Download Static Hedging and Pricing American Exotic Options

Static Hedging and Pricing American
Exotic Options
San-Lin Chung, Pai-Ta Shih*, and Wei-Che Tsai
Presenter: Pai-Ta Shih
National Taiwan University
1
Outlines






The main contributions
Introduction
Formulation of the static hedging portfolio
The hedge performance of static versus dynamic
hedging of an American up-and-out put option
Efficiency of the static hedge portfolio for pricing
American exotic options
Conclusions
2
The main contributions
1.
We show how to construct static hedge portfolios for American barrier
options and floating strike lookback options due to the complexity of
boundary conditions of the option.
2.
We investigate the hedging performance of the proposed method and
compare with that of the dynamic hedge strategy.
3.
This article proposes an efficient numerical method for pricing
American barrier options and floating strike lookback options.
3
Introduction

Pricing and hedging American-style exotic options is an important
yet difficult problem in the finance literature.

Lattice methods: Boyle and Lau (1994), Ritchken (1995),
Cheuk and Vorst (1996), and Chung and Shih (2007)

Analytical approximation formulae: Gao, Huang, and
Subrahmanyam (2000), AitSahlia, Imhof, and Lai (2003), and
Chang, Kang, Kim, and Kim (2007)
4
Introduction

Extend the seminal works of Derman, Ergener, and Kani (1995)
and Carr, Ellis, and Gupta (1998) to several ways:
1. Beyond the Black-Scholes model:

Andersen, Andreasen, and Eliezer (2002), Fink (2003),
Nalholm and Poulsen (2006a), and Takahashi and
Yamazaki (2009)
2. Several replication methods:

the risk-minimizing method of Siven and Poulsen (2009)

the line segments method of Liu (2010)
5
Introduction
3. Beyond the barrier options:

American options (Chung and Shih (2009))

European Asian options (Albrecher, Dhaene, Goovaerts, and
Schoutens (2005))

European Installment options (Davis, Schachermayer, and
Tompkins (2001))
6
Introduction

This paper utilizes the static hedge portfolio (SHP) approach to price
and/or hedge American exotic options.

At least three advantages:
1. The proposed method is applicable for more general stochastic
processes, but the existing numerical or analytical approximation
methods may be difficult to extend them to other stochastic processes,
e.g. the constant elasticity of variance (CEV) model of Cox (1975).
7
Introduction
2.
The hedge ratios, such as delta and theta, are automatically derived
at the same time when the static hedge portfolio is formed.
3.
When the stock price and/or time to maturity instantaneously
change, the recalculation of the prices and hedge ratios for the
American exotic options under the proposed method is quicker
than the initial computational time because there is no need to
solve the static hedge portfolio again.
8
Formulation of the static hedging portfolio

We demonstrate how to construct the SHP for an American upand-out put option and the SHP for an American floating strike
lookback put option under the Black-Scholes model.
9
The static hedge portfolio for an American
up-and-out put
10
The static hedge portfolio for an American
up-and-out put

Our static hedge portfolio starts with one unit of the corresponding
European option to match the terminal condition of the AUOP.

At time t(n-1), three conditions imply that
0  P E  H , X ,  , r , q, T  tn 1  
wn 1 P E  H , Bn 1 ,  , r , q, T  tn 1   wˆ n 1C E  H , H ,  , r , q, T  tn 1  ,
X  Bn 1  P E  Bn 1 , X ,  , r , q, T  tn 1  
wn 1 P E  Bn 1 , Bn 1 ,  , r , q, T  tn 1   wˆ n 1C E  Bn 1 , H ,  , r , q, T  tn 1  ,
1   Ep  Bn 1 , X ,  , r , q, T  tn 1  
ˆ n 1 cE  Bn 1 , H ,  , r , q, T  tn 1 
wn 1 Ep  Bn 1 , Bn 1 ,  , r , q, T  tn 1   w
11
The static hedge portfolio for an American
up-and-out put
Figure 1. The static hedge portfolio for an AUOP option
12
The static hedge portfolio for an American
up-and-out put

Using similar procedures, we work backward to determine the
number of units of the European option.

