Black–Scholes Pricing, d1 and d2, and Model Assumptions
Summary
The document introduces Black–Scholes as a model for calculating a theoretical option price and asks how to interpret its d1 and d2 terms. The response says the model gives an appropriate price only if its assumptions hold. It does not explain d1 or d2 directly, instead pointing to an external explanation, so it offers limited detail on the mathematics or on worked examples.
The answer stresses that model prices need not match market prices because Black–Scholes assumes lognormal price behavior, zero bid–ask spreads, and market equilibrium. It also raises uncertainty about the risk-free rate, using possible US government funding problems to illustrate that even a supposedly safe benchmark can carry credit, maturity, and inflation concerns. These are caveats rather than a procedure for adjusting model output to market value; the response does not provide empirical evidence or a practical calibration method.
Key ideas
- Black–Scholes produces a theoretical option value conditional on its assumptions.
- The document asks about d1 and d2 but defers their explanation to an outside source.
- The answer identifies lognormal price behavior, frictionless liquidity, and equilibrium as model assumptions.
- A market price can diverge from the theoretical value when those assumptions fail.
- The choice of risk-free benchmark can itself be uncertain and may involve other risks.
Tags
Full text
# Help with a research paper on the Black-Scholes equation # Help with a research paper on the Black-Scholes equation I am currently a senior in high school who has been tasked with writing a research paper on a math topic of our choice. I knew I wanted to research some sort of financial model but I was told most company models such as the DCF and CAPM were not advanced enough to research. Que the Black-Scholes Option Pricing Equation. I started researching derivative equations and one of the most widely used models that I could find was the BS. I just have a couple questions regarding this model for anyone who has experience working with it. - From my research I've gathered that the BS is used to find the appropriate price of a call premium, is this correct? - What do d1 and d2 represent in the equation, I believe I have found that d2 represents the risk-free rate of the option but I could use some clarity here. 3.I have to set up a series of examples that apply the BS equation. I have read that the BS reflects the theoretical price of a call premium and not that practical one, would it be wrong to set up an example solving for the call premium using the BS, what other processes does this value have to go through to accurately reflect the market value of a call option? I've only been reading over this topic for a couple of days now but it is quite interesting and I apologize if I incorrectly used some of the terminology or do not fully have a grasp of this concept. ## Answer by Dave Harris (score 0) https://quant.stackexchange.com/a/33324 1) If all the assumptions of Black-Scholes held in nature, then yes. 2)The answer to this can be rather long, so I found someone who already wrote something about it. It can be found at https://financetrainingcourse.com/education/wp-content/uploads/2011/03/Understanding.pdf 3) Black-Scholes assumes log-normality. This is not even remotely true. It also assumes infinite liquidity, that is the bid-ask spread is zero, always. There is also an assumption that the markets are in equilibrium inherited from the Capital Asset Pricing Model from which it can be derived. As such, when markets are not in equilibrium the model is silent about appropriate pricing. In the immediate future, you need to solve for a risk-free rate, but it is quite possible that Congress will close the government next month due to an inability to decide on spending. It is not impossible that it will default on the national debt. You would need an alternate asset, one likely not denominated in dollars. The only US companies with nearly risk-free status is Microsoft and Johnson & Johnson, but the debt has maturity risk and inflation risk because it is not immediately due.
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