Valuation of options using lottery ticket securities

Problem. Valuation of derivatives in incomplete markets requires specification of risk tolerance of agents. The standard way of specifying this is via utility functions and indifference pricing. A one parameter utility function is derived and from this Hodges indifference prices are calculated.

The limitations of this approach are:

• utility functions are not observable and it is difficult to derive the utility function of an investor from observable items
• utility functions mix the concepts of utility and risk aversion
• utility functions do not allow for risk-seeking behavior
• using utility pricing for derivative pricing requires the derivation of a market utility. However it is unclear how this market utility function arises from the individual utility function of individual investors. One standard assumption is to assume that all market traders have a standard utility function, but if this were the case there would be no reason for markets to exist.

We propose an alternate method of specifying risk behavior using a lottery ticket security, and use this risk profile framework to price standard European options. A lottery ticket security is a security with that has a gain or less of x with probability p. We then specify an agents risk profile by determining what the agent is willing to pay or loss for the security and we specify this with a function

$\pi (p,x)$

We include the follow constraints for the lottery security function which describe the limits of the payoff in cases where there is complete certainty.

$\pi (0,x)=0$
$\pi (p,0)=0$
$\pi (1,x)=x$

And we include the following conditions that assert a larger payoff as either probability or payoff increases:

${\frac {\partial \pi }{\partial p}}>0$
${\frac {\partial \pi }{\partial x}}>0$

The risk-neutral payoff function can be specified as follows

$\pi (p,x)=px$

and we make the assertion that payoffs become risk neutral as the value at risk becomes small.

$\lim _{x\rightarrow 0}\pi (p,x)=px$

To value a payoff for a standard option, we need to derive the value of an option of f(u) given that the probability distribution function for the payoff is p(u). To do this let consider an lottery ticket security that pays one dollar if a coin reads heads and two dollars if the coin reads tails. These two securities can be represented as

$\pi _{head}(0.5,1)$
$\pi _{tail}(0.5,2)$

Combining these securities we have a certain payoff of $1 with a 0.5 probability of another $1 payoff or

$1.0+\pi _{(}0.5,1)$

Applying this argument to the continuous limit we derive the pricing function for options in the $\int _{0}^{\infty }{\frac {\partial \pi }{\partial x}}(P(0<f<x),x)dx-\int _{-\infty }^{0}{\frac {\partial \pi }{\partial x}}(P(x<f<0),x)dx$