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In computer or mathematics jargon, a binary number is one which is given a value of either 0 or 1 and nothing else; in the case of derivatives, a binary option, sometimes referred to as a digital option (hereon we use only the term binary), is an option which pays either an asset out at expiry, or nothing at all based on whether or not the option expires in the money.
Types of Binary Options || Cash-or-Nothing Closed Form || Asset-or-Nothing Closed Form || American Style - One Touch Binary || Advanced Techniques || Known Names & Variants || References
In computer or mathematics jargon, a binary number is one which is given a value of either 0 or 1 and nothing else; in the case of derivatives, a binary option, sometimes referred to as a digital option (hereon we use only the term binary), is an option which pays either an asset out at expiry, or nothing at all based on whether or not the option expires in the money. The payoff structure for a binary is characterised as being discontinuous and these types of exotic options come in one of four formats, or which only the first two will be discussed in this section:
Which you will notice is the same payoff as that of a cash-or-nothing binary.
Yet another type of binary option, a gap option is actually quite similar to a binary-or-nothing binary in terms of its payoff:
5) Cash-or-Nothing Binary / Digitals (Reiner & Rubinstein) 1991 Reiner & Rubinstein's classic 1991 paper introduces these options and a set of closed form analytical formulas which can be applied to the pricing of these options which give payoffs as shown above within a Black-Scholes framework. For standard cash or nothing binaries, which are highly path-dependent, the pricing formula is simple:
For the respective European call and put formulae:
Where:
K is the predetermined cash amount, S is the asset price, X is the strike price, r is the risk free rate, D is the dividend yield,
6) Asset-or-Nothing Binary / Digitals (Cox & Rubinstein) 1985 European Asset-or-Nothing binaries are similar to cash-or-nothing binaries, with the exception that asset binaries, at expiry, will payout either 0 or the asset - compared to cash binaries, where the payout is either 0 or a prescribed cash amount.
Asset-or-nothing binaries can be priced using valuation formulae presented by Cox & Rubinstein (1985) as follows:
Where:
The variables are the same as given for cash-or-nothing binaries.
7) American Style Binary Options (Reiner & Rubinstein) 1991 Like most other options, binaries come in both European style as well as American style early exercise types. American style binary options are often referred to as "one-touch binary/digitals" or "binary-at-hit" and can be priced using a set of formulas given by Reiner & Rubinstein (1991):
Where:
8) Binomial Method / Trinomial Method As with many other exotic options, we can make use of a binomial method in order to solve the pricing problem on a tree by simulating the asset path on a binomial tree and discounting the payoff. However, similar to the case of barrier options, one must be careful when implementing binomial or trinomial trees for options with a discontinuous payoff.
9) Partial Differential Equation With any derivative, we can find a pricing solution by solving the partial differential equation which is associated with the derivative. Common techniques used to solve PDEs involve the use of finite differences.
What many do not realize is that typical lattice methods such as binomial and trinomial trees are in essence, the same as the explicit finite difference technique. But because of various issues related to the use of implicit or explicit finite difference methods, many have accepted the use of the Crank-Nicolson scheme; for example, see Pooley, Vetzal & Forsyth (2001), as this gives rise to quadratic convergence resulting in faster convergence and greater accuracy in results.
Wystup, Schmock & Shreve (2001) consider a binary option under leveraged constraints in order to replicate a portfolio to produce pricing formulas that give better results than the Black-Scholes type equations which tend to undervalue binary options because of difficulties associated with hedging. They extend basic results of binary options to the case of one-touch or American binary options under leverage constraints and even to cases where binary options are combined with barriers to form binary barriers. An overview of binary barriers will appear within its respective section shortly. Returning back to our previous section on the partial differential equation, we can consider an advanced technique such as a multi-step second order backward difference technique (Becker 1998). It is an extension of the standard explicit finite difference method, and we recommend the paper for any interested readers.
Pooley, Vetzal & Forsyth (2001) consider several other techniques for valuing digital options using PDE techniques as well as consideration of methods which incorporate the discontinuous payoff seen in digital options. For other interesting reading, see Heston & Zhou (2000), Becker (1998) and Rannacher (1984).
Binary Options
Additional/Useful List of resources Papers: |
represents the volatility T is the time to maturity and N(x) is the cumulative normal distribution function on x.
