Download Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew

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
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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”.
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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.
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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
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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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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.
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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
– Extremes removal
MSFT Apr 16th
S: 30.66
K: 40.00
P: 9.40 (9.314)
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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
• Problem raised before mid-term:
Negative probabilities from market data
• Improvement: Denoise
Throw away the corresponding volatility value that
has negative market implied density.
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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
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
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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.
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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.
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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.
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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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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
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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)
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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.
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Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew
Thank You!