2. Copyright 2015 Dr. LAM Yat-fai 5
Black-Scholes model, 1973
Only be exercised at maturity
Constant volatility
Constant risk-free rate
Market is efficient
Currency rate follows a geometric Brownian
motion
No transaction costs and taxes
( ) ( ) ( ) tdS t S t rdt S t dWσ= +
Copyright 2015 Dr. LAM Yat-fai 6
Black-Scholes formulas
S0: Market rate of underlying currency
σ: Constant volatility for maturity T
rd: Constant domestic risk-free rate for maturity T
rf: Constant foreign risk-free rate for maturity T
K: Strike rate
T: Maturity
c(t) : European call price at time t
P(t) : European put price at time t
S0, σ, rd, rf, K,Τ → c(t) , p(t)
Copyright 2015 Dr. LAM Yat-fai 7
Black-Scholes formulas
( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( )
( )
( )
( ) ( )
1
2
2
1
2
1
2 1
2
exp
exp
exp
exp
ln
2
1
( ) exp
22
f
d
d
f
d f
x
c t S t r T t d t
K r T t d t
p t K r T t d t
S t r T t d t
S t
r r T t
K
where d t
T t
d t d t T t
s
x dsφ
π
σ
σ
σ
−∞
= − − Φ
− − − Φ
= − − Φ −
− − − Φ −
+ − + −
=
−
= − −
Φ = −
∫
The most well know but
also the most inaccurate
option valuation model.
Hence “BS formulas”.
Copyright 2015 Dr. LAM Yat-fai 8
Volatility smile
Implied volatility is not a constant
Black-Scholes model is subject to critical
model error
3. Copyright 2015 Dr. LAM Yat-fai 9
Advanced option valuation models
Heston (1993), Stochastic volatility model
Diffusion type stochastic volatility model
Closed form solution with integration of complex function
Madan, Carr and Chang’s (1998), Variance Gamma
model
Pure jump model
Tedious closed form solution
Duan (1995) GARCH model
Monte Carlo simulation
Calibration of several
market implied parameters
required before any
valuation
Copyright 2015 Dr. LAM Yat-fai 10
An ideal currency option
valuation model
In algebra closed form solution
Fast computation
Market data readily available
Minimum overhead
Very few input parameters
Effective implementation
Copyright 2015 Dr. LAM Yat-fai 11
Outline
Currency option valuation
Vanna-Volga method
Volatility recovery
Copyright 2015 Dr. LAM Yat-fai 12
Vanna-Volga method
Castagna and Mercurio (2007)
Black-Scholes constant volatility is a latent variable
For each standard maturity, there are 3 liquidly traded
characteristic options with transparent market prices
Dynamic hedging with characteristic options
Neutralize first and second order sensitivities to form an
instant risk-free portfolio
Currency option with an arbitrary strike rate can then be
valuated
4. Copyright 2015 Dr. LAM Yat-fai 13
Delta, Vega, Vanna and Volga
( )
( )
( )
( )
2
2
2
BS
BS
BS
V t
Delta
S
V t
Vega
V t
Vanna
S
V t
Volga
σ
σ
σ
∂
=
∂
∂
=
∂
∂
=
∂ ∂
∂
=
∂
Copyright 2015 Dr. LAM Yat-fai 14
Vanna-Volga formula
( )
( )
( )
( )
( )
( )
( )
( )
( )
1 2
3
3 32
1
1 2
3 32 2
1 1 1 2
1 2
3
3 3
1 2
ln ln ln ln
ln ln ln ln
ln ln
ln ln
BS BS
BS BS
BS BS
BS BS
BS
BS
BS
BS
V t V tK KK K
K K K K
w t w t
V t V tK KK K
K K K K
V t K K
K K
w t
V t K K
K K
σ σ
σ σ
σ
σ
∂ ∂
⋅ ⋅ ⋅ ⋅
∂ ∂
= =
∂ ∂
⋅ ⋅ ⋅ ⋅
∂ ∂
∂
⋅ ⋅
∂
=
∂
⋅ ⋅
∂
( ) ( ) ( ) ( ) ( )
3
1
VV BS Market BS
k k k
k
V t V t w t V t V t
=
= + − ∑
Copyright 2015 Dr. LAM Yat-fai 15
Practical implementation
Hedging error
σBS = σATM
Characteristics options
V1: Left 25-Delta Call Option
V2: ATM Call Option
V3: Right 25-Delta Call Option
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
3
0
1
25 25 25
25 25 25
exp
T
Market BS
d k k k
k
VV BS Market BS
Left Left Left
Market BS
Right Right Right
T r T t V t V t dw t
V t V t w t V t V t
w t V t V t
ε
=
= − −
= + −
+ −
∑∫
Copyright 2015 Dr. LAM Yat-fai 16
Outline
Currency option valuation
Vanna-Volga method
Volatility recovery
5. Copyright 2015 Dr. LAM Yat-fai 17
Volatility smile recovery
Calculate the selected currency option value
