๐
= โ๐โ๐,
๐2๐ยฑ๐๐พ๐๐
=1
๐พ๐2โ๐ (47)
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
12
and the identity
๐น๐ก,๐๐(๐+) = ๐พ๐(๐โ) (48)
Further using the partial derivatives
๐๐ถ๐,๐พ๐๐
=๐๐๐,๐พ,๐
๐๐+๐๐๐,๐พ,๐
๐๐
๐๐
๐๐,
๐๐ถ๐,๐พ๐๐พ
=๐๐๐,๐พ,๐
๐๐พ+๐๐๐,๐พ,๐
๐๐
๐๐
๐๐พ
๐2๐ถ๐,๐พ๐๐พ2
=๐2๐๐,๐พ,๐
๐๐พ2+ 2
๐2๐๐,๐พ,๐
๐๐พ๐๐
๐๐
๐๐พ+๐๐๐,๐พ,๐
๐๐
๐2๐
๐๐พ2+๐2๐๐,๐พ,๐
๐๐2(๐๐
๐๐พ)2
(49)
the local volatility given by implied volatility can be derived from (32) as
๐๐,๐พ2 =
๐๐ถ๐,๐พ๐๐
+ ๐๐๐พ๐๐ถ๐,๐พ๐๐พ
โ ๐๐๐ถ๐,๐พ
๐พ2
2๐2๐ถ๐,๐พ๐๐พ2
=
๐๐๐๐+๐๐๐๐(๐๐๐,๐พ๐๐
+ ๐๐๐พ๐๐๐,๐พ๐๐พ
) + ๐๐๐พ๐๐๐๐พ
โ ๐๐๐
๐พ2
2(๐2๐๐๐พ2
+ 2๐2๐๐๐พ๐๐
๐๐๐,๐พ๐๐พ
+๐๐๐๐๐2๐๐,๐พ๐๐พ2
+๐2๐๐๐2
(๐๐๐๐พ)2
)
=
๐๐๐น๐ก,๐ฮฆ(๐+) +๐พ๐(๐โ)๐
2โ๐+๐๐๐๐(๐๐๐๐+ ๐๐๐พ
๐๐๐๐พ) โ ๐๐๐พฮฆ(๐โ) โ ๐๐๐
๐พ2
2๐(๐โ)
๐พ๐โ๐+ ๐พ2
๐(๐โ)๐+๐
๐๐๐๐พ
+๐พ2
2๐๐๐๐๐2๐๐๐พ2
+๐พ2
2๐+๐โ๐พ๐(๐โ)โ๐
๐(๐๐๐๐พ)2
=
๐๐๐๐(๐2๐+๐๐๐๐+ ๐๐๐พ
๐๐๐๐พ)
12๐๐
๐๐๐๐+๐พ๐+๐โ๐
๐๐๐๐๐๐๐๐พ
+๐พ2
2๐๐๐๐๐2๐๐๐พ2
+๐พ2๐+๐โ2๐
๐๐๐๐(๐๐๐๐พ)2
=๐2 + 2๐๐ (
๐๐๐๐+ ๐๐๐พ
๐๐๐๐พ)
1 + 2โ๐๐พ๐+๐๐๐๐พ
+ ๐+๐โ๐๐พ2 (๐๐๐๐พ)2
+ ๐๐๐พ2๐2๐๐๐พ2
(50)
Numerical methods, e.g. PDE or Monte Carlo simulation, often demand a local volatility
function constructed on a 2D grid to perform pricing. In these methods, it is often more numerically
stable and convenient to work with a spatial dimension in log-strike or in log-moneyness.
2.3.1. Formula in Log-strike
The local volatility formula in log strike ๐ฅ = ln๐พ can be derived from (50)
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
13
๐๐,๐ฅ2 =
๐2 + 2๐๐ (๐๐๐๐+ ๐๐
๐๐๐๐ฅ)
1 + 2โ๐๐+๐๐๐๐ฅ+ ๐+๐โ๐ (
๐๐๐๐ฅ)2
+ ๐๐ (๐2๐๐๐ฅ2
โ๐๐๐๐ฅ)
=๐2 + 2๐๐ (
๐๐๐๐+ ๐๐
๐๐๐๐ฅ)
1 + (๐๐ โ 2๐๐)๐๐๐๐ฅ+ (๐2
๐2โ๐2๐2
4 ) (๐๐๐๐ฅ)2
+ ๐๐ (๐2๐๐๐ฅ2
โ๐๐๐๐ฅ)
=๐2 + 2๐๐ (
๐๐๐๐+ ๐๐
๐๐๐๐ฅ)
(1 โ๐๐๐๐๐๐ฅ)2
โ (๐๐2๐๐๐๐ฅ)2
+ ๐๐๐2๐๐๐ฅ2
(51)
providing that we have the following identities
๐๐
๐๐พ=๐๐
๐๐ฅ
๐๐ฅ
๐๐พ=1
๐พ
๐๐
๐๐ฅ,
๐2๐
๐๐พ2=๐
๐๐ฅ(1
๐พ
๐๐
๐๐ฅ)๐๐ฅ
๐๐พ=1
๐พ2(๐2๐
๐๐ฅ2โ๐๐
๐๐ฅ) ,
๐๐ฅ
๐๐พ=1
๐พ
๐ยฑ =โ๐
๐โ๐ยฑ๐โ๐
2, ๐ = ln
๐พ
๐น๐ก,๐
(52)
2.3.2. Formula in Log-moneyness
We may also want to change the spatial variable to log-moneyness ๐. Defining a new quantity,
implied total variance ๐ฃ๐,๐, which is equivalent to ๐๐,๐พ2 ๐, the Black-Scholes call price that is equivalent
to (45) then transforms into
๐๐,๐,๐ฃ = ๐น๐ก,๐(ฮฆ(๐+) โ exp(๐)ฮฆ(๐โ)) with ๐ยฑ =โ๐
โ๐ฃ๐,๐ยฑโ๐ฃ๐,๐
2 (53)
