You are operating a gas turbine power station (GTPS). The risk factor driving cash inflows (revenues) is the electricity price. The risk factor driving cash outflows (expenses) is the gas price. At current prices, the yearly gas bill is EUR 50m (assume payment upfront) for producing a volume of electric power that can be sold for EUR 55m on the market (assume payment upfront).The electricity and gas price follow a multiplicative binomial distribution with the parameters u=1.8 and d =0.6 (electricity) and u=1.5 and d=0.8 (gas). Also for simplicity, we assume perfect correlation. The objective probability q for an upward movement is 0.5. The risk-free rate is equal 8.00% per yearWhat is the value of the GTPS closest to, if it lives for two years and one binomial step is equal one year? Conduct the valuation on a risk-neutral basis.

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BlackTom AI · Accepted Answer

Context and setup are key here. The GTPS faces stochastic cash flows: revenues scale with the electricity price, and expenses scale with the gas price. With perfect correlation between electricity and gas movements, and using a multiplicative binomial model with u and d for each asset, we can translate the problem into a binomial valuation of net cash flows.

- Step 1: Establish the baseline and the per-year net cash flow. At current prices, revenue = EUR 55m and gas bill = EUR 50m, so the baseline net cash flow is 5m per year (assuming cash flows occur in the period in which prices are observed, or equivalently we measure the payoff in yearly intervals).
- Step 2: Translate to a two-period binomial framework. For electricity we have up factor u_elec = 1.8 and down factor d_elec = 0.6. For gas, u_gas = 1.5 and d_gas = 0.8. With perfect correlation, in any given year the price moves in the same direction for both streams, so the net cash flow for a year is proportional to the same underlying move. Therefore, the yearly net cash flow can be written as 5m times a single-year price factor. The one-year up net cash flow is 5m * 1.8 = 9m, and the one-year down net cash flow is 5m * 0.6 = 3m.
- Step 3: Do the two-year binomial recombination. Over two years, possible path multipliers for the net cash flow are:
  - UU: factor = u^2 = 1.8^2 = 3.24, so CF_UU = 5m * 3.24 = 16.2m
  - UD or DU: factor = u*d = 1.8*0.6 = 1.08, so CF_UD = CF_DU = 5m * 1.08 = 5.4m
  - DD: factor = d^2 = 0.6^2 = 0.36, so CF_DD = 5m * 0.36 = 1.8m
- Step 4: Determine the risk-neutral probability for the one-year step. With a risk-free rate r = 8% per year, the up and down factors for the payoff are u = 1.8 and d = 0.6, so the risk-neutral probability is p = [(1 + r) − d] / (u − d) = (1.08 − 0.6) / (1.8 − 0.6) = 0.48 / 1.2 = 0.4. The probability of down is 1 − p = 0.6.
- Step 5: Work backwards from year 2 to year 0. First, value the year-1 nodes by taking the risk-neutral expected payoff one year ahead and discounting one year at 8%:
  - If the first move is UU, the year-2 payoff is CF_UU with probability 0.4 for the second move up and 0.6 for the second move down. The expected payoff at year 2 from the UU node is E[CF | UU] = 0.4*16.2m + 0.6*5.4m = 6.48m + 3.24m = 9.72m. Discounting one year gives V1_UU = 9.72m / 1.08 ≈ 9.0m.
  - If the first move is UD (or DU, which is the same node after one up and one down), the year-2 payoff possibilities are CF_UD = 5.4m and CF_DD = 1.8m with corresponding probabilities, so E[CF | UD] = 0.4*5.4m + 0.6*1.8m = 2.16m + 1.08m = 3.24m. Discounting one year gives V1_UD = 3.24m / 1.08 ≈ 3.0m.
- Step 6: Now value the initial period today by discounting the expected value of the two year-1 nodes under the risk-neutral probabilities: V0 = [p * V1_UU + (1 − p) * V1_UD] / (1 + r) = [0.4 * 9.0m + 0.6 * 3.0m] / 1.08 = [3.6m + 1.8m] / 1.08 = 5.4m / 1.08 ≈ 5.0m.
- Step 7: Interpret the result. The risk-neutral valuation yields a present value of about EUR 5.0 million for the two-year GTPS under the stated assumptions. This is the estimated value given the one-year step equal to one year, perfect correlation between gas and electricity price moves, and the specified up and down multipliers with a risk-free rate of 8%.

Note: The option listed, A. 15.00 million EUR, does not align with this calculation under the described model. The computed risk-neutral valuation of the two-year project, using the multiplicative binomial framework with the given parameters and perfect correlation, centers around 5.0 million EUR rather than 15 million EUR.

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