conversion from implied volatility to local volatility
TRANSCRIPT
Local Volatility Models
Copyright Š 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
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
2
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
Letâs consider the following đ-dimensional stochastic spot process đđĄ â âđ driven by an đ-
dimensional Brownian motion đđĄ whose correlation matrix đ is given by đđđĄ = đđđĄđđđĄâ˛
đđđĄđĂ1
= đ´(đĄ, đđĄ)đĂ1
đđĄ1Ă1
+ đľ(đĄ, đđĄ)đĂđ
đđđĄđĂ1
(1)
We derive the dynamics of â , where â:âđ âśâ in this case is a scalar-valued Borel-measurable
function only on variable đđĄ
đâ(đđĄ)1Ă1
= đ˝â1Ăđ
đđđĄđĂ1
+1
2đđđĄ
â˛
1ĂđđťâđĂđ
đđđĄđĂ1
= đ˝âđ´đđĄ + đ˝âđľđđđĄ +1
2đđđĄ
â˛đľâ˛đťâđľđđđĄ (2)
where đ˝â is the 1 Ă đ Jacobian (i.e. the same as gradient if â is a scalar-valued function) and đťâ the
đ Ăđ Hessian (with subscripts of đ now denoting the indices of vector components)
[đ˝â]đ =đâ
đđđ and [đťâ]đđ =
đ2â
đđđđđđ (3)
Expanding the expression in (2), we have
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
3
đâ =âđâ
đđđđ´đđđĄ
đ
đ=1
+âđâ
đđđ
đ
đ=1
âđľđđđđđ
đ
đ=1
+1
2â
đ2â
đđđđđđâđľđđđđđđľđđđđĄ
đ
đ=1
đ
đ,đ=1
= (âđâ
đđđđ´đ
đ
đ=1
+1
2â
đ2â
đđđđđđđ´đđ
đ
đ,đ=1
)đđĄ +âđâ
đđđ
đ
đ=1
âđľđđđđđ
đ
đ=1
(4)
where đ´ = đľđđľâ˛ is the đ Ăđ instantaneous variance-covariance matrix of đđ . Integrating on both
sides of (4) from đĄ to đ, we have
â(đđ) â â(đđĄ) = ⍠(âđâ
đđđđ´đ
đ
đ=1
+1
2â
đ2â
đđđđđđđ´đđ
đ
đ,đ=1
)đđ˘đ
đĄ
+⍠âđâ
đđđ
đ
đ=1
âđľđđđđđ
đ
đ=1
đ
đĄ
(5)
Taking expectation on both sides of (5), we get (using notation đźđĄ[â] = đź[â|âąđĄ])
LHS = đźđĄ[â(đđ)] â â(đđĄ) = âŤâđŚđđ,đŚ|đĄ,đĽđđŚđş
â âđĽ
RHS = đźđĄ [⍠(âđâ
đđđđ´đ
đ
đ=1
+1
2â
đ2â
đđđđđđđ´đđ
đ
đ,đ=1
)đđ˘đ
đĄ
] + đźđĄ [⍠âđâ
đđđ
đ
đ=1
âđľđđđđđ
đ
đ=1
đ
đĄ
]â
=0
= ⍠âđźđĄ [đâ
đđđđ´đ]
đ
đ=1
đđ˘đ
đĄ
+1
2⍠â đźđĄ [
đ2â
đđđđđđđ´đđ]
đ
đ,đ=1
đđ˘đ
đĄ
(6)
where đđ,đŚ|đĄ,đĽ is the transition probability density having đđ = đŚ at đ given đđĄ = đĽ at đĄ (i.e. if we solve
the equation (1) with the initial condition đđĄ = đĽ â âđ , then the random variable đđ = đŚ â đş has a
density đđ,đŚ|đĄ,đĽ in the đŚ variable at time đ). Differentiating (6) with respect to đ on both sides, we have
⍠âđŚđđđ,đŚ|đĄ,đĽđđ
đđŚđş
=âđźđĄ [đâ
đđđđ´đ]
đ
đ=1
+1
2â đźđĄ [
đ2â
đđđđđđđ´đđ]
đ
đ,đ=1
=ââŤđâđŚđđŚđ
đ´đđđ,đŚ|đĄ,đĽđđŚđş
đ
đ=1
+1
2â âŤ
đ2âđŚđđŚđđđŚđ
đ´đđđđ,đŚ|đĄ,đĽđđŚđş
đ
đ,đ=1
(7)
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
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If we assume đş ⥠âđ and also assume the probability density đ and its first derivatives đđ/đđŚđ vanish
at a higher order of rate than â and đâ/đđŚđ as đŚđ â Âąâ â đ = 1,⯠,đ, then we can integrate by parts
for the right hand side of (7), once for the first integral and twice for the second
âŤđâđŚđđŚđ
đ´đđđđŚđş
= ⍠âđŚđ´đđ|đŚđ =ââ+â
â =0
đđŚâđđşâđ
â⍠âđŚđ(đ´đđ)
đđŚđđđŚ
đş
and
âŤđ2âđŚđđŚđđđŚđ
đ´đđđđđŚđş
= âŤđâđŚđđŚđ
đ´đđđ|đŚđ=ââ
+â
â =0
đďż˝Ě ďż˝đďż˝Ě ďż˝đ
ââŤđâđŚđđŚđ
đ(đ´đđđ)
đđŚđđđŚ
đş
= â⍠âđŚđ(đ´đđđ)
đđŚđ|đŚđ=ââ
+â
â =0
đďż˝Ě ďż˝đďż˝Ě ďż˝đ
+⍠âđŚđ2(đ´đđđ)
đđŚđđđŚđđđŚ
đş
where ⍠(â)đďż˝Ě ďż˝đďż˝Ě ďż˝đ
= ⍠âŻâŤ ⍠âŻâŤ(â)đđŚ1â
âŻđđŚđâ1â
đđŚđ+1â
âŻđđŚđâ
(8)
Plugging the results of (8) into (7), we have
⍠âđŚđđ
đđđđŚ
đş
= ââ⍠âđŚđ(đ´đđ)
đđŚđđđŚ
đş
đ
đ=1
+1
2â ⍠âđŚ
đ2(đ´đđđ)
đđŚđđđŚđđđŚ
đş
đ
đ,đ=1
âšâŤ âđŚ (đđ
đđ+â
đ(đ´đđ)
đđŚđ
đ
đ=1
â1
2â
đ2(đ´đđđ)
đđŚđđđŚđ
đ
đ,đ=1
)đđŚđş
= 0
(9)
By the arbitrariness of â, we conclude that for any đŚ â đş
đđ
đđ+â
đ(đ´đđ)
đđŚđ
đ
đ=1
â1
2â
đ2(đ´đđđ)
đđŚđđđŚđ
đ
đ,đ=1
= 0, đ´ = đľđđľâ˛ (10)
This is the Multi-dimensional Fokker-Planck Equation (a.k.a. Kolmogorov Forward Equation) [1]. In
this equation, the đĄ and đĽ are held constant, while the đ and đŚ are variables (called âforward variablesâ).
In the one-dimensional case, it reduces to
đđ
đđ+đ(đ´đ)
đđŚâ1
2
đ2(đľ2đ)
đđŚ2= 0 (11)
where đ´ = đ´(đ, đŚ) and đľ = đľ(đ, đŚ) are then scalar functions.
