Two Little Vannas Go to Market
We look for a coherent way to extend the Vanna-Volga method for European options to Barrier options. By making use of standard semi-static replication results, we show that a coherent adjustment to Black-Scholes price is made of two separate components: the initial Vanna-Volga adjustment for the replicating portfolio and the expected Vanna-Volga adjustment for the same portfolio at the time the barrier is hit times the hit probability. We then compare these results with the way the Vanna-Volga method is commonly extended to barrier options, namely by multiplying the Vanna-Volga adjustment by the probability that the barrier will not be hit during the life of the option. By direct comparison, we show that the standard method can result in significant mispricing and smile-risk misrepresentation.