Derivatives Analytics with Python: Data Analysis, Models, Simulation, Calibration and Hedging

CHAPTER 11 Model Calibration

11.1 Introduction

Historically, theoretical valuation has been seen as a process where a number of input parameters of a model or formula determine the price/value of a derivative instrument. The fundamental assumption behind this reasoning is that you can observe, in principle, anything in the markets that finally determines the price/value of a security—markets should be, after all, informationally efficient. A major example is the Black-Scholes-Merton formula which takes as input six variables—initial price level of the underlying, the underlying’s volatility, the strike price of the option at hand, time-to-maturity, short rate and maybe dividends paid by the underlying. If you put in numerical values for the six variables, the formula returns a value for the option at hand.

However, if “the market is always right”, what does a model price/value mean which deviates from an observable market value? As earlier chapters discuss, market-based valuation refers to the process where more complex derivatives are valued “in consistency” with observed market prices of plain vanilla derivatives. Therefore, today’s valuation practice requires in the first place that valuation models be capable of replicating observed market values of vanilla products sufficiently well.

This chapter is concerned with the calibration of the general market model to observed market quotes for such vanilla products, i.e. European call options in particular. In comparison to the ...