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
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1.2. Kolmogorov Backward Equation
Letâs express conditional expectation of â(đđĄ) by đ(đĄ, đđĄ) = đźđĄ[â(đđ)]. Because for đ ⤠đĄ ⤠đ
we have
đ(đ, đđ) = đźđ[â(đđ)] = đźđ[đźđĄ[â(đđ)]] = đźđ[đ(đĄ, đđĄ)] (12)
the đ(đĄ, đđĄ) is a martingale by the tower rule (i.e. If â holds less information than đ˘ , then
đź[đź[đ|đ˘]|â] = đź[đ|â]). The dynamics of the đ(đĄ, đđĄ) is given by
đđ =đđ
đđĄđđĄ + đ˝đ
1Ăđ
đđđĄđĂ1
+1
2đđđĄ
â˛
1ĂđđťđđĂđ
đđđĄđĂ1
=đđ
đđĄđđĄ + đ˝đđ´đđĄ + đ˝đđľđđđĄ +
1
2đđđĄ
â˛đľâ˛đťđđľđđđĄ
(13)
where đ˝đ is the Jacobian (i.e. the same as gradient if đ is a scalar-valued function) and đťđ the Hessian of
đ with respect to đ (with subscripts denoting the indices of vector components)
đ˝đ = (đđ
đđ1âŻ
đđ
đđđ) , đťđ =
(
đ2đ
đđ12 âŻ
đ2đ
đđ1đđđ⎠⹠âŽđ2đ
đđđđđ1âŻ
đ2đ
đđđ2 )
(14)
Expanding (13), we have
đđ = (đđ
đđĄ+â
đđ
đđđđ´đ
đ
đ=1
+1
2â
đ2đ
đđđđđđđ´đđ
đ
đ,đ=1
)đđĄ +âđđ
đđđ
đ
đ=1
âđľđđđđđ
đ
đ=1
(15)
Since đ(đĄ, đđĄ) is a martingale, the đđĄ-term must vanish, which gives
đđ
đđĄ+â
đđ
đđđđ´đ
đ
đ=1
+1
2â
đ2đ
đđđđđđđ´đđ
đ
đ,đ=1
= 0 (16)
This is the multi-dimensional Feynman-Kac formula1.
Using the transition probability density đđ,đŚ|đĄ,đĽ, we can write the expectation as
1 https://en.wikipedia.org/wiki/Feynman-Kac_formula
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
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đđĄ,đĽ = đźđĄ[â(đđ)] = âŤâđŚđđ,đŚ|đĄ,đĽđđŚđş
(17)
The formula (16) defines that
đ
đđĄâŤâđŚđđđŚđş
+âđ´đđ
đđĽđâŤâđŚđđđŚđş
đ
đ=1
+1
2â đ´đđ
đ2
đđĽđđđĽđâŤâđŚđđđŚđş
đ
đ,đ=1
= 0
âšâŤ âđŚ (đđ
đđĄ+âđ´đ
đđ
đđĽđ
đ
đ=1
+1
2â đ´đđ
đ2đ
đđĽđđđĽđ
đ
đ,đ=1
)đđŚđş
= 0
(18)
By the arbitrariness of â, we have
đđ
đđĄ+âđ´đ
đđ
đđĽđ
đ
đ=1
+1
2â đ´đđ
đ2đ
đđĽđđđĽđ
đ
đ,đ=1
= 0, đ´ = đľđđľâ˛ (19)
This is the multi-dimensional Kolmogorov Backward Equation. In this equation, the đ and đŚ are held
constant, while the đĄ and đĽ are variables (called âbackward variablesâ). In the 1-D case, it reduces to
đđ
đđĄ+ đ´
đđ
đđĽ+1
2đľ2đ2đ
đđĽ2= 0 (20)
where đ´ = đ´(đĄ, đĽ) and đľ = đľ(đĄ, đĽ) 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
đđđĄđđĄ= đđĄđđĄ + đ(đĄ, đđĄ)đďż˝Ěďż˝đĄ , đđĄ = đđĄ â đđĄ (21)
with cash rate đđĄ and dividend rate đđĄ (or foreign cash rate for FX).
2.1. Local Volatility by Vanilla Call
Under the assumption of deterministic đđĄ , the European (vanilla) call option price can be
expressed as a function of maturity time đ and strike đž
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
7
đđ,đž|đĄ,đĽ = ďż˝Ěďż˝đĄ[đˇđĄ,đ(đđ â đž)+] = đˇđĄ,đ⍠(đŚ â đž)đđ,đŚ|đĄ,đĽđđŚ
â
đž
(22)
where đˇđĄ,đ = exp (â⍠đđ˘đđ˘đ
đĄ) is the deterministic discount factor and đđ,đŚ|đĄ,đĽ is the transition
probability density having spot đđ = đŚ at đ given initial condition đđĄ = đĽ at đĄ. Differentiating (22) with
respect to đž, we have the first order and second order partial derivative
đđđ,đž|đĄ,đĽđđž
= âđˇđĄ,đ⍠đđ,đŚ|đĄ,đĽđđŚâ
đž
,đ2đđ,đž|đĄ,đĽđđž2
= đˇđĄ,đđđ,đž|đĄ,đĽ (23)
which gives the transition probability density function by
đđ,đž|đĄ,đĽ =1
đˇđĄ,đ
đ2đđ,đž|đĄ,đĽđđž2
(24)
The (24) is also known as Breeden-Litzenberger formula.
