# Finding the Probability Distribution Implied in Option Prices

### Juan Martínez

#### 12/04/2022

Prices are said to contain a good deal of information but, as this is drastically compressed into a single figure, it’s often hard to unravel the original elements involved in an asset’s pricing. Here derivatives come to the rescue, by providing us with many more price series related to the same underlying, we can get more angles on the market’s view of that asset and find more success in the information extraction task.

In this post, we will to try to reverse engineer the asset pricing process so as to obtain the expectations on that asset’s performance that correspond to observed market prices. The first step in reverse engineering is to have some notions of forward engineering, so let’s take a brief look at asset pricing.

## Some Intuitions on Asset Pricing

#### Price and Value

Price is what you pay, value is what you get.

Warren Buffet (citing Benjamin Graham)

Given the above, if value is higher than price, market participants are encouraged to buy, pushing the price upwards. The opposite occurs when value is lower and hence market pressures tend to move price towards value, also called utility. Despite this apparent convergence these are clearly distinct concepts, price is an objective measure, same for all, while value is subjective, it depends on each individual’s preferences. In strict financial terms, a transaction is a zero sum game (one wins when the other loses); however, in terms of utility, both parties can profit leading to a win-win situation.

This means there are two potential reasons for executing a transaction:

• 1. A discrepancy between one’s expectation on an asset’s performance and the market’s expectation.
• 2. A discrepancy between one’s way of measuring value given an asset’s performance and the market’s way.

Hence, a trade need not incorporate additional information on an asset’s expected performance, it may just incorporate information regarding the trader preferences.

Now, if everyone has a particular measure of value (also called utility function), to what value is market price expected to converge? We can imagine some sort of market utility function, which would be a market power weighted aggregation of each market participant’s preferences. In financial markets, one thing we can say about this function is that it is risk averse (risk has negative value), unlike in lotteries or casinos (where risk increases utility). Deriving the specifics of this risk aversion is another story, one that goes beyond the scope of this post.

#### Risk Neutrality

Still, we can make simplifying assumptions to avoid the obstacle of risk preferences and later see how to modify results to get closer to reality. Let’s consider that everyone has the same utility function and that this is risk neutral. Risk neutral means being indifferent to the distribution of possible outcomes and caring only about the expected payoff. For instance, having a 50% chance of winning 100$or losing$50 is the same as having 100% chance of getting 25\$ (mathematically: 0.5 · 100 + 0.5 · (-50) = 1 · 25).

The main advantage of the risk neutral framework (we could also have assumed any other type of risk preference) is that since we are being agnostic to uncertainty (risk) we don’t need to bother measuring it.

#### Conclusion

Now we have a framework for pricing assets: an asset price should be its value and its value is its expected payoff. The only thing we need to know to start the reverse engineering process is this expected payoff. For that we will have to specify what kind of asset we have at hands, here we will be dealing with options.

## Information in Option Prices

#### Definitions

For the sake of clarity, let’s start with some brief definitions. As its name implies, an option gives the possibility of executing a prearranged transaction. Given this optionality, the owner will only execute when conditions are favourable (i.e., the transaction is profitable), keeping any potential gains and avoiding losses.

To have a concrete formulation for the payoff we will have to be more specific regarding the nature of the option contract. Let’s stick to European options, as they are the most basic type but also one of the most commonly traded (if not the most depending on the market). A European option gives the possibility of buying (call option) or selling (put option) an underlying at a prespecified future date (maturity date, $$T$$) and for a predetermined price (strike price, $$K$$). We will just consider calls since an analogous argument can be made for puts.

#### Expected Payoff

At maturity, one would exercise the buying option whenever the strike price ($$K$$) is cheaper than market price ($$S_T$$), since then it is possible to sell the underlying in the market earning the difference in price. Otherwise the contract is allowed to expire.

Therefore, the payoff will be $$S_T – K$$ when that figure is positive and 0 otherwise. This can be mathematically expressed as $$\max(S_T – K, 0)$$. As previously stated, call price ($$C$$) should equal expected payoff (for simplicity we assume a 0 risk-free rate, we will come back to this later):

$$C = \mathrm{E}[\max(S_T – K, 0)].$$

#### Probabilities Make Their Way

Note that the mathematical expectation $$\mathrm{E}$$ is nothing but the probability ($$p$$) weighted average of all possible outcomes:

$$C = \mathrm{E}[\max(S_T – K, 0)] = \sum_{S_{T,i}>K} (S_{T,i}-K)p(S_{T,i}).$$

We are starting to see how prices are related to the probability distribution of the underlying price at maturity $$p$$. From now on we will denote it $$p_{S_T}$$ to be more explicit.