with the Vanna-Volga method
Use the Black-Scholes formula to back out
the implied volatility using numerical some
methods
Compare with the implied volatility with
volatility published by Bloomberg
Copyright 2015 Dr. LAM Yat-fai 18
Root search algorithm
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( ) ( )
3
1
1
2
2
1
2 1
exp
exp
ln
2
VV BS Market BS
k k k
k
VV
f
d
d f
V t V t w t V t V t
V t S t r T t d t
K r T t d t
S t
r r T t
K
where d t
T t
d t d t T t
σ
σ
σ
=
= + −
= − − Φ
− − − Φ
+ − + −
=
−
= − −
∑
Copyright 2015 Dr. LAM Yat-fai 19
Linear approximation
( ) ( )
( )
( )
[ ]
( ) ( ) ( )
( )
( )
[ ]
3
1
332
1 1 2
1 2 3
3 3 3 32 2
1 1 1 2 1 2
ln ln ln lnln ln
ln ln ln ln ln ln
BS
VV BS
BS
BS
BS
kVV BS
k k BS
k BS
V t
V t V t
t
V t
V t V t w t
t
KK K KKK
K K K KK K
K K K KK K
K K K K K K
σ σ
σ
σ σ
σ
σ σ σ σ
=
∂
≈ + ⋅ −
∂
∂
≈ + ⋅ ⋅ −
∂
⋅ ⋅⋅
≈ + +
⋅ ⋅ ⋅
∑
Copyright 2015 Dr. LAM Yat-fai 20
Quadratic approximation (1)
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
( )
( )
( )
( )
( )
( )
2
2
2
3
2
1 21
2
1
2
1
2
BS BS
VV BS
bs bs
BS BS
BS
k
k BS
BSVV BS
k BS
k
k BS
BS
V t V t
V t V t
V t
t
V t V t w t
V t
t
σ σ σ σ
σ σ
σ σ
σ
σ σ
σ
=
∂ ∂
≈ + ⋅ − + ⋅ −
∂ ∂
∂
⋅ −
∂ ≈ + ∂ + ⋅ − ∂
∑
6. Copyright 2015 Dr. LAM Yat-fai 21
Quadratic approximation (2)
( ) ( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
23
2
2
1
2
2
0
1
2
1
2
bs bs
BS BS
k k
k k BS k BS
k BS BS
BS
BS
BS
BS
C B A
V t V t
A w t
t t
V t
B
V t
C
σ σ σ σ
σ σ σ σ
σ σ
σ
σ
=
− + − + =
∂ ∂
= − ⋅ − + ⋅ −
∂ ∂
∂
=
∂
∂
=
∂
∑
Copyright 2015 Dr. LAM Yat-fai 22
Three recovery methods
Root searching algorithm
Definition of implied volatility
Linear approximation
Very simple linear formula
Quadratic approximation
Solving quadratic equation in closed form
Copyright 2015 Dr. LAM Yat-fai 23
Three volatility ranges
Middle
Extreme
Wing
Copyright 2015 Dr. LAM Yat-fai 24
Three scenarios
Major currencies, regular market
EUR, GBP, CHF, SEK, CAD, JPY, AUD
157 Fridays during the period from 2010 to 2012
Major currencies, stress market
52 Fridays during the period from July 2008 to
June 2009
CNY, regular market
157 Fridays during the period from 2010 to 2012
7. Copyright 2015 Dr. LAM Yat-fai 25
Back testing summary
CNY, normalExtreme
Quadratic
approximation
Major currency,
stress
Wing
Linear
approximation
Major currency,
normal
MiddleRoot search
ScenariosSmile region
Recovery
method
Copyright 2015 Dr. LAM Yat-fai 26
Findings (1)
Quadratic approximation and root search
algorithm outperform linear approximation
for wing and extreme regions
Quadratic approximation performs equally
well as root search algorithm
Not much differentiation for middle region
among three methods
Copyright 2015 Dr. LAM Yat-fai 27
Findings (2)
The quadratic approximation is the most
efficient and effective in terms of
Absolute percentage error: Average, standard
deviation
Closed form solution
The linear approximation could be a good
choice for the middle region
Very simple linear closed form solution
Copyright 2015 Dr. LAM Yat-fai 28
Domain of application
FailedFailedAdequateCNY
FailedMarginalAdequateStress
MarginalAdequateAdequate
Major
currency
ExtremeWingMiddle
8. Copyright 2015 Dr. LAM Yat-fai 29
Conclusions
Is the Vanna-Volga method an accurate
approach to valuate currency options?
Yes
How robust is the Vanna-Volga method in
recovering a volatility smile?
Quadratic approximation is robust
Copyright 2015 Dr. LAM Yat-fai 30
Extensions for research
Value-at-risk
Dynamic hedging
Exotic currency options
Stochastic Vanna-Volga method
A combination the best of Vanna-Volga method
and Heston’s stochastic volatility model
Copyright 2015 Dr. LAM Yat-fai
Q & A
http://sites.google.com/site/quanrisk