Taking the first derivative of đđ,đž|đĄ,đĽ in (22) with respect to đ, we find
đđđ,đž|đĄ,đĽđđ
= âđđđđ,đž|đĄ,đĽ + đˇđĄ,đ⍠(đŚ â đž)đđđ,đŚ|đĄ,đĽđđ
đđŚâ
đž
= âđđđđ,đž|đĄ,đĽ + đˇđĄ,đ⍠(đŚ â đž)(1
2
đ2(đđ,đŚ2 đŚ2đđ,đŚ|đĄ,đĽ)
đđŚ2âđ(đđđŚđđ,đŚ|đĄ,đĽ)
đđŚ)đđŚ
â
đž
(25)
using the Kolmogorov Forward Equation (11)
đđđ,đŚ|đĄ,đĽđđ
=1
2
đ2(đđ,đŚ2 đŚ2đđ,đŚ|đĄ,đĽ)
đđŚ2âđ(đđđŚđđ,đŚ|đĄ,đĽ)
đđŚ (26)
Applying integration by parts to the integrals on the right hand side of (25) yields
⍠(đŚ â đž)đ(đđđŚđđ,đŚ|đĄ,đĽ)
đđŚđđŚ
â
đž
= (đŚ â đž)đđđŚđđ,đŚ|đĄ,đĽ|đŚ=đžâ
â =0
â⍠đđđŚđđ,đŚ|đĄ,đĽđđŚâ
đž
= âđđ (⍠(đŚ â đž)đđ,đŚ|đĄ,đĽđđŚâ
đž
+ đžâŤ đđ,đŚ|đĄ,đĽđđŚâ
đž
) =đđđž
đˇđĄ,đ
đđđ,đž|đĄ,đĽđđž
âđđđđ,đž|đĄ,đĽđˇđĄ,đ
and
(27)
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
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⍠(đŚ â đž)đ2(đđ,đŚ
2 đŚ2đđ,đŚ|đĄ,đĽ)
đđŚ2đđŚ
â
đž
= (đŚ â đž)đ(đđ,đŚ
2 đŚ2đđ,đŚ|đĄ,đĽ)
đđŚ|đŚ=đž
â
â =0
ââŤđ(đđ,đŚ
2 đŚ2đđ,đŚ|đĄ,đĽ)
đđŚđđŚ
â
đž
= đđ,đŚ2 đŚ2đđ,đŚ|đĄ,đĽ|đŚ=đž
â= đđ,đž
2 đž2đđ,đž|đĄ,đĽ =đđ,đž2 đž2
đˇđĄ,đ
đ2đđ,đž|đĄ,đĽđđž2
where we have đđ,â|đĄ,đĽ = 0 and đđđ,â|đĄ,đĽ/đđŚ = 0 assuming the probability density đđ,đŚ|đĄ,đĽ and its first
derivative vanish at a higher order of rate as đŚ â â. Plugging (27) into (25), we find
đđđ,đž|đĄ,đĽđđ
= âđđđđ,đž|đĄ,đĽ +đđ,đž2 đž2
2
đ2đđ,đž|đĄ,đĽđđž2
â đđđžđđđ,đž|đĄ,đĽđđž
+ đđđđ,đž|đĄ,đĽ
=đđ,đž2 đž2
2
đ2đđ,đž|đĄ,đĽđđž2
â đđđžđđđ,đž|đĄ,đĽđđž
â đđđđ,đž|đĄ,đĽ
(28)
and eventually the Dupire formula for the local volatility đđ,đž expressed in terms of vanilla call price
đđ,đž2
2=
đđđ,đž|đĄ,đĽđđ
+ đđđžđđđ,đž|đĄ,đĽđđž
+ đđđđ,đž|đĄ,đĽ
đž2đ2đđ,đž|đĄ,đĽđđž2
(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
đśđ,đž|đĄ,đĽ = ďż˝Ěďż˝đĄ[(đđ â đž)+] = ⍠(đŚ â đž)đđ,đŚ|đĄ,đĽđđŚ
â
đž
=đđ,đž|đĄ,đĽđˇđĄ,đ
(30)
with đđ,đž|đĄ,đĽ given in (22) (note that (30) is true only if the interest rate is deterministic). Following a
similar derivation, we find that
đđśđ,đž|đĄ,đĽđđž
= â⍠đđ,đŚ|đĄ,đĽđđŚâ
đž
,đ2đśđ,đž|đĄ,đĽđđž2
= đđ,đž|đĄ,đĽ and
đđśđ,đž|đĄ,đĽđđ
= ⍠(đŚ â đž)đđđ,đŚ|đĄ,đĽđđ
đđŚâ
đž
=1
2đđ,đž2 đž2
đ2đśđ,đž|đĄ,đĽđđž2
+ đđ(đśđ,đž|đĄ,đĽ â đžđđśđ,đž|đĄ,đĽđđž
)
(31)
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
9
Therefore the Dupire formula for đđ,đž expressed in terms of forward call price reads
đđ,đž2
2=
đđśđ,đž|đĄ,đĽđđ
+ đđđžđđśđ,đž|đĄ,đĽđđž
â đđđśđ,đž|đĄ,đĽ
đž2đ2đśđ,đž|đĄ,đĽđđž2
(32)
2.2.1. Local Variance as a Conditional Expectation of Instantaneous Variance
The forward call (30) can be expressed as
đśđ,đž|đĄ,đĽ = ďż˝Ěďż˝đĄ[(đđ â đž)+] = ďż˝Ěďż˝đĄ[đ˝(đđ â đž)(đđ â đž)] = ďż˝Ěďż˝đĄ[đ˝(đđ â đž)đđ] â đžďż˝Ěďż˝đĄ[đ˝(đđ â đž)] (33)
where đ˝ is the Heaviside step function1. Differentiating once with respect to đž, we get
đđśđ,đž|đĄ,đĽđđž
= â⍠đđ,đŚ|đĄ,đĽđđŚâ
đž
= âďż˝Ěďż˝đĄ[đ˝(đđ â đž)] âš ďż˝Ěďż˝đĄ[đ˝(đđ â đž)đđ] = đśđ,đž|đĄ,đĽ â đžđđśđ,đž|đĄ,đĽđđž
(34)
Differentiating again with respect to đž, we have
đ2đśđ,đž|đĄ,đĽđđž2
= đđ,đž|đĄ,đĽ = ďż˝Ěďż˝đĄ[đż(đđ â đž)] (35)
where đż is the Dirac delta function2. Applying Itoâs Lemma to the terminal payoff of the option gives
the identity
đ(đđ â đž)+ = đ˝(đđ â đž)đđđ +
1
2đż(đđ â đž)đđ
2đđ2đđ
= đ˝(đđ â đž)đđđđđđ +1
2đż(đđ â đž)đđ
2đđ2đđ + đ˝(đđ â đž)đđđđđďż˝Ěďż˝đ
(36)
Taking conditional expectations on both sides gives
đđśđ,đž|đĄ,đĽ = đďż˝Ěďż˝đĄ[(đđ â đž)+] = đđďż˝Ěďż˝đĄ[đ˝(đđ â đž)đđ]đđ +
1
2ďż˝Ěďż˝đĄ[đż(đđ â đž)đđ
2đđ2]đđ
âšđđśđ,đž|đĄ,đĽđđ
= đđďż˝Ěďż˝đĄ[đ˝(đđ â đž)đđ] +1
2ďż˝Ěďż˝đĄ[đż(đđ â đž)đđ
2đđ2]
(37)
Notice that
1 Heaviside step function: đ˝(đĽ) = {
0, đĽ < 01, đĽ ⼠0
2 Dirac delta function can be viewed as the derivative of the Heaviside step function: đż(đĽ) =đđ˝(đĽ)
đđĽ= {â, đĽ = 00, đĽ â 0
,
which is also constrained to satisfy the identity: ⍠đż(đĽ)đđĽâ
= 1.
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
10
ďż˝Ěďż˝đĄ[đż(đđ â đž)đđ2đđ2] = đž2ďż˝Ěďż˝đĄ[đż(đđ â đž)đđ
2] (38)
and from Bayesâ rule
ďż˝Ěďż˝đĄ[đż(đđ â đž)đđ2] = ďż˝Ěďż˝đĄ[đđ
2|đđ = đž]ďż˝Ěďż˝đĄ[đż(đđ â đž)] (39)
we can derive from (37)
đđśđ,đž|đĄ,đĽđđ
= đđđśđ,đž|đĄ,đĽ â đđđžđđśđ,đž|đĄ,đĽđđž
+1
2đž2đ2đśđ,đž|đĄ,đĽđđž2
ďż˝Ěďż˝đĄ[đđ2|đđ = đž]
âšďż˝Ěďż˝đĄ[đđ
2|đđ = đž]
2=
đđśđ,đž|đĄ,đĽđđ
+ đđđžđđśđ,đž|đĄ,đĽđđž
â đđđśđ,đž|đĄ,đĽ
đž2đ2đśđ,đž|đĄ,đĽđđž2
(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 đđ being equal to the strike price đž [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
đ = lnđž
đšđĄ,đ where
đđ
đđž=1
đž,
đđ
đđ= âđđ (41)
We want to express the Dupire formula in the (đ, đ)-plane using call option price đśđ,đ (short for đśđ,đ|đĄ,đ§
for đ§ = lnđĽ
đšđĄ,đ) equivalent to the forward call đśđ,đž (short for đśđ,đž|đĄ,đĽ). Note that although the đśđ,đ and
đśđ,đž are equivalent, they are two different functions. The conversion from (đ, đž)-plane to (đ, đ)-plane
is achieved by using the following partial derivatives derived by chain rule
đđśđ,đžđđ
=đđśđ,đđđ
+đđśđ,đđđ
đđ
đđ=đđśđ,đđđ
â đđđđśđ,đđđ
đđśđ,đžđđž
=đđśđ,đđđ
đđ
đđž+đđśđ,đđđ
đđ
đđž=1
đž
đđśđ,đđđ
đ2đśđ,đžđđž2
=đ
đđ(đđśđ,đđđ
)đđ
đđž+đ
đđ(đđśđ,đđđ
)đđ
đđž=1
đž
đ
đđ(1
đž
đđśđ,đđđ
) =1
đž2(đ2đśđ,đđđ2
âđđśđ,đđđ
)
(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 đ
đđ,đ2
2=
đđśđ,đđđ
â đđđśđ,đ
đ2đśđ,đđđ2
âđđśđ,đđđ
(43)
where đđ,đ is the local volatility in (đ, đ)-plane equivalent to đđ,đž.
