# How to Build Valuation Models Like Black-Scholes

Options valuation may be a difficult task. Consider the following example: IBM stock was trading at $155 in January 2015, and you anticipated it to rise in the following year. You want to acquire a call option on IBM stock with an ATM strike price of $155, hoping to gain from large percentage returns based on a low option cost (option premium) relative to a high buy price for the shares.

Today, a number of ready-made approaches for valuing options are available, such the Black-Scholes model and the binomial tree model, which may offer rapid results. But what are the fundamental causes and ideas that drive such value models? Can anything comparable, based on the principle of these models, be prepared?

The building blocks, underlying principles, and elements that may be utilized as a framework to develop a valuation model for an asset such as options are covered in this section, along with a side-by-side comparison to the beginnings of the Black-Scholes(BS) model.

This article does not seek to criticize the BS model’s assumptions or any other elements (which is a completely other issue); rather, it tries to explain the basic notion of the Black-Scholes model, as well as the concept of valuation model creation.

## The World Before Black-Scholes

The equilibrium-based Capital Asset Pricing Model (CAPM) was frequently used prior to Black-Scholes. Profits and risks were balanced depending on the investor’s desire, i.e. a high risk-taking investor was paid with (the possibility for) larger returns in a corresponding proportion.

The BS model is based on CAPM. “I applied the Capital Asset Pricing Model to every instant in a warrant’s existence, for every potential stock price and warrant value,” says Fischer Black. Unfortunately, theCAPM was unable to meet the warrant (option) price criterion.

Black-Scholes was the first model, based on the notion of arbitrage, to usher in a paradigm shift away from risk-based approaches (such asCAPM).The CAPM stock return notion was replaced by the knowledge that a correctly hedged position would earn a risk-free rate in this new BS model advancement. This removed the risk and return fluctuations and developed the notion of arbitrage, in which values are based on risk-neutral assumptionsâ€”a hedged (risk-free) position should result in a risk-free rate of return.

## The Development of Black-Scholes

Let us begin by defining the issue, quantifying it, and constructing a framework for its resolution. We will continue our example of evaluating an ATM call option on IBM with a strike price of $155 and a one-year expiration date.

According to the fundamental definition of a call option, the payment stays 0 unless the stock price reaches the strike price level. After then, the reward grows linearly (i.e., a one-dollar increase in the underlying will provide a one-dollar payoff from the call option).

Assuming the buyer and seller agree on a fair valuation (including the zero price), the theoretical fair pricing for this call option is:

- If the underlying strike price is $0, the call option price is $0 as well (redgraph)
- If underlying >= strike, call option price = (underlyingâ€”strike) (blue graph)

This indicates the option’s inherent value and seems ideal from the perspective of a call option buyer. Both the buyer and the seller have a reasonable value in the red zone (zero price to seller, zero payoff to buyer).However, the valuation problem begins in the blue zone, when the buyer benefits from a positive return while the seller suffers a loss (provided that the underlying price goes above the strike price).With zero pricing, the buyer has an edge over the seller. Pricing must be more than zero in order to pay the seller for the risk he is incurring.

In the first situation (red graph), the seller potentially receives zero price and the buyer has no payout potential (fair to both).In the latter situation (blue graph), the seller must pay the buyer the difference between the underlying and the strike. The seller’s risk extends over the course of a year. For example, if the underlying stock price rises dramatically (say, to $200 in four months), the seller must pay the buyer the $45.

Thus, it boils down to:

- Will the underlying price cross the strike price?
- If it does, how high can the underlying price go (since this determines the buyer’s payoff)?

This highlights the seller’s large risk, which begs the question, “Why would someone sell such a call if they don’t receive anything for the risk they’re taking?”

Our goal is to come at a single price that the seller should charge the customer that can compensate him for the entire risk he is incurring over a yearâ€”in both the zero payment zone (red) and the linear payment region (green) (blue).The price should be reasonable and agreeable to both the buyer and the seller. If not, the person who is at a disadvantage due to paying or obtaining an unjust price will refrain from participating in the market, negating the aim of the trading enterprise. The Black-Scholes model attempts to determine this fair pricing by taking into account the stock’s consistent price volatility, the time value of money, the option’s strike price, and the time until the option expires. Let’s look at how we may approach evaluating this for our case using our own techniques, similar to the BS model.

## How to Evaluate Intrinsic Value In Blue Region?

There are many approaches for predicting projected price change in the future over a specific time frame:

- Similar price moves of the same timeframe in the recent past may be examined. The historical IBM closing price shows that in the previous year (January 2, 2014 to December 31, 2014), the price declined to $160.44 from $185.53, a 13.5% decrease. Can we deduce that IBM’s price will fall by -13.5%?
- A more thorough examination reveals that it reached an annual high of $199.21 (on April 10, 2014) and a yearly low of $150.5. (on Dec.16,2014).Based on the beginning date of January 2, 2014, and the closing price of $185.53, the percentage change ranges from +7.37% to -18.88%. The variance range now seems to be substantially broader than the previously computed fall of 13.5%.

