Empirical Option Pricing

Transcription

1 Empirical Option Pricing

2 Holes in Black& Scholes Overpricing Price pressures in derivatives and underlying Estimating volatility and VAR

3 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

4 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

5 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?

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

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

8 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)?

9 Holes in Black& Scholes Overpricing Price pressures in derivatives and underlying Estimating volatility and VAR

10 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:

11 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

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

13 13 15 May 2016 Matti Suominen (Aalto)

14 Evidence that structured products are more overpriced (and more complex the more so). Source: Thomas Ruf, UBC, Dynamics of Overpricing in Structured products

15 CONCLUSION 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.

16 Holes in Black& Scholes Overpricing Price pressures in derivatives and underlying Estimating volatility and VAR

17 Goldman Roll (Roslander, 2014) May 2016

18 Commodities Limits to Arbitrage and Hedging: Evidence from Commodity Markets Viral V. Acharya, Lars A. Lochstoer and Tarun Ramadorai - Equilibrium model of commodity markets in which speculators are capital constrained - Commodity producers have hedging demands for commodity futures. - Increases (decreases) in producers hedging demand (speculators.riskcapacity) increase hedging costs via price-pressure on futures, reduce producers inventory holdings, and thus spot prices. - Consistent with their model, producers default risk forecasts futures returns, spot prices, and inventories in oil and gas market data from

19 Matti Suominen (Aalto) May 2016

20 Inefficiencies in the stock market: Monthly Return Reversals at NYSE On average 28% of daily returns revert within a month (24% within a week). - Temporary price movements associated with these return reversals have increased daily return volatility by 20% Question: Do stock market ineffieciencies affect options pricing?

21 Order imbalances in stock market affect the options implied volatility: Here DOTS = distance between option implied stock price (average from bid and ask for a synthetic share) and traded stock price Goncalves-Pinto 2014

22 Holes in Black& Scholes Overpricing Price pressures in derivatives and underlying Estimating volatility and VAR

23 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

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

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

26 Estimating Volatilities and Correlations

27 Standard Approach to Estimating Volatility Define σ 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

28 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 σ n = u m i = 1 n i

29 Weighting Scheme Instead of assigning equal weights to the observations we can set σ m i= 1 2 m n = α i iu 2 = 1 n i where α i = 1

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

31 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

32 Example of GARCH Conditional and Unconditional T-Bill Volatility Standard Deviation /3/93 4/13/ 93 7/22/ 93 10/30/93 2/7/94 5/18/ 94 8/26/ 94 12/4/ 94 3/14/ 95 6/22/ 95 9/30/ 95 1/8/96 4/17/ 96 Date

33 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

34 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

35 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 σ = σ + σ + 2σ σ ρ A + B A B A B AB