Consider a risky asset (stock, commodity, a unit of energy) with the price S(t), where t ∈ [0, T], for a given T > 0. Consider an option with the payoff

F_{u} = Φ(u(·), S(·)) —– (1)

This payoff depends on a control process u(·) that is selected by an option holder from a certain class of admissible controls U. The mapping Φ : U × S → R is given; S is the set of paths of S(t). All processes from U has to be adapted to the current information flow, i.e., adapted to some filtration F_{t} that describes this information flow. We call the corresponding options controlled options.

For simplicity, we assume that all options give the right on the corresponding payoff of the amount F_{u} in cash rather than the right to buy or sell stock or commodities.

Consider a risky asset with the price S(t). Let T > 0 be given, and let g : R → R and f : R × [0, T] → R be some functions. Consider an option with the payoff at time T

F_{u} = g(∫_{0}^{T }u(t) f (S(t), t)dt) —– (2)

Here u(t) is the control process that is selected by the option holder. The process u(t) has to be adapted to the filtration F_{t} describing the information flow. In addition, it has to be selected such that

∫_{0}^{T} u(t)dt = 1

A possible modification is the option with the payoff

F_{u} = ∫_{0}^{T} u(t) f(S(t), t)dt + (1 – ∫_{0}^{T} u(t)dt) f(S(T), T)

In this case, the unused u(t) are accumulated and used at the terminal time. Let us consider some examples of possible selection of f and g. We denote x^{+} =^{∆} max(0, x)

Important special cases are the options with g(x) = x, g(x) = (x − k)^{+}, g(x) = (K − x)^{+},

g(x) = min(M, x), where M > 0 is the cap for benefits, and with

f(x, t) = x, f(x, t) = (x − K)^{+}, f(x, t) = (K − x)^{+} —– (3)

or

f(x, t) = e^{r(T−t)}(x − K)^{+}, f(x, t) = e^{r(T−t)}(K − x)^{+} —– (4)

where K > 0 is given and where r > 0 is the risk-free rate. Options (3) correspond to the case when the payments are made at current time t ∈ [0, T], and options (4) correspond to the case when the payment is made at terminal time T. This takes into account accumulation of interest up to time T on any payoff.

The option with payoff (2) with f(x, t) ≡ x represents a generalization of Asian option where the weight u(t) is selected by the holder. It needs to be noted that an Asian option , which is also called an average option, is an option whose payoff depends on the average price of the underlying asset over a certain period of time as opposed to at maturity. The option with payoff (2) with g(x) ≡ x represents a limit version of the multi-exercise options, when the distribution of exercise time approaches a continuous distribution. An additional restriction on |u(t)| ≤ const would represent the continuous analog of the requirement for multi-exercise options that exercise times must be on some distance from each other. For an analog of the model without this condition, strategies may approach delta-functions.

These options can be used, for instance, for energy trading with u(t) representing the quantity of energy purchased at time t for the fixed price K when the market price is above K. In this case, the option represents a modification of the multi-exercise call option with continuously distributed payoff time. For this model, the total amount of energy that can be purchased is limited per option. Therefore, the option holder may prefer to postpone the purchase if she expects better opportunities in future.