An Analytical Approximation for Pricing VWAP Options

Growing Presence of Real Options in Global Financial Markets

“…Recently, Funahashi and Kijima (2017) apply the chaos expansion technique to derive a unified approximation method for pricing VWAP options, which shows reasonably high accuracy of the second-order chaos expansion approximation for practical use. Compared to previous works, Funahashi and Kijima's (2017) method is applicable to the local volatility model, not just for the geometric Brownian motion case. In this chapter, we adopt the methodology of Funahashi and Kijima (2017) to value the VWAP options embedded in Eq.…”

“…Compared to previous works, Funahashi and Kijima's (2017) method is applicable to the local volatility model, not just for the geometric Brownian motion case. In this chapter, we adopt the methodology of Funahashi and Kijima (2017) to value the VWAP options embedded in Eq. (7).…”

“…The sample period is further subdivided into two intervals to examine the stationarity of the sample statistics: the period prior to the 30-day volume-weighted averaging period and the 30-day volume-weighted averaging period. For comparison purposes, in addition to testing Funahashi and Kijima's (2017) VWAP model, we also use Black and Scholes' (1973) plain vanilla model and Levy's (1992) AP model in the empirical tests.…”

“…As indicated in Eq. 7, DirecTV stock's intrinsic value is determined by several factors embedded in the option components: (1) AT&T's stock price, (2) the time to maturity, (3) the risk-free rate, (4) AT&T's trading volume, (5) the volatility of AT&T's stock price, (6) DirecTV's cost of equity, and (7) several parameters used in Funahashi and Kijima's (2017) VWAP model, which assumes that stock price follows the local volatility model and trading volume is formulated by a mean-reverting process. The parameters used by Funahashi and Kijima include gamma that is the volatility of AT&T's trading volume, theta that is the long-term average of the AT&T's trading volume, and rho that is the correlation between AT&T's stock price and trading volume.…”

“…2. The calculation of E[e ÀrT S Ã T ] is detailed in Theorem 3.1 of Funahashi and Kijima (2017). For the Black and Scholes (1973) option model, S* is simply the current stock price.…”