The partial derivatives of ๐๐,๐,๐ฃ can be derived as
๐๐๐,๐,๐ฃ๐๐ฃ
= ๐น๐ก,๐ (๐ฮฆ(๐+)
๐๐+
๐๐+๐๐ฃ
โ exp(๐)๐ฮฆ(๐โ)
๐๐โ
๐๐โ๐๐ฃ) = ๐น๐ก,๐๐(๐+) (
๐๐+๐๐ฃ
โ๐๐โ๐๐ฃ) =
๐น๐ก,๐๐(๐+)
2โ๐ฃ
๐2๐๐,๐,๐ฃ๐๐ฃ2
=๐๐๐,๐,๐ฃ๐๐ฃ
(โ1
2๐ฃโ ๐+
๐๐+๐๏ฟฝฬ๏ฟฝ) =
๐๐๐,๐,๐ฃ๐๐ฃ
(โ1
2๐ฃโ (โ
๐
โ๐ฃ+โ๐ฃ
2) (
๐
2โ๐ฃ3+
1
4โ๐ฃ))
=๐๐๐,๐,๐ฃ๐๐ฃ
(๐2
2๐ฃ2โ1
2๐ฃโ1
8)
(54)
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
14
๐๐๐,๐,๐ฃ๐๐
= ๐น๐ก,๐ (๐(๐+)๐๐+๐๐
โ exp(๐)ฮฆ(๐โ) โ exp(๐)๐(๐โ)๐๐โ๐๐) = โ๐น๐ก,๐ exp(๐)ฮฆ(๐โ)
๐2๐๐,๐,๐ฃ๐๐2
= โ๐น๐ก,๐ exp(๐)ฮฆ(๐โ) + ๐น๐ก,๐exp(๐)๐(๐โ)
โ๐ฃ=๐๐๐,๐,๐ฃ๐๐
+ 2๐๐๐,๐,๐ฃ๐๐ฃ
๐2๐๐,๐,๐ฃ๐๐๐๐ฃ
=๐
๐๐(๐น๐ก,๐๐(๐+)
2โ๐ฃ) =
๐๐๐,๐,๐ฃ๐๐ฃ
๐+๐๐+๐๐
=๐๐๐,๐,๐ฃ๐๐ฃ
(1
2โ๐
๐ฃ)
๐๐๐,๐,๐ฃ๐๐
= (ฮฆ(๐+) โ exp(๐)ฮฆ(๐โ))๐๐น๐ก,๐๐๐
= ๐๐๐๐,๐,๐ฃ
We may connect the local volatility ๐๐,๐ to the implied total variance ๐ฃ๐,๐ via two steps. Firstly
we bridge the ๐๐,๐ to ๐๐,๐,๐ฃ by (43) using the chain rule
๐๐ถ๐,๐๐๐
=๐๐๐,๐,๐ฃ๐๐
+๐๐๐,๐,๐ฃ๐๐ฃ
๐๐ฃ
๐๐,
๐๐ถ๐,๐๐๐
=๐๐๐,๐,๐ฃ๐๐
+๐๐๐,๐,๐ฃ๐๐ฃ
๐๐ฃ
๐๐
๐2๐ถ๐,๐๐๐2
=๐
๐๐(๐๐๐,๐,๐ฃ๐๐
+๐๐๐,๐,๐ฃ๐๐ฃ
๐๐ฃ
๐๐) +
๐
๐๐ฃ(๐๐๐,๐,๐ฃ๐๐
+๐๐๐,๐,๐ฃ๐๐ฃ
๐๐ฃ
๐๐)๐๐ฃ
๐๐
=๐2๐๐,๐,๐ฃ๐๐2
+๐2๐๐,๐,๐ฃ๐๐๐๐ฃ
๐๐ฃ
๐๐+๐๐๐,๐,๐ฃ๐๐ฃ
๐2๐ฃ
๐๐2+๐2๐๐,๐,๐ฃ๐๐ฃ๐๐
๐๐ฃ
๐๐+๐2๐๐,๐,๐ฃ๐๐ฃ2
(๐๐ฃ
๐๐)2
=๐2๐๐,๐,๐ฃ๐๐2
+ 2๐2๐๐,๐,๐ฃ๐๐๐๐ฃ
๐๐ฃ
๐๐+๐๐๐,๐,๐ฃ๐๐ฃ
๐2๐ฃ
๐๐2+๐2๐๐,๐,๐ฃ๐๐ฃ2
(๐๐ฃ
๐๐)2
(55)
This gives the local volatility expressed in terms of derivatives of ๐๐,๐,๐ฃ
๐๐,๐2 =
2(๐๐๐,๐,๐ฃ๐๐
+๐๐๐,๐,๐ฃ๐๐ฃ
๐๐ฃ๐๐โ ๐๐๐๐,๐,๐ฃ)
๐2๐๐,๐,๐ฃ๐๐2
+ 2๐2๐๐,๐,๐ฃ๐๐๐๐ฃ
๐๐ฃ๐๐+๐๐๐,๐,๐ฃ๐๐ฃ
๐2๐ฃ๐๐2
+๐2๐๐,๐,๐ฃ๐๐ฃ2
(๐๐ฃ๐๐)2
โ๐๐๐,๐,๐ฃ๐๐
โ๐๐๐,๐,๐ฃ๐๐ฃ
๐๐ฃ๐๐
(56)
Secondly we substitute the partial derivatives in (54) into (56) and reach the final equation
๐๐,๐2 =
2(๐๐๐ +๐๐๐๐ฃ๐๐ฃ๐๐โ ๐๐๐)
๐๐๐๐+ 2
๐๐๐๐ฃ+ 2
๐๐๐๐ฃ(12โ๐๐ฃ)๐๐ฃ๐๐+๐๐๐๐ฃ๐2๐ฃ๐๐2
+๐๐๐๐ฃ(๐2
2๐ฃ2โ12๐ฃโ18)(๐๐ฃ๐๐)2
โ๐๐๐๐โ๐๐๐๐ฃ๐๐ฃ๐๐
โน ๐๐,๐2 =
๐๐ฃ๐๐
1 โ๐๐ฃ๐๐ฃ๐๐+14 (๐2
๐ฃ2โ1๐ฃโ14)(๐๐ฃ๐๐)2
+12๐2๐ฃ๐๐2
(57)
2.3.3. Equivalency in Formulas
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15
The ๐๐,๐พ2 in (50) is in fact equivalent to the ๐๐,๐
2 in (57). This can be shown as follows
๐๐,๐2 =
๐๐ฃ๐๐
1 โ๐๐ฃ๐๐ฃ๐๐+ (
๐2
4๐ฃ2โ14๐ฃโ116) (๐๐ฃ๐๐)2
+12๐2๐ฃ๐๐2
=๐2 + 2๐๐ (
๐๐๐๐+ ๐๐พ
๐๐๐๐พ)
1 โ 2๐๐๐พ๐๐ฃ๐๐๐๐พ
+ (๐2
๐ฃ2โ1๐ฃโ14)(๐๐๐พ
๐๐๐๐พ)2
+ ๐๐พ2 ((๐๐๐๐พ)2
+๐๐พ๐๐๐๐พ
+ ๐๐2๐๐๐พ2
)
=๐2 + 2๐๐ (
๐๐๐๐+ ๐๐พ
๐๐๐๐พ)
1 + (1 โ 2๐๐ฃ) ๐๐๐พ
๐๐๐๐พ
+ (๐2
๐ฃโ๐ฃ4)๐๐พ2 (
๐๐๐๐พ)2
+ ๐๐๐พ2๐2๐๐๐พ2
=๐2 + 2๐๐ (
