Alternative Methods to Estimate Implied Variance: Review and Comparison

Transcription

1 Alternative Methods to Estimate Implied Variance: Review and Comparison Cheng-Few Lee, Yibing Chen, John Lee July 7, 05

2 Abstract he 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. hen we show how the MALAB 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. (03) 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, MALAB computer program approach and approximation method derived by Ang, Jou et al. Also, this paper presents the empirical results of China EF 50 options which were new in the financial markets.

3 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 aylor expansion or inverse function to calculate the analytical solutions for the ISD. In Section C, we show how the MALAB computer program can be used to estimate implied variance. his 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. (03) can be used to estimate implied variance under the case of continuous dividends. his 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, MALAB computer program approach and approximation method derived by Ang, Jou et al. he results are presented in Section E. Also, this paper presents the empirical results of China EF 50 options which were new in the financial markets. Section F summarizes the paper.

4 Framework Summary review several alternative methods to estimate implied variance. classify methods into two different estimation routines: numerical search methods and closed-form derivation approaches. Section B Section C show how the MALAB computer program can be used to estimate implied variance discuss how the approximation method derived by Ang, Jou et al. (03) can be used to estimate implied variance under the case of continuous dividends. also discuss: a pair of options Section D Section E comparison of alternative methods: empirical results cases from US individual stock options cases from China EF 50 options

5 Alternative methods to estimate implied variance Numerical Search rial and error Latane and Rendleman (976) Choose an initial point, iterative algorithm Manaster and Koehler (98) Closed-form Derivation aylor Series Expansion First-order expansion: Brenner and Subrahmanyam (988); Corrado and Miller (996) Second-order expansion: Chance (996) hird-order expansion: Li (005) Inverse Function Estimate parameters by regression: Lai, Lee et al. (99)

6 Numerical search method Latane and Rendleman (976) r C SN( d ) Xe N( d ) d ln( S X) ( r ) d d rail and error Within ±0.00 of the observed actual call price S=current market price of the underlying stock; X=exercise price; r=continuous constant interest rate; =remaining life of the option

7 Numerical search method C f( S, X, r,, ) f( ) r lim f( ) max(0, S Xe ) 0 Manaster and Koehler (98), choose an initial point strictly monotone increasing lim f( ) S Ensure: a positive solution of implied standard deviation * r max(0, S Xe ) C S Mean-Value heorem. Let f be a continuous function on the closed interval[ ab, ], and can be differentiable on the open interval ( ab, ), wherea b. here exists some c ( a, b ) such that: f '( c) f( b) f( a) b a initial point * * n f n C f n ( ) '( ( ) ) '( )( ) f '( ) * * n f n n n ln S X r

8 Closed-Form Derivation: aylor Series Expansion First-Order aylor Series Expansion: Brenner and Subrahmanyam (988) r S Xe At-the-money N( d) N(0) N '(0) d d o( d) N( d) d N( d) N( d) C S C S Note that Brenner and Subramanyam s method can only be used to estimate implied standard deviation from atthe-money or at least not too far in- or out-of-the-money options.

9 Closed-Form Derivation: aylor Series Expansion First-Order aylor Series Expansion: Brenner and Subrahmanyam (988) z N( z) ( z ) 6 3 d r d C S( ) Xe ( ) d ln( S X) ( r ) ln( S K) ln( S K) ln( S K) C S( ) K( ) d d ( S K) [( C ( S K)] ( S K)ln( S K) 0

10 Closed-Form Derivation: aylor Series Expansion Second-Order aylor Series Expansion: Chance (996) * * C S * * C C C At-the-money call Brenner and Subrahmanyam s ISD * * X X X * * * * * * * * C * C * C * C * C * * * * * * * * C ( X ) ( X ) ( ) ( ) ( X ) X X X * * a( ) b( ) c 0 a * C * b * * C C ( * X ) * * * X * * * C * C * ( ) ( ) * * c C C X X X X

11 Closed-Form Derivation: aylor Series Expansion hird-order aylor Series Expansion: Li (005) 3 3 r d d d d C S( ) Xe ( ) z 8z z Where C S z cos 3 cos 3 3

12 Closed-Form Derivation: Regression Method Lai, Lee et al. (99) C S C X Nd ( ) e r N( d ) C d N [( )] S r C d d N [ e ( )] X C r C {[ N ( ) N [ e ( )]} S X C C C ( ) S ( ) X SS XX S X Regression r it S t X it it C S ' e X S [ N ( ) N ( ' )] X his alternative approach would work best for index options, where there are many simultaneous quotes.

