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Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew Mentor : Chris Prouty Members : Ping An, Dawei Wang, Rui Yan, Shiyi Chen, Fanda Yang, Che Wang 2010 Modeling Program Team 2, School of Mathematics, UMN present. All rights reserved. Version: 20100116 Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew Outline • Background • Assumptions • Our Model - Workflow - Cubic Spline Interpolation - Market Implied Distribution - Implied Volatility Calculation - Extending Data - Denoise MID • Test (Monte Carlo) & Result - BS Test - MID PDF RVs Test • Improvement - Volatility Surface • Conclusion 2 Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew Background • Before the Black Monday in 1987, the volatility smile looks like this: • After Black Monday, we have a volatility smile, but if the market moves, the seller of the option may lose or make money on the option account even he/she has already deltahedged. So the seller needs to sell/buy extra underlying to remain delta-neutral. Our model is to calculate how much is the extra delta that we need to take into consideration, and we called it “skew delta”. 3 Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew Assumptions • It is possible to borrow and lend cash at a known constant risk-free interest rate. • The price follows a Geometric Brownian motion with constant drift and volatility. • There are no transaction costs. • The stock does not pay a dividend. • All securities are perfectly divisible (i.e. it is possible to buy any fraction of a share). • There are no restrictions on short selling. • There is no arbitrage opportunity. • The price follows a implied motion that we can know from our MID method. • The volatility smile doesn’t change its shape or increase/decrease, it just moves paralleled to the left or right. 4 Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew Our Model - Workflow IV, CS, Ext MID, Denoise Plug In Random Path Expected Volatility Integration Plug In Expected S Plug In Plug In Plug In Plug In Daily P&L Calculation Calculate P&L to test 5 Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew Our Model - Implied Volatility Calculation • Newton’s method – Fast – Local convergence • Bisection – Self-determine starting points 6 Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew Our Model - Implied Volatility Calculation • Newton’s method – Fast – Local convergence • Bisection – Self-determine starting points Opt price f(x) Vol. 7 Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew Our Model - Implied Volatility Calculation • Newton’s method – Fast – Local convergence • Bisection – Self-determine starting points – Extremes removal MSFT Apr 16th S: 30.66 K: 40.00 P: 9.40 (9.314) 8 Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew Our Model - Cubic Spline Interpolation • Volatility Skew: The variation of implied volatility with strike price • Cubic Splines : Method To approximate a function continuously when we are only given a sample of the values. 9 Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew Our Model - Cubic Spline Interpolation The conditions of Cubic Splines: 10 Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew Our Model - Extending Data • Using Least Squares Method to extend the skew curve • Least Square Assumption: The best fitting curve has the least square error: 11 Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew Our Model - Extending Data • The unknown coefficients a, b and c must yield zero first derivatives 12 Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew Our Model - Extending Data • Example: K=[5,7.5,10,12.5,15,17.5,20,22.5], vol=[1.22,1.2,0.9,0.82,0.74,0.6,0.58,0.5] 13 Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew Our Model - Market Implied Distribution 14 Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew Our Model - Market Implied Distribution 15 Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew Our Model - Market Implied Distribution 16 Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew Our Model - Denoise MID • Problem raised before mid-term: Negative probabilities from market data • Improvement: Denoise Throw away the corresponding volatility value that has negative market implied density. 17 Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew Our Model - Market Implied Distribution 18 Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew Our Model - Market Implied Distribution 19 Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew Test (Monte Carlo) & Result S • Generate the underlying price process • 1.B-S Model 2.Return rate distribution σ • Get every day skew curve from the assumption • Calculate implied volatility P, δ, ν δ′ P&L • Calculate Option Price, B-S Delta and Vega • Calculate Skew-Delta from the formula get new Delta • “Skew-Delta=Vega*(ES-S)/(Evol-vol)” • Calculate P&L of the path • Get the statistics: Mean and Standard Deviation 20 Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew Test (Monte Carlo) & Result - BS Generate Underlying Price 1. B-S Model dS = S ( µdt + σdw) Simulate 1 Million times Mean of P&L Mean (BS Delta) Mean of P&L Var (BS Delta) 0.001029 0.00025853 Mean of P&L Mean (New Delta) 0.0914 Mean of P&L Var 184710 (New Delta) The reason we have such a bad result is that we use the B-S assumption to generate the underlying price process. 21 Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew Test (Monte Carlo) & Result - Distribution 2. Return rate distribution Model The distribution of the return rate of underlying price is available, so we can get the CDF. Generate one random number x in [0,1], use the inverse CDF function get the number y. Then y follows the given distribution. 22 Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew Test (Monte Carlo) & Result - Distribution 2. Return rate distribution Model Use the random return rate to generate the underlying price process. St +1 = St × e random _ return _ rate Simulate 100 thousand times Mean of P&L Mean (BS Delta) 0.012278 Mean of P&L Std 0.0000548 (BS Delta) Mean of P&L Mean (New Delta) 0.011251 Mean of P&L Std (New Delta) 0.0024492 The new Mean of P&L is better than the older one at the cost of the worse standard deviation. 23 Test (Monte Carlo) & Result Result: Advantage: better mean of P&L (long run) Disadvantage: worse stand deviation of P&L (short run) Maybe a good news for traders who want to hedge options in a long run! 24 Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew Improvement - Volatility Surface • In addition to volatility skew, we can plot the 3-D Volatility Surface: variation of implied volatilities with strike price and time to maturity. • Referring to cubic splines Method to interpolate between volatilities with different maturities and same Strike Prices 25 Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew Improvement - Volatility Surface Data from Options which underlying is S&P DEP RECEIPTS (SPY) 26 Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew Conclusion From this modeling program, we’ve learnt: • • • • • Try to create a model to improve the B-S model if there is a volatility smile. Why we need to use bisection method rather than Newton's method to get the volatility smile from the market, and why sometimes both these two methods can’t work. How to interpolate, extend or eliminate some points from a given data in order to maintain its information in the greatest degree but still qualify our standard. How to know the market movement of the future if we know today’s market information. How to generate a bunch of random numbers that follows a given Probability Density Function. And we’ve also learnt: • • How to work as a team to focus on a problem, discuss and solve it. How to break programming task into pieces for everybody, define the standard and make it up at last. 27 Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew Thank You!