๐๐๐๐+ ๐๐พ
๐๐๐๐พ)
1 + 2โ๐๐พ๐+๐๐๐๐พ
+ ๐+๐โ๐๐พ2 (๐๐๐๐พ)2
+ ๐๐๐พ2๐2๐๐๐พ2
= ๐๐,๐พ2
(58)
where by definition we have
๐ = ln๐พ
๐น๐ก,๐, ๐ฃ = ๐2๐, ๐ยฑ =
ln๐น๐ก,๐๐พยฑ๐2๐2
๐โ๐=โ๐
โ๐ฃยฑโ๐ฃ
2, ๐+๐โ =
๐2
๐ฃโ๐ฃ
4 (59)
and also have the identities
๐๐ฃ
๐๐=๐(๐2๐)
๐๐= ๐2 + 2๐๐ (
๐๐
๐๐+๐๐
๐๐พ
๐๐พ
๐๐) = ๐2 + 2๐๐ (
๐๐
๐๐+ ๐๐พ
๐๐
๐๐พ)
๐๐ฃ
๐๐=๐(๐2๐)
๐๐=๐(๐2๐)
๐๐
๐๐
๐๐+๐(๐2๐)
๐๐พ
๐๐พ
๐๐= 2๐๐
๐๐
๐๐พ
๐๐พ
๐๐= 2๐๐๐พ
๐๐
๐๐พ
๐2๐ฃ
๐๐2=๐ (2๐๐๐พ
๐๐๐๐พ)
๐๐
๐๐
๐๐+๐ (2๐๐๐พ
๐๐๐๐พ)
๐๐พ
๐๐พ
๐๐= 2๐๐พ (๐
๐๐
๐๐พ+ ๐พ
๐๐
๐๐พ
๐๐
๐๐พ+ ๐๐พ
๐2๐
๐๐พ2)
= 2๐๐พ2 ((๐๐
๐๐พ)2
+๐
๐พ
๐๐
๐๐พ+ ๐
๐2๐
๐๐พ2)
(60)
considering the fact that in (๐, ๐)-plane the ๐ and ๐พ are no longer independent
๐๐พ
๐๐=๐(๐น๐ก,๐ exp(๐))
๐๐= ๐๐๐พ,
๐๐พ
๐๐=๐(๐น๐ก,๐ exp(๐))
๐๐= ๐พ (61)
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16
3. LOCAL VOLATILITY: PDE BY FINITE DIFFERENCE METHOD
In this chapter, we will present a PDE based local volatility model, in which the local volatility
surface is constructed as a 2-D function that is piecewise constant in maturity and piecewise linear in
log-moneyness (for equity) or delta (for FX). Due to great similarity between FX and equity processes,
our interest lies primarily in the context of equity derivatives, the conclusions and formulas drawn from
our discussion here are in general applicable to FX products with minor changes. In contrast to the
traditional way to construct the local volatility by estimating highly sensitive and numerically unstable
partial derivatives in Dupire formulas, this method relies heavily on solving forward PDEโs to calibrate a
parametrized local volatility surface to vanilla option prices in a bootstrapping manner. Once the local
volatility surface is calibrated, the backward PDE can then be used to price exotic options (e.g. barrier
options) that are in consistent with the market observed implied volatility surface.
Before proceeding to the PDEโs, it is important to have an overview of the date conventions for
equity and equity options. The date conventions for FX products are defined in a similar manner.
3.1. Date Conventions of Equity and Equity Option
The diagram illustrates the date definitions for an equity and its associated option. The quantities
appeared in the diagram are listed in Table 1.
