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Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew
Mentor : Chris Prouty
Members : Ping An, DaweiWang, Rui Yan, Shiyi Chen,
Fanda Yang, Che Wang
2010 Modeling Program Team 2, School of Mathematics, UMN present.All rights reserved.
Version: 20100116
Outline
• Background• Assumptions• Our Model
- Workflow - Implied Volatility Calculation- Cubic Spline Interpolation - Extending Data- Market Implied Distribution - Denoise MID
• Test (Monte Carlo) & Result- BS Test - MID PDF RVs Test
• Improvement- Volatility Surface
• Conclusion2
Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew
Background
• 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 delta-hedged. 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
• Before the Black Monday in 1987, the volatility smile looks like this:
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.
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• 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.
Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew
Our Model - Workflow
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Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew
Random Path Expected SExpected Volatility
Calculate P&L to test
IV, CS, Ext MID, Denoise
Integration
Plug In
Plug In
Plug In
Plug InPlug In
Plug In
Daily P&L Calculation
Our Model - Implied Volatility Calculation
• Newton’s method– Fast– Local convergence
• Bisection– Self-determine starting points
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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
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f(x)Opt price
Vol.
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
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MSFT Apr 16th
S: 30.66 K: 40.00P: 9.40 (9.314)
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.
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Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew
Our Model - Cubic Spline Interpolation
The conditions of Cubic Splines:
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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:
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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
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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]
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Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew
Our Model - Market Implied Distribution
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Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew
Our Model - Market Implied Distribution
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Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew
Our Model - Market Implied Distribution
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Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew
Our Model - Denoise MID
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Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew
• Problem raised before mid-term: Negative probabilities from market data
• Improvement: DenoiseThrow away the corresponding volatility value that has negative market implied density.
Our Model - Market Implied Distribution
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Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew
Our Model - Market Implied Distribution
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Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew
Test (Monte Carlo) & Result
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Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew
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• Calculate Option Price, B-S Delta and Vega
δ′• Calculate Skew-Delta from the formula get new Delta• “Skew-Delta=Vega*(ES-S)/(Evol-vol)”
P&L• Calculate P&L of the path• Get the statistics: Mean and Standard Deviation
Test (Monte Carlo) & Result - BSGenerate Underlying Price 1. B-S Model
Simulate 1 Million times
The reason we have such a bad result is that we use the B-S assumption to generate the underlying price process. 21
)dd(d wtSS σµ +=
Mean of P&L Mean (BS Delta) 0.001029
Mean of P&L Var (BS Delta) 0.00025853
Mean of P&L Mean (New Delta) 0.0914
Mean of P&L Var (New Delta) 184710
Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew
Test (Monte Carlo) & Result - Distribution2. 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.
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Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew
Test (Monte Carlo) & Result - Distribution2. Return rate distribution Model
Use the random return rate to generate the underlying price process.
Simulate 100 thousand times
The new Mean of P&L is better than the older one at the cost of the worse standard deviation.
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Mean of P&L Mean (BS Delta) 0.012278
Mean of P&L Std (BS Delta) 0.0000548
Mean of P&L Mean (New Delta) 0.011251
Mean of P&L Std (New Delta) 0.0024492
ratereturnrandomtt eSS __
1 ×=+
Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew
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!
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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
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Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew
Improvement - Volatility Surface
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Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew
Data from Options which underlying is S&P DEP RECEIPTS (SPY)
ConclusionFrom 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!
Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew