Outline
Derivation of Risk-Neutral Density
Derivatives pricing in action
Implied Distribution of CEZ
The end
Risk-Neutral Options Pricing,Implied
Distribution of CEZ
Martin Cícha1
MODERNÍ NÁSTROJE PRO FINANČNÍ ANALÝZU A
MODELOVÁNÍ
1
Supported by IGA VSE, # 26/08
Martin Cícha
Risk-Neutral Options Pricing,Implied Distribution of CEZ
Outline
Derivation of Risk-Neutral Density
Derivatives pricing in action
Implied Distribution of CEZ
The end
Derivation of Risk-Neutral Density
Binomial Tree
Continuous World
Derivatives pricing in action
Implied Distribution of CEZ
Volatility Smile
Implied Distribution
The end
Martin Cícha
Risk-Neutral Options Pricing,Implied Distribution of CEZ
Outline
Derivation of Risk-Neutral Density
Derivatives pricing in action
Implied Distribution of CEZ
The end
Binomial Tree
Continuous World
Real World
I
Real world → P measure
I
Constant grow rate µ
I
Constant noise σ
I
All jumps equally likely → p =
I
Irrelevant for correct pricing
Martin Cícha
1
2
Risk-Neutral Options Pricing,Implied Distribution of CEZ
Outline
Derivation of Risk-Neutral Density
Derivatives pricing in action
Implied Distribution of CEZ
The end
Binomial Tree
Continuous World
Risk-Neutral World
I
Goal: finding measure
Q under which
the process of
n
o
−1
discounted stock Bt St , t ≥ 0 is a martingale
I
New measure: qt =
I
Stock value → St =
I
Xn → number of up-jumps, binomially distributed
down
St exp(r δt−St+δt
)
up
down
St+δt −St+δt
√ 2X −n n
√
µ+σ t
n
S0 e
Martin Cícha
Risk-Neutral Options Pricing,Implied Distribution of CEZ
Outline
Derivation of Risk-Neutral Density
Derivatives pricing in action
Implied Distribution of CEZ
The end
Binomial Tree
Continuous World
Continuous World
I
Transition from discrete to continuous time
I
δt → 0, n → ∞
I
Applying central limit theorem
I
St log-normally distributed under Q
√
1 2
S = S e((r − 2 σ )t+σ tZ )
I
t
0
Martin Cícha
Risk-Neutral Options Pricing,Implied Distribution of CEZ
Outline
Derivation of Risk-Neutral Density
Derivatives pricing in action
Implied Distribution of CEZ
The end
General Pricing Formula
I
General pricing formula: V0 = EQ BT−1 X
I
Goal: determining density of discounted claim X
I
Transformation of random variable ST
I
Using MATLAB Symbolic Toolbox
Martin Cícha
Risk-Neutral Options Pricing,Implied Distribution of CEZ
Outline
Derivation of Risk-Neutral Density
Derivatives pricing in action
Implied Distribution of CEZ
The end
Call Option
I
European call: right, but not obligation, to buy the stock at
specified price (strike) at time t = T
I
Claim: X = (ST − k )+
I
Option value: V (S0 , T ) = BT−1
Martin Cícha
R∞
0
x f (x)dx
Risk-Neutral Options Pricing,Implied Distribution of CEZ
Outline
Derivation of Risk-Neutral Density
Derivatives pricing in action
Implied Distribution of CEZ
The end
Put Option
I
European put: right, but not obligation, to sell the stock at
specified price (strike) at time t = T
I
Claim: X = (k − ST )+
I
Value: V (S0 , T ) = BT−1
R∞
0
Martin Cícha
x f (x)dx
Risk-Neutral Options Pricing,Implied Distribution of CEZ
Outline
Derivation of Risk-Neutral Density
Derivatives pricing in action
Implied Distribution of CEZ
The end
Arbitrary Derivative
I
European style
I
n
n
o
o
Claim: X = min max 1.3S0 , 0.9 SST0 , 1.8S0
I
Numerical integration
Martin Cícha
Risk-Neutral Options Pricing,Implied Distribution of CEZ
Outline
Derivation of Risk-Neutral Density
Derivatives pricing in action
Implied Distribution of CEZ
The end
Volatility Smile
Implied Distribution
Volatility Smile
I
Implied or historical volatility used in pricing formulas
I
Implied volatility is unobservable variable
I
Nonlognormality can be often observed in the market
I
Volatility smiles are used to allow for nonlognormality
I
Volatility: function of strike and time to maturity
Martin Cícha
Risk-Neutral Options Pricing,Implied Distribution of CEZ
Outline
Derivation of Risk-Neutral Density
Derivatives pricing in action
Implied Distribution of CEZ
The end
Volatility Smile
Implied Distribution
Volatility Smile on CEZ call
I
Call warrants on CEZ stock traded on Boerse Stuttgart
denominated in EUR as of 30 April 2009 used
I
American exercising style
I
Dividend paying
Martin Cícha
Risk-Neutral Options Pricing,Implied Distribution of CEZ
Outline
Derivation of Risk-Neutral Density
Derivatives pricing in action
Implied Distribution of CEZ
The end
Volatility Smile
Implied Distribution
Implied Volatility
I
V0 = k ω1 (δk ) − S0 ω2 δS
0


