EaR Sensitive Control

In section 3.3.6 we introduced the total option value as the fair value plus or minus a risk spread dependent on whether one is the option buyer or seller. We explained that this value represents adequate lower and upper bounds for a risk adjusted swing option value. First, our risk spread is not based on mere variance like many risk-return models that fail to describe the behavior in the tails of non-normal distributions. This would have significantly reduced the spectrum of price processes for our option model. At the same time we introduced with EaR a risk measure that follows the popular concept of VaR. The option holder can express his risk aversion by choosing his acceptable loss probability. It states that (1 − β) percent of the value fluctuations will be compensated by the extra spread either as a discount (buyer’s option) or an extra fee (seller’s option). We think that the percentile is a more intuitive way to capture the risk aversion of an investor compared to a utility function or a risk/return weighting factor. For this reason we will rely our benchmark for different risk controlling strategies on our total option value and introduce the following definition:

Definition 3.4.1 (EaR-Efficient Option Value). An option value ˜C_β^e:= ˜C_β^e(x₀, g₀, δ) is EaR efficient if the underlying exercise policy ˜π₀^∗ can reduce the risk premium measured as EaR qβ(˜) with fixed β < 0.5 more than the inherent deduction in option value ˜C0:= ˜C0(x₀, g0, δ) compared to a risk neutral valuation C₀:= C₀(x₀, g₀, δ) and q_β()

C˜_β^e := ˜C₀+ q_β(˜) s.t.

C0− ˜C₀^e< qβ(˜) − qβ(),

(3.56)

where x₀ is the current spot price, g₀ and F₀ are defined in equation 3.33 with j = 0 and q_β() as described in equation 3.54.

Translated to our buyer’s and seller’s option we can write C₀− ˜C₀ < q_β^B(˜) − q^B_β()

C0− ˜C0 < q_β^S() − q^S_β(˜). (3.57) The quantiles of the seller’s option q_β^S() and q_β^S(˜) are flipped as we look at the right tail (recall from equation 3.55 that q_β^S := q_1−β). A risk sensitive dispatch policy ˜π0 (from now on we mark risk sensitive values with a tilde) in the sense of definition 3.4.1 should raise the new buyer’s total option value compared to C_β^B and lower the new seller’s total option value compared to C_β^S to a more attractive sales price such that the inevitable reduction of real option value will be over-compensated by the mitigation in EaR (see equation 3.55 for the definition of C_β^B/S). The distribution of an EaR efficient option is tighter as shown in Figure 3.9. Let us illustrate this concept in context of our business case with our retail, trading and

Figure 3.9.: Benchmark for a risk sensitive policy

generation unit. If the generation department could offer a production strategy that would lower the price risk and still assure an adequate return then the trading department will be willing to accept a smaller discount on the price for the generated electricity denoted as the new buyer’s swing option ˜C_β^B. The generation department in return could achieve a higher profit without increasing the risk for the entire organization. At the opposite side the trading unit could sell the risk sensitive policy as a swing option with special exercise conditions.

Again, if those conditions still ensure a reasonable return and lower the risk fee at the same time, then the new lower option price ˜C_β^S becomes more attractive to an industry customer and therefore the retail department would be willing to accept the price.

In case of our swing option with N exercise rights and initial price X₀ = x₀ we can write two optimization problems, for buyer and seller separately, that finds the most EaR efficient option value

C˜₀^B,e(x₀, N ) = max

˜ πB

{ ˜C₀(x₀, N ) + q_β(˜)}

C˜₀^S,e(x₀, N ) = max

π˜_S { ˜C₀(x₀, N ) − q_1−β(˜)},

(3.58)

with ˜C₀ and q_β(˜) as defined in equations 2.3 and 3.54 respectively and ˜π_B/S = {˜τ_N^B/S(0, N ), ..., ˜τ₁^B/S(0, N )} is a sequence of exercise times. The two equations penalize the swing option value with the EaR figure. Recall that the buyer’s and seller’s EaR are on opposite sides of the loss distribution which explains the change in sign. As the two equations stand for two separate optimization problems the resulting two optimal policies ˜π_B^∗ and ˜π_S^∗ might not be consistent. In other words, for the same hour one policy might exercise, but the other does

not. Translated to our business case we could imagine a situation where the retail department asks for power according to strategy ˜π₀^S, but the trading unit cannot provide electricity from the generation department because it did not exercise his option according to ˜π^B₀. Still this is not an issue, since the extra fee q_β^S(˜) compensates the cost to buy the relevant power from the market. The same is true in the opposite direction where the trading unit exercises according to ˜π^B₀, but the retail department does not. Then the trading unit might need to sell the power to a lower price to the market, but the compensation fee q^B_β(˜) compensates the loss.

Let us now look at lower and upper bounds for these risk adjusted option values. The lower bound of these two optimization problems will be derived from the risk neutral policy according to equation 3.55. For ˜C₀^B,e this is immediately C_β^B as C₀+ q_β happens to be the definition of the buyer’s option (see equation 3.55). For the seller’s option we get C₀− q_1−β = C₀− q_β^S = C₀− (C_β^S− C₀) = 2C₀− C_β^S. Thus, an EaR efficient buyer’s and seller’s option C˜_β^B/S,e has to be above these values

C˜_β^B,e> C^! _β^B

C˜_β^S,e> 2C^! ₀− C_β^S.

(3.59)

We can also provide an upper bound for the seller’s option. Recall that we are actually interested in lowering the seller’s option value C_β^S. Therefore an upper bound is more intuitive, especially for our numerical analysis in section 3.4.3. ˜C_β^S becomes automatically cheaper due to the drop in fair value by C₀− ˜C₀. An efficient option value must be even cheaper by at least yet another C₀− ˜C0 resulting from the reduction in the quantile q_1−β(˜). Only then we have the desired compensation of risk vs. return

C˜_β^S,e< C^! _β^S− 2(C₀− ˜C₀). (3.60) We will use this benchmark in section 3.4.3. Formally, equation 3.58 introduces a quantile based measure in our initial optimization problem. Both summands, the fair value C₀and the quantile q_β(), are linked by their common distribution. Usually the first summand reduces more relative to the second summand i.e. a mitigation in risk premium entails a declining fair value. Furthermore, the quantile is not a linear operator as discussed in section 3.4. Standard linear or dynamic programming are therefore not applicable. Another difficulty arises from the fact that  is the remaining spot exposure after a Forward hedge which can only be calculated after the exercise strategy ˜π_B and ˜π_S is known. To illustrate the problem in context of our example, we need to assess the impact of each hour’s dispatch decision on the Forward hedge that covers the entire delivery period. However, we can calculate the latter only after we know the entire schedule/option value. Thirdly, we need to compute the quantile conditional on today’s price X₀ = x₀. Especially the latter motivated us to introduce a heuristic that is based on quantile regression. The heuristic will not work with the objective function in equation 3.58, but rather rely on a direct comparison with the risk neutral policy according to equations 3.59 and 3.60. The heuristic will still be based on our dynamic programming framework, but focus on a direct and permanent comparison of the expectation vs. the risk adjusted policy during each stage of the backward iteration. Before we lay out the heuristic in detail in section 3.4.3 we will briefly review the methodology of quantile regression and in particular look for a proper calibration of the respective regression function to our swing option framework.