Empirical Option Pricing. Matti Suominen

Transcription

1 Empirical Option Pricing Matti Suominen

2 Put-Call Parity Arguments Put-call parity p +S 0 e -dt = c +EX e r T holds regardless of the assumptions made about the stock price distribution It follows that p mkt -p bs =c mkt -c bs

3 Implied Volatilities The implied volatility calculated from a European call option should be the same as that calculated from a European put option when both have the same strike price and maturity The same is approximately true of American options

4 Volatility Smile A volatility smile shows the variation of the implied volatility with the strike price The volatility smile should be the same whether calculated from call options or put options With Black & Scholes assumptions, there implied volatility should be same irrespective of the strike price. Is it?

5 The Volatility Smile for Foreign Currency Options Why could this be? Implied Volatility Strike Price

6 The Volatility Smile (Smirk) for Equity Options Implied Volatility Strike Price

7 Possible Causes of Volatility Smile (Holes in Black & Scholes) Asset price exhibit jumps rather than continuous change Volatility for asset price being stochastic (One reason for a stochastic volatility in the case of equities is the relationship between volatility and leverage) => Other than normal distribution for returns Prices are not right due to market participants limited ability to do arbitrage (no one can borrow sufficiently at risk free rate)?

8 A puzzle: Empirical returns from issuing options Driessen & Maenhout (2004) Large monthly payoffs from writing Out of The Money Puts (OTM) and At The Money (ATM) straddles:

9 Appendix: Dynamic hedging vs. buying commodity options Recent thesis by Riikka Tuominen Cumulative Return from Dynamic Hedging of 3-Month Options ($/ton) Thousands Daily adjustment Weekly adjustment

10 Does supply and demand affect options prices? (Garleanu, Pedersen, Poteshman, RFS, 2009)

11 11 6 February 2018 Matti Suominen (Aalto)

12 COCLUSION It seems that options, especially out of the money put options are overpriced due to high demand. Similar evidence can be found related to structured products that provide option like features.

13 Volatility Term Structure In addition to calculating a volatility smile, traders also calculate a volatility term structure This shows the variation of implied volatility with the time to maturity of the option

14 Volatility Term Structure The volatility term structure tends to be downward sloping when volatility is high and upward sloping when it is low

15 Example of a Volatility Surface Strike Price month month month year year year

16 Estimating Volatilities and Correlations

17 Standard Approach to Estimating Volatility Define s n as the conditional volatility per day between day n-1 and day n, as estimated at end of day n-1 Define S i as the value of market variable at end of day i Define u i = ln(s i /S i-1 ) σ 2 n = 1 m 1 u = 1 m m i=1 m i=1 u n i (u n i u) 2

18 Simplifications Usually Made Define u i as (S i -S i-1 )/S i-1 Assume that the mean value of u i is zero Replace m-1 by m This gives 2 1 m 2 s n = å u m i = 1 n - i

19 Weighting Scheme Instead of assigning equal weights to the observations we can set s m i= 1 2 m n = å a i iu 2 = 1 n-i where å a i = 1

20 Correlations Define u i =(U i -U i-1 )/U i-1 and v i =(V i -V i-1 )/V i-1 Also s u,n : daily vol of U calculated on day n-1 s v,n : daily vol of V calculated on day n-1 cov n : covariance calculated on day n-1 The correlation is cov n /(s u,n s v,n )

21 Volatility modelling: GARCH In practice conditional volatility σ t seems to follow a GARCH (1,1) process where σ 2 n = a+b * σ 2 n-1 + c * u n-1 2 u n-12 = ((S n -S n-1 )/S n-1 ) 2 Variations: GARCH, E-GARCH, GARCH in volume

22 Example of GARCH

23 These measures are important both in derivatives pricing but also in risk measures such as VAR Value at Risk (VAR) is an important measure of risk It is defined as the Loss such that a greater loss occurs only e.g. at 5% probability This measure can be adjusted to make full use of conditional volatility and correlation structures

24 Applying VAR Investment 10m in Microsoft shares σ A = 32% σ d = 2% 200K N(-2.23) = 1% Hence 99% probabilty that this investment does not lose more than 2.33σ d = 466K in one day 466K is the Value at Risk at 99% confidence level

25 Applying VAR σ d = 2% 200K Hence σ 10d = 10 2% and 10d value at risk is 10*466 = 1473K Two asset case: 2 2 s = s + s + 2s s r A + B A B A B AB

26 Executive Stock Options Stock options issued as part of a new incentivebased compensation plan. Should employee choose the cash bonus or the option grant? 26

