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## 5 Applications 

## 5.1 Unknown Variance 

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? 

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 

<img src='content_image/398120.jpg'>

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. 

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. 

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). 

<img src='content_image/398121.jpg'>

<img src='content_image/398111.jpg'>

