Options Trading Financial Mathematics For Dummies (Pricing, Risk & Strategies)

You might have heard the legendary degenerates of Wall Street Bets that traded option with insane amount of money in hopes of generating absurd amount of returns. Hence the purpose of this article, to reduce your risk and hedge your positions against those who used wall street as a giant casino by basicly taking their money in front of their eyes in broad daylight. A.K.A Legal Theft!

Pricing Models — Relative Value and Arbitrage-Free:

Relative value pricing is a common technique used by investors and traders to evaluate the attractiveness of different securities based on their relative value. For example, a relative value trader might compare the price-earnings ratio (P/E) of two stocks in the same industry to determine which one is cheaper or more expensive.

Alternatively, a relative value investor might compare the dividend yield of two stocks to see which one offers a higher income stream. Relative value pricing can also be applied to other asset classes, such as currencies, commodities, or derivatives.

One example of relative value pricing in the fixed income market is the swap spread arbitrage. A swap spread is the difference between the fixed rate of an interest rate swap and the yield of a Treasury bond with the same maturity. A swap spread arbitrage involves buying (selling) a Treasury bond and entering into a pay-fixed (receive-fixed) swap with the same maturity, if the swap spread is too high (low) relative to its historical average. The arbitrageur expects to profit from the convergence of the swap spread to its fair value.

Arbitrage-free pricing is based on the assumption that there are no arbitrage opportunities in the market, meaning that there are no transactions that can generate a riskless profit with zero net investment. This assumption implies that the prices of all securities are consistent with each other and reflect their true values. Arbitrage-free pricing models use mathematical techniques to derive the fair prices of securities based on their cash flows and risk characteristics.

Black-Scholes Model

One example of arbitrage-free pricing is the Black-Scholes model for European options. The Black-Scholes model uses a partial differential equation (PDE) to determine the fair price of an option as a function of the underlying asset price, the strike price, the time to maturity, the risk-free interest rate, and the volatility of the underlying asset. The PDE is derived by applying the principle of no-arbitrage to a portfolio consisting of an option and a delta-hedged position in the underlying asset.

How the Black-Scholes Model Works

Black-Scholes posits that instruments, such as stock shares or futures contracts, will have a lognormal distribution of prices following a random walk with constant drift and volatility. Using this assumption and factoring in other important variables, the equation derives the price of a European-style call option.

The Black-Scholes equation requires five variables. These inputs are volatility, the price of the underlying asset, the strike price of the option, the time until expiration of the option, and the risk-free interest rate. With these variables, it is theoretically possible for options sellers to set rational prices for the options that they are selling.

Furthermore, the model predicts that the price of heavily traded assets follows a geometric Brownian motion with constant drift and volatility. When applied to a stock option, the model incorporates the constant price variation of the stock, the time value of money, the option’s strike price, and the time to the option’s expiry.

Black-Scholes Assumptions

The Black-Scholes model makes certain assumptions:

• No dividends are paid out during the life of the option.
• Markets are random (i.e., market movements cannot be predicted).
• There are no transaction costs in buying the option.
• The risk-free rate and volatility of the underlying asset are known and constant.
• The returns of the underlying asset are normally distributed.
• The option is European and can only be exercised at expiration.

While the original Black-Scholes model didn’t consider the effects of dividends paid during the life of the option, the model is frequently adapted to account for dividends by determining the ex-dividend date value of the underlying stock. The model is also modified by many option-selling market makers to account for the effect of options that can be exercised before expiration.

The Black-Scholes Model Formula

The mathematics involved in the formula are complicated and can be intimidating. Fortunately, you don’t need to know or even understand the math to use Black-Scholes modeling in your own strategies. Options traders have access to a variety of online options calculators, and many of today’s trading platforms boast robust options analysis tools, including indicators and spreadsheets that perform the calculations and output the options pricing values.

The Black-Scholes call option formula is calculated by multiplying the stock price by the cumulative standard normal probability distribution function. Thereafter, the net present value (NPV) of the strike price multiplied by the cumulative standard normal distribution is subtracted from the resulting value of the previous calculation.

​C=SN(d1​)−Ke−rtN(d2​)

where:

d1​=σs​t​lnKS​+(r+2σv2​​)t​

and

d2​=d1​−σs​t​

and where:

C=Call option price

S=Current stock (or other underlying) price

K=Strike price

r=Risk-free interest rate

t=Time to maturity

N=A normal distribution​

Limitations of the Black-Scholes Model

Though the Black-Scholes model is widely use, there are still some drawbacks to the model; some of the drawbacks are listed below.

• Limits Usefulness: As stated previously, the Black-Scholes model is only used to price European options and does not take into account that U.S. options could be exercised before the expiration date.
• Lacks Cashflow Flexibility: The model assumes dividends and risk-free rates are constant, but this may not be true in reality. Therefore, the Black-Scholes model may lack the ability to truly reflect the accurate future cashflow of an investment due to model rigidity.
• Assumes Constant Volatility: The model also assumes volatility remains constant over the option’s life. In reality, this is often not the case because volatility fluctuates with the level of supply and demand.
• Misleads Other Assumptions: The Black-Scholes model also leverages other assumptions. These assumptions include that there are no transaction costs or taxes, the risk-free interest rate is constant for all maturities, short selling of securities with use of proceeds is permitted, and there are no risk-less arbitrage opportunities. Each of these assumptions can lead to prices that deviate from actual results.