−δ ∆t   Pe
= e
N1 d1 
ω1 (δk )
−δ∆t
k



P∗
+
1

+e
−δ 2∆t
k

−δ∆t
Pe


N2 −d1 



P∗

Pe
, d1 

1

+e
−δ τ ∆t
k

Pe


Nn 
−d1 
−δ 2∆t

v 
ut
u 1 

T
; −t


P∗
2
−δ∆t
P∗





, . . . , −d1 

−δ(n−1)∆t
Pe
P∗
n −1
1



∗ e−δ ∆t N d  P
1
1
P1


t



, d1
−δτ
Pe

; Ωn 

t
1
k

P
− e−δ ∆t N1 d2 
d







r
∗ e−δ∆
∗ e−δ∆
P
∗ −δ 2∆t ; − t  −
 + e−δ 2∆t N −d 
,d
P1 e
2
1
1
∗
∗
T
P
P
1
k
+ ... +
1

r
∗ e−δ∆
∗ e−δ∆
P
∗ −δ 2∆t ; − t 
 + e−δ 2∆t N −d 
,d
P e
2
2
2
∗
∗
1
T
P
P
t


1

t
1
d

1
= P1∗ − 1,
ln(G) + σ τ

2
= 


√
σ2 τ

d1 (G)
I
2

see [P. Carr, “The Valuation of American Exchange Options with Application to Real Options”] for details
Martin Cícha
Risk-Neutral Options Pricing,Implied Distribution of CEZ
Outline
Derivation of Risk-Neutral Density
Derivatives pricing in action
Implied Distribution of CEZ
The end
Volatility Smile
Implied Distribution
Implied Distribution
I
I
I
I
I
I
I
Derived from the market prices of warrants
Two-lognormal mixture distribution used
Minimisation problem → find parameters of two-lognormal
mixture so that the difference between observed prices
and implied prices is minimal.
Trading volume used as weights
Problem: resulting distribution fulfills the definition of
density (integrated to one)
To avoid arbitrage opportunities: mean of implied
distribution must be equal to forward price of CEZ
KNITRO third-party libraries and MATLAB Optimization
Toolbox used
Martin Cícha
Risk-Neutral Options Pricing,Implied Distribution of CEZ
Outline
Derivation of Risk-Neutral Density
Derivatives pricing in action
Implied Distribution of CEZ
The end
Volatility Smile
Implied Distribution
Implied Density of CEZ
I
Implied density of CEZ for set of times to maturity τ
Martin Cícha
Risk-Neutral Options Pricing,Implied Distribution of CEZ
Outline
Derivation of Risk-Neutral Density
Derivatives pricing in action
Implied Distribution of CEZ
The end
Volatility Smile
Implied Distribution
Observed vs Implied Option Prices
I
Observed vs implied prices by two-lognormal mixture
Martin Cícha
Risk-Neutral Options Pricing,Implied Distribution of CEZ
Outline
Derivation of Risk-Neutral Density
Derivatives pricing in action
Implied Distribution of CEZ
The end
Volatility Smile
Implied Distribution
What for?
I
Pricing the options on CEZ not traded on the market
I
Searching for mispriced options
I
Problem: bid-offer spread
Martin Cícha
Risk-Neutral Options Pricing,Implied Distribution of CEZ
Outline
Derivation of Risk-Neutral Density
Derivatives pricing in action
Implied Distribution of CEZ
The end
Thank you.
Thank you
Martin Cícha
Ministry of Finance
Debt and Financial Assets Management
Letenska 15
180 00 Prague 1
Czech Republic
[email protected]
Martin Cícha
University of Economics Prague
Department of Statistics and Probability
W. Churchill Sq. 4
130 67 Prague 3
Czech Republic
[email protected]
Risk-Neutral Options Pricing,Implied Distribution of CEZ