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 đĄ as of today, we can define the forward price as
đšđĄ,đ = đđĄ exp(⍠đđ˘đđ˘đ
đĄ
) (44)
The forward call price in Black-Scholes model is then given by
đđ,đž,đ = đšđĄ,đÎŚ(đ+) â đžÎŚ(đâ) with đÂą =lnđšđĄ,đđž
đđ,đžâđÂąđđ,đžâđ
2, đ = đ â đĄ (45)
where đđ,đž is the Black-Scholes implied volatility derived from market quotes of vanilla options and ÎŚ
the standard normal cumulative density function. Its partial derivatives can be derived as
đđđ,đž,đ
đđ= đđđšđĄ,đÎŚ(đ+) + đšđĄ,đđ(đ+)
đđ+đđ
â đžđ(đâ)đđâđđ
= đđđšđĄ,đÎŚ(đ+) +đžđ(đâ)đ
2âđ
đđđ,đž,đ
đđž= đšđĄ,đđ(đ+)
đđ+đđž
â đžđ(đâ)đđâđđž
â ÎŚ(đâ) = âÎŚ(đâ),đ2đđ,đž,đ
đđž2=đ(đâ)
đžđâđ
đđđ,đž,đ
đđ= đšđĄ,đđ(đ+)
đđ+đđ
â đžđ(đâ)đđâđđ
= đžđ(đâ)âđ,đ2đđ,đž,đ
đđ2=đ+đâđžđ(đâ)âđ
đ
đ2đđ,đž,đ
đđđđž=đ(đâ)đ+
đ
(46)
where we have used
đđÂąđđ
=đđ
đâđâđâ2đ,
đđÂąđđž
= â1
đžđâđ,
đđÂąđđ
= âđâđ,
đ2đÂąđđžđđ
=1
đžđ2âđ (47)
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
12
and the identity
đšđĄ,đđ(đ+) = đžđ(đâ) (48)
Further using the partial derivatives
đđśđ,đžđđ
=đđđ,đž,đ
đđ+đđđ,đž,đ
đđ
đđ
đđ,
đđśđ,đžđđž
=đđđ,đž,đ
đđž+đđđ,đž,đ
đđ
đđ
đđž
đ2đśđ,đžđđž2
=đ2đđ,đž,đ
đđž2+ 2
đ2đđ,đž,đ
đđžđđ
đđ
đđž+đđđ,đž,đ
đđ
đ2đ
đđž2+đ2đđ,đž,đ
đđ2(đđ
đđž)2
(49)
the local volatility given by implied volatility can be derived from (32) as
đđ,đž2 =
đđśđ,đžđđ
+ đđđžđđśđ,đžđđž
â đđđśđ,đž
đž2
2đ2đśđ,đžđđž2
=
đđđđ+đđđđ(đđđ,đžđđ
+ đđđžđđđ,đžđđž
) + đđđžđđđđž
â đđđ
đž2
2(đ2đđđž2
+ 2đ2đđđžđđ
đđđ,đžđđž
+đđđđđ2đđ,đžđđž2
+đ2đđđ2
(đđđđž)2
)
=
đđđšđĄ,đÎŚ(đ+) +đžđ(đâ)đ
2âđ+đđđđ(đđđđ+ đđđž
đđđđž) â đđđžÎŚ(đâ) â đđđ
đž2
2đ(đâ)
đžđâđ+ đž2
đ(đâ)đ+đ
đđđđž
+đž2
2đđđđđ2đđđž2
+đž2
2đ+đâđžđ(đâ)âđ
đ(đđđđž)2
=
đđđđ(đ2đ+đđđđ+ đđđž
đđđđž)
12đđ
đđđđ+đžđ+đâđ
đđđđđđđđž
+đž2
2đđđđđ2đđđž2
+đž2đ+đâ2đ
đđđđ(đđđđž)2
=đ2 + 2đđ (
đđđđ+ đđđž
đđđđž)
1 + 2âđđžđ+đđđđž
+ đ+đâđđž2 (đđđđž)2
+ đđđž2đ2đđđž2
(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 đĽ = lnđž can be derived from (50)
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
13
đđ,đĽ2 =
đ2 + 2đđ (đđđđ+ đđ
đđđđĽ)
1 + 2âđđ+đđđđĽ+ đ+đâđ (
đđđđĽ)2
+ đđ (đ2đđđĽ2
âđđđđĽ)
=đ2 + 2đđ (
đđđđ+ đđ
đđđđĽ)
1 + (đđ â 2đđ)đđđđĽ+ (đ2
đ2âđ2đ2
4 ) (đđđđĽ)2
+ đđ (đ2đđđĽ2
âđđđđĽ)
=đ2 + 2đđ (
đđđđ+ đđ
đđđđĽ)
(1 âđđđđđđĽ)2
â (đđ2đđđđĽ)2
+ đđđ2đđđĽ2
(51)
providing that we have the following identities
đđ
đđž=đđ
đđĽ
đđĽ
đđž=1
đž
đđ
đđĽ,
đ2đ
đđž2=đ
đđĽ(1
đž
đđ
đđĽ)đđĽ
đđž=1
đž2(đ2đ
đđĽ2âđđ
đđĽ) ,
đđĽ
đđž=1
đž
đÂą =âđ
đâđÂąđâđ
2, đ = ln
đž
đšđĄ,đ
(52)
2.3.2. Formula in Log-moneyness
We may also want to change the spatial variable to log-moneyness đ. Defining a new quantity,
implied total variance đŁđ,đ, which is equivalent to đđ,đž2 đ, the Black-Scholes call price that is equivalent
to (45) then transforms into
đđ,đ,đŁ = đšđĄ,đ(ÎŚ(đ+) â exp(đ)ÎŚ(đâ)) with đÂą =âđ
âđŁđ,đÂąâđŁđ,đ
2 (53)
The partial derivatives of đđ,đ,đŁ can be derived as
đđđ,đ,đŁđđŁ
= đšđĄ,đ (đÎŚ(đ+)
đđ+
đđ+đđŁ
â exp(đ)đÎŚ(đâ)
đđâ
đđâđđŁ) = đšđĄ,đđ(đ+) (
đđ+đđŁ
âđđâđđŁ) =
đšđĄ,đđ(đ+)
2âđŁ
đ2đđ,đ,đŁđđŁ2
=đđđ,đ,đŁđđŁ
