Evaluating options can be tricky. Consider the following scenario: In January 2015, IBM stock was trading at $155, and you expect it to go higher in the next year. You intend to purchase a call option on IBM stock at an ATM strike price of $155. Compared to stocks purchased at a high purchase price, based on a smaller option cost (option fee), you want to benefit from a high percentage return.

Today, there are several different off-the-shelf methods for evaluating options—including the Black-Scholes model and the binary tree model—that can provide quick answers. However, what are the underlying factors and driving concepts that led to this valuation model? Based on the concepts of these models, can something similar be prepared?

Here, we cover the building blocks, basic concepts, and factors that can be used as a framework for building valuation models for assets (such as options), and provide a side-by-side comparison with the Black-Scholes (BS) model.

This article does not intend to challenge the assumptions of the BS model or any other factors (this is a completely different topic); on the contrary, it aims to explain the basic concepts of the Black-Scholes model and the idea of valuation model development.

## The world before Blake Scholes

Before Black-Scholes, the Equilibrium-based Capital Asset Pricing Model (CAPM) was widely adopted. Return and risk balance each other, based on investor preferences, that is, high-risk investors expect to obtain higher returns (potential) in similar proportions.

The root of the BS model lies in CAPM. According to Fischer Black: “I apply the capital asset pricing model to every moment in the life cycle of a warrant, for every possible stock price and warrant value.” Unfortunately, CAPM cannot satisfy the requirements of warrant (option) pricing. Require.

Black-Scholes is still the first model based on the concept of arbitrage, a paradigm shift from a risk-based model (such as CAPM). This new BS model development replaces the CAPM stock return concept, acknowledging that perfect hedging positions will receive a risk-free interest rate. This eliminates changes in risk and return and establishes the concept of arbitrage, where valuation is based on the assumption of a risk-neutral concept-hedging (risk-free) positions should result in a risk-free rate of return.

## The development of Black-Scholes

Let’s start by identifying the problem, quantifying the problem, and developing a framework for its solution. Let’s continue our example and evaluate IBM’s ATM call option with a strike price of $155 and a one-year expiration.

According to the basic definition of a call option, unless the stock price reaches the strike price level, the return is zero. After this level, the return increases linearly (that is, an increase of $1 in the underlying will provide a return of $1 in the call option).

Assuming that the buyer and seller agree to a fair valuation (including zero price), the theoretical fair price of the call option is:

- Call option price = 0 USD, if the underlying <strike price (red chart)
- Call option price = (underlying-strike price), if underlying >= strike price (blue chart)

This represents the intrinsic value of the option, which looks perfect from the perspective of the call option buyer. In the red area, both buyers and sellers have a fair valuation (zero price to the seller and zero benefit to the buyer). However, the valuation challenge starts in the blue area, because the buyer has the advantage of a positive return, while the seller suffers a loss (provided that the underlying price is higher than the strike price). This is where the buyer has an advantage over the zero-price seller. The pricing needs to be non-zero to compensate for the risk taken by the seller.

In the former case (red chart), theoretically, the price received by the seller is zero and the buyer’s return potential is zero (fair to both). In the latter case (blue chart), the difference between the price of the subject matter and the strike is paid by the seller to the buyer. The seller’s risk spans a full year duration. For example, the price of the underlying stock may be very high (for example, reaching $200 in four months), and the seller needs to pay the buyer a difference of $45.

Therefore, it boils down to:

- Will the price of the underlying securities cross the strike price?
- If so, how high can the underlying price rise (because this will determine the buyer’s profit)?

This shows that the sellers have taken a lot of risk, which begs the question-why would anyone sell such a call option if they did not get any reward for the risk they took?

Our goal is to determine the single price that the seller should charge the buyer, which can compensate him for the overall risk taken in a year-in the zero payment area (red) and linear payment area (blue). The price should be fair , Both buyers and sellers can accept. If not, then those who are disadvantaged in paying or accepting unfair prices will not participate in the market, thus going against the purpose of the transaction business. The Black-Scholes model aims to establish the fair price by considering the constant price change of the stock, the time value of money, the exercise price of the option, and the expiration time of the option. Similar to the BS model, let’s see how to use our own method to evaluate our example.

## How to evaluate the intrinsic value of the blue area?

There are several methods that can be used to predict the expected future price changes within a given time frame:

- One can analyze recent similar price changes of the same duration. IBM’s historical closing price shows that in the past year (January 2, 2014 to December 31, 2014), the price fell from US$185.53 to US$160.44, a decrease of 13.5%. Can we get a price trend of -13.5%? For IBM?
- A more detailed inspection revealed that it hit an annual high of US$199.21 (April 10, 2014) and an annual low of US$150.5 (December 16, 2014). Based on the starting date of January 2, 2014 and the closing price of US$185.53, the percentage change has changed from +7.37% to -18.88%. Now, compared to the 13.5% drop calculated earlier, the range of change appears to be much larger.

Similar analysis and observations can be made on historical data. To continue the development of our pricing model, let us assume this simple method to measure future price changes.