ฮ๐,๐ ๐ก๐,๐ ๐ก๐,๐ ๐ก๐,๐ ฮ๐,๐
๐ก
ฮ๐,๐ ๐ก๐,๐ ฮ๐,๐
๐๐ = ๐ก๐,๐ โ ๐ก0 ๐ก๐,๐ ๐๐,๐
๐๐ = ๐ก๐,๐ โ ๐ก0 ๐ก๐,๐ ๐๐,๐ ๐ก0
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17
Table 1. Dates of Equities and Options
attribute symbol description remark/example
trade date ๐ก0 on which the equity/option is traded today
equity spot lag ฮ๐,๐ equity premium settlement lag 3D
equity spot date ๐ก๐,๐ on which the equity premium is settled ๐ก๐,๐ = ๐ก0โฮ๐,๐
equity maturity date ๐ก๐,๐ equity maturity date ๐ก๐,๐ = ๐ก0โ1๐
equity pay lag1 ฮ๐,๐ lag between ๐ก๐,๐ and ๐ก๐,๐ e.g. same as ฮ๐,๐
equity pay date ๐ก๐,๐ on which the equity payoff is settled ๐ก๐,๐ = ๐ก๐,๐โฮ๐,๐
๐-th dividend ๐๐ dividend payment amount
๐-th ex- div. date ๐ก๐,๐ ex-dividend date
๐-th div. pay date ๐ก๐,๐ dividend pay date
option spot lag ฮ๐,๐ option premium settlement lag 2D
option spot date ๐ก๐,๐ on which the option is settled ๐ก๐,๐ = ๐ก0โฮ๐,๐
option maturity date ๐ก๐,๐ option maturity date ๐ก๐,๐ = ๐ก0โ1๐
option pay lag ฮ๐,๐ lag between ๐ก๐,๐ and ๐ก๐,๐ e.g. same as ฮ๐,๐
option pay date ๐ก๐,๐ on which the equity payoff is settled ๐ก๐,๐ = ๐ก๐,๐โฮ๐,๐
day rolling โ rolling with convention โfollowingโ Following
calendar defining business days and holidays US / UK / HK
As most of the quantities are self-explanatory, our discussion focuses more on the treatment of
equity dividends.
3.2. Deterministic Dividends
In our example, we can assume both the short rate and the dividend rate are deterministic and
continuous, e.g. time-dependent ๐๐ข and ๐๐ข as in (21). the equity forward in this case can be calculated by
๐น(๐ก0, ๐ก๐,๐) = ๐(๐ก0)๐๐(๐ก๐,๐ , ๐ก๐,๐)
๐๐(๐ก๐,๐ , ๐ก๐,๐) where
๐๐(๐ก, ๐) = exp(โโซ ๐๐ข๐๐ข๐
๐ก
) , ๐๐(๐ก, ๐) = exp(โโซ ๐๐ข๐๐ข๐
๐ก
)
(62)
In a more realistic implementation, we may assume the underlying equity issues a series of
discrete dividends with fixed amounts in a foreseeable future. It is obvious that the equity spot still
follows the SDE (21) with ๐๐ข = 0 in between two adjacent ex-dividend dates (There is discontinuity in
1 Equity settlement delay
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
18
spot process on ex-dividend dates that demands special treatment. This will be discussed in detail in due
course). With fixed dividends, the equity forward becomes
๐น(๐ก0, ๐ก๐,๐) =๐(๐ก0) โ โ ๐๐๐๐(๐ก๐,๐ , ๐ก๐,๐)๐
๐๐(๐ก๐,๐ , ๐ก๐,๐) for ๐ก0 < ๐ก๐,๐ โค ๐ก๐,๐ (63)
where ๐๐ is the fixed amount of the ๐-th dividend issued on ex-dividend date ๐ก๐,๐.
Discrete dividend can also be modeled as proportional dividend. It assumes that at each ex-
dividend date, the dividend payment will result in a price drop in equity spot proportional to the spot
level. For example, the equity spot before and after the dividend fall has the relationship
๐(๐ก๐,๐ + ฮ) = ๐(๐ก๐,๐ โ ฮ)(1 โ ๐๐) (64)
where ฮ denotes an infinitesimal amount of time and ๐๐ the proportional dividend rate at ex-dividend
date ๐ก๐,๐. By this relationship, we can write the equity forward as
๐น(๐ก0, ๐ก๐,๐) = ๐(๐ก0)โ (1 โ ๐๐)๐
๐๐(๐ก๐,๐ , ๐ก๐,๐) for ๐ก0 < ๐ก๐,๐ โค ๐ก๐,๐ (65)
Sometimes it is often more convenient to approximate the fixed dividends by proportional
dividends. The conversion can be achieved by equating the equity forward in (63) and in (65), such that
โ(1โ ๐๐)
๐
= 1 โ1
๐(๐ก0)โ๐๐๐๐(๐ก๐,๐ , ๐ก๐,๐)
๐
for ๐ก0 < ๐ก๐,๐ โค ๐ก๐,๐ (66)
The proportional dividend ๐๐ can then be bootstrapped from a series of fixed dividends ๐๐ starting from
the first ex-dividend date.
3.3. Forward PDE
In the following, our derivation is based on the spot process ๐๐ก defined in (21) and its variants.
For example, depending on the application we may write the SDE (21) in terms of log-spot ๐๐ข = ln ๐๐ข
or in terms of centered log-spot ๐ง๐ข = ln๐๐ข
๐น๐ก,๐ข
๐๐๐ข = (๐๐ข โ1
2๐(๐ข, ๐)2) ๐๐ข + ๐(๐ข, ๐)๐๏ฟฝฬ๏ฟฝ๐ข and ๐๐ง๐ข = โ
1
2๐(๐ข, ๐ง)2๐๐ข + ๐(๐ข, ๐ง)๐๏ฟฝฬ๏ฟฝ๐ข (67)
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
19
where ๐(๐ข, ๐) and ๐(๐ข, ๐ง) are the local volatility function of ๐ and ๐ง, respectively.