Finally, the value of the n-point static hedge portfolio at time 0 is
obtained as follows:
13
The static hedge portfolio for an American
floating strike lookback put

We use the underlying asset as the numeraire and express the price of
an American floating strike lookback put option.
14
The hedge performance of static versus dynamic
hedging of an American up-and-out put option

According to CBOE’s equity option product specifications, the strike
price interval is $2.5 when the strike price is between $5 and $25, $5
when the strike price is between $25 and $200, and $10 when the
strike price is over $200. We consider two types of SHPs.

The first SHP is formed by exactly following the previous
procedure and we call this portfolio is termed “SHP with
nonstandard strikes”.

The second portfolio utilizes European put options with standard
15
strikes and thus is called “SHP with standard strikes”.
S0
Table 1. The Static Hedge Portfolio (SHP)
for an American Up-and-Out Put
Quantity of
European
Call
Panel A. SHP with nonstandard strikes
Quantity of
Expiration
European
Strike
Strike
(months)
Put
Value for =
100
-0.004415
110
0.046494
80.238013
2
-0.002381
-0.011848
110
0.047374
80.810245
4
-0.015407
-0.026374
110
0.048418
81.621783
6
-0.057712
-0.057201
110
0.052649
82.799137
8
-0.182052
-0.125989
110
0.007167
84.558963
10
-0.588633
-0.109803
110
0.166623
88.061471
12
-0.276521
1
100
12
6.003998
Net

Parameters: S0 = 100, X = 100,
H = 110, r = 4%, q = 0, sigma =
0.2, and T = 1.
The benchmark value 4.890921
uses Ritchken’s trinomial
lattice method with 52,000 time
steps.
4.881292
Panel B. SHP with standard strikes
Quantity of
Expiration
European
Strike
(months)
Put
Quantity of
European
Call
Strike
-0.004334
110
0.041485
80
2
-0.002391
-0.011677
110
0.043703
80
4
-0.016377
-0.026126
110
0.042927
80
6
-0.062652
-0.057018
110
0.101540
80
8
-0.175055
-0.126408
110
0.044267
85
10
-0.545784
-0.109983
110
0.118888
90
12
-0.321408
1
100
12
6.003998
Net

Value for =
100
4.880331
16
Figure 2. The Mismatch Values on the Boundary for the
Static Hedge Portfolios


Parameters: S0 = 100, X = 100, H = 110, r = 4%, q = 0, sigma = 0.2, and T = 1.
The accurate early exercise boundary is calculated from the Black-Scholes model of
Ritchken (1995) with 52,000 time steps per year.
17
Figure 3. The profit and loss distributions of two SHPs and
the dynamic hedging strategy



Parameters: S0 = 100, X =
100, H = 110, r = 4%, q = 0,
sigma = 0.2, and T = 1.
We run 50,000 Monte Carlo
simulations with time
discretized to 52,000 time
steps per year to construct the
profit-and-loss distribution.
At every knock-out stock
price time point or early
exercise decision, we
liquidate the static hedging
portfolio (SHP).
18
Table 2. Hedging Performance of the SHPs and the DHPs

Parameters: S0 = 100, X =
100, H = 110, r = 4%, q =
0, sigma = 0.2, and T = 1.

We adopt four risk
measures used by
Siven and Poulsen
(2009) to evaluate
the hedging
performance.

For example, n=130
means that we
dynamically adjust deltahedged portfolios every
400/52000 time.
19
Efficiency of the static hedge portfolio for pricing
American exotic options

In this section, we will evaluate the numerical
efficiency of the SHP approach for pricing American
barrier options and American lookback options,
respectively.