Similar analyses and findings may be made using historical data. To continue developing our pricing model, let’s use this easy way to predict future price fluctuations.

Assume that IBM grows at 10% every year (based on data from the last 20 years). If previous trends repeat, basic statistics show that the likelihood of the IBM stock price fluctuating about +10% is far larger than the probability of the IBM price climbing 20% or falling 30%. An overall predicted return on IBM’s stock price in a one-year time period may be determined as a weighted average of probabilities and related returns by collecting comparable historical data points with probability values. Assume, for example, that IBM’s historical pricing data reflects the following moves:

- (-10%) in 25% of times,
- +10% in 35% of times,
- +15% in 20% of times,
- +20%in 10%of times,
- +25% in 5%of timesand
- (-15%) in 5% of times.

As a result, the weighted average (or Expected Value) is:

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

That means, for every dollar invested, the IBM stock is anticipated to return +6.5% over the next year. With a one-year horizon and a purchase price of $155, an investor may anticipate a net return of 155*6.5% = $10.075.

This, however, is for the stock return. We need to explore for call options with comparable projected returns.

Based on a zero payout of the call below the strike price (current $155 – ATM call), all negative movements will result in zero payoffs, while all positive moves above the strike price will result in an equal payback. Thus, the anticipated return on the call option is:

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

That is, for every $100 invested in this option, one may expect to get $9.75. (based on the above assumptions).

However, this is still limited to a fair appraisal of the intrinsic value of the option and does not accurately account the risk incurred by the option seller for the huge swings that may occur in the interim (in the case of above-mentioned intrayear high and low prices).Aside from the intrinsic worth, what price can the buyer and seller agree on such that the seller is appropriately rewarded for the risk he is assuming over the one-year time frame?

These fluctuations may be very large, and the seller may have his own idea of how much he wants to be rewarded for them. The Black-Scholes model is based on European-style options, which are not exercised before the expiration date. As a result, it is unaffected by intermediate price movements and based its value on trading days from end to finish.

This volatility is significant in determining option pricing in real-time trading. The often seen blue payout function is the payoff at expiration date. In reality, the option price (pink line) is always more than the reward (blue graph), suggesting that the seller paid a premium to compensate for his risk-taking ability. This is why the option price is frequently referred to as the option “premium,” denoting the risk premium.

This may be included into our valuation model, based on how much volatility the stock price is predicted to have and how much estimated value it would produce.

The Black-Scholes model accomplishes it efficiently (within its own assumptions, of course) as follows:

C = S N ( d 1 ) X e r T N ( d 2 ) beginningaligned &textC = textS times textN (textd 1) – textX times e-rT textN (textd 2) endingaligned â€‹â€‹

The BS model implies that stock price fluctuations are lognormally distributed, which explains the use of N(d1) and N(d2) (d2).

- In the first portion, S represents the stock’s current price.
- N(d1) represents the likelihood of the current stock price change.

If this option expires in the money and the buyer exercises it, he will get one share of the underlying IBM stock. If the trader exercises it today, the S*N(d1) indicates the option’s present day anticipated value.

In the second part, X indicates the strike price.

- N(d2) is the likelihood that the stock price will rise over the strike price.
- As a result, X*N(d2) is the anticipated value of the stock price staying above the strike price.

Because the Black-Scholes model assumes European-style options in which exercise is only available at the end, the anticipated value given by X*N(d2) above should be discounted for the time value of money. As a result, the final portion is multiplied by an exponential term equal to the rate of interest throughout the time period.

The net difference between the two terms represents the option’s current price (wherein the second term is discounted)

Such price movements may be more properly incorporated in our framework in a variety of ways:

- Extending the range to tighter intervals to accommodate intraday/intrayear price movements improves projected return estimates.
- Inclusion of current market data since it represents current activities (similar to implied volatility)
- Expected returns on the expiration date, which may be discounted back to the present day for realistic values and decreased further from the current day value

As a result, there is no limit to the assumptions, approaches, and customisation that may be used for quantitative analysis. A self-developed model may be worked on depending on the asset to be exchanged or investment to be considered. It is vital to highlight that the volatility of price movements varies greatly among asset classesâ€”equities have volatility skew, whereas forex has volatility frownâ€”and users should integrate the relevant volatility patterns into their models. Any model has assumptions and limitations, and informed implementation of models in real-world trading circumstances may provide superior outcomes.

## The Bottom Line

When complex assets join the market, or even simple assets enter complicated forms of trade, quantitative modeling and analysis become essential for valuation. Unfortunately, no mathematical model is without flaws and assumptions. The ideal strategy is to reduce assumptions to a minimum and be aware of the implicit downsides, which may help draw the limits on model use and applicability.

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