(â1
2đŁâ đ+
đđ+đďż˝Ěďż˝) =
đđđ,đ,đŁđđŁ
(â1
2đŁâ (â
đ
âđŁ+âđŁ
2) (
đ
2âđŁ3+
1
4âđŁ))
=đđđ,đ,đŁđđŁ
(đ2
2đŁ2â1
2đŁâ1
8)
(54)
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
14
đđđ,đ,đŁđđ
= đšđĄ,đ (đ(đ+)đđ+đđ
â exp(đ)ÎŚ(đâ) â exp(đ)đ(đâ)đđâđđ) = âđšđĄ,đ exp(đ)ÎŚ(đâ)
đ2đđ,đ,đŁđđ2
= âđšđĄ,đ exp(đ)ÎŚ(đâ) + đšđĄ,đexp(đ)đ(đâ)
âđŁ=đđđ,đ,đŁđđ
+ 2đđđ,đ,đŁđđŁ
đ2đđ,đ,đŁđđđđŁ
=đ
đđ(đšđĄ,đđ(đ+)
2âđŁ) =
đđđ,đ,đŁđđŁ
đ+đđ+đđ
=đđđ,đ,đŁđđŁ
(1
2âđ
đŁ)
đđđ,đ,đŁđđ
= (ÎŚ(đ+) â exp(đ)ÎŚ(đâ))đđšđĄ,đđđ
= đđđđ,đ,đŁ
We may connect the local volatility đđ,đ to the implied total variance đŁđ,đ via two steps. Firstly
we bridge the đđ,đ to đđ,đ,đŁ by (43) using the chain rule
đđśđ,đđđ
=đđđ,đ,đŁđđ
+đđđ,đ,đŁđđŁ
đđŁ
đđ,
đđśđ,đđđ
=đđđ,đ,đŁđđ
+đđđ,đ,đŁđđŁ
đđŁ
đđ
đ2đśđ,đđđ2
=đ
đđ(đđđ,đ,đŁđđ
+đđđ,đ,đŁđđŁ
đđŁ
đđ) +
đ
đđŁ(đđđ,đ,đŁđđ
+đđđ,đ,đŁđđŁ
đđŁ
đđ)đđŁ
đđ
=đ2đđ,đ,đŁđđ2
+đ2đđ,đ,đŁđđđđŁ
đđŁ
đđ+đđđ,đ,đŁđđŁ
đ2đŁ
đđ2+đ2đđ,đ,đŁđđŁđđ
đđŁ
đđ+đ2đđ,đ,đŁđđŁ2
(đđŁ
đđ)2
=đ2đđ,đ,đŁđđ2
+ 2đ2đđ,đ,đŁđđđđŁ
đđŁ
đđ+đđđ,đ,đŁđđŁ
đ2đŁ
đđ2+đ2đđ,đ,đŁđđŁ2
(đđŁ
đđ)2
(55)
This gives the local volatility expressed in terms of derivatives of đđ,đ,đŁ
đđ,đ2 =
2(đđđ,đ,đŁđđ
+đđđ,đ,đŁđđŁ
đđŁđđâ đđđđ,đ,đŁ)
đ2đđ,đ,đŁđđ2
+ 2đ2đđ,đ,đŁđđđđŁ
đđŁđđ+đđđ,đ,đŁđđŁ
đ2đŁđđ2
+đ2đđ,đ,đŁđđŁ2
(đđŁđđ)2
âđđđ,đ,đŁđđ
âđđđ,đ,đŁđđŁ
đđŁđđ
(56)
Secondly we substitute the partial derivatives in (54) into (56) and reach the final equation
đđ,đ2 =
2(đđđ +đđđđŁđđŁđđâ đđđ)
đđđđ+ 2
đđđđŁ+ 2
đđđđŁ(12âđđŁ)đđŁđđ+đđđđŁđ2đŁđđ2
+đđđđŁ(đ2
2đŁ2â12đŁâ18)(đđŁđđ)2
âđđđđâđđđđŁđđŁđđ
âš đđ,đ2 =
đđŁđđ
1 âđđŁđđŁđđ+14 (đ2
đŁ2â1đŁâ14)(đđŁđđ)2
+12đ2đŁđđ2
(57)
2.3.3. Equivalency in Formulas
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
15
The đđ,đž2 in (50) is in fact equivalent to the đđ,đ
2 in (57). This can be shown as follows
đđ,đ2 =
đđŁđđ
1 âđđŁđđŁđđ+ (
đ2
4đŁ2â14đŁâ116) (đđŁđđ)2
+12đ2đŁđđ2
=đ2 + 2đđ (
đđđđ+ đđž
đđđđž)
1 â 2đđđžđđŁđđđđž
+ (đ2
đŁ2â1đŁâ14)(đđđž
đđđđž)2
+ đđž2 ((đđđđž)2
+đđžđđđđž
+ đđ2đđđž2
)
=đ2 + 2đđ (
đđđđ+ đđž
đđđđž)
1 + (1 â 2đđŁ) đđđž
đđđđž
+ (đ2
đŁâđŁ4)đđž2 (
đđđđž)2
+ đđđž2đ2đđđž2
=đ2 + 2đđ (
đđđđ+ đđž
đđđđž)
1 + 2âđđžđ+đđđđž
+ đ+đâđđž2 (đđđđž)2
+ đđđž2đ2đđđž2
= đđ,đž2
(58)
where by definition we have
đ = lnđž
đšđĄ,đ, đŁ = đ2đ, đÂą =
lnđšđĄ,đđžÂąđ2đ2
đâđ=âđ
âđŁÂąâđŁ
2, đ+đâ =
đ2
đŁâđŁ
4 (59)
and also have the identities
đđŁ
đđ=đ(đ2đ)
đđ= đ2 + 2đđ (
đđ
đđ+đđ
đđž
đđž
đđ) = đ2 + 2đđ (
đđ
đđ+ đđž
đđ
đđž)
đđŁ
đđ=đ(đ2đ)
đđ=đ(đ2đ)
đđ
đđ
đđ+đ(đ2đ)
đđž
đđž
đđ= 2đđ
đđ
đđž
đđž
đđ= 2đđđž
đđ
đđž
đ2đŁ
đđ2=đ (2đđđž
đđđđž)
đđ
đđ
đđ+đ (2đđđž
đđđđž)
đđž
đđž
đđ= 2đđž (đ
đđ
đđž+ đž
đđ
đđž
đđ
đđž+ đđž
đ2đ
đđž2)
= 2đđž2 ((đđ
đđž)2
+đ
đž
đđ
đđž+ đ
đ2đ
đđž2)
(60)
considering the fact that in (đ, đ)-plane the đ and đž are no longer independent
đđž
đđ=đ(đšđĄ,đ exp(đ))
đđ= đđđž,
đđž
đđ=đ(đšđĄ,đ exp(đ))
đđ= đž (61)
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
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 PDEâs 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 PDEâs, 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.
