1. Implied Volatility
An Alternative to Black-Scholes
How to estimate Implied Volatility?
Flow chart for the Newton-Raphson
Practical Applications & Next Steps
Implied Volatility for Options on Futures Using
the Cox-Ross-Rubinstein (CRR) Model
Xin Fang, Qinlin Li, Jose Luis Rodriguez
Loyola University Chicago
Quinlan School of Business
June 25, 2015
Xin Fang, Qinlin Li, Jose Luis Rodriguez Implied Volatility for Options on Futures

2. Implied Volatility
An Alternative to Black-Scholes
How to estimate Implied Volatility?
Flow chart for the Newton-Raphson
Practical Applications & Next Steps
Overview
Implied Volatility
An Alternative to Black-Scholes
How to estimate Implied Volatility?
Flow chart for the Newton-Raphson
Practical Applications & Next Steps
Xin Fang, Qinlin Li, Jose Luis Rodriguez Implied Volatility for Options on Futures

3. Implied Volatility
An Alternative to Black-Scholes
How to estimate Implied Volatility?
Flow chart for the Newton-Raphson
Practical Applications & Next Steps
Option Pricing for for European options - Black-Scholes
C = SN(d1) − Ke−rT
N(d2)
Where d1, is given by:
d1 =
ln( S
K ) + (r + 0.5σ2)T
σ
√
T
And d2 is determine as:
d2 = d1 − σ
√
T
1. Here, all inputs are observable except σ.
2. Setting the above formula equal to the market price of the call
option and solving for σ gives the implied volatility (forward
looking).
Xin Fang, Qinlin Li, Jose Luis Rodriguez Implied Volatility for Options on Futures

4. Implied Volatility
An Alternative to Black-Scholes
How to estimate Implied Volatility?
Flow chart for the Newton-Raphson
Practical Applications & Next Steps
An Alternative to Black-Scholes
There exists a discrete-time analog to the continuous time
Black-Scholes model, the binomial model.
Figure: This model can handle early exercise (American Options)
Xin Fang, Qinlin Li, Jose Luis Rodriguez Implied Volatility for Options on Futures

5. Implied Volatility
An Alternative to Black-Scholes
How to estimate Implied Volatility?
Flow chart for the Newton-Raphson
Practical Applications & Next Steps
How to estimate Implied Volatility?
Two standard approaches
1. Make simplifying assumptions to the Black-Scholes model,
enabling one to solve for σ, by expanding the expression
around a point K = SerT , using Taylor Series or similar
(CM,BCS,BS).
2. Use an iterative procedure (e.g., Newton-Raphson) to update
estimate of the implied vol. Relies crucially on a reasonable
first guess.
We rely on (1) above to inform our initial volatility guess.
Xin Fang, Qinlin Li, Jose Luis Rodriguez Implied Volatility for Options on Futures

6. Implied Volatility
An Alternative to Black-Scholes
How to estimate Implied Volatility?
Flow chart for the Newton-Raphson
Practical Applications & Next Steps
CRR Model - Newton-Raphson Flow Chart
Start
Calculate Initial Volatility Guess
CRR Model computes option price
compare converges
New Guess
stop
Market Price
NO
YES
Xin Fang, Qinlin Li, Jose Luis Rodriguez Implied Volatility for Options on Futures

7. Implied Volatility
An Alternative to Black-Scholes
How to estimate Implied Volatility?
Flow chart for the Newton-Raphson
Practical Applications & Next Steps
BCS Model Manager Graph
Xin Fang, Qinlin Li, Jose Luis Rodriguez Implied Volatility for Options on Futures

8. Implied Volatility
An Alternative to Black-Scholes
How to estimate Implied Volatility?
Flow chart for the Newton-Raphson
Practical Applications & Next Steps
BCS Model Kernel Graph
Xin Fang, Qinlin Li, Jose Luis Rodriguez Implied Volatility for Options on Futures

9. Implied Volatility
An Alternative to Black-Scholes
How to estimate Implied Volatility?
Flow chart for the Newton-Raphson
Practical Applications & Next Steps
Practical Applications
1. Implied volatility outperforms time-series models based on
historical data for the purposes of forecasting volatility.
2. Volatility is an important input into VAR and other models.
Relevant to all money managers.
3. Using CMEs S&P500 futures options (minis) we have the
highest quality data thereby maximizing efficacy.
Xin Fang, Qinlin Li, Jose Luis Rodriguez Implied Volatility for Options on Futures

10. Implied Volatility
An Alternative to Black-Scholes
How to estimate Implied Volatility?
Flow chart for the Newton-Raphson
Practical Applications & Next Steps
Next Steps
1. Program the Newton-Raphson algorithm to run in the DFE.
2. Bring in a time dimension to the problem (estimating a vol
surface instead of a smile).
3. Migrate all calculations to fixed point.
4. Consider other approaches that might better exploit the DFE
(e.g., Monte Carlo).
5. Create something akin to the VIX using CME contracts?
Xin Fang, Qinlin Li, Jose Luis Rodriguez Implied Volatility for Options on Futures