13 MALAB approach to estimate implied variance C M jt, j, t j, t( 0) 0 ( 0 ) j, t C C e C F tj, r r d / Xe N ( d) Xe e 0 0 olerance level.00 Inputs: Price - Current price of the underlying asset. Strike - Strike (i.e., exercise) price of the option. Rate - Annualized continuously compounded riskfree rate of return over the life of the option, expressed as a positive decimal number. ime - ime 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, ime, Value, Limit, Yield, olerance, Class) Volatility = blsimpv(90, 95, 0.03, 0.5, 5,[],0.05,[], {'Call'}) Volatility = blsimpv(90, 95, 0.03, 0.5, 5,[],0.05,[], true)

14 Approximation approach to estimate implied variance Ang, Jou et al. (009) C S ' N( d ) KN( d ) d ln( S X) ( r q) ln( S' K) d d BS model with dividends N( L ' ) Let L ' ln( S ' K) N( L ') N '( L ') N ''( L ')( ) e N( L ') N '( L ')( )[ ln( S ' K) 4] e N( L ' ) N( L ') N '( L ') N ''( L ')( ) e N( L ') N '( L ')( )[ ln( S ' K) 4] e [8( S ' K) ( S ' K)ln( S '/ K)] 8 ( C S ' K) ln( S '/ K)[( S ' K)(6 (ln( S '/ K)) ) 4( S ' K)ln( S '/ K)] 0

15 Approximation approach to estimate implied variance Ang, Jou et al. (009) Where a 4 b b ac a 8( S ' K) ( S ' K)ln( S ' K) Call option case b 8 ( C S ' K) c ln( S ' K)[( S ' K)(6 ln( S ' K)) ] 4( S ' K)ln( S ' K)] a 4 b b ac Where a 8( S ' K) ( S ' K)ln( S ' K) Put option case b 8 ( P K S ') c ln( S ' K)[( S ' K)(6 ln( S ' K) )] 4( S ' K)ln( S ' K)]

16 Approximation approach to estimate implied variance A pair of two call options C C N(ln( S ' K ) )( K K ) C C N(ln( S ' K ) )( K K ) N [( C C ) ( K K )] ln( S ' K ) N [( C C ) ( K K )] ln( S ' K ) N [( C C ) ( K K )]( ) ln( S ' K ) ln( S ' K ) 0 N (( C C ) ( K K )) [ N (( C C ) ( K K ))] ln( S ' K K )

17 Approximation approach to estimate implied variance A pair of two put options P P ( K K ) N(ln( S ' K ) )( K K ) P P ( K K ) N(ln( S ' K ) )( K K ) N (( P P ) ( K K ) ) ln( S ' K ) N (( P P ) ( K K ) ) ln( S ' K ) N [( P P ) ( K K ) ]( ) ln( S ' K ) ln( S ' K ) 0 N (( P P ) ( K K ) ) [ N (( P P ) ( K K ) )] ln( S ' K K )

18 Some empirical results Cases from US Individual Stock Options Security ID icker Company Name SIC Code Industry 0594 AAPL Apple Inc. 357 Electronic Computers XOM Exxon Mobil Corporation 9 Petroleum Refining 8 GOOGL Google Inc Information Retrieval Services 0755 MSF Microsoft Corporation 737 Prepackaged Software JNJ Johnson & Johnson 834 Pharmaceutical Preparations 953 WFC Wells Fargo & Company 60 State Commercial Banks 0569 GE General Electric Company 35 Steam, Gas, and Hydraulic urbines, and urbine Engine 860 WM Wal-Mart Stores Inc. 533 Variety Stores 0968 CVX Chevron Corporation 9 Petroleum Refining 094 PG he Procter & Gamble Company 84 Soap and Other Detergents 0936 JPM JPMorgan Chase & Co. 6 Security Brokers, Dealers & Flotation Companies 668 VZ Verizon Communications Inc. 48 Radiotelephone Communications PFE Pfizer Inc. 834 Pharmaceutical Preparations 0676 IBM International Business Machines Corporation 357 Electronic Computers A&, Inc. 48 Radiotelephone Communications relative large market values

19 Cases from US Individual Stock Options icker IV-matlab IV-approximation IV-regression AAPL XOM GOOGL MSF JNJ No positive solution WFC No positive solution GE WM CVX PG No positive solution JPM VZ PFE IBM No positive solution

20 Some empirical results Cases from China EF 50 Options Option icker Exercise Price Expiration Date IV-matlab IVapproximation SH SH SH SH SH SH SH SH SH SH SH SH SH SH SH SH SH SH SH SH SH SH SH SH SH SH SH SH In Chinese financial market, there were no stock options in the exchange until February, 05. Now, the only traded options in China are EF 50 options.

21 HANKS! Q&A