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Notes
- 1.
American Depository Receipts are issued by foreign companies on US exchanges.
- 2.
In many texts, options are categorized into simply first generation options (vanilla type) and second generation (exotics). This further categorization is intended to give an additional perspective for readers to explore specific areas of exotic options.
- 3.
The drift can be considered as the growth rate of the asset.
- 4.
Also known as a Wiener process. In short, this represents the “uncertainty” or “randomness” present in the underlying asset.
- 5.
Boundary conditions refer to the set of conditions (initial and final) that define the limits to the underlying asset and the option. For example, if the value of an option is the function of the underlying asset S and the time to maturity T, a boundary condition could be V(0,T) = 0. that is, the value of the option is zero at maturity if the asset price is 0.
- 6.
Moneyness is a generalized term used to describe the “profitability” of an option at any given time. An option which is in-the-money has positive intrinsic value, that is, the underlying stock price is greater than the strike price and positive time value. Out-of-the-money call options have zero intrinsic value as the underlying stock price is less than the strike price.
- 7.
Assumes daily monitoring of the barrier. In practice, the barrier can be monitored over any period – daily, weekly, monthly, or even continuously.
- 8.
These probabilities are only approximate. The prices should be adjusted for discounting effects in order to reflect true probabilities.
- 9.
In fact, H() is a Heaviside function, which takes the value of 1 when the argument is true (i.e., the binary strike has been reached) and zero otherwise.
- 10.
The reason for this lies in the fact that geometric rate options have an underlying price process which is lognormal, whereas arithmetic rate options do not.
- 11.
For more on the market entry/exit timing problem in relation to lookback options, see Kat and Heynen (1994).
- 12.
- 13.
There are of course many ways one can express their views in a trading strategy. The following are a generalization of possible instruments.
- 14.
The bid-offer spread refers to the difference between the price at which you can sell at and the price at which you can buy at in the market. When someone says a stock is trading at $24.53, they are usually referring to the mid-market price. A tight bid-offer spread could be $24.52–$24.54 (i.e. you can buy at $24.54 or sell at $24.52).
- 15.
1 basis point refers to 0.01%, a common denomination used in quoting prices for options.
- 16.
Tree methods refer to the class of methods which value an option in discrete time through a lattice. Examples of such methods include Binomial trees (see Cox et al. 1979).
- 17.
Finite difference schemes discretely evaluate the PDE at each time/asset step and are considered to be fast and efficient in pricing options (see Duffy 2006).
- 18.
By the law of large numbers, one would expect that simulating many paths will give us the expected payoff in a risk-neutral world. Discounting this payoff over the life of the option gives the option value.
- 19.
In recent years, variance reduction techniques and the use of low discrepancy sequences have greatly improved the speed and accuracy of MCS methods.
- 20.
- 21.
Note that the Gamma has been scaled to show that the maximum Gamma occurs at-the-money.
- 22.
An ongoing debate amongst practitioners is the usefulness of second (or higher order) Greeks when investing in options. There is no simple answer to the debate but having a grasp of higher order Greeks is more likely to help in assessing the portfolio than not.
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Cheng, K. (2009). Investing in Exotic Options. In: Krishnamurti, C., Vishwanath, R. (eds) Investment Management. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88802-4_15
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