Letโs denote the forward temporal variable by ๐ข for ๐ก < ๐ข < ๐ , the spatial variable by log-
moneyness ๐ = ln๐พ
๐น๐ก,๐ข (as in (41)) and the spot by ๐ง = ln
๐๐ข
๐น๐ก,๐ข. Given that ๐ง๐ก = 0 , the value of a
normalized forward call can be defined as
๐๐ข,๐|๐ก,๐ง =๐ถ๐ข,๐|๐ก,๐ง๐น๐ก,๐ข
=๏ฟฝฬ๏ฟฝ[(๐๐ข โ ๐พ)
+|๐ก, ๐๐ก]
๐น๐ก,๐ข (68)
Let ๐๐ข,๐ be the local volatility function of variable ๐ equivalent to ๐๐,๐พ. We can derive the forward PDE
for ๐๐ข,๐|๐ก,๐ง from (43)
๐๐ข,๐2
2=๐น๐ก,๐ข
๐๐๐ข,๐|๐ก,๐ง๐๐ข
+ ๐๐ข๐น๐ก,๐ข๐๐ข,๐|๐ก,๐ง โ ๐๐ข๐ถ๐ข,๐|๐ก,๐ง
๐น๐ก,๐ข๐2๐๐ข,๐|๐ก,๐ง๐๐2
โ ๐น๐ก,๐ข๐๐๐ข,๐|๐ก,๐ง๐๐
=
๐๐๐ข,๐|๐ก,๐ง๐๐ข
๐2๐๐ข,๐|๐ก,๐ง๐๐2
โ๐๐๐ข,๐|๐ก,๐ง๐๐
โน๐๐๐ข,๐|๐ก,๐ง๐๐ข
=๐๐ข,๐2
2(๐2๐๐ข,๐|๐ก,๐ง๐๐2
โ๐๐๐ข,๐|๐ก,๐ง๐๐
)
(69)
with initial condition
๐๐ก,๐|๐ก,๐ง =๐ถ๐ก,๐|๐ก,๐ง๐น๐ก,๐ก
=๏ฟฝฬ๏ฟฝ [(๐๐ก โ ๐น๐ก,๐ก๐
๐)+| ๐ก, ๐๐ก]
๐น๐ก,๐ก= (1 โ ๐๐)+ (70)
using the partial derivatives
๐๐๐ข,๐|๐ก,๐ง๐๐ข
=1
๐น๐ก,๐ข
๐๐ถ๐ข,๐|๐ก,๐ง๐๐ข
โ ๐๐ข๐๐ข,๐|๐ก,๐ง , ๐๐๐ข,๐|๐ก,๐ง๐๐
=1
๐น๐ก,๐ข
๐๐ถ๐ข,๐|๐ก,๐ง๐๐
, ๐2๐๐ข,๐|๐ก,๐ง๐๐2
=1
๐น๐ก,๐ข
๐2๐ถ๐ข,๐|๐ก,๐ง๐๐2
(71)
The PDE (69) appears drift-less and provides more robust calibration stability at low volatility and/or
high drift due to the โtransparencyโ of drift in the PDE.
3.3.1. Treatment of Deterministic Dividends
A (discrete) dividend pay-out will typically result in a drop in equity price on the ex-dividend
date. Suppose that time ๐ข is the ex-dividend date, the no-arbitrage condition states that at ๐ข+ the time
right after the ex-dividend date (e.g. the difference between ๐ข and ๐ข+ can be infinitesimal), we must
have
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
20
๐๐ข+ = ๐๐ข โ ๐๐ข (72)
where ๐๐ข is the value of dividend issued at ๐ข (note that in a rigorous setup the value must take into
account the discounting effect due to dividend payment delay). Since a forward is expectation of spot
under risk neutral measure1, we may write
๐น๐ก,๐ข+ = ๏ฟฝฬ๏ฟฝ๐ก[๐๐ข+] = ๏ฟฝฬ๏ฟฝ๐ก[๐๐ข โ ๐๐ข] = ๐น๐ก,๐ข โ ๏ฟฝฬ๏ฟฝ๐ก[๐๐ข] (73)
Under the assumption that ๐๐ข is a fixed amount, it reads
๐น๐ก,๐ข+ = ๐น๐ก,๐ข โ ๐๐ข (74)
In our finite difference method, the spatial grid for log-moneyness ๐ is assumed uniform such that ๐๐ โ
๐๐โ1 is constant for all ๐. Dividend payment causes discontinuity in the underlying spot. Evolving the
forward PDE (69) from initial time ๐ก produces a state vector ๐๐ข,๐|๐ก,๐ง at time ๐ข. Immediately after the
issuance of dividend at time ๐ข+, the spot and forward drop the same ๐๐ข amount and hence the state
vector ๐๐ข+,๐|๐ก,๐ง must be realigned to reflect the dividend fall. This can be done using the option no-
arbitrage condition, such that
๐ถ๐ข+,๐|๐ก,๐ง = ๏ฟฝฬ๏ฟฝ๐ก [(๐๐ข+ โ ๐พ)+] = ๏ฟฝฬ๏ฟฝ๐ก [(๐๐ข โ ๐๐ข โ ๐น๐ก,๐ข+๐
๐)+] = ๏ฟฝฬ๏ฟฝ๐ก [(๐๐ข โ ๐น๐ก,๐ข๐
๏ฟฝฬ๏ฟฝ)+] = ๐ถ๐ข,๏ฟฝฬ๏ฟฝ|๐ก,๐ง
where ๏ฟฝฬ๏ฟฝ = ln๐น๐ก,๐ข+๐
๐ + ๐๐ข
๐น๐ก,๐ข
(75)
Subsequently we can use ๏ฟฝฬ๏ฟฝ to interpolate from the ๐๐ข,๐|๐ก,๐ง state vector and transform the interpolated
value to form ๐๐ข+,๐|๐ก,๐ง vector by
๐๐ข+,๐|๐ก,๐ง =๐ถ๐ข+,๐|๐ก,๐ง
๐น๐ก,๐ข+=๐ถ๐ข,๏ฟฝฬ๏ฟฝ|๐ก,๐ง
๐น๐ก,๐ข
๐น๐ก,๐ข๐น๐ก,๐ข+
=๐น๐ก,๐ข๐น๐ก,๐ข+
๐๐ข,๏ฟฝฬ๏ฟฝ|๐ก,๐ง (76)
If the dividend is proportional, we must have spot price ๐๐ข+ = ๐๐ข(1 โ ๐๐ข) for a rate ๐๐ข and
hence forward price ๐น๐ก,๐ข+ = ๐น๐ก,๐ข(1 โ ๐๐ข) before and after the dividend fall. Because we can show that
1 Strictly speaking, a forward on time ๐ spot is an expectation of the spot under ๐-forward measure, i.e. ๐น๐ก,๐ =
๏ฟฝฬ๏ฟฝ๐ก๐[๐๐]. However since the interest rate is assumed deterministic, the ๐-forward measure coincides with the risk
neutral measure.
