In the Black & Scholes formula it is assumed that the volatility is constant. However, using historical option prices one can uniquely solve for the implied volatility. The implied volatility can then be used in the Black & Scholes formula to retrieve the market prices. In practice, options with the same underlying asset but different strikes/maturities require different implied volatilities. This is inconsistent since the implied volatility should not depend on the specifications of the contract. Thus the implied volatility is a function of the strike price. The plot of the implied volatility against the strike price is often refered to as volatility smile or market skew.
A way to handle these skews is to use local volatility models developed by Dupire. In local volatility models it is assumed that the volatility depends on the current stock price and time. The idea is to calibrate the local volatility model to market prices of liquid European options. In other words the local volatility function is varied until the theoretical prices match the actual market prices of the option. When the calibration is finished, the model is fit to correctly reproduce all the market prices of options for all strikes and maturities. Thus the local volatility model provides a method of pricing options in the presence of market skews.
One can also introduce a stochastic process for the volatility itself. In that case the volatility process is neither constant (as in Black & Scholes) nor deterministic (as in local volatility models). These models are called stochastic volatility models. One particular stochastic volatility model is the SABR model (derived by Hagan) where the asset price and the volatility are correlated. The SABR model is handy because there exists a closed form approximation for the implied volatility. The main reason for choosing the SABR model in this thesis is due to the fact that it is widely used in the financial industry.
The advantage of using local and stochastic volatility is that it provides us with additional information compared to using the implied volatility. The implied volatility skew tells us information about the distribution of the underlying asset at maturity. This is satisfactory when working with vanilla options but to price path dependent options (i.e. barrier options) it is also neccessary to know whether the path of the underlying asset has hit the option barrier. Both local and stochastic volatility approaches have their pitfalls.
The dynamic behavior of skews predicted by local volatility models is the opposite of the behavior observed in the market: when the price of the underlying asset decreases the model suggests that the smile shifts to higher prices and vice versa. Stochastic volatility on the other hand, introduces another source of randomness (i.e. the volatility of the volatility) which leads to an incomplete market meaning that hedging using the underlying asset and the risk free interest rate is not sufficient.
Contents
1 Introduction
2 Theory
- 2.1 Notation
2.2 Local volatility
2.3 Stochastic volatility
2.4 Monte Carlo simulation
2.5 Barrier options
3 Data
4 Calibration
- 4.1 Dupire’s formula
4.2 Local volatility - Tikhonov regularization
4.3 Local volatility - the Crank-Nicholson scheme
4.4 Stochastic volatility - SABR
5. Calibration results
- 5.1 Local volatility
5.2 Stochastic volatility
6 Pricing barrier options
- 6.1 Preliminary conditions
6.2 Barrier option prices - tables
6.3 Barrier option prices - graphs
7 Analysis and Discussion
8 Conclusions
9 References
Download
Investigation of the effect of using stochastic and local volatility when pricing barrier options
