# Pricing and Greeks Simplex Options Black Scholes formula

Modern option pricing theory began when Black and Scholes (1973) and Merton (1973) published their groundbreaking papers on pricing vanilla options. In the same year, the first organized options exchange in the world, the Chicago Board of Options Exchange (CBOE), was established. In the ensuing decades, numerous papers have been published to extend the Black-Scholes model, and countless exotic options have been designed, issued and traded in the markets.

Exotic options are tailor-made to satisfy investors with special requirements on risk and reward. For example, path-dependent options have payoffs that depend critically on the price history of the underlying asset, and they are structured to address that risk. Rainbow options, in contrast, have payoffs that depend on multiple underlying assets, and they are designed to capture multi- factor risks. Although each type of options has been studied intensively in the literature, the options that combine both features (i.e., path dependency and multiple underlying assets) have received few analytical treatments.

This paper contributes to the option pricing theory by analyzing a broad class of path-dependent rainbow options called simplex options and pricing them. This notion is general and includes many known options as special examples, like vanilla options (Black and Scholes 1973, Merton 1973), reset options (Gray and Whaley 1999), rainbow options (Johnson 1987), discrete lookback options (Heynen and Kat 1995), and numerous other options not covered in this paper. Furthermore, it also inspires the design of new path-dependent rainbow options, such as forward-start rainbow options and discrete lookback rainbow options. The building blocks of simplex options will be called the simplex expectations, which can be easily evaluated by our extended Black-Scholes (EBS) formula. These concepts are new to the literature, to which we turn in the following paragraphs.

An expectation is simplex if it is of the form

## ,

where Q is the risk-neutral measure, Sp(tq) is the price of the p-th underlying asset at monitoring time tq, I is the indicator function, and inside the joint event are all pairwise price-comparisons. The EBS formula extends the celebrated Black-Scholes formula to evaluate simplex expectations.

To give a taste of what the EBS formula looks like, consider the following simplex expectation with three underlying assets:

## .

Its EBS formula, similar to the famous Black-Scholes formula, is

where

## .

Above, r is the risk-free rate, N is the standard normal cumulative distribution function (CDF), is the volatility of the a-th underlying asset's return process, and is the correlation between the a-th and b-th assets' returns. This paper offers a general theorem that easily yields the EBS formula for any simplex expectation.

An option is called simplex if its risk-neutral expected payoff is a linear sum of simplex expectations. For example, the vanilla call, with underlying asset S, maturity date T, strike price K and payoff function max(S(T)-K,0), is simplex since its risk-neutral expected payoff,

## ,

is a linear sum of simplex expectations after we associate K, the strike price determined at the initial time 0, with S0(0), the price of the riskless asset S0 at time 0. Many path-dependent rainbow options turn out to be simplex. The primary focus of this paper is the pricing and Greeks of simplex options that include a broad class of complex path-dependent rainbow options.

The underlying asset's prices of path-dependent options may be continuously monitored or discretely monitored. This paper focuses on the method of discrete monitoring, more widely adopted in practice. (We shall drop the phrase “path-dependent” whenever the feature of discrete monitoring is included, which already implies path dependency.)

Numerous studies have been conducted on discretely monitored options and rainbow options. Nonetheless, few attempts are made to price options that combine both features by analytical approaches. Table 1 summarizes the analytical results related to discretely monitored and rainbow options. Simplex options include the options in Table 1 and fill the vacuum in its lower-right quadrant.

We next go over each of the table's four quadrants in turn. For the upper-left quadrant of Table 1, vanilla options are priced by Black and Scholes (1973) and Merton (1973). The original Black-Scholes model assumes that there is only one risky underlying asset in the market. The Black-Scholes-Merton pricing formulas for vanilla options are computationally simple and do not require investors' risk preferences. These characteristics contribute greatly to their popularity.

An option on two or more risky assets is referred to as a rainbow option by Rubinstein (1991). Rainbow options occupy the lower-left quadrant of Table 1. The options that exchange one underlying asset for another are first studied by Margrabe (1978). Stulz (1982) prices the options on the minimum or the maximum of two assets by solving partial differential equations. The options on the maximum or the minimum of several assets are considered by Johnson (1987). Different from those in Stulz (1982), the pricing techniques in Johnson (1987) are from Margrabe (1978).

The upper-right quadrant of Table 1 contains discretely monitored options. One example is the discrete lookback option whose payoff depends on the discretely monitored maximum or minimum historical price. Heynen and Kat (1995) value these kinds of options by the martingale pricing method and give closed-form formulas in terms of multinormal CDFs. Another example is the reset option whose strike price can be reset based on the underlying asset's prices on the reset dates. Gray and Whaley (1999) derive a pricing formula for the reset put option with one reset date. Cheng and Zhang (2000) price the reset option with multiple reset dates. Finally, Liao and Wang (2003) value the reset option with multiple strike resets and reset dates.

The lower-right quadrant of Table 1 features discretely monitored rainbow options, whose payoffs each may depend on every price of every underlying asset at every monitoring time. Due to their complexity, analytical results for such options are rare. With the earlier unpublished works on discrete lookback rainbow options and forward-start rainbow options, this paper fills the vacuum in this quadrant with new exotic options and offers their EBS solutions.

Although they have complex external features, simplex options have the simple essence of being linear on the events called simplex. Equipped with the proposed EBS formula, simplex options are easily priced and their pricing formulas are systematically formalized. Furthermore, analytical approaches to their Greeks and numerical issues of applying simplex options will be discussed.

We remark that the wonderful idea of pricing complex options by simpler ones appears as well in Ingersoll (2000), where the building blocks are digital options and only single risky asset is taken into account. Contrast to previous literature, simplex options feature a broad class of discretely monitored rainbow options with Black-Scholes-like pricing formulas. We also remark that an option with a Black-Scholes-like formula is not necessarily simplex. Both continuous geometric Asian options (Angus 1999) and geometric average trigger reset options (Dai et al. 2005) have Black-Scholes-like formulas, but neither are simplex.

In summary, the contributions of this paper are:

a new option class called simplex options with analytical formulas and Greeks;

the extended Black-Scholes (EBS) formula for simplex expectations;

the valuation of new exotic options that are discretely monitored rainbow options.

This paper is organized as follows. Section 2 formally investigates the settings, fundamental concepts, theoretical results and numerical issues of the theory of simplex options. Section 3 gives concrete examples of pricing simplex options by the EBS formula. Section 4 concludes the paper.

Supplementary Background Knowledge

A financial option entitles its holder to buy or sell a risky asset for a certain price (the strike price) on a future date (the maturity date). For example, a vanilla call option on TSMC with the strike price of 60 NT dollars and with the maturity date on November 17 gives its holder the right, but not the obligation, to buy 2,000 shares of TSMC, each of which is worth 60 NT dollars. It is clear to see that the call option on TSMC will be valuable if the stock price of TSMC is higher than the strike price (60 NT dollars) on the maturity date (November 17); however, the option will be worthless if the stock price drops to a price lower than the strike price on the maturity date.

Greeks of a financial option measures the sensitivities of the option value to a change in factors, such as the underlying asset’s price, the time to maturity and the strike price, on which the option value is dependent. Being often denoted by Greek letters, these measurements of sensitivities are called Greeks.

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