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

CHAPTER 2 What is Market-Based Valuation?

2.1 Options and their Value

An equity option represents the right to buy (call) or sell (put) a unit of the underlying stock at a prespecified price (strike) at a predetermined date (European option) or over a determined period of time (American option). Some options are settled in actual stocks; most options, like those on an index, are settled in cash. People or institutions selling options are called option writers. Those buying options are called option holders.

For a European call option on an index with strike 8,000 and index level of 8,200 at maturity, the option holder receives the difference 8, 200 − 8, 000 = 200 (e.g. in EUR or USD) from the option writer. If the index level is below the strike, say at 7,800, the option expires worthless and the writer does not have to pay anything. We can formalize this via the so-called inner value (or intrinsic value or payoff)—from the holder’s viewpoint—of the option

where T is the maturity date of the option, S_T the index level at this date and K represents the strike price. We can now use Python for the first time and plot this inner value function.

A script could look like:

 # # European Call Option Inner Value Plot # 02_MBV/inner_value_plot.py # # (c) Dr. Yves J. Hilpisch # Derivatives Analytics with Python # import numpy as np import matplotlib as mpl import matplotlib.pyplot as ...