5 Applications 5.1 Unknown Variance 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.5 1 1.5 2 2.5 3 put price rel strike Figure 5: Log-normal probability distribution for the standa rd deviation (Black-Scholes volatility). Figure 6: Example put prices for certain standard deviation (green line) and for a log-normal probabil- ity distribution for the standard deviation (blue line). 12 We assume here that there are no dividends and all other parameters are known, i.e. expected return, time to expiry, and risk-free rate. One could now consider introducing additional “Greeks” for the parameters of the distribution of the standard deviation. If, as in the example, a log-normal distribution was used then the resulting Greeks follow directly from differentiating (2). Figures 5, 6, and 7 give a graphical example of above relationship. Figure 5 shows a probability dis- tribution describing our beliefs about what the stan- dard deviation (Black-Scholes ‘volatility’) could be. As the standard deviation is a positive quantity we consider the log-standard deviation motivated by the scale-invariance argument [1]. The second order max- imum entropy distribution for the standard deviation is then, just as for the price in the Black-Scholes case, a log-normal distribution. The resulting put option valuations and the implied volatility skew are pre- sented in figures 6 and 7. where f ( x ) is the pay-off function, V $^{bs}$( r, σ ) is the Black-Scholes price for given risk-free rate and second moment σ 2 (‘variance’), and p ( σ | I ) is expressing our beliefs about the possible values of the (square-root of the) second moment. In a probabilistic framework this can be dealt with easily by marginalizing over the second moment pa- rameter (Black-Scholes ‘volatility’). Starting with the definition of the derivatives value as the present value of the expected cash-flows 12 we find Let us assume we accept the risk-free rate of return as our expected future asset return, but we are not cer- tain about the second moment (variance) for a certain date. How can we value an European option under such circumstances? V = e − rt ∫ ∞ 0 d x f ( x ) p ( x | I ) = e − rt ∞ 0 d σ p ( σ | I ) ∫ ∞ 0 d x f ( x ) p ( x | σI ) = ∫ ∞ 0 d σ p ( σ | I ) V $^{bs}$( r, σ ) , (2) 0 0.005 0.01 0.015 0.02 0.025 0.03 0 0.5 1 1.5 2 prob volatility 7