# Compound option

Published05 Mar Abstract Considering the uncertainty of a financial market includes two aspects: risk and vagueness; in this paper, fuzzy sets theory is applied to model compound option imprecise input parameters interest rate and volatility. For eachthe -level set of fuzzy prices is obtained according to the fuzzy arithmetics and the definition of fuzzy-valued function.

## Compound option - Wikipedia

We apply a defuzzification method based on crisp possibilistic mean values of the fuzzy interest rate and fuzzy volatility to obtain the crisp possibilistic mean value of compound option price.

Finally, we present a numerical analysis to illustrate the compound option pricing under fuzzy environment. Compound option Compound options are options with other options as underlying assets. Since Geske [ 1 ] derived the closed form pricing formula using the method of partial differential compound option for the first time, some scholars have extended the pricing model and proposed some new pricing methods.

For example, [ 2 ] used the martingale approach and the expectation of a truncated bivariate normal variables to prove the pricing formula for 2-fold compound options, respectively.

The work of [ 3 ] extended the Geske model to a multivariate normal integral for the valuation of a compound real option. The work of compound option 4 — 6 ] extended the Geske model to n-fold compound options.

The work of [ 78 ] introduced time dependent volatility and an interest rate to the pricing model of compound options. Fouque and Han [ 9 ] proposed perturbation approximation to compute the prices of compound options.

There are few literature studied compound option pricing under jump-diffusion model, stochastic volatility model, or stochastic interest rate model, such as [ 10 — 12 ]. Compound option is widely employed in the field of financial derivatives pricing, for instance, American put option [ 13 ], sequential exchange options [ 14 ], and sequential American exchange property options [ 15 ].

### Compound option

Compound option is also widely used in the real options; examples include project valuation of new drug application [ 16 ], valuation of multistage BOT projects [ 17 ], and decision-making in petroleum exploration [ 18 ]. The literature mentioned above studied compound option within stochastic framework.

The uncertainty of the compound option market includes two aspects: risk and vagueness, and the two parts could not substitute each other. In the real financial market, due to market fluctuations and human errors, some parameters such as the interest rate and volatility sometimes cannot be recorded or collected precisely.

### Compound Option

The risk uncertainty could be modeled by compound option probability theory; the vagueness could be modeled by a fuzzy methodology, the fuzzy sets theory provides an appropriate tool for tackling this kind of uncertainty. Therefore, the fuzzy sets theory proposed by Zadeh [ 19 ] has been widely used in the option pricing recently.

The existing literature on option pricing under the fuzzy stochastic model mainly studied the European option, based on the Black-Scholes model. For example, Yoshida [ 20 ] introduced fuzzy logic to the stochastic financial model and discussed the valuation of the European options with the uncertainty of both randomness and fuzziness. Wu [ 21 ] considered the fuzzy pattern of the Black-Scholes formula by fuzzing the interest rate, volatility, and stock price in his compound option when the arithmetics in the Black-Scholes formula are replaced by fuzzy arithmetic.

### Compound Option Pricing under Fuzzy Environment

The work of [ 2223 ] presented a sensitivity analysis based on the Black-Scholes formula. The work of [ compound option ] introduced a crisp weighted possibilistic mean value Black-Scholes option pricing formula.

There are only few papers that studied American options or exotic options pricing within the Black-Scholes framework, such as [ 25 — 27 ], and few papers for alternative models with jumps [ 28 — 30 ].

As far as we know, there is no literature research on compound option pricing under fuzzy environment; this paper will consider both the risk and vagueness to study compound option option pricing.

1. This option has two elements: 1 the upfront premium and 2 the strike premium which will have to be paid later if the compound right is exercised.
2. Compound Options
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The main compound option of this paper is that we present the -level set of fuzzy prices for each and give a sensitivity analysis of the crisp possibilistic mean value of compound option price with respect compound option the core value of fuzzy interest rate and fuzzy volatility. The remainder of the paper is organized as follows.

Compound option Section 2the notions of fuzzy numbers and the arithmetics of fuzzy numbers are introduced.

In Section 3the pricing formula for compound option under stochastic model is introduced. Section 4 presents the fuzzy price, -level set of fuzzy prices, and the crisp possibilistic mean value of compound option price. In Section 5a numerical analysis is performed.

Finally, the conclusions are stated in Section 6. Fuzzy Numbers In this section we follow the notations and concepts introduced in Wu [ 2131 ].

## Compound Option Definition

Let be the set of all real numbers. Then a fuzzy subset of is defined by its membership function. We denote by.