# Using Black-Scholes equations to “buy” stocks

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.

Thanks,

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An equity represents ownership of a company and may be thought of as a derivative on the cash flow. For this reason, equities are valued through discounted cash-flow (DCF) analysis.

An option is a right, though not an obligation, to buy or sell an asset at a fixed price at some point in the future. As per Black-Scholes, the value of an at-the-money option is principally determined by the time to expiration and the volatility of the underlying's price.

Equities are very different from options:

• The volatility of the cash flows cannot be modeled by Brownian motion. Instead, volatility is represented by the discounting factor in DCF used to determine the present value.
• Equities don't have an expiration, so their value can't simply decrease to zero over time.
• Equities don't really confer a right to the cash flow; there is a whole series of corporate governance on that one.

In short, the features of equities and options are so vastly different that their valuation techniques must be distinct.

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I prefer richardh's answer...viewing equity as a perpetual option on assets is certainly interesting, even though it's operationally difficult. –  Brian B Nov 12 '12 at 18:46

You can look at equity as a call option on the firm. In theory this illustrates the differences between holding equity or debt.

The quick and dirty is that equity holders own the firm, but only after the debt holders are repaid. If you have a simple levered firm with one outstanding debt issue, it as though the equity holders have a call option on the firm with strike equal the face value of debt with expiration equal to the debt maturity date.

You can use the technique to price equity, but it still requires you to value the underlying firm and do the calculation for all outstanding debt. A lot of research tests these pricing models, but there are so many assumptions wrapped up in it, I would have a really hard time using the "call option" price to say that the firm's equity is under- or over-priced.

Regardless, it's an interesting way of looking at the relation between equity and debt. If you're interested in learning more, Damodaran has a good reference on his website.

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This question has been answered many times over already, though hopefully this will provide a bit more insight. If I understand your question correctly, you're basically asking if you can use BSM as a trading indicator.

So let's think about what it really means to be trading an option. Every single variable (i.e. price of underlying asset, strike price, risk-free rate, and time until expiration) in the model is observable in the market, except volatility. So what does this mean? Mostly when we're trading an option, we're trading a difference in volatility. So let's assume you believe that your interpretation of what volatility is, is better than what's currently being priced by the market. That doesn't get us too far if we only want to trade the underlying.

Now some people may look at ratios of calls/puts or call/put volume as well. I would definitely warn against this type of indicator as well. You're basing your indicator on the assumption that, say people only buy calls if they think the direction of the underlying is going to rise. This is simply untrue, those positions may be there for a market maker to hedge inventory risk.

Hope this helps!

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Yes, this certainly is valid point of view, if you are thinking about purchase the stock outright rather than with options. In theory, if the Implied Volatility of calls was significantly higher the the Implied Volatility for puts, buying the stock would be a better bet than shorting it. This is particularly true for the At-the-Money calls and puts. You could also use the expiration month as a guide to the time frame. Longer time frames indicate more of a trust in fundamentals.

-Ralph Winters

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The B-S formula is not such interesting for you as its derivation. This formula is based on the B-S model.

(i) Suppose that the model nicely describe real-life - remember that price movements are assumed to be random and unpredictable so it does not contradict with EMH.

If the model is true, the derivation of the B-S equation tells you how that if you short an option - you can perfectly (non-randomly!) hedge yourself with a given strategy (delta-hedging). What does it mean? It means that if on the real market the price of the option do not coincide with a theoretical value - then using delta-hedging you can make and the maturity on your bank account more money then you need to exercise an option.

Roughly speaking, if real price of the option does not coincide with that which B-S formula gives you - you can make a profit.

Problems? The model is not perfect, i.e. using delta-hedging you are still risky - e.g. in the B-S delta-hedging there are no transaction costs which is artificial.

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As the BSM gives a call price as function of Stock price, volatility and other inputs, c(0) = BSM[Stock, Strike, volatility, riskfree rate, term], it seems to me you could use it in an analogous way to implied volatility (i.e., implied vol is the volatility input that produce a model output = traded market price).

The analogous use, i think, would be to input/assume a volatility, then given an observed (traded) option price, simply iterate to solve for the Stock price that calibrates the BSM = the traded option price; i.e., conditional on a given volatility assumption, this would be the fair stock/asset price implied by the BSM.

... of course, in addition to the inherent model risk, you have to assume a volatility [that is NOT the implied volatility] so, i think it's Merton-esque in that you are hinging it on the hard to estimate volatility.