Although prices are certainly discrete (nobody pays a millionth of a cent), working in continuous domain makes little difference and will be more convenient further ahead.

$$C = \mathrm{E}[\max(S_T – K, 0)] = \int_{S_T>K} (S_T-K)p_{S_T}(S_T)dS = \int_{S_T=K}^{\infty} (S_T-K)p_{S_T}(S_T)dS_T.$$

We are almost there, the term $$p_{S_T}(S_T)dS_T$$ is what we are looking for but $$(S_T – K)$$ is in our way. The most straightforward way to get rid of it is by derivating with respect to $$K$$, as we want to leave everything else untouched. There’s a small caveat, since the lower bound of the integral depends on $$K$$, we need to apply the Leibniz integral rule which states that the derivative of the integral is the integral of the derivative plus an additional term. Fortunately, in this particular case (namely the integrand being proportional to $$S_T-K$$) this extra term happens to be 0 so we can get what we were looking for.

$$\frac{\partial C}{\partial K} = \int_{S_T=K}^{\infty} \frac{\partial}{\partial K} (S_T-K)p_{S_T}(S_T)dS_T = -\int_{S_T=K}^{\infty} p_{S_T}(S_T)dS_T = -p(S_T > K)$$

In plain English, the sensitivity of the option price to variations in strike depends on the probability of the underlying price at maturity being higher than the strike. When this probability is 0, the call price will be insensitive to changes in the strike; when it’s 1, price will change in the same amount (and opposite direction) as the strike.

#### Probability Distribution Implied in Options Prices

So the derivative has left us with the probability that the underlying price at maturity is above the strike. In statistics, it’s more common to work with the complementary probability; that is, $$p(S_T \leq K)$$. This is called the cumulative density function (CDF) and we will denote it with uppercase $$P_{S_T}$$. The relationship between the two probabilities is straightforward:

$$P_{S_T}(K) = p(S_T \leq K) = 1 – p(S_T > K).$$

Therefore, we can substitute and rearrange a little bit:

$$\frac{\partial C}{\partial K} = P_{S_T}(K) – 1.$$

The derivative of the CDF is the PDF (probability density function, i.e. lowercase $$p_{S_T}(K)$$) which is what we’ve been looking for all along:

$$\frac{\partial^2 C}{\partial K^2} = p_{S_T}(K)$$

In summary, by taking the second derivative (the curvature) of the call price with respect to the strike, we get the probability density of the asset price at maturity being equal to that very same strike.

Another way to say this is that the sensitivity of option price with respect to changes in strike varies itself as we alter the strike, and the magnitude of these variations will be proportional to the probability of the underlying price at maturity being exactly equal to the strike.

## Just a Handful of Qualifications

For a conceptual understanding the above suffices, however, there are a few details we have overlooked which are important for practical purposes.

#### Risk-Free Rate

Note that when we expressed the following:

$$C = \mathrm{E}[\max(S_T – K, 0)],$$

we have on one side of the equation the present price and on the other side a future payoff. To be rigorous we should bring those future cashflows to present time by discounting them with the risk-free rate $$r$$:

$$C = \frac{1}{1+r} \mathrm{E}[\max(S_T – K, 0)].$$

#### Numerical Differentiation

We actually don’t have an analytical expression of the function $$C=f(K)$$, instead we have a set (hopefully a big one) of prices $$C$$ and strikes $$K$$. Thus we will have to resort to numerical differentiation:

$$p_{S_T}(K) = \frac{\partial^2 C}{\partial K^2} \simeq \frac{C(K – \Delta K) + C(K + \Delta K) – 2C(K)}{\left(\Delta K \right)^2}.$$

#### Interpolation

It’s advisable to first interpolate prices in order to have smoother results (especially when strikes are too far apart from each other or prices are noisy). Actually, instead of interpolating prices it’s best to interpolate implied volatilities for sensibility reasons, but that’s a discussion for another post (see  if you’d rather not wait for that).

#### Parametric Alternatives

We have discussed the non-parametric approach as it involves the least amount of assumptions. Nevertheless, when data are unreliable, parametric methods, being less flexible, are also more robust. This too is discussed in .

## Back to Reality: Risk Aversion

Finally, so far we got what’s called the risk neutral implied distribution, not the actual (often called physical) probability distribution. It’s feasible to convert from risk neutral probabilities to physical probabilities but one needs the market’s utility function or degree of risk aversion (the relevant keyword here is Stochastic Discount Factor, which is the way utility is actually expressed in the context of asset pricing) and it’s not straightforward to derive (again makes a good topic for another post).

Still, in low risk aversion environments risk neutral probabilities can be a good enough proxy. Also, they can provide useful bounds on actual probabilities. Note that risk neutral probabilities are the ones a risk neutral investor would need to consider in order to reach the same price as the actual risk averse market. Since under the risk neutral framework the impact of adverse events is not overweighted, their probability is overestimated instead so as to reach the same expected value.

## Wrapping Up

Rather than building a model to price options, we have described the process of using option prices to derive the model implied in them. This provides us with a rather comprehensive description of the market’s view on an asset’s future performance. Conceptually the derivation is quite straightforward, in practice there are some more difficulties that can be overcome but are substantial enough to deserve another post. See you then!

## References

 Clews, R., Panigirtzoglou, N. and Proudman, J., 2000. Recent developments in extracting information from options markets. Bank of England Quarterly Bulletin.