European Call Option Boundary Conditions in Truncated Domains

Summary

The document presents asymptotic boundary conditions used when numerically solving generalized Black–Scholes models on a truncated price domain. At the low-price boundary, the call value tends to zero. At the high-price boundary, it approaches the discounted value of the underlying price less the discounted strike, with time-varying interest rates and dividend yields represented by integrals over the option horizon.

The author asks what these limits mean financially and what hedging strategy a numerical method represents at the artificial boundaries. The document supplies the boundary expressions and defines the underlying variables, but contains no answers to those questions. As a result, it introduces a useful numerical-pricing concept without explaining boundary hedging, implementation choices, or how the conditions behave under particular model assumptions.

Key ideas

• Truncated-domain option pricing uses limiting values at the low- and high-price boundaries.
• The call value approaches zero at the low-price boundary in the stated setup.
• At the high-price boundary, the call approaches discounted underlying value minus discounted strike.
• The document raises questions about the financial interpretation and hedging implications but does not answer them.

Tags

# Assymptotic behaviors of European options


# Assymptotic behaviors of European options












In some of the numerical works on Black-Scholes generalized models, the boundary conditions on the truncated domain taken from the asymptotic behaviors of European call options, which is given by $$\lim_{x \to -\infty} u(\tau,x)=0,~\text{and}~\lim_{x \to \infty}u(\tau,x)-K(e^{x-\int_0^\tau D(s)ds}-e^{-\int_0^\tau r(s)ds}),$$ where, $u(\tau,x)$ is the option's value depending on the price variable $x$ and time variable $\tau$ with the strike price $K$. The parameters $r$ and $D$ represents the interest rate and dicividend yield repectively.

My doubt:

- What precisely the financial importance of these asymptotic behaviors.

- What type of hedging strategy is being processed at the truncated boundaries in numerical approaches.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.