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Financial Transmission Rights Pricing

Adapting Black-Scholes Option Theory for Electricity Market Derivatives

Mickias Ambaye | Algorithmic Trading In Python, DAT-4876 | June 16, 2025

Black-Scholes

C=S0N(d1)−Ke−r(T−t)N(d2)C = S_0 N(d_1) - K e^{-r(T-t)} N(d_2)

Jump-Diffusion

dXt=κ(μ−Xt)dt+σ dBt+dJtdX_t = \kappa(\mu - X_t)dt + \sigma \, dB_t + dJ_t

Seasonality

f(t)=α+βDt+γcos⁡ ⁣(2π(t+τ)365)f(t) = \alpha + \beta D_t + \gamma \cos\!\left(\frac{2\pi(t+\tau)}{365}\right)

1.Introduction

Financial Transmission Rights (FTRs) are advanced financial instruments in electricity markets that hedge against transmission congestion costs. Their pricing relies on adaptations of financial theory, including the Black-Scholes model (Patiño-Echeverri & Morel, 2006). This paper explores how options theory principles have been adapted for electricity markets, where traditional pricing models struggle due to the non-storable nature of electricity and transmission network constraints (Nomikos & Soldatos, 2010).

The development of FTR pricing approaches marks a crucial nexus, where the operational realities of electricity transmission collide with traditional derivatives pricing theory. FTRs derive their value from locational marginal price (LMP) differences across transmission networks, which creates pricing complexities that call for new theoretical frameworks.

2.Theoretical Foundation: Black-Scholes Model and Its Limitations

The Black-Scholes model (1973) revolutionized derivatives pricing with a risk-neutral framework assuming geometric Brownian motion for asset prices (Asati, Gangele, & Kumar, 2024). This process assumes the logarithm of asset prices follows a normal distribution with constant drift and volatility parameters.

However, the Black-Scholes framework's underlying assumptions are fundamentally broken by the features of power markets. Geometric Brownian motion is insufficient to explain the high volatility, mean reversion characteristics, and frequent price spikes of electricity prices (ISO New England, 2025). Additionally, the conventional arbitrage opportunities that serve as the theoretical basis for Black-Scholes pricing are eliminated due to the non-storable nature of power (Alghanim, 2019).

Black-Scholes Call Option

C(S,t)=S0N(d1)−Ke−r(T−t)N(d2)C(S,t) = S_0 N(d_1) - K e^{-r(T-t)} N(d_2)

where C is the option value, S₀ is the current asset price, K is the strike price, r is the risk-free rate, T-t is time to expiration, and N(d₁), N(d₂) are cumulative standard normal distribution functions.

3.FTR Market Structure and Mechanics

FTRs provide their holders with the ability to pay or receive the price differences at specified injection and withdrawal points within the transmission network. The LMP at any given location represents the marginal cost of delivering an additional megawatt of load at that specific moment (ISO New England, 2025).

The simultaneous feasibility test ensures total FTRs do not exceed transmission capacity of the network, with power transfer distribution factors (PTDFs) staying within line limits.

FTR Revenue

FTR Revenue=MW×(LMPsink−LMPsource)×Hours\text{FTR Revenue} = MW \times (LMP_{\text{sink}} - LMP_{\text{source}}) \times \text{Hours}

where MW represents the contract quantity, and the LMP difference captures the congestion component between source and sink nodes.

Figure 1. Locational Marginal Prices across multiple transmission nodes over a 24-hour period

Locational Marginal Prices (LMP), Multiple Nodes

4.Adaptation of Black-Scholes Theory for FTR Pricing

The most thorough adaptation of options theory for FTR valuation is Patiño-Echeverri and Morel's groundbreaking work. Their methodology expands on the Black-Scholes paradigm by tackling the basic issue that FTRs, despite not being options, act as hedging tools against unpredictable future congestion costs. The most important realization is that FTRs reduce risk in a manner akin to insurance policies, in which policyholders pay a premium to reduce their exposure to fluctuating congestion charges.

