Rainbow option

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Rainbow option is a derivative exposed to two or more sources of uncertainty, [1] as opposed to a simple option that is exposed to one source of uncertainty, such as the price of underlying asset.

Contents

The name of rainbow comes from Rubinstein (1991), [2] who emphasises that this option was based on a combination of various assets like a rainbow is a combination of various colors. More generally, rainbow options are multiasset options, also referred to as correlation options, or basket options. Rainbow can take various other forms but the combining idea is to have a payoff that is depending on the assets sorted by their performance at maturity. When the rainbow only pays the best (or worst) performing asset of the basket, it is also called best-of (or worst-of). Other popular options that can be reformulated as a rainbow option are spread and exchange options. [3]

Overview

Rainbow options are usually calls or puts on the best or worst of n underlying assets. [4] Like the basket option, which is written on a group of assets and pays out on a weighted-average gain on the basket as a whole, a rainbow option also considers a group of assets, but usually pays out on the level of one of them. [5]

A simple example is a call rainbow option written on FTSE 100, Nikkei and S&P 500 which will pay out the difference between the strike price and the level of the index that has risen by the largest amount of the three. [5]

Another example is an option that includes more than one strike on more than one underlying asset with a payoff equivalent to largest in-the-money portion of any of the strike prices. [6]

Alternatively, in a more complex scenario, the assets are sorted by their performance at maturity, for instance, a rainbow call with weights 50%, 30%, 20%, with a basket including FTSE 100, Nikkei and S&P 500 pays 50% of the best return (at maturity) between the three indices, 30% of the second best and 20% of the third best. [3]

The options are often considered a correlation trade since the value of the option is sensitive to the correlation between the various basket components.

Rainbow options are used, for example, to value natural resources deposits. Such assets are exposed to two uncertainties—price and quantity.

Some simple options can be transformed into more complex instruments if the underlying risk model that the option reflected does not match a future reality. In particular, derivatives in the currency and mortgage markets have been subject to liquidity risk that was not reflected in the pricing of the option when sold.

Payoff

Rainbow options refer to all options whose payoff depends on more than one underlying risky asset; each asset is referred to as a color of the rainbow. [3]

Examples of these include: [7]

Thus, the payoffs at expiry for rainbow European options are:

Pricing and valuation

Rainbow options are usually priced using an appropriate industry-standard model (such as Black–Scholes) for each individual basket component, and a matrix of correlation coefficients applied to the underlying stochastic drivers for the various models.

While there are some closed-form solutions for simpler cases (e.g. two-color European rainbows), [11] semi-analytic solutions, [12] and analytical approximations, [13] [14] [15] the general case must be approached with Monte Carlo or binomial lattice methods. For bibliography see Lyden (1996). [16]

Related Research Articles

The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the parabolic partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price given the risk of the security and its expected return. The equation and model are named after economists Fischer Black and Myron Scholes; Robert C. Merton, who first wrote an academic paper on the subject, is sometimes also credited.

In financial mathematics, put–call parity defines a relationship between the price of a European call option and European put option, both with the identical strike price and expiry, namely that a portfolio of a long call option and a short put option is equivalent to a single forward contract at this strike price and expiry. This is because if the price at expiry is above the strike price, the call will be exercised, while if it is below, the put will be exercised, and thus in either case one unit of the asset will be purchased for the strike price, exactly as in a forward contract.

An interest rate cap is a type of interest rate derivative in which the buyer receives payments at the end of each period in which the interest rate exceeds the agreed strike price. An example of a cap would be an agreement to receive a payment for each month the LIBOR rate exceeds 2.5%.

In finance, the style or family of an option is the class into which the option falls, usually defined by the dates on which the option may be exercised. The vast majority of options are either European or American (style) options. These options—as well as others where the payoff is calculated similarly—are referred to as "vanilla options". Options where the payoff is calculated differently are categorized as "exotic options". Exotic options can pose challenging problems in valuation and hedging.

Strike price Options fixed price to exercise it on the expiration date

In finance, the strike price of an option is a fixed price at which the owner of the option can buy, or sell, the underlying security or commodity. The strike price may be set by reference to the spot price, which is the market price of the underlying security or commodity on the day an option is taken out. Alternatively, the strike price may be fixed at a discount or premium.

In finance, the binomial options pricing model (BOPM) provides a generalizable numerical method for the valuation of options. Essentially, the model uses a "discrete-time" model of the varying price over time of the underlying financial instrument, addressing cases where the closed-form Black–Scholes formula is wanting.

In mathematical finance, the Greeks are the quantities representing the sensitivity of the price of derivatives such as options to a change in underlying parameters on which the value of an instrument or portfolio of financial instruments is dependent. The name is used because the most common of these sensitivities are denoted by Greek letters. Collectively these have also been called the risk sensitivities, risk measures or hedge parameters.

In finance, moneyness is the relative position of the current price of an underlying asset with respect to the strike price of a derivative, most commonly a call option or a put option. Moneyness is firstly a three-fold classification: if the derivative would have positive intrinsic value if it were to expire today, it is said to be in the money; if it would be worthless if expiring with the underlying at its current price it is said to be out of the money, and if the current underlying price and strike price are equal, it is said to be at the money. There are two slightly different definitions, according to whether one uses the current price (spot) or future price (forward), specified as "at the money spot" or "at the money forward", etc.

