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Revisiting basic models
Black & Scholes
This review is widely inspired from the work of John C. Hull, published in his book, "Options, Futures, and Other Derivatives".
It is a synthetic summary of the basic principles developed by Black and Scholes.
Binomial
This part reviews the Cox, Ross & Rubinstein Model, also called the Binomial model.
The assumption that underlies this numerical procedure for pricing European and American options is that price movements of underlying assets follow a multiplicative binomial process over discrete periods of time.
Exotic options
Barrier
The additional feature of barrier options is that the contract may be terminated or activated if the underlying trades at a pre-determined level (the trigger or barrier), on or before the expiry date. For an option that is terminated, the trigger level is the "knock-out" level and for one that is activated, the "knock-in" level. As the option may not survive until, or may never be activated before expiry, there is a reduction in the premium cost relative to the European style option.
Binary
Binary options are also called "all-or-nothing options", since they correspond to a straight bet, on whether the underlying value would be above or below a predetermined level, before or at expiry. Unlike vanilla options, they have a non-linear payoff profile, i.e. they either pay out a fixed amount or nothing. This pay-off can either be expressed as ‘cash’ or ‘asset’.
Acumen covers different types of binary options:
- Path independent. For this type, conditions are expressed only on the strike(s): the option premium depends on the underlying relative position at expiry compared to the strike(s) level(s); that is above for a "call", and below for a "put".
- Path dependent, which can present barrier features, simple or double. They can either pay out as soon as the trigger level is reached at "hit", or at "expiry".
Hull & White
Modeling and picing swaptions
Hull and White model enables to build the term structure of interest rate which is widely used by Acumen to price multi-callable structures.
The strongest underlying assumption of the model is that discount factors of all maturity are perfectly correlated between them, and their value at a given time depend on the instantaneous short rate value. The latter is represented by the Ornstein-Uhlenbeck process.
The model is implemented analytically and numerically, by using a mean reversion tree fitted to the market term structure. Both approaches require calibration.
The document explains how to price European, Bermudan and American swaptions.
Callable bonds
Callable, putable and extendible bond structures have embedded option features. In terms of pricing, the structure is stripped into a bond issue and an option on that bond. That way, each component can be approached individually.
Option features can be of European, American, or Bermudan style.
We look more in details how to price these options with the Hull&White term structure model.
Exotic caps & floors
An insight into Digital, CMS/CMT and in Advanced Caps and Floors.
The pricing of these structures is achieved by unrolling Hull and White (analytical and numerical), and extended Black and Scholes (in order to take convexity into consideration).
Volatility Panorama
Volatility types
Family tree of the different volatility surfaces, and analysis of their structure.
Flat Forward Volatility
Caps and Floors are quoted as flat yearly implied volatility.
Although the flat level can be used directly in plain vanilla structures, the pricing of more complex interest rate products requires the determination of a forward volatility for each caplet.
The document describes the mechanism and algorithm implemented in Acumen in order to build a forward surface backed out of a flat.
Strangle & Risk reversal Mechanism
A volatility surface per currency pair is needed to price fx options at all strikes.
A surface is fully defined when:
- The Call volatility for all deltas is known
- The volatility of a Put can be deducted from the volatility of a Call
The difficulty is that the entire surface is not available in the market. Only ATM volatility is quoted.
Acumen offers an algorithm to build the entire surface using two indicators:
- The strangle level (kurtosis). It gives the incremental volatility necessary to price an OTM option.
- The risk reversal level (skew). It gives the difference in the volatility between a Call and a Put with same delta.