偏導関数

最近リスク管理を学びながらオプションのギリシア文字の計算式を考えています。
A chapter in The Professional Risk Manager's Handbook mentions a relationship between gamma(d^2C/dS^2) and vega(dC/dsigma).
vega = -(gamma*(T-t)). However, my cross-checks of the formulas for gamma and vega do not yield such a relationship. Thus, either the gamma and vega individual formulas are not valid, or else this relationship does not hold.
gamma = N'(d1)/(sigma*S*sqrt(T-t))
vega = sigma*N'(d1)*sqrt(T-t)
Both of the above are formulas for a European calls.
Thus, vega = gamma*(T-t)*sigma^2*S^2
So, in order for vega=-(gamma*(T-t)) to hold sigma^2*S^2 must equal -1.

Another area of interest is the Generalized Sharpe Ratio. The standard Sharpe Ratio is (mu-r)/sigma, where mu is the expected return of the risky asset, r is the risk-free return, and sigma is the standard deviation of the risky asset. However, this requires that returns on the risky asset be normally distributed. The Generalized Sharpe Ratio requires only that the utility funtion be exponential. GSR = sqrt(-2*ln(-EU*/lambda)). Here lambda is the risk tolerance, and EU* is the utility when the allocation to the risky asset maximizes some sort of return-risk criterion. When the returns on the risky asset are normally distributed and the mean-variance criterion is maximized, the GSR becomes the standard Sharpe Ratio.