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
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๐๐ข+,๐|๐ก,๐ง =๏ฟฝฬ๏ฟฝ๐ก [(๐๐ข+ โ ๐พ)
+]
๐น๐ก,๐ข+=(1 โ ๐๐ข)๏ฟฝฬ๏ฟฝ๐ก [(๐๐ข โ ๐น๐ก,๐ข๐
๐)+]
๐น๐ก,๐ข(1 โ ๐๐ข)=๏ฟฝฬ๏ฟฝ๐ก [(๐๐ข โ ๐น๐ก,๐ข๐
๐)+]
๐น๐ก,๐ข= ๐๐ข,๐|๐ก,๐ง (77)
the state vector remains unchanged before and after the issuance of dividend.
With continuous dividend ๐๐ข, the realignment of state vector is unnecessary because there is no
discontinuity in equity spot.
3.4. Backward PDE
Again we assume the spot follows the SDE (21). Without loss of generality, letโs denote
๐บ(๐๐|๐พ) an arbitrary payoff function with parameter ๐พ, whose value is contingent on ๐๐ at time ๐. One
example of such function would be the payoff function of a call option: ๐บ(๐๐|๐พ) = (๐๐ โ ๐พ)+. Let
๐๐ข,๐ฅ|๐,๐พ be the expectation of the function ๐บ(๐๐|๐พ) at time ๐ข with spatial variable ๐ฅ = ๐๐ข, which can be
written as
๐๐ข,๐ฅ|๐,๐พ = ๏ฟฝฬ๏ฟฝ[๐บ(๐๐|๐พ)|๐ข, ๐ฅ] = โซ๐บ(๐ฆ|๐พ)๐๐,๐ฆ|๐ข,๐ฅ๐๐ฆโ
(78)
where the transition probability ๐๐,๐ฆ|๐ข,๐ฅ follows the Kolmogorov backward equation (20)
๐๐๐,๐ฆ|๐ข,๐ฅ๐๐ข
= ๐๐ข๐ฅ๐๐๐,๐ฆ|๐ข,๐ฅ๐๐ฅ
+๐๐ข,๐ฅ2 ๐ฅ2
2
๐2๐๐,๐ฆ|๐ข,๐ฅ๐๐ฅ2
(79)
In turn, we can derive the backward PDE for the ๐๐ข,๐ฅ|๐,๐พ such that
๐๐๐ข,๐ฅ|๐,๐พ๐๐ข
= โซ ๐บ(๐ฆ|๐พ)๐๐๐,๐ฆ|๐ข,๐ฅ๐๐ข
๐๐ฆโ
= โโซ ๐บ(๐ฆ|๐พ) (๐๐ข๐ฅ๐๐๐,๐ฆ|๐ข,๐ฅ๐๐ฅ
+๐๐ข,๐ฅ2 ๐ฅ2
2
๐2๐๐,๐ฆ|๐ข,๐ฅ๐๐ฅ2
)๐๐ฆโ
= โ๐๐ข๐ฅ๐๐๐ข,๐ฅ|๐,๐พ๐๐ฅ
โ๐๐ข,๐ฅ2 ๐ฅ2
2
๐2๐๐ข,๐ฅ|๐,๐พ๐๐ฅ2
(80)
with terminal condition
๐๐,๐ฅ|๐,๐พ = ๐บ(๐ฅ|๐พ) (81)
3.4.1. PDE in Centered Log-spot
Assuming the spatial variable is ๐ง๐ข = ln๐ฅ
๐น๐ก,๐ข at time ๐ข, we may write ๐๐ข,๐ง|๐,๐ in the (๐ข, ๐ง)-plane
equivalent to ๐๐ข,๐ฅ|๐,๐พ. The backward PDE (80) can then be transformed into
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
22
๐๐๐ข,๐ง|๐,๐๐๐ข
= โ๐๐ข,๐ง2
2(๐2๐๐ข,๐ง|๐,๐๐๐ง2
โ๐๐๐ข,๐ง|๐,๐๐๐ง
) (82)
with terminal condition
๐๐,๐ง|๐,๐ = ๐บ(๐น๐ก,๐๐๐ง|๐น๐ก,๐๐
๐) (83)
by using the following partial derivatives derived from the chain rule
๐๐ง
๐๐ฅ=1
๐ฅ,
๐๐ง
๐๐ข= โ๐๐ข,
๐๐๐ข,๐ฅ|๐,๐พ๐๐ข
=๐๐๐ข,๐ง|๐,๐๐๐ข
+๐๐๐ข,๐ง|๐,๐๐๐ง
๐๐ง
๐๐ข=๐๐๐ข,๐ง|๐,๐๐๐ข
โ ๐๐ข๐๐๐ข,๐ง|๐,๐๐๐ง
๐๐๐ข,๐ฅ|๐,๐พ๐๐ฅ
=๐๐๐ข,๐ง|๐,๐๐๐ง
๐๐ง
๐๐ฅ=1
๐ฅ
๐๐๐ข,๐ง|๐,๐๐๐ง
,๐2๐๐ข,๐ฅ|๐,๐พ๐๐ฅ2
=1
๐ฅ2(๐2๐๐ข,๐ง|๐,๐๐๐ง2
โ๐๐๐ข,๐ง|๐,๐๐๐ง
)
(84)
3.4.1.1. Treatment of Deterministic Dividends
With fixed dividend ๐๐ข, we have
๐๐ข+ = ๐๐ข โ ๐๐ข and ๐น๐ก,๐ข+ = ๐น๐ก,๐ข โ ๐๐ข (85)
The no arbitrage condition shows that for the spatial grid ๐ง
๐๐ข,๐ง|๐,๐ = ๏ฟฝฬ๏ฟฝ[๐บ(๐๐|๐น๐ก,๐๐๐)|๐ข, ๐น๐ก,๐ข๐
๐ง] = ๏ฟฝฬ๏ฟฝ[๐บ(๐๐|๐น๐ก,๐๐๐)|๐ข+, ๐น๐ก,๐ข๐
๐ง โ ๐๐ข]
= ๏ฟฝฬ๏ฟฝ[๐บ(๐๐|๐น๐ก,๐๐๐)|๐ข+, ๐น๐ก,๐ข+๐
๏ฟฝฬ๏ฟฝ] = ๐๐ข+,๏ฟฝฬ๏ฟฝ|๐,๐ where ๏ฟฝฬ๏ฟฝ = ln๐น๐ก,๐ข๐
๐ง โ ๐๐ข๐น๐ก,๐ข+
(86)
It is likely that if ๐ง is sufficiently small (e.g. at lower boundary of spatial grid) we may end up with
๐น๐ก,๐ข๐๐ง โ ๐๐ข < 0, which makes the ๏ฟฝฬ๏ฟฝ not well defined. A solution is to floor it to a small positive number,
e.g. taking max(10โ10, ๐น๐ก,๐ข๐๐ง โ ๐๐ข) . This is valid because equity spot must be positive and the
๐๐ข+,๏ฟฝฬ๏ฟฝ|๐,๐ flattens as ๏ฟฝฬ๏ฟฝ goes to negative infinity. After the special treatment, we can use the ๏ฟฝฬ๏ฟฝ to
interpolate from the ๐๐ข,๐ง|๐,๐ state vector and convert the interpolated value into vector ๐๐ข+,๐ง|๐,๐.
