クリクエットオプションの値

最近複数のオプションの累計によってペイオフを決めるエキゾチック°オプションを学んでいます。株価指数などオプションの代わりに使用される事があるらしいです。This paper is rife with problems, as it relies on a characteristic function representation of payoffs in order to express dV/dr, dV/dsigma, and other Greeks. However, the value of this characteristic function and its relation to the payoff density is unclear, other than that it is the inverse Fourier transform, phi(t) = INTEGRAL(exp(itx)f(x)dx).
The equation phi(t)=1+it*INTEGRAL(exp(itx)Q(Z>x)dx) cannot be proven, and its relation to Fubini's Theorem, which involves multiple and repeated integrals, makes no sense. In fact, the above expression for the characteristic can be derived using integration by parts, beginning with phi(t)= INTEGRAL(exp(itx)*(1-f(x))dx, as INTEGRAL(1-f(x))dx=-Q(Z>x), where Q is the probability distribution. The connection to Fubini rains unknown.
Another problematic aspect is the assertion that the returns "R" are lognormal, as it is customary in derivatives (options) trading to assume that the price is lognormal and the returns normal.
The vega derivation based on dphi/dsigma, where phi is the characteristic function and sigma the underlying's volatility. 微分の方法が不明です。特に-sigma^2*deltaの項の由来がわかりません。