Transcript Changes in Partnership Interest

อาจารย์
มธ. อธิบายการใช้
โมเดลของ ...........
ผู้ช่วยศาสตราจารย์ ดร.สุ ลกั ษมณ์ ภัทรธรรมมาศ
ผู้อานวยการโครงการปริญญาโททางการเงิน (MIF)
ภาควิชาการเงิน คณะพาณิชยศาสตร์ และการบัญชี
มหาวิทยาลัยธรรมศาสตร์
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GARCH Option Pricing Model of Duan(1995)
Asset returns follow the generalized autoregressive
conditional heteroskedastic (GARCH) process.
GARCH (1,1) model is the most commonly used
GARCH process.
St
1
rt  ln
 rf   ht  ht   t ;
St 1
2
ht  0  1 t21   2 ht 1;
t
N  0, ht 
0  0, 1  0,  2  0
St is the stock price at time t
rf is the constant one-period risk-free rate of return
λ is constant unit risk premium
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GARCH (1,1) Model
To ensure covariance stationary of the GARCH (1,1)
process
1  2  1
The stationarity conditions are important to ensure
that the moments of the normal distribution are finite.
See Greene (2003) for more explanation.
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GARCH (1,1) Model Estimation
Maximum likelihood: Select likelihood function
 t2 
1
ln L    ln  2   ln ht  
2
ht 
t 1
T
Estimated parameters from GARCH option Prices—
Daily.xls written by Khanthavit (2007).
ื่ พารามิเตอร์
ชอ
ค่าตัง้ ต ้น
ค่าพารามิเตอร์
Lambda
a0
a1
b1
0.220600
3.196279
1.657971
-0.467191
0.048664
0.000037
0.063998
0.752433
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From GARCH process to LRNVR
Duan (1995) used the locally risk-neutral valuation
relationship (LRNVR) to derive GARCH option
pricing.
St
1
rt  ln
 rf  ht   t ;
St 1
2

ht  0  1  t 1   ht 1

t
2
N  0, ht 
  2 ht 1
This is to ensure that the one-period ahead
conditional variance is invariant with respect to a
change to the risk-neutralized pricing measure.
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The Terminal Asset Price
The terminal asset price is as follows:
1


St 1  St exp  rf  ht 1   t 1 
2


1


St  2  St 1 exp  rf  ht  2   t  2 
2


T
1 T


ST  St exp T  t  rf   hs    s 
2 s t 1
s  t 1


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Option Pricing Model: Call Option
European call option with exercise price X maturing
at time T has to time-t value equal to
Ct  exp rf T  t   E Q  Max ST  X , 0.00 | t 
For GARCH(1,1) model, Xt and ht+1 serve as the
sufficient statistics for Θt.
The delta of the call option equals to
 ST

  exp rf T  t   E  1ST  X | t 
 St

C
t
Q
1ST  X is an indicator function, i.e. equals 1if ST ≥ X and
equals 0 otherwise.
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Option Pricing Model: Put Option
European put option with exercise price X maturing at
time T has to time-t value can be derived from putcall parity relationship.
Ct  X exp rf T  t   Pt  St
Pt  Ct  St  X exp rf T  t 
The delta of the put option equals to
tP  tC 1.00
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Advantage and Disadvantage of GARCH Option
Pricing Model
Volatility is observable from discrete asset price data
and only a few parameters need to be estimated even
in a long time series of options records.
Unfortunately, the analytic solution for the GARCH
option price is not available because the conditional
distribution over more than one period cannot be
analytically derived.
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Option Pricing Model Estimation
Monte Carlo simulation can be used. (GARCH option
Prices—Daily.xls)
Simulate 10,000 times to get 10,000 values of possible
terminal asset prices.
t
N  0, ht 
Get 10,000 possible option values.
Use expected value as the option price.
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Option Pricing Model Estimation :Alternative
Heston, S., and S. Nandi, 2000, “A Closed Form
GARCH Option Pricing Model, ” The Review of
Financial Studies, 13, 585-625.
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Points for Consideration
Duan (1995) pointed out that this model can only be
used for the valuation of individual equity options.
The market portfolio is expected to be highly
correlated with aggregate assumption and the returns
will not follow a GARCH process.
How often do we need to estimate the parameters for
GARCH model?
When should we estimate the parameters from the
option price directly?
What model would be the best?
Other GARCH-type model such as EGARCH and
NGARCH could be explored in the future.
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ขอขอบคุณ
ผู้ช่วยศาสตราจารย์ ดร.สุ ลกั ษมณ์ ภัทรธรรมมาศ
ผู้อานวยการโครงการปริญญาโททางการเงิน (MIF)
คณะพาณิชยศาสตร์ และการบัญชี มหาวิทยาลัยธรรมศาสตร์
10 กันยายน 2550
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