The 23rd Annual Conference on Pacific Basin Finance, Economics, Accounting, and Management Cheng-Few Lee, Yibing Chen, John Lee July 17, 2015 Alternative.

The 23rd Annual Conference on Pacific Basin Finance, Economics, Accounting, and Management

Cheng-Few Lee, Yibing Chen, John Lee

July 17, 2015

Alternative Methods to Estimate Implied Variance:

Review and Comparison


Abstract

The main purpose of this paper is to review and compare alternative methods for estimating implied variance. In this paper, we first review several alternative methods to estimate implied variance. Then we show how the MATLAB computer program can be used to estimate implied variance based upon the Black-Scholes model. In addition, we also discuss how the approximation method derived by Ang, Jou et al. (2013) can be used to estimate implied variance and implied stock price per share. Real world data from US individual stock options are used to compare the estimation results using three typical alternative methods: regression method proposed by Lai, Lee et al, MATLAB computer program approach and approximation method derived by Ang, Jou et al. Also, this paper presents the empirical results of China ETF 50 options which were new in the financial markets.


Introduction

It is well known that implied variance estimation is important for evaluating option pricing. In this paper, we first review several alternative methods to estimate implied variance in Section B. We classify them into two different estimation routines: numerical search methods and closed-form derivation approaches. Closed-form derivation approaches took use of either Taylor expansion or inverse function to calculate the analytical solutions for the ISD.

In Section C, we show how the MATLAB computer program can be used to estimate implied variance. This computer program is based upon the Black-Scholes model using Newton-Raphson method.

In Section D, we discuss how the approximation method derived by Ang, Jou et al. (2013) can be used to estimate implied variance under the case of continuous dividends. This approximation method can also estimate implied volatility from two options with the same maturity, but different exercise prices and values.

In Section E, real data from American option markets are used to compare the performances of three typical alternative methods: regression method proposed by Lai, Lee et al, MATLAB computer program approach and approximation method derived by Ang, Jou et al. The results are presented in Section E. Also, this paper presents the empirical results of China ETF 50 options which were new in the financial markets. Section F summarizes the paper.


Framework Summary

Section B

• review several alternative methods to estimate implied variance.

• classify methods into two different estimation routines: numerical search methods and closed-form derivation approaches.

• discuss how the approximation method derived by Ang, Jou et al. (2013) can be used to estimate implied variance under the case of continuous dividends.

• also discuss: a pair of options

• comparison of alternative methods: empirical results

• cases from US—individual stock options

• cases from China—ETF 50 options

Section C

Section D Section E

• show how the MATLAB computer program can be used to estimate implied variance


Alternative methods to estimate implied variance

Numerical Search Closed-form Derivation

Trial and errorLatane and Rendleman (1976)

Taylor Series ExpansionFirst-order expansion: Brenner and Subrahmanyam (1988); Corrado and Miller (1996)Second-order expansion: Chance (1996)Third-order expansion: Li (2005)

Choose an initial point, iterative algorithmManaster and Koehler (1982)

Inverse Function

Estimate parameters by regression: Lai, Lee et al. (1992)


Numerical search method

1 2( ) ( )rTC SN d Xe N d-= -

2

1

1ln( ) ( )

2S X r T

dT

s

s

+ +=

2 1d d Ts= -

S=current market price of the underlying stock;X=exercise price;r=continuous constant interest rate;T=remaining life of the option

Trail and error

Within ±0.001 of the observed actual call price

Latane and Rendleman (1976)


Numerical search method Manaster and Koehler (1982), choose an initial point

( , , , , ) ( )C f S X r T fs s= =

0lim ( ) max(0, )rTf S Xes

s+

-

®= -

lim ( )f Ss

s®¥

=

Mean-Value Theorem. Let f be a continuous function on the closed interval[ , ]a b , and can be differentiable on

the open interval ( , )a b , wherea b< . There exists some ( , )c a bÎ such that:

( ) ( )'( )

f b f af c

b a

-=

-

strictly monotone increasing

max(0, )rTS Xe C S-- < <

Ensure: a positive solution of implied standard deviation*s

* *1

* *

( ) '( (1 ) )1 1

'( )'( )( )

n n n

nn n n

f C f

ff

s s s l s l s

ss s s s s

+ - - + -= - = -

- -

maximize

21

2ln

SrT

X Ts

æ ö÷ç= ÷+ç ÷ç ÷è ø

initial point


Closed-Form Derivation: Taylor Series Expansion

First-Order Taylor Series Expansion: Brenner and Subrahmanyam (1988)

