Risk-indifference Pricing of American-style Contingent Claims

This paper establishes a general framework for risk-indifference pricing of American-style contingent claims under asymmetric information, demonstrating its consistency with no-arbitrage principles and characterizing these prices via reflected backward stochastic differential equations to enable deep learning-based numerical implementation in stochastic volatility models.

The Gist

The Big Picture: The "Fair Price" Puzzle

Imagine you are at a flea market trying to sell a used, high-tech gadget (an American Option). The catch? The buyer can decide to buy it anytime today, tomorrow, or next week, but you (the seller) don't know when they will pull the trigger.

In a perfect world (a "complete market"), there is one mathematically perfect price for this gadget. But in the real world, markets are messy and incomplete. There isn't just one price; there's a huge range of prices where no one is technically "cheating."

So, how do you find the fair price?
This paper proposes a method called Risk-Indifference Pricing. Think of it as finding the "reservation price."

• For the Buyer: It's the price where they are just as happy buying the gadget and managing the risk, as they are keeping their money in their pocket and doing nothing.
• For the Seller: It's the price where they are just as happy selling the gadget and managing the risk, as they are keeping the gadget and doing nothing.

If the price is too high, the buyer walks away. If it's too low, the seller walks away. The "fair price" is the sweet spot where neither side feels they are losing out compared to doing nothing.

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The Twist: The "Exercise" Problem

The tricky part about American options is the timing.

• The Buyer holds the "remote control." They can press the button (exercise the option) whenever they want.
• The Seller is blind to the remote. They don't know when the buyer will press the button. They only know that it will happen eventually.

Most previous math models assumed the seller knew the buyer's strategy or that they both had the same information. This paper argues: "No, that's not realistic." The seller needs a pricing model that accounts for the fact that they are flying blind regarding the buyer's timing.

The Solution: "Risk-Indifference" with a Safety Net

The authors use a concept called Convex Risk Measures. Imagine this as a "Risk Calculator."

• If you hold a risky asset, the calculator tells you how much "pain" (risk) you might feel in the future.
• The goal is to find a price where the "pain" of holding the asset is exactly balanced by the "pain" of not having the asset.

The Seller's Dilemma (The "Blind" Seller)

Since the seller doesn't know when the buyer will exercise the option, they have to prepare for the worst-case scenario.

• Analogy: Imagine you are a landlord selling a lease on a house. The tenant can move out anytime. You don't know when. To price the lease fairly, you have to assume the tenant might move out at the worst possible moment for you.
• The paper creates a mathematical framework where the seller calculates the price based on the worst possible exercise time, while still allowing them to trade in the market to hedge (protect) themselves.

The Buyer's Advantage (The "Remote" Holder)

The buyer knows exactly when they want to exercise. They will choose the moment that minimizes their risk.

• Analogy: The tenant knows exactly when they want to move out. They will pick the date that saves them the most money.

The Mathematical Magic: "BSDE-R-BSDEs"

This is the most technical part, but here is the simple version:
To calculate these prices, the authors use a complex equation called a Backward Stochastic Differential Equation (BSDE).

• Normal BSDE: Like looking at a movie in reverse. You know the ending (the final price) and work backward to figure out the price today.
• The New Twist (Reflected BSDE): Because the buyer can stop the game at any time, the equation has a "ceiling" or a "floor" (a reflection boundary). If the price hits this line, the equation bounces back.

The Paper's Breakthrough:
They discovered that for American options, the "ceiling" isn't a fixed number. The ceiling is itself another complex equation!

• Analogy: Imagine you are trying to calculate the height of a bouncing ball. But the floor the ball bounces off of is also moving up and down based on a different set of rules.
• They call this a BSDE-R-BSDE (Backward Stochastic Differential Equation reflected at a Backward Stochastic Differential Equation).
• Why it matters: This structure perfectly captures the reality that even after the option is exercised, the seller still has risk (holding a "zero" contract) until the final maturity date. The "reflection boundary" accounts for this lingering risk.

The "Deep Learning" Engine

Solving these "equations inside equations" is incredibly hard for traditional computers. It's like trying to solve a Rubik's cube that changes shape while you are twisting it.

The authors used Deep Learning (Artificial Intelligence) to solve this.

• The Metaphor: Instead of trying to write a single, perfect formula to solve the puzzle, they trained a neural network (a digital brain) to learn the pattern.
• They used a method called RDBDP (Reflected Deep Backward Dynamic Programming).
• How it works: The AI simulates thousands of possible futures (what if the stock goes up? what if it crashes? what if the buyer exercises early?). It learns the best strategy for the seller and buyer in each scenario and calculates the fair price.