Îđ,đ đĄđ,đ đĄđ,đ đĄđ,đ Îđ,đ
đĄ
Îđ,đ đĄđ,đ Îđ,đ
đđ = đĄđ,đ â đĄ0 đĄđ,đ đđ,đ
đđ = đĄđ,đ â đĄ0 đĄđ,đ đđ,đ đĄ0
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
17
Table 1. Dates of Equities and Options
attribute symbol description remark/example
trade date đĄ0 on which the equity/option is traded today
equity spot lag Îđ,đ equity premium settlement lag 3D
equity spot date đĄđ,đ on which the equity premium is settled đĄđ,đ = đĄ0âÎđ,đ
equity maturity date đĄđ,đ equity maturity date đĄđ,đ = đĄ0â1đ
equity pay lag1 Îđ,đ lag between đĄđ,đ and đĄđ,đ e.g. same as Îđ,đ
equity pay date đĄđ,đ on which the equity payoff is settled đĄđ,đ = đĄđ,đâÎđ,đ
đ-th dividend đđ dividend payment amount
đ-th ex- div. date đĄđ,đ ex-dividend date
đ-th div. pay date đĄđ,đ dividend pay date
option spot lag Îđ,đ option premium settlement lag 2D
option spot date đĄđ,đ on which the option is settled đĄđ,đ = đĄ0âÎđ,đ
option maturity date đĄđ,đ option maturity date đĄđ,đ = đĄ0â1đ
option pay lag Îđ,đ lag between đĄđ,đ and đĄđ,đ e.g. same as Îđ,đ
option pay date đĄđ,đ on which the equity payoff is settled đĄđ,đ = đĄđ,đâÎđ,đ
day rolling â rolling with convention âfollowingâ 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 đđ˘ and đđ˘ as in (21). the equity forward in this case can be calculated by
đš(đĄ0, đĄđ,đ) = đ(đĄ0)đđ(đĄđ,đ , đĄđ,đ)
đđ(đĄđ,đ , đĄđ,đ) where
đđ(đĄ, đ) = exp(â⍠đđ˘đđ˘đ
đĄ
) , đđ(đĄ, đ) = exp(â⍠đđ˘đđ˘đ
đĄ
)
(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 đđ˘ = 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
đš(đĄ0, đĄđ,đ) =đ(đĄ0) â â đđđđ(đĄđ,đ , đĄđ,đ)đ
đđ(đĄđ,đ , đĄđ,đ) for đĄ0 < đĄđ,đ ⤠đĄđ,đ (63)
where đđ is the fixed amount of the đ-th dividend issued on ex-dividend date đĄđ,đ.
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
đ(đĄđ,đ + Î) = đ(đĄđ,đ â Î)(1 â đđ) (64)
where Î denotes an infinitesimal amount of time and đđ the proportional dividend rate at ex-dividend
date đĄđ,đ. By this relationship, we can write the equity forward as
đš(đĄ0, đĄđ,đ) = đ(đĄ0)â (1 â đđ)đ
đđ(đĄđ,đ , đĄđ,đ) for đĄ0 < đĄđ,đ ⤠đĄđ,đ (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
â(1â đđ)
đ
= 1 â1
đ(đĄ0)âđđđđ(đĄđ,đ , đĄđ,đ)
đ
for đĄ0 < đĄđ,đ ⤠đĄđ,đ (66)
The proportional dividend đđ can then be bootstrapped from a series of fixed dividends đđ starting from
the first ex-dividend date.
3.3. Forward PDE
In the following, our derivation is based on the spot process đđĄ defined in (21) and its variants.
For example, depending on the application we may write the SDE (21) in terms of log-spot đđ˘ = ln đđ˘
or in terms of centered log-spot đ§đ˘ = lnđđ˘
đšđĄ,đ˘
đđđ˘ = (đđ˘ â1
2đ(đ˘, đ)2) đđ˘ + đ(đ˘, đ)đďż˝Ěďż˝đ˘ and đđ§đ˘ = â
1
2đ(đ˘, đ§)2đđ˘ + đ(đ˘, đ§)đďż˝Ěďż˝đ˘ (67)
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
19
where đ(đ˘, đ) and đ(đ˘, đ§) are the local volatility function of đ and đ§, respectively.
Letâs denote the forward temporal variable by đ˘ for đĄ < đ˘ < đ , the spatial variable by log-
moneyness đ = lnđž
đšđĄ,đ˘ (as in (41)) and the spot by đ§ = ln
đđ˘
đšđĄ,đ˘. Given that đ§đĄ = 0 , the value of a
normalized forward call can be defined as
đđ˘,đ|đĄ,đ§ =đśđ˘,đ|đĄ,đ§đšđĄ,đ˘
=ďż˝Ěďż˝[(đđ˘ â đž)
+|đĄ, đđĄ]
đšđĄ,đ˘ (68)
Let đđ˘,đ be the local volatility function of variable đ equivalent to đđ,đž. We can derive the forward PDE
for đđ˘,đ|đĄ,đ§ from (43)
đđ˘,đ2
2=đšđĄ,đ˘
đđđ˘,đ|đĄ,đ§đđ˘
+ đđ˘đšđĄ,đ˘đđ˘,đ|đĄ,đ§ â đđ˘đśđ˘,đ|đĄ,đ§
đšđĄ,đ˘đ2đđ˘,đ|đĄ,đ§đđ2
â đšđĄ,đ˘đđđ˘,đ|đĄ,đ§đđ
=
đđđ˘,đ|đĄ,đ§đđ˘
đ2đđ˘,đ|đĄ,đ§đđ2
âđđđ˘,đ|đĄ,đ§đđ
âšđđđ˘,đ|đĄ,đ§đđ˘
=đđ˘,đ2
2(đ2đđ˘,đ|đĄ,đ§đđ2
âđđđ˘,đ|đĄ,đ§đđ
)
(69)
with initial condition
đđĄ,đ|đĄ,đ§ =đśđĄ,đ|đĄ,đ§đšđĄ,đĄ
=ďż˝Ěďż˝ [(đđĄ â đšđĄ,đĄđ
đ)+| đĄ, đđĄ]
đšđĄ,đĄ= (1 â đđ)+ (70)
using the partial derivatives
đđđ˘,đ|đĄ,đ§đđ˘
=1
đšđĄ,đ˘
đđśđ˘,đ|đĄ,đ§đđ˘
â đđ˘đđ˘,đ|đĄ,đ§ , đđđ˘,đ|đĄ,đ§đđ
=1
đšđĄ,đ˘
đđśđ˘,đ|đĄ,đ§đđ
, đ2đđ˘,đ|đĄ,đ§đđ2
=1
đšđĄ,đ˘
đ2đśđ˘,đ|đĄ,đ§đđ2
(71)
The PDE (69) appears drift-less and provides more robust calibration stability at low volatility and/or
high drift due to the âtransparencyâ 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 đ˘ is the ex-dividend date, the no-arbitrage condition states that at đ˘+ the time
right after the ex-dividend date (e.g. the difference between đ˘ and đ˘+ can be infinitesimal), we must
have
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
20
đđ˘+ = đđ˘ â đđ˘ (72)
where đđ˘ is the value of dividend issued at đ˘ (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
đšđĄ,đ˘+ = ďż˝Ěďż˝đĄ[đđ˘+] = ďż˝Ěďż˝đĄ[đđ˘ â đđ˘] = đšđĄ,đ˘ â ďż˝Ěďż˝đĄ[đđ˘] (73)