1 1 1 11 1

( ) (0) '(0) ( )2 2

N d N N d d odp

= + + = + +L

rTS Xe-= At-the-money

1 11 1 1 1

( )2 22 2 2

N d d Tsp p

» + = +

2 11 1

( ) 1 ( )2 2 2

N d N d Tsp

» - = -

2

S TC

s

p= 2C

S T

ps =

Note that Brenner and Subramanyam’s method can only be used to estimate implied standard deviation from at-the-money or at least not too far in- or out-of-the-money options.

Limitation



31 1( ) ( )

2 62

zN z z

p= + - +L

First-Order Taylor Series Expansion: Brenner and Subrahmanyam (1988)

1 11 1( ) ( )2 22 2

rTd d TC S Xe

s

p p- -

= + - +

2 2

1

1 1ln( ) ( ) ln( )

2 2S X r T S K T

dT T

s s

s s

+ + += =

2 1d d Ts= -

2 ( ) [(2 2 2 ( )] 2( )ln( ) 0T S K T C S K S K S Ks s p p+ - - - + - =

2 21 1ln( ) ln( )1 12 2( ) ( )

2 22 2

S K T S K TC S K

T T

s s

s p s p

+ -= + - +



Second-Order Taylor Series Expansion: Chance (1996)

** 2C

S T

ps = Brenner and Subrahmanyam’s ISD

At-the-money call

* *C C CD = -

* *X X XD = -

* *s s sD = -

* 2 * * 2 * 2 ** * * 2 * * 2 * *

* *2 * *2 * *

1 1( ) ( ) ( ) ( ) ( )

2 2C C C C C

C X X XX X X

s s ss s s

¶ ¶ ¶ ¶ ¶D = D + D + D + D + D D

¶ ¶ ¶ ¶ ¶ ¶

* 2 *( ) ( ) 0a b cs sD + D + =

2 *

*2

12

Ca

s

¶=

¶

* 2 **

* * *( )

C Cb X

Xs s

¶ ¶= + D

¶ ¶ ¶

* 2 ** * * 2

* *2

1( ) ( )

2C C

c C C X XX X

¶ ¶= - + D + D

¶ ¶


Third-Order Taylor Series Expansion: Li (2005)


3 31 1 2 21 1

( ) ( )2 22 6 2 2 6 2

rTd d d dC S Xe

p p p p-= + - - + -

22 2 1 68

2z z

T T z

as = - -

2 C

S

pa =

11 3cos cos

3 32z

a-é æ öù÷çê ú= ÷ç ÷ç ÷ê úè øë û

Where


Closed-Form Derivation: Regression Method

1( )C

N dS

¶=

¶

2( )rTCe N d

X-¶

= -¶

11 [( )]

Cd N

S- ¶

=¶

12 1 [ ( )]rT Cd d T N e

Xs - ¶

= - = -¶

1 1{[ ( ) [ ( )]}rTC CN N e T

S Xs - -¶ ¶= - -

¶ ¶

( ) ( ) S XC C

C S X S XS X

b b¶ ¶

= + = +¶ ¶ RegressionRegression ' rT

it S t X it itC S e Xa b b e-= + + +

1 1[ ( ) ( ' )]S XN N Ts b b- -= - -) ) This alternative approach would work best

for index options, where there are many simultaneous quotes.

Lai, Lee et al. (1992)


MATLAB approach to estimate implied variance

,, , 0 0 0 ,( ) ( )

Tj tM T

j t j t j t

CC C es s s s

s

é ù¶ê ú- = - +ê ú¶ê úë û

21

, / 21( )

2

Ft j dr r

CXe N d Xe et t t

ts p

-- -¶

¢= =¶

1 0

0

.001s s

s

-<

Tolerance level

Inputs: Price - Current price of the underlying asset.Strike - Strike (i.e., exercise) price of the option.Rate - Annualized continuously compounded risk-free rate of return over the life of the option, expressed as a positive decimal number.Time - Time to expiration of the option, expressed in years.Value - Price (i.e., value) of a European option from which the implied volatility of the underlying asset is derived.

