Adjusting the Black- Scholes Framework in the …Adjusting the Black- Scholes Framework in the...

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


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:


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.


Our Model - Workflow

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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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• Bisection– Self-determine starting points

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f(x)Opt price

Vol.




• Bisection– Self-determine starting points– Extremes removal

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MSFT Apr 16th

S: 30.66 K: 40.00P: 9.40 (9.314)


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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Our Model - Cubic Spline Interpolation

The conditions of Cubic Splines:

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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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• The unknown coefficients a, b and c must yield zero first derivatives

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• 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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Our Model - Market Implied Distribution

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Our Model - Denoise MID

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• Problem raised before mid-term: Negative probabilities from market data

• Improvement: DenoiseThrow away the corresponding volatility value that has negative market implied density.


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Test (Monte Carlo) & Result

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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


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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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 ×=+


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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Improvement - Volatility Surface

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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


Thank You!


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