Assume that IBM is rising by 10% per year (based on historical data for the past 20 years). Basic statistics show that assuming the historical pattern is repeated, the probability that IBM’s stock price fluctuates around +10% will be much higher than the probability that IBM’s stock price will rise by 20% or fall by 30%. Collecting similar historical data points with probability values, the overall expected return of the IBM stock price in a one-year time frame can be calculated as a weighted average of the probability and related returns. For example, suppose IBM’s historical price data indicates the following trends:

- (-10%) In 25% of the cases,
- +10% in 35% of the cases,
- +15% in 20% of cases,
- +20% in 10% of cases ,
- +25% in 5% of the time and
- (-15%) within 5% of the time.

Therefore, the weighted average (or expected value) becomes:

(-10%*25% + 10%*35% + 15%*20% + 20%*10% + 25%*5% – 15%*5%)/100% = 6.5%

In other words, on average, the price of IBM stock is expected to return +6.5% per dollar within one year. If someone buys IBM stock for a one-year period and a purchase price of $155, the net return can be expected to be 155*6.5% = $10.075.

However, this is for stock returns. We need to find similar expected returns for call options.

Based on the zero return of the call option below the strike price (existing $155-ATM call option), all negative changes will generate zero return, and all positive changes above the strike price will generate the same return. Therefore, the expected return of a call option is:

（-0% *25%+10%*35%+15%*20%+20%*10%+25%*5%—— 0 %*5%）/100%= 9.75%

In other words, for every $100 invested in this option, one can expect $9.75 (based on the above assumptions).

However, this is still limited to the fair valuation of the intrinsic quantity of the option, and it does not correctly capture the risk (in the case of high and low prices during the above-mentioned year) that the option seller may take on possible high volatility. In addition to the intrinsic value, what price can the buyer and the seller agree on so that the seller can be fairly compensated for the risks they take within one year?

These fluctuations may vary greatly, and the seller may have his own explanation for how much compensation he hopes to receive. The Black-Scholes model assumes European-style options, that is, they are not exercised before the expiry date. Therefore, it is not affected by the fluctuation of the intermediate price and is valued based on the end-to-end trading day.

In actual intraday trading, this volatility plays an important role in determining option prices. The blue return function we usually see is actually the return on the maturity date. In fact, the option price (pink chart) is always higher than the return (blue chart), indicating the price that the seller is taking to compensate for his risk-taking ability. This is why option prices are also referred to as option “premiums”-essentially referring to risk premiums.

This can be included in our valuation model, depending on the expected volatility of the stock price and the expected value generated.

The Black-Scholes model effectively (of course, within its own assumptions) is as follows:

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begin{aligned} &text{C} = text{S} 次 text{N} ( text{d}_1 ) – text{X } 次 e^{-rT} text{N} ( text{d}_2 ) \ end{aligned}

Ç = S × Ñ ( D . 1 ) – X- × Ë – – [R & lt Ť Ñ ( D 2 )

The BS model assumes a lognormal distribution of stock price changes, which proves that it is reasonable to use N(d1) and N(d2).

- In the first part, S represents the current price of the stock.
- N(d1) represents the probability of the current price change of the stock.

If the option becomes an in-the-money option and the buyer is allowed to exercise the option, he will receive a copy of the underlying IBM stock. If the trader exercises it today, then S*N(d1) represents the current expected value of the option.

In the second part, X represents the strike price.

- N(d2) represents the probability that the stock price is higher than the strike price.
- So X*N(d2) represents the expected value of the stock price staying above the strike price .

Since the Black-Scholes model assumes that European options can only be exercised at the end, the expected value represented by X*N(d2) above should be discounted against the time value of money. Therefore, the last part is multiplied by the index term and raised to the interest rate over a period of time.

The net difference between the two items represents the price value of the option as of today (the second item is discounted)

In our framework, there are several ways to more accurately include such price changes:

- Further refine the calculation of expected returns by expanding the scope to more granular intervals to include intra-day/intra-year price changes
- Contains current market data because it reflects current activity (similar to implied volatility)
- The expected return on the maturity date can be rolled back to the current date based on the actual valuation, and further reduced from the current date value

Therefore, we see that there is no limit to the selection of assumptions, methods, and customization for quantitative analysis. Depending on the asset to be traded or the investment to be considered, a self-developed model may be used. It is important to note that the price changes of different asset classes are very volatile-stocks have volatility skewness, foreign exchange has volatility-users should add applicable volatility patterns to their models. Assumptions and shortcomings are an integral part of any model, and knowledgeable model application in real-world trading scenarios can produce better results.

## Bottom line

As complex assets enter the market, and even ordinary assets enter complex trading forms, quantitative modeling and analysis have become necessary conditions for valuation. Unfortunately, any mathematical model does not have a series of shortcomings and assumptions. The best way is to keep assumptions to a minimum and pay attention to hidden shortcomings. This helps to distinguish between the use and applicability of the model.