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Review of Derivatives Research

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01-07-2015

The valuation of forward-start rainbow options

Authors: Chun-Ying Chen, Hsiao-Chuan Wang, Jr-Yan Wang

Published in: Review of Derivatives Research | Issue 2/2015

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Abstract

This paper studies the valuation and hedging problems of forward-start rainbow options (FSROs). By combining the characteristics of both multiple assets and forward-start feature, this new type of derivative has many potential applications, for instance, to incorporate the reset provision in rainbow options for investors or hedgers or design more effective executive compensation plans. The main contribution of this paper is a novel martingale pricing technique for options whose payoffs are associated with multiple assets and time points. Equipped with this technique, the analytic pricing formula and the formulae of the delta and gamma of the FSRO are first derived. We conduct numerical experiments to verify these formulae and examine the characteristics of the FSRO’s price and Greek letters. To demonstrate the importance and general applicability of the proposed technique, we also apply it to deriving the pricing formula for the discrete-sampling lookback rainbow options.

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Theoretically speaking, both of these two techniques can be classified as martingale pricing methods.

More concretely, to evaluate $E^Q[S_a(s)\cdot I(S_b(u)\le S_c(v))]$, where $Q$ denotes the risk-neutral probability measure, $S_i(z)$ denotes the price of the asset $i$ at the time point $z$, and $I(\cdot )$ is defined as an indicator function.

Although this paper considers only the non-dividend-paying case for simplicity, all results are straightfoward to be extended to underlying assets with constant dividend yields.

Although it is widely accepted that jumps are able to explain the empirical regularities of derivative pricing, this paper focuses on pure diffusion processes. This is because incorporating additional jumps not only complicates the problem substantially but also obscures the contribution of this paper, which proposes a technique to tackle the evaluation of the expected value of an asset price at a specified time point conditional on the comparison results between individual asset prices (or the corresponding Brownian motions) evolving up to different time points. Even so, we highly appreciate the anonymous referee to mention this point.

Nevertheless, we can analyze qualitatively the possible impacts of adding jumps on the values of FSRPOs. Since the reset provision can turn out-of-the-money options to become at-the-money options on reset dates, the reset provision can partially eliminate the effect of unfavorable jump movements. As for put options on the minimum of multiple assets, through the diversification effect, additional jumps could lower the expected value of $\min [S_1(T),\ldots ,S_n(T)]$. This is because if there is any asset with a net negative jump movement, this harmful effect will be retained by the minimum function. On the other hand, the realized value of $\min [S_1(T),\ldots ,S_n(T)]$ rises only when all assets are with net positive jump movements. Since the FSRPO considered in this paper is essentially based on a minimum put options plus the reset option for the strike price, due to the above analysis, it can be expected that the additional jump processes will exhibit a generally positive impact on the value of the FSRPO.

For example, Eqs. (A.5) and (A.6) in Liao and Wang (2003) can be rewritten with our notation system as follows.

$$\begin{aligned} \frac{dR_1}{dQ}=\exp \left( \sigma _1 W_1^Q (T)-\sigma _1^2 T/2\right) , \text{ and } dW_1^{R_1}(z)=dW_1^Q (z)-\sigma _1 dt. \end{aligned}$$

Since they price a single-asset reset option, the volatility of the only asset is denoted as $\sigma _1$. With $\sigma _1$ as the kernel in the Girsanov theorem, an equivalent probability measure $R_1$ and the Brownian motion under it, $dW_1^{R_1}(z)$, can be defined. Note that the probability measure $R_1$ is defined with respect to the single asset but independent of any specified time point.

The results in this figure are consistent with those in Fig. 5 of Gray and Whaley (1999).

Baxter, M., & Rennie, A. (2000). Financial calculus. Cambridge: Cambridge University Press.

Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. The Journal of Political Economy, 81, 637–659.CrossRef

Cheng, W. Y., & Zhang, S. (2000). The analytics of reset options. The Journal of Derivatives, 8, 59–71.CrossRef

Curnow, R. N., & Dunnett, C. W. (1962). The numerical evaluation of certain multivariate normal integrals. The Annals of Mathematical Statistics, 33, 571–579.CrossRef

Genz, A. (1992). Numerical computation of multivariate normal probabilities. Journal of Computational and Graphical Statistics, 1, 141–149.

Gray, S. F., & Whaley, R. E. (1999). Reset put options: Valuation, risk characteristics, and an application. Australian Journal of Management, 24, 1–20.CrossRef

Johnson, H. (1987). Options on the maximum or the minimum of several assets. Journal of Financial and Quantitative Analysis, 22, 277–283.CrossRef

Liao, S. L., & Wang, C. W. (2003). The valuation of reset options with multiple strike resets and reset dates. Journal of Futures Markets, 23, 87–107.CrossRef

Margrabe, W. (1978). The value of an option to exchange one asset for another. The Journal of Finance, 33, 177–186.CrossRef

Ouwehand, P., & West, G. (2006). Pricing rainbow options. Wilmott Magazine, 5, 74–80.

Rubinstein, M. (1991). Pay now, choose later. Risk, 4, 13.

Stulz, R. (1982). Options on the minimum or the maximum of two risky assets: Analysis and applications. Journal of Financial Economics, 10, 161–185.CrossRef

Title: The valuation of forward-start rainbow options
Authors: Chun-Ying Chen
Hsiao-Chuan Wang
Jr-Yan Wang
Publication date: 01-07-2015
Publisher: Springer US
Published in: Review of Derivatives Research / Issue 2/2015
Print ISSN: 1380-6645
Electronic ISSN: 1573-7144
DOI: https://doi.org/10.1007/s11147-014-9105-0