Why Risk-Neutral Pricing Works for Replicable Black–Scholes Options

# Why the Black-Scholes formula can be used in the real world?


# Why the Black-Scholes formula can be used in the real world?












The BS formula is deduced using the risk neutral measure. Why can it be used in the real world?

## Answer by Matt Wolf (score 10)

https://quant.stackexchange.com/a/17078

The short answer is:

As long as a derivative can be perfectly replicated via hedging in the underlying asset then the price of the derivative should be independent of investors' risk aversion and hence the application of risk-neutral probabilities and discounting of the future expected payoff under risk neutral probability leads to the same price of the derivative as an application of real-world probabilities would.

Longer answer:

The above has been proven and shown by Black and Scholes' seminal work (along with Robert Merton). Note that several conditions have to be met in order to assure one can move into a risk-neutral pricing framework:

- Derivative can be replicated by trading in the underlying asset(s) and trading in the money market.

- No arbitrage requirement

- Complete Market

- Law of one price, among others

Please note that contrary to the statement of another user one can perfectly price derivatives via real-world probabilities. But the problem is that one would have to have knowledge of how to discount each expected future payoff, meaning, one would have to know investors' risk preferences. That is virtually impossible or at least very difficult, hence, the whole application of risk-neutral pricing is in order to simplify the work of pricing derivative securities.

Note of caution: One cannot use risk-neutral pricing to assess the future value of a derivative. The reason for that is that the underlying assets do not grow at a risk-free rate, generally a risk-premium has to be applied.

## Answer by Abramo (score 7)

https://quant.stackexchange.com/a/17066

The risk-neutral probability is used as a convenient mathematical tool but, strictly speaking, it is not a necessary ingredient for the BS formula.

In fact, the formula can be derived by computing the expectation (under the physical probability) of the option payoff, properly discounted with the right stochastic discount factor:

$$ p_t = E_t[\;m_{t,T}\; f(S_T)\,] $$

where $f$ is the payoff function of the derivative, a function of the realisation at maturity time $T$ of the underlying, i.e. $S_T$. The BS formula for a call option is obtained by choosing the corresponding payoff function (exercise: what is it?).

The stochastic discount factor (SDF) $m_{t, T}$ is a positive random variable which discounts for both time and risk. In every possible state of the world encompassed by the expectation operator, the value of the $SDF$ in that state gives a weight to the payoff. For instance, a good payoff in a very bad state is highly valuable: the SDF will give a very strong weight to that state, which ultimately leads to an higher price.

The effect of the SDF may as well be described by means of the risk-neutral probability: after all, it all boils down to give the right weights (probabilities) to each state. Hence we can write

$$ p_t = E_t[\;m_{t,T}\; f(S_T)\,] = E^Q[\, e^{-r(T-t)} f(S_T) \,] $$

where $Q$ is the risk-neutral probability, which accounts for the risk discounting.

## Answer by berkorbay (score 2)

https://quant.stackexchange.com/a/17059

If we are talking about BS world, one plausible explanation is every move in the risk-free world is "perfectly hedgeable". In other terms if you consider the drift term $\mu = r + \lambda\sigma$ BS formula allows you to reduce $\lambda$ to zero by constantly hedging with the underlying. That is why you can get a correct price with $r$ under BS world.

However "real" real world is different and even in theoretical incomplete markets there are more than one risk-free measure. With some models, there is no perfect hedge. Good luck with that.

## Answer by Kumar (score 1)

https://quant.stackexchange.com/a/17052

Risk-neutral probabilities and physical probabilities agree on what is possible and what is impossible. Also, a hedge under Risk-neutral probability works almost surely and so does the hedge under physical probability.

## Answer by vonjd (score 1)

https://quant.stackexchange.com/a/17061

Because BS is about derivatives and not about the underlying. In a way if you priced derivatives with real world measures (all else being equal) you would double count risk preferences because these are already included in the underlying - think about it this way (beware, oversimplification ahead):

You want to price a derivative on gold, a gold certificate. The product just pays the current price of an ounce in $.

Now, how would you price it? Would you think about your risk preferences? No, you won't, you would just take the current gold price and perhaps add some spread. Therefore the risk preferences did not matter (=risk neutrality) because this product is derived (= derivative) from an underlying product (=underlying).

This is because all of the different risk preferences of the market participants are already included in the price of the underlying and the derivative can be hedged with the underlying continuously (at least this is what is taken for granted in a BS world). As soon as the price of the gold certificate diverges from the original price traders would just buy/sell the underlying and sell/buy the certificate to pocket a risk free profit - and the price will soon come back again.

So, you see, the basic concept of risk neutrality is quite natural and easy to grasp.