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From what I understand, Black-Scholes equation in finance is used to price options which are a contract between a potential buyer and a seller. Can I use this mathematical framework to "buy" a stock? I do not have the choice using options in the market I am dealing with -- I either buy something or I don't. So I was wondering if B-S be used to decide to buy a stock, the next day, taking its last price, volatility and other necessary variables into account.


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In the original Black-Scholes paper, the last chapter describes alternative applications of their equation, I remember one of them was how stock holders in a company can be interpreted as actually having an option on the company's assets which were themselves stocks in another company. If you have access to JSTOR, here's the article. –  Raskolnikov Mar 17 '11 at 22:40
Also, I forgot to mention that there is a stack for quantitative finance questions. –  Raskolnikov Mar 17 '11 at 22:56
Yes, and if you want to take into account the risk of bankruptcy, you can model the company as a barrier option... –  Apollo Mar 17 '11 at 22:59
Also asked here quant.stackexchange.com/questions/764/… –  BB_ML Apr 9 '11 at 16:52
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2 Answers

up vote 7 down vote accepted

The Black-Scholes models assumes the price of the underlying asset (stock price) is given. It therefore could not tell you if the stock price is over-/under-priced. Risk-neutral pricing also won't give you any information about the likely drift of the stock in the future --- by definition, under the risk-neutral measure, the expected value of any tradeable asset is a martingale process; thus the expected value of any stock (under the martingale measure) is just the current price discounted by the risk-free rate.

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Can I use the closing price from previous day as "the price of the underlying asset"? –  BB_ML Mar 17 '11 at 22:33
Another one: since the market I work in does not offer options, can I use hypothetical prices (5% increase, 10% increase) as test cases, scenarios, to decide on buying decisions? I apologize if this question is very odd for finance, I do not know the terminology very well. –  BB_ML Mar 17 '11 at 22:36
Short answer is that Black-Scholes is not likely to help you. You may want to look into models of company fundamentals. B-S is a model for pricing derivative instruments (assuming features of the stock, get value of options). [Note that nothing I say should be construed as giving financial advice. I am not a financial advisor.] –  Apollo Mar 17 '11 at 22:40
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I'm not an expert, though I'm studying stochastic calculus right now in one of my classes. The Black-Scholes price gives an exact solution to European Call option prices, subject to fixed risk free rates, constant volatilities, and other assumptions. It can be used as one tool (out of many) to hedge a portfolio, such that, at any time, regardless of the change in price of the underlying, the change in value of the portfolio will be positive, subject to other assumptions like, for example, the option price vs underlying price curve doesn't straighten out (it maintains convexity as time goes on). So, you can use the Black-Scholes equation to create a delta-neutral portfolio, in theory.

In actuality, it's much much more difficult than this. Everyone already knows the pricing formulae, so in theory, all of the profit has been arbitraged away. Any advantage will last for a few seconds (imagine 100 other people just as smart or smarter than you trying to find a pricing mistake). Secondly, even if an investor did have a strategy, the execution of that strategy requires considerable skill. Simply buying or selling a stock incurs price slippage, trading costs, feedback, etc. There are entire departments in the banks devoted to trade execution alone.

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