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... actually, the worst part of the model risk is probably that by using BSM you assume the future asset price is lognormaly distributed –  David Harper Mar 24 '11 at 0:36

I'm trying to understand what you are asking:

• you cannot use options directly, only buying stock
• thus, you want to use options as an indicator on whether or not you should buy a stock

I don't think the BS model is a great indicator for stock direction because it's based of not knowing where the stock may go. That said, a lot of people use action in the options market for suggestions about the stock movement.

One such indicator is the put/call ratio. Greater put buying suggest downside risk and call buying suggests upside movement.

You may want to also consider heavily traded strike prices as an indicator to what level the stock might trade to. If your stock is trading at 100 and there's heavy open interest at the 120 calls, the theory suggests that "smart money" knows something.

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This is basically how one would use BlackScholes to "purchase a stock"...

double optionPrice = blackScholes(stock, strike, volatility, rate, time);
if (optionPurchased) {
if (stockPrice < thresholdPrice){
ExerciseOption(PurchaseStock());
}
}


Note that the purchase of the option would be on a futures exchange, not the regular stock market. Also note the double if/then clauses. First you have to purchase the option, then use it to evaluate whether you want to proceed with the purchase. There are a couple different evaluation functions you could use to set the thresholdPrice.

http://sourceforge.net/projects/chipricingmodel/

ps. for what it's worth, I used to work at the Booth School of Business in Chicago, as a tutor in the computing labs, helping MBA students do Black-Scholes pricing for assignments.

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Options are not traded on the futures exchange. In the US, options are primarily traded on CBOE, PHLX, ISE, and ARCA; futures are mostly traded on CME and CBOT. Options are traded on options exchanges; futures are traded on futures exchanges. They are two totally different asset classes. –  chrisaycock Apr 7 '11 at 17:46
What of it? It doesn't change the fact that Black-Scholes isn't used to purchase the stock in of itself. And who has said anything about the original poster being located in the US? Black-Scholes applies to European and Asian markets as well as the US. –  BioinformaticsGal Apr 7 '11 at 18:35
The question is effectively about whether BS helps provide a mathematical framework for the decision to purchase stock. You have answered a different question: whether one can use option purchases (which the poster said he cannot perform) to effectively purchase stock. –  Brian B Nov 12 '12 at 18:44

I am not sure any of the other answers mentioned this but the main reason you should not use an option model to buy/sell the underlying (BS or other) is that the option models are more about market-making in options and hedging using the underlying rather than forecasting the underlying.

The layman way to understand this is that: using an option model, you do not care about forecasting the right price of the underlying as your job is to market-make options and your loss and profit are related to the efficiency of the hedge which is NOT directly correlated with the efficiency of forecasting your underlying.

This is why most of option models start with the hypothesis of no arb as they are trying to hedge using replication.

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So, reading the question, I see two parts: first, can Black-Scholes be part of the buying process? and second, is Black-Scholes, by itself, sufficient to evaluate a stock and make a decision on whether to purchase it?

The answer to the first, is yes. The answer to the second, is basically no. Black-Scholes generates the price of an option on a futures market, not a value of a stock on a regular stock market. That being said, the futures market are intimately related to stock markets, and an option can be considered an "if-then" clause which defines the availability of a stock.

Black-Scholes is basically telling you how much it's going to cost to have an option to say yes-or-no to proceed with a purchase, at a certain day in the future, at today's price. (It's a little like getting on a waiting list to purchase scalped tickets for a sold-out concert.) Whether or not you want to go ahead with the purchase has to be based on some other function. But Black-Scholes can certainly be part of the process, if you want to be that sophisticated.

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What do futures have to do with options, Black-Scholes, or even the question? The OP is asking whether BS can be used to make a purchasing decision. Your answer doesn't address how that could be possible. And your mention of futures doesn't make any sense. –  chrisaycock Apr 6 '11 at 23:56
@chisaycock... Uh, a 'future' is an exchange-traded derivative. And an 'option' is a type of futures contract. Black-Scholes isn't used to make purchasing decisions unto itself, so the original question is nonsensical. It's not possible to use a Black-Scholes to make a purchasing decision on a stock, other than to use it as an 'if/then' clause, regarding the availability of stock in question. It's only used to determine the price of an option; not whether one should purchase it or not. For that you need a threshold. –  BioinformaticsGal Apr 7 '11 at 14:53
An option is not a type of futures contract. A futures contract is a standardized, exchange-traded forward. –  chrisaycock Apr 7 '11 at 17:44
I mispoke. I meant to say that a future is a type of option. en.wikipedia.org/wiki/Option_(finance) –  BioinformaticsGal Apr 7 '11 at 18:32
A future is not a type of option, either. They are two completely different asset classes. The only feature they even have in common is that they (along with swaps) are classified as "derivatives". –  chrisaycock Apr 7 '11 at 23:35