FTR Fair Price

P=Cˉ+HP = \bar{C} + H

The fair price P of an FTR must be higher than the expected value of congestion costs C̄ by a premium H.

Premium Calculation (Options-Theoretic)

H=E[max⁡(C−P,0)]⋅e−r(T−t)H = E[\max(C - P, 0)] \cdot e^{-r(T-t)}

Mirrors the Black-Scholes call option structure, where the holder benefits when the underlying cost C exceeds the fixed price P.

5.Stochastic Process Modeling

Adapting Black-Scholes requires advanced modeling of congestion costs to reflect the mean-reverting and spike-prone behavior of electricity prices. Unlike geometric Brownian motion in Black-Scholes, electricity prices use mean-reverting jump-diffusion processes.

The mean-reverting term (κ) reflects electricity prices' tendency to revert to long-term equilibrium, while jumps (dJₜ) account for sudden price spikes caused by outages or demand surges. Volatility (σ) directly impacts the risk premium (H), as higher volatility increases the value of hedging (Alghanim, 2019).

Mean-Reverting Jump-Diffusion

dXt=κ(μ−Xt)dt+σ dBt+dJtdX_t = \kappa(\mu - X_t)dt + \sigma \, dB_t + dJ_t

κ captures mean reversion speed, μ is the long-term mean, σ is volatility, dBₜ is Brownian motion, and dJₜ is the jump component.

Revenue Adequacy Constraint

∑iFTR_Revenuei≤Total_Congestion_Revenue\sum_{i} \text{FTR\_Revenue}_i \leq \text{Total\_Congestion\_Revenue}

Total FTR revenue is capped by actual congestion revenues generated in the system.

6.Advanced Modeling Considerations

Several additional factors affect the underlying components of the power market modeling:

Seasonality Model

f(t)=α+β⋅Dt+γ⋅cos⁡(2π(t+τ)365)f(t) = \alpha + \beta \cdot D_t + \gamma \cdot \cos\left(\frac{2\pi(t + \tau)}{365}\right)

α, β, γ, and τ capture baseline levels, weekend effects, and annual seasonality. Dₜ represents a binary variable for weekends and holidays.

Compound Poisson Jump Process

dJt=Yk dNtdJ_t = Y_k \, dN_t

Nₜ is a Poisson process with intensity λ, and Yₖ represents normally distributed jump magnitudes.

Figure 2. Logarithm of electricity price with and without seasonality component over one year

Logarithm of Price w/ & w/o Seasonality Component

7.Conclusion

The Black-Scholes framework, adapted for FTRs, effectively captures their hedging value but requires adjustments for electricity market complexities (mean reversion, spikes, transmission constraints). While the Patiño-Echeverri and Morel model provides a strong foundation, challenges like parameter uncertainty, revenue adequacy, and market power highlight the need for ongoing refinement. As grids evolve with intermittent renewables such as wind and solar, which increase congestion volatility, FTR pricing models need to incorporate these non-dispatchable energy dynamics to ensure effective risk management and accurate price discovery.

References

[1] ISO New England. (2025). Financial Transmission Right (FTR).

[2] Patiño-Echeverri, D., & Morel, B. (2006). An options theory method to value electricity Financial Transmission Rights. Carnegie Mellon CEIC.

[3] PJM Interconnection. (2025). Locational marginal pricing: Real-time & day-ahead LMP chart.

[4] Nomikos, N. K., & Soldatos, O. A. (2010). Analysis of model implied volatility for jump diffusion models. City, University of London.

[5] Asati, S., Gangele, R. K., & Kumar, B. (2024). The Black-Scholes Model: A comprehensive analysis.

[6] Alghanim, R. K. (2019). Modelling electricity prices and pricing derivatives with Markov regime switching models. University of Manchester.

[7] Morel, B., & Patiño-Echeverri, D. (2005). Electricity financial transmission rights as hedging instruments. Carnegie Mellon University.

[8] Rudkevich, A. et al. (2015). Preserving revenue adequacy in FTR markets with changing topology. FERC Technical Conference.

[9] MathWorks. (n.d.). Simulating electricity prices with mean-reversion and jump-diffusion.