Lookback options, in the terminology of finance, are a type of exotic option with path dependency, among many other kind of options. The payoff depends on the optimal underlying asset's price occurring over the life of the option. The option allows the holder to "look back" over time to determine the payoff. There exist two kinds of lookback options: with floating strike and with fixed strike.

In finance, an exotic option is an option which has features making it more complex than commonly traded vanilla options. Like the more general exotic derivatives they may have several triggers relating to determination of payoff. An exotic option may also include non-standard underlying instrument, developed for a particular client or for a particular market. Exotic options are more complex than options that trade on an exchange, and are generally traded over the counter (OTC).

Rational pricing is the assumption in financial economics that asset prices - and hence asset pricing models - will reflect the arbitrage-free price of the asset as any deviation from this price will be "arbitraged away". This assumption is useful in pricing fixed income securities, particularly bonds, and is fundamental to the pricing of derivative instruments.

In finance, a foreign exchange option is a derivative financial instrument that gives the right but not the obligation to exchange money denominated in one currency into another currency at a pre-agreed exchange rate on a specified date. See Foreign exchange derivative.

A variance swap is an over-the-counter financial derivative that allows one to speculate on or hedge risks associated with the magnitude of movement, i.e. volatility, of some underlying product, like an exchange rate, interest rate, or stock index.

The backspread is the converse strategy to the ratio spread and is also known as reverse ratio spread. Using calls, a bullish strategy known as the call backspread can be constructed and with puts, a strategy known as the put backspread can be constructed.

In finance, a spread option is a type of option where the payoff is based on the difference in price between two underlying assets. For example, the two assets could be crude oil and heating oil; trading such an option might be of interest to oil refineries, whose profits are a function of the difference between these two prices. Spread options are generally traded over the counter, rather than on exchange.

Option (finance) Right to buy or sell a certain thing at a later date at an agreed price

In finance, an option is a contract which conveys to its owner, the holder, the right, but not the obligation, to buy or sell an underlying asset or instrument at a specified strike price on or before a specified date, depending on the style of the option. Options are typically acquired by purchase, as a form of compensation, or as part of a complex financial transaction. Thus, they are also a form of asset and have a valuation that may depend on a complex relationship between underlying asset value, time until expiration, market volatility, and other factors. Options may be traded between private parties in over-the-counter (OTC) transactions, or they may be exchange-traded in live, orderly markets in the form of standardized contracts.

In finance, a volatility swap is a forward contract on the future realised volatility of a given underlying asset. Volatility swaps allow investors to trade the volatility of an asset directly, much as they would trade a price index. Its payoff at expiration is equal to

A basket option is a financial derivative, more specifically an exotic option, whose underlying is a weighted sum or average of different assets that have been grouped together in a basket. A basket option is similar to an index option, where a number of stocks have been grouped together in an index and the option is based on the price of the index, but differs in that the members and weightings of an index can change over time while those in a basket option do not.

Synthetic position

In finance, a synthetic position is a way to create the payoff of a financial instrument using other financial instruments.


In finance, an option on realized variance is a type of variance derivatives which is the derivative securities on which the payoff depends on the annualized realized variance of the return of a specified underlying asset, e.g. stock index, bond, exchange rate, etc. Another liquidated security of the same type is variance swap, which is, in other words, the futures contract on realized variance.

References

  1. ↑ "What Does Rainbow Option Mean?". investopedia.com . Retrieved 2014-02-12.
  2. 1 2 Rubinstein, Mark. "Somewhere over the rainbow." Risk 4.11 (1991): 61-63.
  3. 1 2 3 Benhamou, Eric. Rainbow options
  4. ↑ "Supported Equity Derivatives". mathworks.com. Retrieved 2014-02-12.
  5. 1 2 Choudhry, Moorad. Bond and money markets: strategy, trading, analysis. Butterworth-Heinemann, 2003. p.838
  6. ↑ Taleb, Nassim. Dynamic hedging: managing vanilla and exotic options. Vol. 64. John Wiley & Sons, 1997. p.384
  7. ↑ Ouwehand, Peter, and Graeme West. "Pricing rainbow options." Wilmott magazine 5 (2006): 74-80.
  8. 1 2 3 4 5 Stulz, RenéM. "Options on the minimum or the maximum of two risky assets: analysis and applications." Journal of Financial Economics 10.2 (1982): 161-185.
  9. 1 2 3 4 5 Johnson, Herb. "Options on the maximum or the minimum of several assets." Journal of Financial and Quantitative Analysis 22.3 (1987): 277-283.
  10. 1 2 Margrabe, William. "The value of an option to exchange one asset for another." The journal of finance 33.1 (1978): 177-186
  11. ↑ Rubinstein, Mark. Exotic options. No. RPF-220. University of California at Berkeley, 1991. URL:http://www.haas.berkeley.edu/groups/finance/WP/rpf220.pdf
  12. ↑ Austing, Peter. Smile Pricing Explained. Springer, 2014.
  13. ↑ Alexander, Carol, and Aanand Venkatramanan. "Analytic Approximations for Multi‐Asset Option Pricing." Mathematical Finance 22.4 (2012): 667-689.
  14. ↑ Hull, John C. Options, futures, and other derivatives. Eighth ed. Prentice Hall, 2012. p.588
  15. ↑ Wystup, Uwe. "FX Basket Options Valuation with Smile." (2009).
  16. ↑ Lyden, Scott. "Reference check: a bibliography of exotic options models." The Journal of Derivatives 4.1 (1996): 79-91.
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