As shown in Schoutens et al. (2004), different dynamics for the underlying asset price are able to achieve a similarly good calibration to the market prices of European plain vanilla options. Consequently, these models are able to generate very similar marginal distributions for all future points in time. However, their path-behaviour differs thus often yielding significantly different prices for exotic options. In this post, we provide some intuition for the sign of the price differences between local and stochastic volatility models for barrier and American binary options. These arguments are not novel and we refer to e.g. Baker et al. (2004) for a more in-depth and rigorous discussion.

We start with a few assumptions:

- The local volatility model is calibrated to the implied volatility surface generated by the stochastic volatility model.
- The implied volatility skew follows a typical equity index pattern which is downward sloping in the strike price.
- Interest rates and dividends are zero.

Assumption 1 allows us to isolate the pure model difference. It ensures that both models generate exactly the same marginal distributions for all times-to-maturity. Consequently, they also agree on the European plain vanilla option price surface.

The price differences for barrier options between local and stochastic volatility models can mostly be attributed to the different hitting probabilities that they generate. We thus first focus on the valuation of American binary options. These contracts have a fixed notional payoff at the first hitting time of the underlying asset price to the contractual barrier.

A proxy semi-static hedge for a short position in an American binary option corresponds to buying twice the notional amount in European binary options with the same maturity and barrier; see e.g. Gatheral (2006), p. 112f. We unwind this position upon the first hitting time of the underlying asset to the barrier. In all pure diffusion models, the underlying asset price is exactly equal to the barrier at this time. The European binary option then roughly has a 50% probability of ending in-the-money and thus the value of the hedge portfolio is approximately equal to notional of the American binary option. We now make the previous statement more precise as the devil lies in the details.

A static super-replication strategy for a European binary call (put) options in turn involves buying a European call (put) spread with the same maturity. The long strike is slightly below (above) the barrier and the short strike is equal to the barrier. A European binary option thus has a very high skew exposure. A steeper, more negatively sloped implied volatility smile makes the European binary call (put) more expensive (cheaper)

Since the hedge position in the European binary option has to be unwound upon the first hitting time, it follows that the model implied forward skews are the main determinants of the relative prices. Stochastic volatility models usually generate self-similar forward implied volatility smiles that resembles the spot implied volatility smiles. However, forward implied volatility smiles under the local volatility model flatten out as the maturity increases; see Baker et al. (2004). Consequently, the conditional value of the European binary call (put) option upon the first hitting time is higher (lower) in the stochastic volatility model. Since we are unwinding this position, it follows that the American binary call (put) option is cheaper (more expensive) in the stochastic volatility model. Alternatively, we could think of this as having to buy less (more) European binary call (put) options for the semi static hedge under the stochastic volatility model in the first place.

From the above discussion, it follows that the value of an up & out call (down & out put) option is generally more expensive (cheaper) under the stochastic volatility model.

**References**

Baker, Glyn, Reimer Beneder and Alex Zilber (2004) “FX Barriers with Smile Dynamics,” Working Paper, available at SSRN: http://ssrn.com/abstract=964627

Gatheral, Jim (2006) *The Volatility Surface: A Practitioner’s Guide*: Wiley Finance

Schoutens, Wim, Erwin Simons and Jurgen Tistaert (2004) “A Perfect Calibration! Now What?,” Wilmott, pp. 67-78