In the former case, we consider pricing AUOP options
under the CEV model of Cox (1975).
In the latter case, we consider valuing American floating
strike lookback put options under the BS model to
demonstrate that the proposed method is applicable for
other types of exotic options beyond barrier options.
20
Figure 4. The Convergence of the SHP Values to the AUOP
Price under CEV model



We refer to the parameter setting
of Gao, Huang, and
Subrahmanyam (2000) and
Chang, Kang, Kim, and Kim
(2007) with the following
parameters: S = 40, X = 45, H =
50, r = 4.48%, σ = 0.683990, and
q = 0.
We set
, and take beta
parameter (β=4/3) from Schroder
(1989).
The benchmark accurate
American up-and-out put option
price which uses Boyle and Tian’s
method with 52,000 time steps is
about 5.358829.
21
Table 3. The Early Exercise Boundary of
the AUOP under CEV Model
T-t
0
1/52
1/26
3/52
1/13
5/52
3/26
7/52
2/13
9/52
5/26
11/52
3/13
1/4
7/26
15/52
4/13
17/52
.
.
.
1/2
3/4
1
Benchmark Value
45.0000
42.6295
41.9174
41.4324
41.0384
40.7337
40.4305
40.2149
40.0000
39.7859
39.6152
39.4449
39.3175
39.1481
39.0214
38.8950
38.7688
38.6848
.
.
.
37.8932
37.2350
36.8275
SHP
45.0000
42.7451
41.8710
41.4148
41.0268
40.7064
40.4312
40.1893
39.9730
39.7770
39.5980
39.4330
39.2802
39.1380
39.0049
38.8803
38.7630
38.6525
.
.
.
37.8836
37.2154
36.8276
Difference (%)
0.0000%
0.2712%
-0.1107%
-0.0424%
-0.0284%
-0.0670%
0.0018%
-0.0635%
-0.0676%
-0.0222%
-0.0434%
-0.0301%
-0.0949%
-0.0260%
-0.0422%
-0.0378%
-0.0149%
-0.0836%
.
.
.
-0.0253%
-0.0527%
0.0002%


The “Benchmark Value” is
obtained by the Boyle and Tian
method with 52,000 time steps of
the early exercise price of an
AUOP option.
The number of nodes matched on
the early exercise boundary ( n )
in the SHP method is 52 per year.
22

S0  0.5
Table 4. The Valuation of American Up-and-Out Put Options
under the CEV Model


Parameters: X = 45, H =
50, r = 4.48%, q = 0.
The “Benchmark Value”
shows the numerical
results of option values
from the Boyle and Tian
method with 52,000 time
steps.
23
Table 4. The Valuation of American Up-and-Out Put
Options under the CEV Model


Parameters: X = 45, H =
50, r = 4.48%, q = 0.
The “Benchmark
Value” shows the
numerical results of
option values from the
Boyle and Tian method
with 52,000 time steps.
24
Table 4. The Valuation of American Up-and-Out Put
Options under the CEV Model




Parameters: X = 45, H
= 50, r = 4.48%, q = 0.
The root-mean-squared
absolute error (RMSE)
and the root-meansquared relative error
(RMSRE) are
presented in this Table.
Table 4 indicates that
the numerical
efficiency of our SHP
method is comparable
to the tree method of
Boyle and Tian (1999).
One advantage of the
SHP method is the
recalculation of the
25
American exotic option
prices.
Pricing American floating strike lookback put
options under the Black-Scholes model

The numerical results are based on the parameter
setting of Chang, Kang, Kim, and Kim (2007) and
are reported in Figure 5 and Table 5.


Parameters: S0 = 50, y0
= M0/S0 = 1.02.
The “Benchmark Value”
shows the numerical
results of option values
from the Babb’s method
with 52,000 time steps.
26
Table 5. The Valuation of American Floating Strike
Lookback Put Options under the Black-Scholes Model


Parameters: S0 = 50, y0
= M0/S0 = 1.02.
The “Benchmark Value”
shows the numerical
results of option values
from the Babb’s method
with 52,000 time steps.
27
Table 5. The Valuation of American Floating Strike
Lookback Put Options under the Black-Scholes Model


Parameters: S0 = 50, y0
= M0/S0 = 1.02.
The “Benchmark Value”
shows the numerical
results of option values
from the Babb’s method
with 52,000 time steps.
28
Conclusions

We successfully construct a static hedge portfolio to match the terminal
and boundary conditions of American barrier options and lookback
options.

We show that while the average profit and loss values of all hedge
portfolios are similar, the SHPs are far less risky than the DHPs no matter
which risk measure is used.

We conduct detailed efficiency analyses and show that the proposed
method is as efficient as the numerical methods for pricing American
barrier options under the CEV model and American lookback options
under the Black-Scholes model.
29