Under the assumption that đđ˘ is a fixed amount, it reads
đšđĄ,đ˘+ = đšđĄ,đ˘ â đđ˘ (74)
In our finite difference method, the spatial grid for log-moneyness đ is assumed uniform such that đđ â
đđâ1 is constant for all đ. Dividend payment causes discontinuity in the underlying spot. Evolving the
forward PDE (69) from initial time đĄ produces a state vector đđ˘,đ|đĄ,đ§ at time đ˘. Immediately after the
issuance of dividend at time đ˘+, the spot and forward drop the same đđ˘ amount and hence the state
vector đđ˘+,đ|đĄ,đ§ must be realigned to reflect the dividend fall. This can be done using the option no-
arbitrage condition, such that
đśđ˘+,đ|đĄ,đ§ = ďż˝Ěďż˝đĄ [(đđ˘+ â đž)+] = ďż˝Ěďż˝đĄ [(đđ˘ â đđ˘ â đšđĄ,đ˘+đ
đ)+] = ďż˝Ěďż˝đĄ [(đđ˘ â đšđĄ,đ˘đ
ďż˝Ěďż˝)+] = đśđ˘,ďż˝Ěďż˝|đĄ,đ§
where ďż˝Ěďż˝ = lnđšđĄ,đ˘+đ
đ + đđ˘
đšđĄ,đ˘
(75)
Subsequently we can use ďż˝Ěďż˝ to interpolate from the đđ˘,đ|đĄ,đ§ state vector and transform the interpolated
value to form đđ˘+,đ|đĄ,đ§ vector by
đđ˘+,đ|đĄ,đ§ =đśđ˘+,đ|đĄ,đ§
đšđĄ,đ˘+=đśđ˘,ďż˝Ěďż˝|đĄ,đ§
đšđĄ,đ˘
đšđĄ,đ˘đšđĄ,đ˘+
=đšđĄ,đ˘đšđĄ,đ˘+
đđ˘,ďż˝Ěďż˝|đĄ,đ§ (76)
If the dividend is proportional, we must have spot price đđ˘+ = đđ˘(1 â đđ˘) for a rate đđ˘ and
hence forward price đšđĄ,đ˘+ = đšđĄ,đ˘(1 â đđ˘) before and after the dividend fall. Because we can show that
1 Strictly speaking, a forward on time đ spot is an expectation of the spot under đ-forward measure, i.e. đšđĄ,đ =
ďż˝Ěďż˝đĄđ[đđ]. However since the interest rate is assumed deterministic, the đ-forward measure coincides with the risk
neutral measure.
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
21
đđ˘+,đ|đĄ,đ§ =ďż˝Ěďż˝đĄ [(đđ˘+ â đž)
+]
đšđĄ,đ˘+=(1 â đđ˘)ďż˝Ěďż˝đĄ [(đđ˘ â đšđĄ,đ˘đ
đ)+]
đšđĄ,đ˘(1 â đđ˘)=ďż˝Ěďż˝đĄ [(đđ˘ â đšđĄ,đ˘đ
đ)+]
đšđĄ,đ˘= đđ˘,đ|đĄ,đ§ (77)
the state vector remains unchanged before and after the issuance of dividend.
With continuous dividend đđ˘, 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, letâs denote
đş(đđ|đž) an arbitrary payoff function with parameter đž, whose value is contingent on đđ at time đ. One
example of such function would be the payoff function of a call option: đş(đđ|đž) = (đđ â đž)+. Let
đđ˘,đĽ|đ,đž be the expectation of the function đş(đđ|đž) at time đ˘ with spatial variable đĽ = đđ˘, which can be
written as
đđ˘,đĽ|đ,đž = ďż˝Ěďż˝[đş(đđ|đž)|đ˘, đĽ] = âŤđş(đŚ|đž)đđ,đŚ|đ˘,đĽđđŚâ
(78)
where the transition probability đđ,đŚ|đ˘,đĽ follows the Kolmogorov backward equation (20)
đđđ,đŚ|đ˘,đĽđđ˘
= đđ˘đĽđđđ,đŚ|đ˘,đĽđđĽ
+đđ˘,đĽ2 đĽ2
2
đ2đđ,đŚ|đ˘,đĽđđĽ2
(79)
In turn, we can derive the backward PDE for the đđ˘,đĽ|đ,đž such that
đđđ˘,đĽ|đ,đžđđ˘
= ⍠đş(đŚ|đž)đđđ,đŚ|đ˘,đĽđđ˘
đđŚâ
= â⍠đş(đŚ|đž) (đđ˘đĽđđđ,đŚ|đ˘,đĽđđĽ
+đđ˘,đĽ2 đĽ2
2
đ2đđ,đŚ|đ˘,đĽđđĽ2
)đđŚâ
= âđđ˘đĽđđđ˘,đĽ|đ,đžđđĽ
âđđ˘,đĽ2 đĽ2
2
đ2đđ˘,đĽ|đ,đžđđĽ2
(80)
with terminal condition
đđ,đĽ|đ,đž = đş(đĽ|đž) (81)
3.4.1. PDE in Centered Log-spot
Assuming the spatial variable is đ§đ˘ = lnđĽ
đšđĄ,đ˘ at time đ˘, we may write đđ˘,đ§|đ,đ in the (đ˘, đ§)-plane
equivalent to đđ˘,đĽ|đ,đž. The backward PDE (80) can then be transformed into
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
22
đđđ˘,đ§|đ,đđđ˘
= âđđ˘,đ§2
2(đ2đđ˘,đ§|đ,đđđ§2
âđđđ˘,đ§|đ,đđđ§
) (82)
with terminal condition
đđ,đ§|đ,đ = đş(đšđĄ,đđđ§|đšđĄ,đđ
đ) (83)
by using the following partial derivatives derived from the chain rule
đđ§
đđĽ=1
đĽ,
đđ§
đđ˘= âđđ˘,
đđđ˘,đĽ|đ,đžđđ˘
=đđđ˘,đ§|đ,đđđ˘
+đđđ˘,đ§|đ,đđđ§
đđ§
đđ˘=đđđ˘,đ§|đ,đđđ˘
â đđ˘đđđ˘,đ§|đ,đđđ§
đđđ˘,đĽ|đ,đžđđĽ
=đđđ˘,đ§|đ,đđđ§
đđ§
đđĽ=1
đĽ
đđđ˘,đ§|đ,đđđ§
,đ2đđ˘,đĽ|đ,đžđđĽ2
=1
đĽ2(đ2đđ˘,đ§|đ,đđđ§2
âđđđ˘,đ§|đ,đđđ§
)
(84)
3.4.1.1. Treatment of Deterministic Dividends
With fixed dividend đđ˘, we have
đđ˘+ = đđ˘ â đđ˘ and đšđĄ,đ˘+ = đšđĄ,đ˘ â đđ˘ (85)
The no arbitrage condition shows that for the spatial grid đ§
đđ˘,đ§|đ,đ = ďż˝Ěďż˝[đş(đđ|đšđĄ,đđđ)|đ˘, đšđĄ,đ˘đ
đ§] = ďż˝Ěďż˝[đş(đđ|đšđĄ,đđđ)|đ˘+, đšđĄ,đ˘đ
đ§ â đđ˘]
= ďż˝Ěďż˝[đş(đđ|đšđĄ,đđđ)|đ˘+, đšđĄ,đ˘+đ
ďż˝Ěďż˝] = đđ˘+,ďż˝Ěďż˝|đ,đ where ďż˝Ěďż˝ = lnđšđĄ,đ˘đ
đ§ â đđ˘đšđĄ,đ˘+
(86)
It is likely that if đ§ is sufficiently small (e.g. at lower boundary of spatial grid) we may end up with
đšđĄ,đ˘đđ§ â đđ˘ < 0, which makes the ďż˝Ěďż˝ not well defined. A solution is to floor it to a small positive number,
e.g. taking max(10â10, đšđĄ,đ˘đđ§ â đđ˘) . This is valid because equity spot must be positive and the
đđ˘+,ďż˝Ěďż˝|đ,đ flattens as ďż˝Ěďż˝ goes to negative infinity. After the special treatment, we can use the ďż˝Ěďż˝ to
interpolate from the đđ˘,đ§|đ,đ state vector and convert the interpolated value into vector đđ˘+,đ§|đ,đ.
