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For the Black-Scholes model my feeling is that the volatility parameter is like sweeping stuff under the rug.

Are there models which improve on the volatility aspect of Black-Scholes by adding other parameters (I'm guessing things like the distribution of past returns, or perhaps some measure of debt load held by the company).

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An option's price is based on the volatility of the underlying asset (ignoring the difference between the strike and spot, the time to expiration, etc). The kinds of parameters you are asking for will impact cash flow, and therefore affect the price of the underlying. You'd still have to show how debt load affects volatility, which doesn't get rid of volatility. See this other question. –  chrisaycock Feb 8 '11 at 4:53
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The Black-Scholes model assumes that the underlying volatility is constant over the life of the derivative, which is indeed a gross oversimplification. Stochastic Volatility models improve on that assumption by making volatility dependent on additional parameters such as distribution of returns and variance itself. However, the well known stochastic volatility models do not include company fundamentals among their parameters.

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Extensions of the Black-Scholes model usually focus on relaxing one or more assumptions. Some of these generalizations include:

  1. Log normal distribution of returns (e.g., Corrado and Su, 1996)
  2. Continuous trading (e.g., relaxed by Merton, 1976)
  3. Continuous evolution of the share price (e.g., relaxed by Cox-Ross-Rubinstein, 1979)
  4. Constant interest rates (e.g., relaxed by Baksi et al, 1997)
  5. Constant variance on the underlying returns (e.g., relaxed by Heston, 1993)
  6. No dividends (e.g., relaxed by Merton, 1973)
  7. Contentious diffusion of the underlying (e.g., relaxed by Merton, 1976)"

Etc. Etc.

Note: I give example of relative `simple' and extensions.

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