CRR_PRESENTATION

Implied VolatilityAn Alternative to Black-Scholes

How to estimate Implied Volatility?Flow chart for the Newton-RaphsonPractical Applications & Next Steps

Implied Volatility for Options on Futures Usingthe Cox-Ross-Rubinstein (CRR) Model

Xin Fang, Qinlin Li, Jose Luis Rodriguez

Loyola University ChicagoQuinlan School of Business

June 25, 2015

Xin Fang, Qinlin Li, Jose Luis Rodriguez Implied Volatility for Options on Futures



Overview

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−rTN(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 calloption and solving for σ gives the implied volatility (forwardlooking).




An Alternative to Black-ScholesThere exists a discrete-time analog to the continuous timeBlack-Scholes model, the binomial model.

Figure: This model can handle early exercise (American Options)




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 expressionaround a point K = SerT , using Taylor Series or similar(CM,BCS,BS).

2. Use an iterative procedure (e.g., Newton-Raphson) to updateestimate of the implied vol. Relies crucially on a reasonablefirst guess.

We rely on (1) above to inform our initial volatility guess.




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




BCS Model Manager Graph




BCS Model Kernel Graph




Practical Applications

1. Implied volatility outperforms time-series models based onhistorical 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 thehighest quality data thereby maximizing efficacy.




Next Steps

1. Program the Newton-Raphson algorithm to run in the DFE.

2. Bring in a time dimension to the problem (estimating a volsurface 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?


CRR_PRESENTATION

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