With proportional dividend, the conclusion drawn for forward PDE still applies here and the
state vector remains unchanged before and after the dividend fall. With continuous dividend, the
realignment of state vector is unnecessary because there is no discontinuity in equity spot.
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
23
3.4.1.2. Vanilla Call
Due to the duality between the forward and backward PDE, it is evident that vanilla calls (or
puts) must admit the identity: ๐๐ก,๐ง|๐,๐ = ๐๐,๐|๐ก,๐ง๐น๐ก,๐ , where ๐๐ก,๐ง|๐,๐ is the forward call solved from
backward PDE (82) and ๐๐,๐|๐ก,๐ง the normalized forward call solved from forward PDE (69). This
relationship can be used to check the correctness of implementation of the numerical engines of forward
and backward PDE.
3.4.2. PDE in Log-spot
For pricing some exotic options, e.g. barrier options, it is more convenient to use log-spot ๐ =
ln ๐ฅ as the spatial variable. Similarly we can define ๐ = ln๐พ. Letโs denote the (discounted) price of a
derivative product by
๐๐ข,๐|๐,๐ = ๐ท๐ข,๐๐๐ข,๐|๐,๐ = ๏ฟฝฬ๏ฟฝ[๐ท๐ข,๐๐บ(๐๐|๐๐)|๐ข, ๐๐] (87)
By taking into account the discount factor, it must follow the following backward PDE
๐๐๐ข,๐|๐,๐๐๐ข
= ๐๐ข๐๐ข,๐|๐,๐ + ๐ท๐ข,๐๐๐๐ข,๐|๐,๐๐๐ข
= ๐๐ข๐๐ข,๐|๐,๐ + ๐ท๐ข,๐ (โ๐๐ข๐ฅ1
๐ฅ
๐๐๐ข,๐|๐,๐๐๐
โ๐๐ข,๐2
2
1
๐ฅ2(๐2๐๐ข,๐|๐,๐๐๐2
โ๐๐๐ข,๐|๐,๐๐๐
))
= โ๐๐ข,๐2
2
๐2๐๐ข,๐|๐,๐๐๐2
+ (๐๐ข,๐2
2โ ๐๐ข)
๐๐๐ข,๐|๐,๐๐๐
+ ๐๐ข๐๐ข,๐|๐,๐
(88)
where the partial derivatives below have been used
๐๐
๐๐ฅ=1
๐ฅ,
๐๐
๐๐ข= 0,
๐๐๐ข,๐ฅ|๐,๐พ๐๐ข
=๐๐๐ข,๐|๐,๐๐๐ข
+๐๐๐ข,๐|๐,๐๐๐
๐๐
๐๐ข=๐๐๐ข,๐|๐,๐๐๐ข
๐๐๐ข,๐ฅ|๐,๐พ๐๐ฅ
=๐๐๐ข,๐|๐,๐๐๐ข
๐๐ข
๐๐ฅ+๐๐๐ข,๐|๐,๐๐๐
๐๐
๐๐ฅ=1
๐ฅ
๐๐๐ข,๐|๐,๐๐๐
๐2๐๐ข,๐ฅ|๐,๐พ๐๐ฅ2
=๐
๐๐ฅ(1
๐ฅ
๐๐๐ข,๐|๐,๐๐๐
) = โ1
๐ฅ2๐๐๐ข,๐|๐,๐๐๐
+1
๐ฅ
๐2๐๐ข,๐|๐,๐๐๐๐๐ข
๐๐ข
๐๐ฅ+1
๐ฅ
๐2๐๐ข,๐|๐,๐๐๐2
๐๐
๐๐ฅ
=1
๐ฅ2(๐2๐๐ข,๐|๐,๐๐๐2
โ๐๐๐ข,๐|๐,๐๐๐
)
(89)
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
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3.4.2.1. Treatment of Deterministic Dividends
With fixed dividend ๐๐ข, the no arbitrage condition states that
๐๐ข,๐|๐,๐ = ๏ฟฝฬ๏ฟฝ[๐ท๐ข,๐๐บ(๐๐|๐๐)|๐ข, ๐๐] = ๏ฟฝฬ๏ฟฝ[๐ท๐ข+,๐๐บ(๐๐|๐
๐)|๐ข+, ๐๐ โ ๐๐ข]
= ๏ฟฝฬ๏ฟฝ[๐ท๐ข+,๐๐บ(๐๐|๐๐)|๐ข+, ๐
๏ฟฝฬ๏ฟฝ] = ๐๐ข+,๏ฟฝฬ๏ฟฝ|๐,๐ where ๏ฟฝฬ๏ฟฝ = ln(๐๐ โ ๐๐ข)
(90)
Again, extremely small ๐ may result in ๏ฟฝฬ๏ฟฝ that is not well defined, we may floor the difference ๐๐ โ ๐๐ข
to a small positive number, e.g. taking max(10โ10, ๐๐ โ ๐๐ข) . The vector ๐๐ข,๐|๐,๐ can then be
interpolated from the known ๐๐ข+,๐|๐,๐ using the ๏ฟฝฬ๏ฟฝ.