Output:Volatility - Implied volatility of the underlying asset derived from European option prices, expressed as a decimal number. If no solution can be found, a NaN (i.e., Not-a-Number) is returned.

Volatility = blsimpv(Price, Strike, Rate, Time, Value, Limit, Yield, Tolerance, Class)Volatility = blsimpv(90, 95, 0.03, 0.25, 5,[],0.05,[], {'Call'})Volatility = blsimpv(90, 95, 0.03, 0.25, 5,[],0.05,[], true)

Examples


BS model with dividends

1 2' ( ) ( )C S N d KN d= -2

1

ln( ) ( ) ln( ' )2 2S X r qT S K

d TT T

s

ss s

+ + -= = +

2 1d d Ts= -

' ln( ' )L S K Ts=Let

21

1

( ' 2)

( ') '( ') 2 ''( ')( 2) 2

( ') '( ')( 2)[1 ln( ' ) 4]

N L T

N L N L T N L T e

N L N L T S K e

s

s s

s

+

= + + +

= + - +

22

2

( ' 2)

( ') '( ') 2 ''( ')( 2) 2

( ') '( ')( 2)[1 ln( ' ) 4]

N L T

N L N L T N L T e

N L N L T S K e

s

s s

s

-

= - + +

= - + +

2

2

[8( ' ) 2( ' )ln( '/ )] 8 2 (2 ' )

ln( '/ )[( ' )(16 (ln( '/ )) ) 4( ' )ln( '/ )] 0

T S K S K S K T C S K

S K S K S K S K S K

s s p+ - - - - +

+ - + - + =

Approximation approach to estimate implied variance Ang, Jou et al. (2009)


Approximation approach to estimate implied variance Ang, Jou et al. (2009)

2 42

b b acT

as

- ± -=

8( ' ) 2( ' ) ln( ' )a S K S K S K= + - -Where

8 2 (2 ' )b C S Kp= - - +

2ln( ' )[( ' )(16 ln( ' )) ] 4( ' ) ln( ' )]c S K S K S K S K S K= - + - +

Call option case

2 42

b b acT

as

- ± -=

Where 8( ' ) 2( ' ) ln( ' )a S K S K S K= + - -

8 2 (2 ')b P K Sp= - - +

2ln( ' )[( ' )(16 ln( ' ) )] 4( ' ) ln( ' )]c S K S K S K S K S K= - + - +

Put option case


1 2 1 2 12(ln( ' ) 2)( )C C N S K T T K Ks s e= - - - +

Approximation approach to estimate implied variance

A pair of two call options

2 1 2 1 21(ln( ' ) 2)( )C C N S K T T K Ks s e= - - - +

11 2 2 1 1 1[( ) ( )] ln( ' ) 2N C C K K S K T Ts s h- - - = - +

11 2 2 1 2 2[( ) ( )] ln( ' ) 2N C C K K S K T Ts s h- - - = - +

2 11 2 2 1 1 22 [( ) ( )]( ) ln( ' ) ln( ' ) 0T N C C K K T S K S Ks s-+ - - - - =

1 1 2 21 2 2 1 1 2 2 1 1 2(( ) ( )) [ (( ) ( ))] ln( ' )T N C C K K N C C K K S K Ks - -= - - - ± - - +

Solution


Approximation approach to estimate implied variance

A pair of two put options

Solution

1 2 1 2 1 2 12( ) (ln( ' ) 2)( )P P K K N S K T T K Ks s d= + - - - - +

2 1 2 1 2 1 22( ) (ln( ' ) 2)( )P P K K N S K T T K Ks s d= + - - - - +

11 2 2 1 12

(( ) ( ) 1) ln( ' ) 2N P P K K S K T Ts s g- - - + = - +

11 2 2 1 21

(( ) ( ) 1) ln( ' ) 2N P P K K S K T Ts s g- - - + = - +

2 11 2 2 1 1 22 [( ) ( ) 1]( ) ln( ' ) ln( ' ) 0T N P P K K T S K S Ks s-+ - - + - - =