The Results: What Did They Find?

They tested this on a Put Option (a bet that a stock price will go down) using a "Stochastic Volatility" model (where market volatility changes randomly, like real life).

1. The Spread is Small: The difference between what the buyer is willing to pay and what the seller wants to charge is surprisingly small. This suggests that even with different information, the market finds a very tight "fair price."
2. The "Smile": When they plotted the "Implied Volatility" (a measure of how wild the market expects to be), they saw the classic "volatility smile" curve that real traders see every day. This proves their model matches reality.
3. American vs. European: The price difference between an American option (can exercise anytime) and a European option (can only exercise at the end) was small, but noticeable.

Summary: Why Should You Care?

This paper is a bridge between pure math and real-world trading.

1. It fixes a hole in how we price American options by acknowledging that buyers and sellers often have different information.
2. It provides a rigorous, "fair" way to price these assets using risk management principles rather than just guessing.
3. It shows that AI (Deep Learning) is now powerful enough to solve the most complex financial equations that were previously impossible to calculate.

In a nutshell: The authors built a new, smarter calculator for pricing "anytime" options. They realized the seller is flying blind, so they built a model that prepares for the worst. Then, they used a super-smart AI to solve the math, proving that this method works just like the real market does.

Technical Summary

1. Problem Statement

The paper addresses the pricing of American-style contingent claims (options exercisable at any time before maturity) in incomplete markets.

• The Challenge: In incomplete markets, the no-arbitrage principle only provides a wide interval of arbitrage-free prices (super- and sub-hedging bounds) rather than a unique fair price.
• The Gap: While indifference pricing (based on utility maximization or risk measures) is well-established for European options, its application to American options is complex. The primary difficulty lies in the asymmetry of information and decision-making: the buyer chooses the exercise time, while the seller must hedge against this unknown stopping time. Existing literature often lacks rigorous definitions of admissible strategies for the seller that account for this non-anticipativity (i.e., the seller cannot know the buyer's future exercise decision).
• Goal: To establish a rigorous, arbitrage-free definition of risk-indifference prices for both buyers and sellers in a continuous-time setting with potentially different information filtrations, and to provide a computational framework for stochastic volatility models.

2. Methodology

A. General Framework: Risk-Indifference Pricing

The authors define prices based on fully dynamic convex risk measures ($\rho_{s,t}$), which are time-consistent and satisfy monotonicity, cash-invariance, and convexity.

• Different Filtrations: The model allows the buyer ($\mathcal{F}^{buy}$) and seller ($\mathcal{F}^{sell}$) to have different information sets. The seller only observes the market assets and the realization of the exercise time, not the buyer's internal stopping rule.
• Seller's Price ($P^{sell}_t$):
  • The seller must minimize the residual risk over all possible hedging strategies while considering the worst-case exercise time chosen by the buyer.
  • Key Innovation: The authors introduce a set of strategy selection functions ($H'_{sell}$). These functions select a hedging strategy based on the realized exercise time $\tau$, ensuring the strategy is predictable with respect to the enlarged filtration $\mathcal{F}^{sell, \tau}$ but does not anticipate $\tau$ as a random variable.
  • The price is defined as:
    $$P^{sell}_t = \text{essinf}_{H \in H'_{sell}} \text{esssup}_{\tau \in \mathcal{T}^{sell}_{t,T}} \rho_{t,T} \left( \int_t^T h^\tau_s dS_s - \xi_\tau \right) - \check{\rho}_{t,T}(0)$$
• Buyer's Price ($P^{buy}_t$):
  • The buyer minimizes risk by choosing both the optimal exercise time and the optimal hedging strategy.
  • The price is defined as:
    $$P^{buy}_t = \hat{\rho}_{t,T}(0) - \text{essinf}_{\tau \in \mathcal{T}^{buy}_{t,T}} \hat{\rho}_{t,T}(\xi_\tau)$$
• Arbitrage-Free Property: The authors prove that these definitions preclude arbitrage opportunities for both parties. If the filtrations coincide, the difference between the two prices represents a non-negative bid-ask spread.

B. Stochastic Volatility Models & BSDE-R-BSDEs

The paper specializes the general framework to stochastic volatility models (e.g., Heston-like models) where the asset price $S$ and volatility $V$ are driven by correlated Brownian motions.