With proportional dividend ๐๐ข, again the no arbitrage condition shows
๐๐ข,๐|๐,๐ = ๏ฟฝฬ๏ฟฝ[๐ท๐ข,๐๐บ(๐๐|๐๐)|๐ข, ๐๐] = ๏ฟฝฬ๏ฟฝ[๐ท๐ข+,๐๐บ(๐๐|๐
๐)|๐ข+, ๐๐(1 โ ๐๐ข)]
= ๏ฟฝฬ๏ฟฝ[๐ท๐ข+,๐๐บ(๐๐|๐๐)|๐ข+, ๐
๏ฟฝฬ๏ฟฝ] = ๐๐ข+,๏ฟฝฬ๏ฟฝ|๐,๐ where ๏ฟฝฬ๏ฟฝ = ๐ + ln(1 โ ๐๐ข)
(91)
The vector ๐๐ข,๐|๐,๐ can be interpolated from the ๐๐ข+,๐|๐,๐ using the ๏ฟฝฬ๏ฟฝ.
With continuous dividend, the realignment of state vector is unnecessary because there is no
discontinuity in equity spot.
3.5. Local Volatility Surface
This section is devoted to discussing the construction of local volatility surface ๐(๐ข, ๐). There
are various ways to define the local volatility surface. The one that we would like to discuss is a 2-D
function that is piecewise constant in maturity ๐ข and piecewise linear in log-moneyness ๐ = ln๐พ
๐น๐ก,๐ข (or in
delta for FX). The volatility surface comprises a series of volatility smiles ๐๐(๐) for maturity ๐ก < ๐ข1 <
โฏ < ๐ข๐ < โฏ < ๐ข๐ = ๐ . At each maturity ๐ข๐ , volatility smile ๐๐(๐) is constructed by linear
interpolation between log-moneyness pillars ๐๐ = ln๐พ๐
๐น๐ก,๐ข for strikes ๐พ1 < โฏ < ๐พ๐ < โฏ < ๐พ๐ and flat
extrapolation where the volatility values at ๐1 and ๐๐ are used for all ๐ < ๐1 and ๐ > ๐๐, respectively.
The smile ๐๐(๐) constructed at ๐ข๐ is assumed to remain constant over time for any ๐ข between the two
adjacent maturities ๐ข๐โ1 < ๐ข โค ๐ข๐.
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25
Calibration of the local volatility surface is conducted in a bootstrapping manner starting from
the shortest maturity ๐ข1. It is done by solving the forward PDE such that the local volatility surface is
able to reproduce the vanilla call prices at the prescribed log-moneyness pillars ๐๐ for each of the
maturities ๐ข๐. The PDE can be solved using finite difference method1 on a uniform grid defined on log-
moneyness ๐ that extends to ยฑ5 standard deviations of the underlying spot. The choice of boundary
condition has little impact to the solutions of vanilla option prices because at ยฑ5 standard deviations the
transition probability becomes negligibly small. Our application uses linearity boundary condition for its
simplicity. To allow a higher tolerance to market data input and smoother calibration process, the
objective function may include a penalty term to suppress unfavorable concavity of a local volatility
smile. Again, there can be many ways to define the objective function as well as the penalty function. In
this essay, we will only focus on the simplest objective (e.g. at maturity ๐ข๐ ): the least square
minimization of vanilla call prices
argmin๐๐(๐๐)
โ(๐๐ก,๐ง|๐ข๐,๐๐๐ต๐ โ ๐๐ก,๐ง|๐ข๐,๐๐
๐๐ท๐ธ )2
๐
๐=1
(92)
where ๐๐ก,๐ง|๐,๐ is the normalized forward call price defined in (68), the superscript โBSโ denotes the
theoretical price by Black-Scholes model and the โPDEโ denotes the numerical value by forward PDE.
Note that without a penalty term, the minimization can lead to an exact solution given a proper2 implied
volatility surface.
3.6. Barrier Option Pricing
In contrast to the calibration, the pricing of a barrier option relies on the backward PDE (88) in
line with proper terminal condition (i.e. payoff function) and boundary conditions defined by the
characteristics of the barrier option. Barrier options often demand a spatial grid defined on log-spot ๐ =
ln ๐๐ข , which allows an easier fit of time-invariant barrier (e.g. with European or American type of
1 A brief introduction to finite difference method can be found in my notes โIntroduction to Interest Rate Modelsโ,
which can be downloaded from http://www.cs.utah.edu/~cxiong/. 2 A proper implied volatility surface should well behave and admit no arbitrage.
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
26
observation window) into the domain. For example, an up-and-out barrier option would be priced on a
domain with upper bound at the barrier level ๐ where Dirichlet boundary condition is applied (the lower
bound and its boundary condition remain the same as for vanilla options).
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
27
REFERENCES
1. Clark, I., Foreign Exchange Option Pricing - A Practitionerโs Guide, Wiley-Finance, 2011, pp.82
2. Online resource: http://itf.fys.kuleuven.be/~nikos/papers/lect4_localvol.pdf
3. Gatheral, J., The Volatility Surface: A Practitionerโs Guide, Wiley-Finance, 2006, pp. 13-14