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: đđĄ,đ§|đ,đ = đđ,đ|đĄ,đ§đšđĄ,đ , where đđĄ,đ§|đ,đ is the forward call solved from
backward PDE (82) and đđ,đ|đĄ,đ§ 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 đ =
ln đĽ as the spatial variable. Similarly we can define đ = lnđž. Letâs denote the (discounted) price of a
derivative product by
đđ˘,đ|đ,đ = đˇđ˘,đđđ˘,đ|đ,đ = ďż˝Ěďż˝[đˇđ˘,đđş(đđ|đđ)|đ˘, đđ] (87)
By taking into account the discount factor, it must follow the following backward PDE
đđđ˘,đ|đ,đđđ˘
= đđ˘đđ˘,đ|đ,đ + đˇđ˘,đđđđ˘,đ|đ,đđđ˘
= đđ˘đđ˘,đ|đ,đ + đˇđ˘,đ (âđđ˘đĽ1
đĽ
đđđ˘,đ|đ,đđđ
âđđ˘,đ2
2
1
đĽ2(đ2đđ˘,đ|đ,đđđ2
âđđđ˘,đ|đ,đđđ
))
= âđđ˘,đ2
2
đ2đđ˘,đ|đ,đđđ2
+ (đđ˘,đ2
2â đđ˘)
đđđ˘,đ|đ,đđđ
+ đđ˘đđ˘,đ|đ,đ
(88)
where the partial derivatives below have been used
đđ
đđĽ=1
đĽ,
đđ
đđ˘= 0,
đđđ˘,đĽ|đ,đžđđ˘
=đđđ˘,đ|đ,đđđ˘
+đđđ˘,đ|đ,đđđ
đđ
đđ˘=đđđ˘,đ|đ,đđđ˘
đđđ˘,đĽ|đ,đžđđĽ
=đđđ˘,đ|đ,đđđ˘
đđ˘
đđĽ+đđđ˘,đ|đ,đđđ
đđ
đđĽ=1
đĽ
đđđ˘,đ|đ,đđđ
đ2đđ˘,đĽ|đ,đžđđĽ2
=đ
đđĽ(1
đĽ
đđđ˘,đ|đ,đđđ
) = â1
đĽ2đđđ˘,đ|đ,đđđ
+1
đĽ
đ2đđ˘,đ|đ,đđđđđ˘
đđ˘
đđĽ+1
đĽ
đ2đđ˘,đ|đ,đđđ2
đđ
đđĽ
=1
đĽ2(đ2đđ˘,đ|đ,đđđ2
âđđđ˘,đ|đ,đđđ
)
(89)
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
24
3.4.2.1. Treatment of Deterministic Dividends
With fixed dividend đđ˘, the no arbitrage condition states that
đđ˘,đ|đ,đ = ďż˝Ěďż˝[đˇđ˘,đđş(đđ|đđ)|đ˘, đđ] = ďż˝Ěďż˝[đˇđ˘+,đđş(đđ|đ
đ)|đ˘+, đđ â đđ˘]
= ďż˝Ěďż˝[đˇđ˘+,đđş(đđ|đđ)|đ˘+, đ
ďż˝Ěďż˝] = đđ˘+,ďż˝Ěďż˝|đ,đ where ďż˝Ěďż˝ = ln(đđ â đđ˘)
(90)
Again, extremely small đ may result in ďż˝Ěďż˝ that is not well defined, we may floor the difference đđ â đđ˘
to a small positive number, e.g. taking max(10â10, đđ â đđ˘) . The vector đđ˘,đ|đ,đ can then be
interpolated from the known đđ˘+,đ|đ,đ using the ďż˝Ěďż˝.
With proportional dividend đđ˘, again the no arbitrage condition shows
đđ˘,đ|đ,đ = ďż˝Ěďż˝[đˇđ˘,đđş(đđ|đđ)|đ˘, đđ] = ďż˝Ěďż˝[đˇđ˘+,đđş(đđ|đ
đ)|đ˘+, đđ(1 â đđ˘)]
= ďż˝Ěďż˝[đˇđ˘+,đđş(đđ|đđ)|đ˘+, đ
ďż˝Ěďż˝] = đđ˘+,ďż˝Ěďż˝|đ,đ where ďż˝Ěďż˝ = đ + ln(1 â đđ˘)
(91)
The vector đđ˘,đ|đ,đ can be interpolated from the đđ˘+,đ|đ,đ using the ďż˝Ěďż˝.
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 đ(đ˘, đ). 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 đ˘ and piecewise linear in log-moneyness đ = lnđž
đšđĄ,đ˘ (or in
delta for FX). The volatility surface comprises a series of volatility smiles đđ(đ) for maturity đĄ < đ˘1 <
⯠< đ˘đ < ⯠< đ˘đ = đ . At each maturity đ˘đ , volatility smile đđ(đ) is constructed by linear
interpolation between log-moneyness pillars đđ = lnđžđ
đšđĄ,đ˘ for strikes đž1 < ⯠< đžđ < ⯠< đžđ and flat
extrapolation where the volatility values at đ1 and đđ are used for all đ < đ1 and đ > đđ, respectively.
The smile đđ(đ) constructed at đ˘đ is assumed to remain constant over time for any đ˘ between the two
adjacent maturities đ˘đâ1 < đ˘ ⤠đ˘đ.
Changwei Xiong, October 2017 http://www.cs.utah.edu/~cxiong/
25
Calibration of the local volatility surface is conducted in a bootstrapping manner starting from
the shortest maturity đ˘1. 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 đđ for each of the
maturities đ˘đ. The PDE can be solved using finite difference method1 on a uniform grid defined on log-
moneyness đ that extends to Âą5 standard deviations of the underlying spot. The choice of boundary
condition has little impact to the solutions of vanilla option prices because at Âą5 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 đ˘đ ): the least square
minimization of vanilla call prices
argminđđ(đđ)
â(đđĄ,đ§|đ˘đ,đđđľđ â đđĄ,đ§|đ˘đ,đđ
đđˇđ¸ )2
đ
đ=1
(92)
where đđĄ,đ§|đ,đ is the normalized forward call price defined in (68), the superscript âBSâ denotes the
theoretical price by Black-Scholes model and the âPDEâ 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 đ =
ln đđ˘ , 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 âIntroduction to Interest Rate Modelsâ,
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 đ 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 Practitionerâs 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 Practitionerâs Guide, Wiley-Finance, 2006, pp. 13-14