1 1 2 21 2 2 1 1 2 2 1 1 2(( ) ( ) 1) [ (( ) ( ) 1)] ln( ' )T N P P K K N P P K K S K Ks - -= - - - + ± - - + +


Some empirical results

Cases from US—Individual Stock Options

Security ID Ticker Company Name SIC Code Industry

101594 AAPL Apple Inc. 3571 Electronic Computers

104533 XOM Exxon Mobil Corporation 2911 Petroleum Refining

121812 GOOGL Google Inc. 7375 Information Retrieval Services

107525 MSFT Microsoft Corporation 7372 Prepackaged Software

106566 JNJ Johnson & Johnson 2834 Pharmaceutical Preparations

111953 WFC Wells Fargo & Company 6022 State Commercial Banks

105169 GE General Electric Company 3511Steam, Gas, and Hydraulic

Turbines, and Turbine Engine

111860 WMT Wal-Mart Stores Inc. 5331 Variety Stores

102968 CVX Chevron Corporation 2911 Petroleum Refining

109224 PG The Procter & Gamble Company 2841 Soap and Other Detergents

102936 JPM JPMorgan Chase & Co. 6211Security Brokers, Dealers &

Flotation Companies

111668 VZ Verizon Communications Inc. 4812 Radiotelephone Communications

108948 PFE Pfizer Inc. 2834 Pharmaceutical Preparations

106276 IBMInternational Business Machines

Corporation3571 Electronic Computers

109775 T AT&T, Inc. 4812 Radiotelephone Communications

relative large market values


Cases from US—Individual Stock Options

Ticker IV-matlab IV-approximation IV-regressionAAPL 0.387 0.332 0.458XOM 0.208 0.185 0.106

GOOGL 0.376 0.389 0.305MSFT 0.327 0.341 0.298JNJ 0.223 0.216 No positive solutionWFC 0.312 0.301 No positive solutionGE 0.176 0.142 0.224

WMT 0.127 0.124 0.164CVX 0.306 0.285 0.367PG 0.209 0.185 No positive solution

JPM 0.189 0.192 0.135VZ 0.169 0.174 0.247

PFE 0.216 0.208 0.185IBM 0.463 0.457 No positive solution

T 0.186 0.189 0.264


Some empirical results

Cases from China—ETF 50 Options

Option Ticker Exercise Price Expiration Date IV-matlabIV-

approximation10000021.SH 2.20 2015-06-25 0.515 0.50410000022.SH 2.25 2015-06-25 0.486 0.49810000023.SH 2.30 2015-06-25 0.417 0.42310000024.SH 2.35 2015-06-25 0.439 0.42410000025.SH 2.40 2015-06-25 0.489 0.47810000031.SH 2.20 2015-09-24 0.426 0.44210000032.SH 2.25 2015-09-24 0.435 0.44710000033.SH 2.30 2015-09-24 0.417 0.42810000034.SH 2.35 2015-09-24 0.422 0.43210000035.SH 2.40 2015-09-24 0.443 0.45410000045.SH 2.45 2015-06-25 0.428 0.43610000047.SH 2.45 2015-09-24 0.393 0.40810000053.SH 2.50 2015-06-25 0.443 0.42810000055.SH 2.50 2015-09-24 0.415 0.42010000061.SH 2.55 2015-06-25 0.442 0.43810000063.SH 2.55 2015-09-24 0.398 0.41610000069.SH 2.60 2015-06-25 0.420 0.43110000071.SH 2.60 2015-09-24 0.409 0.41210000077.SH 2.65 2015-06-25 0.426 0.41910000079.SH 2.65 2015-09-24 0.416 0.42810000085.SH 2.70 2015-06-25 0.427 0.43410000087.SH 2.70 2015-09-24 0.414 0.41910000093.SH 2.75 2015-06-25 0.417 0.42610000095.SH 2.75 2015-09-24 0.405 0.41110000101.SH 2.80 2015-06-25 0.427 0.44310000103.SH 2.80 2015-09-24 0.409 0.40110000123.SH 2.85 2015-06-25 0.441 0.43210000125.SH 2.85 2015-09-24 0.417 0.422

In Chinese financial market, there were no stock options in the exchange until February, 2015. Now, the only traded options in China are ETF 50 options.


THANKS!Q&A

The 23rd Annual Conference on Pacific Basin Finance, Economics, Accounting, and Management Cheng-Few Lee, Yibing Chen, John Lee July 17, 2015 Alternative.

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