**Calibration of the Hull-White Stochastic Volatility Model**

**Introduction**

**Stochastic**

Model

Model

**Calibration**

Objective of Calibration

Conclusion

Brief Introduction

Black-Scholes assumes a constant volatility, which does not reflect the market

Hull-White incorporates changes in volatility over time according to a geometric Brownian motion

Hull-White model is still a model depending on unknown parameters: it still needs to be calibrated to be used for pricing options on a given date

Calibration of the Hull-White model for S&P 500 options pricing on June 30th 2006:

Derive the S&P 500 implied volatility surface on that same date.

Objective of the project

**Team member:**

Delphine Radet Yang Li

Bochen Liu Yang Bai

Yuliang Zhang Yangbing Li

Delphine Radet Yang Li

Bochen Liu Yang Bai

Yuliang Zhang Yangbing Li

Hull White Stochastic Volatility Model

Thank you!

**Application :**

Implied Volatility Surface

Implied Volatility Surface

Stochastic Stock Price :

Stochastic Variance :

Stock Price and Variance are assumed to be correlated :

where ρ is the correlation, and are independant standard

Brownian motions

thank

2 pricing methods :

2 optimization tools :

2 pricing methods :

Monte Carlo Simulation : simulation of the stock price and the variance over the option life

Black-Scholes pricing with a stochastic volatility distribution

2 optimization tools :

- Excel - SOLVER

- Matlab - Genetic Algorithm

IVY options prices

Data taken from the IVY database:

S&P 500 spot

Risk-free rate (LIBOR 3M)

Dividends

Initial volatility and variance V0 (Implied volatility of very short-term at-the-money option)

Correlation between the index level and its variance (estimated from historical data)

Monte Carlo Pricing (1)

Pricing Method :

Simulation of N sample paths for the variance and the index level:

V(i+1)=V(i) * exp ( (a-b²/2).dt + b sqrt(dt) )

S(i+1)=S(i) * exp ( (r-V(i)/2).dt + sqrt(V(i).dt) ) – div(i)

where and are two correlated standard normal random variables

Expected payoff as the mean of the payoffs obtained from each sample path

Hull White Price as the present value of the expected payoff

Monte Carlo Pricing (2)

Minimization of :

(Hull White Price/Market Price - 1 ) ²

Optimization using Excel SOLVER :

Limitations of Excel : 16 000 sample paths at most

No conclusive results

Optimization using Matlab genetic algorithm:

Limitations : execution time

Using 20 000 sample paths, 41 options and 20 iterations, we get :

a=-0.1430 , b=0.4071 , objective value =0.1133

Determine the parameters a, b and the correlation

ρ that minimize the difference between the Hull White prices and the market prices taken from the IVY database.

Black Scholes with Hull-White volatility (1)

Principles :

Where and are Independent Standard Brownian Motion. If we assume ρ=0, then:

Black Scholes with Hull-White volatility (1)

Principles :

Variance V is a stochastic variable which subjects to

Geometric Brownian Motion. Since σ^2=V

Volatility σ is also Geometric Brownian Motion, and it has the sam distribution with a lognormal distributed variable.

Black Scholes with Hull-White volatility (1)

Principles:

The PDF of x is:

for any fixed a,b, and x>0

We also have Black-Scholes Pricing Model BS(x,K,T),

The price of the option is:

Black Scholes with Hull-White volatility (1)

Principles :

Minimize

by changing a and b

Key assumptions:

ρ= 0

σ0 =0.149242

Dividend yield 1.84% (2006.06.30)

Data was filtered: exclude deep in/out of the money options(153 options used)

Black Scholes with stochastic volatility (2)

Integration with Gauss-Kronrod Numerical method

Optimization using Genetic Algorithm

(coded in MATLAB):

a=-0.0626; b=0.0869;

value of objective function 0.1634

Conclusion on the calibration :

The pricing method is determinant in the calibration process.

The model might not totally reflect market movements : good results can be obtained only for a certain type of options (not too far in/out the money).

Implied Volatility Surface (1)

Generate Implied Volatility:

Minimize

— standard implied volatility given by IVY database, graph 1

— Input_Price is real option price, graph 2

— Input_Price is Hull-White option price, graph 3

{after calibrating the Hull-White, we get the parameters,

and we can price options with any (K,T) }

Gaussian Kernel to smooth the surface

Performance of Implied

Volatility Surface

Implied Volatility Surface (2)

Advantages of using the Hull White Model :

We can get a more continuous volatility surface

than when directly matching Black Scholes

with market data.

Disadvantages :

The Hull White model used to derive the

volatility surface needs to be perfectly calibrated

in order to get an accurate volatility surface,

which is hard to do in practice. This method

consequently lacks precision.

Thanks for your attention!