• Risk Measure Representation: The risk measures are represented via solutions to Backward Stochastic Differential Equations (BSDEs) with a driver $g$.
• The BSDE-R-BSDE Structure:
  • The authors show that the indifference prices can be characterized by a system of BSDEs reflected at BSDEs (BSDE-R-BSDEs).
  • Mechanism: The seller's price involves a Reflected BSDE (RBSDE) where the reflection boundary is not a fixed function but is itself the solution to another BSDE.
  • Economic Interpretation: The reflection boundary $\zeta_u + Y_u$ represents the exercise value ($\zeta_u$) plus the residual risk of holding a zero contract from the exercise time to maturity ($Y_u$). This captures the fact that risk mitigation continues even after the option is exercised.
• Theorems:
  • Theorem 3.8: Characterizes the seller's price via a BSDE-R-BSDE system involving the partial Fenchel-Legendre transform of the driver ($g^*$).
  • Theorem 3.10: Provides a similar characterization for the buyer's price.

C. Numerical Implementation via Deep Learning

Solving BSDE-R-BSDEs analytically is intractable for high-dimensional problems. The authors employ Deep Learning methods.

• Algorithm: They utilize the Reflected Deep Backward Dynamic Programming (RDBDP) scheme (Huré, Pham, Warin).
• Implementation:
  • The problem is discretized backward in time.
  • Two neural networks are trained: one to approximate the boundary condition (the zero-contract risk $Y$) and another to approximate the reflected process (the option price).
  • The boundary condition is solved first and then used as the reflection barrier for the main RBSDE.
  • The implementation uses TensorFlow with Adam optimization and ReLU activation functions.

3. Key Contributions

1. Rigorous Definition of Seller's Price: The paper resolves a long-standing conceptual issue in American option pricing by defining admissible strategies for the seller that respect the non-anticipativity of the exercise time. This generalizes previous discrete-time results (Kühn [40]) to continuous time with general filtrations.
2. BSDE-R-BSDE Characterization: It establishes that risk-indifference prices in stochastic volatility models are solutions to a novel class of equations: BSDEs reflected at BSDEs. This provides a precise mathematical structure linking the exercise decision to the residual risk of the post-exercise period.
3. Arbitrage-Free Consistency: The definitions are proven to be consistent with no-arbitrage principles, ensuring that neither buyer nor seller can generate risk-free profits using the indifference price.
4. Computational Feasibility: The paper demonstrates that these complex, high-dimensional systems can be solved numerically using deep learning, bridging the gap between theoretical risk measure pricing and practical implementation.

4. Results

• Numerical Example: The authors price an American Put Option under a stochastic volatility model (arctangent volatility) using distorted entropic risk measures.
• Bid-Ask Spread: The numerical results show that the absolute difference between the buyer's and seller's prices is small (in dollar terms), supporting the view that these prices correspond to realistic market bid-ask spreads.
• Implied Volatility: When converted to implied volatilities, the difference between buyer and seller prices becomes more pronounced, exhibiting the typical "volatility smile" observed in markets.
• American vs. European: The difference between American and European indifference prices is found to be small but non-negligible, consistent with the early exercise premium.
• Validation: The numerical results align with previous studies using traditional methods (e.g., Sircar and Sturm [52]), validating the deep learning approach.

5. Significance

• Theoretical Advancement: This work fills a critical gap in the literature by providing a unified, mathematically rigorous framework for American option pricing under convex risk measures in incomplete markets. It moves beyond utility maximization to a framework more aligned with modern regulatory capital requirements (risk measures).
• Practical Application: By linking the pricing problem to BSDE-R-BSDEs and solving them via deep learning, the paper offers a viable path for pricing complex American derivatives in high-dimensional settings (e.g., multi-asset or stochastic volatility models) where traditional PDE or Monte Carlo methods struggle.
• Information Asymmetry: The framework explicitly handles scenarios where buyers and sellers have different information (filtrations), a crucial feature for modeling real-world market dynamics, insider trading scenarios, or shadow trading.
• Regulatory Relevance: The use of convex risk measures aligns with current financial regulations (e.g., Basel III/IV) that require banks to hold capital based on risk measures rather than just utility functions.

In summary, the paper successfully bridges advanced stochastic analysis (BSDEs), financial economics (indifference pricing), and modern computational techniques (Deep Learning) to solve a persistent problem in derivative pricing.