27 Option Pricing Without Complications Apply Black-Scholes: S= $18.75, X=$35, r=6.02%, T= 5 years. Need an estimate of firm s volatility, σ: Historical estimate: 42% (most recent 90-days); 83% (max in 1987); 19% (min); 34% (average); 37% (average after crash)? Better: Implied volatility! Remember: C is a function of (S, X, T, r, σ). Given the market value of a call option, one can infer the market s estimate of volatility using Black-Scholes. The implied volatility is the volatility that makes the price from the option pricing model equal to the observed price in the market. 27

28 Implied Volatilities Inferring volatilities from option prices is used very commonly in practice. Implied vol of the company from traded call option prices Expiration of Option Strike X June 1992 July 1992 October 92 January % % 40% 37% 35% % 34% 37% 38% % 35% Longest traded option expires in 20 months. Sally s horizon is 60 months! LEAPS (Long-term Equity AnticiPation Securities) uncommon and at most 3 years. Therefore: be careful when pricing and do a sensitivity analysis! 28

29 Value of Unconditional Stock Option Grant Volatility Estimate (%) Price per Option Value of Grant (3000 Options) 20 $1.30 $3, $2.89 $8, $3.71 $11, $4.55 $13, $5.38 $16,140 The option grant value is most likely much higher in value than the $5,000 cash, BUT: Warrant, not call option! Vesting restriction Taxes Illiquidity/diversification 29

30 Complications I Warrants: At the time the executive stock options are exercised, the company will have to issue NEW shares. This causes dilution. We have seen that the value of a warrant is: W N 0 = s C 0 Ns + Nw 30

31 Complications II Vesting: If employee leaves compamy within 5 years, the options expire worthless. How many of you will stay with your first employer for >5 years? Ø Ø if grant is valued at $11,130, then accept option grant if probability of staying exceeds 45%. [$11,130p + $0(1-p) =$5000]. 45% is the break-even probability in this case. what s the signal you send if you choose cash? Note: In practice, this type of option would typically be structured so as to vest gradually over the 5-year interval and would also be exercisable over a window of 5 to 10 years from the date of the grant. 31

32 Complications III Taxes: Cash bonus is taxed now at 28%, leaves $3,600. Option grant: No taxes now Used to be: At maturity, pay ordinary income tax (28%) on difference between S T and strike of $35. When selling the stock, pay capital gains taxes (28%) on any further increase in stock price above S T or get tax shield on losses. Currently in US: Pay capital gains taxes on S-X once you sell, if not sold within a certain period. 32

33 Illiquidity and Diversification: Complications IV Employee cannot trade the option Banks would be unwilling to lend against it Her human capital is already tied up to with the company: do you also want your financial capital tied up with them?! Ø Can she hedge by writing a call? No market for 5-year calls... Ø Could she synthetically create this position? A synthetic short call requires a short position in the stock... Goldman could do this, not normal employees... Dynamic rebalancing might smell like insider trading... Stock option plans often even prohibit short-selling shares or hedging! 33

34 Bottom Line Reasonable volatility estimates in the 30-35% range imply option grant value of between $8,640 and $11,130. How long will you stay with company? With σ = 35%, even if probability of leaving company is 50%, need a 10% liquidity premium to prefer cash over option. Signaling value might be worth a lot more! 34

35 Average Pay Across 800 Large US Firms

36 Proportion of Each Wage Component

37 Pay for Luck? One aspect of the debate on pay-for-performance is that managers are not rewarded for good performance and not punished for bad performance However, there is a growing concern that some managers are rewarded for luck rather than for performance Also agency problems in options: Example: In January 2003, two weeks before announcing the full year loss for 2002 the company (AK Steel) amended the terms of its annual bonus plan so that bonuses would be pegged to net income excluding special unusual and extraordinary items. (WSJ June 2003)

38 Manipulation of earnings at time of options expiry: SEC Charges Xerox With Fraud Options expire here

39 Compensation and Earnings Restatements Proxy for manipulation is accounting earnings restatements Significantly more restatements if: Higher powered option incentives (a 10% increase in option incentives increases probability of misreporting by 1%; average misreporting 3% of firms) Vested options No effect for restricted stocks, LTIP, Salary+Bonus Firms with earnings restatements reduce option based compensation in following 2 years. Leads to improvements in operating and market performance (Source: Cheng and Farber, 2006, Earnings Restatements, Changes in CEO Compensation, and Firm Performance, Working Paper.)