Binomial Tree Model

Basics of the Binomial Option Pricing Model

With binomial option price models, the assumptions are that there are two possible outcomes — hence, the binomial part of the model. With a pricing model, the two outcomes are a move up, or a move down.The major advantage of a binomial option pricing model is that they’re mathematically simple. Yet these models can become complex in a multi-period model.

In contrast to the Black-Scholes model, which provides a numerical result based on inputs, the binomial model allows for the calculation of the asset and the option for multiple periods along with the range of possible results for each period (see below).

The advantage of this multi-period view is that the user can visualize the change in asset price from period to period and evaluate the option based on decisions made at different points in time. For a U.S-based option, which can be exercised at any time before the expiration date, the binomial model can provide insight as to when exercising the option may be advisable and when it should be held for longer periods.

By looking at the binomial tree of values, a trader can determine in advance when a decision on an exercise may occur. If the option has a positive value, there is the possibility of exercise whereas, if the option has a value less than zero, it should be held for longer periods.

Calculating Price with the Binomial Model

The basic method of calculating the binomial option model is to use the same probability each period for success and failure until the option expires. However, a trader can incorporate different probabilities for each period based on new information obtained as time passes.

A binomial tree is a useful tool when pricing American options and embedded options. Its simplicity is its advantage and disadvantage at the same time. The tree is easy to model out mechanically, but the problem lies in the possible values the underlying asset can take in one period of time. In a binomial tree model, the underlying asset can only be worth exactly one of two possible values, which is not realistic, as assets can be worth any number of values within any given range.

For example, there may be a 50/50 chance that the underlying asset price can increase or decrease by 30 percent in one period. For the second period, however, the probability that the underlying asset price will increase may grow to 70/30.

For example, if an investor is evaluating an oil well, that investor is not sure what the value of that oil well is, but there is a 50/50 chance that the price will go up. If oil prices go up in Period 1 making the oil well more valuable and the market fundamentals now point to continued increases in oil prices, the probability of further appreciation in price may now be 70 percent. The binomial model allows for this flexibility; the Black-Scholes model does not.

Real-World Example of Binomial Option Pricing Model

A simplified example of a binomial tree has only one step. Assume there is a stock that is priced at $100 per share. In one month, the price of this stock will go up by $10 or go down by $10, creating this situation:

• Stock price = $100
• Stock price in one month (up state) = $110
• Stock price in one month (down state) = $90

Next, assume there is a call option available on this stock that expires in one month and has a strike price of $100. In the up state, this call option is worth $10, and in the down state, it is worth $0. The binomial model can calculate what the price of the call option should be today.

For simplification purposes, assume that an investor purchases a one-half share of stock and writes or sells one call option. The total investment today is the price of half a share less the price of the option, and the possible payoffs at the end of the month are:

• Cost today = $50 — option price
• Portfolio value (up state) = $55 — max ($110 — $100, 0) = $45
• Portfolio value (down state) = $45 — max($90 — $100, 0) = $45

The portfolio payoff is equal no matter how the stock price moves. Given this outcome, assuming no arbitrage opportunities, an investor should earn the risk-free rate over the course of the month. The cost today must be equal to the payoff discounted at the risk-free rate for one month. The equation to solve is thus:

• Option price = $50 — $45 x e ^ (-risk-free rate x T), where e is the mathematical constant 2.7183.

Assuming the risk-free rate is 3% per year, and T equals 0.0833 (one divided by 12), then the price of the call option today is $5.11.

The binomial option pricing model presents two advantages for option sellers over the Black-Scholes model. The first is its simplicity, which allows for fewer errors in the commercial application. The second is its iterative operation, which adjusts prices in a timely manner so as to reduce the opportunity for buyers to execute arbitrage strategies.

For example, since it provides a stream of valuations for a derivative for each node in a span of time, it is useful for valuing derivatives such as American options — which can be executed anytime between the purchase date and expiration date. It is also much simpler than other pricing models such as the Black-Scholes model.

Human Risk Aversion

Human risk aversion is influenced by various factors, such as cognitive abilities, genetic factors, life-cycle effects, and behavioral biases.

Some studies have found that higher cognitive abilities are associated with lower risk aversion, as they enable individuals to process complex information and make optimal decisions.

Other studies have suggested that risk aversion has a genetic component, as it is related to the regulation of the serotonergic and dopaminergic pathways in the brain.

Moreover, risk aversion may change over the life cycle, as individuals face different income and wealth levels, health status, and family situations4. Finally, risk aversion may also be affected by behavioral biases, such as loss aversion, framing effects, overconfidence, and myopia.

Greed/Fear

Greed and fear are not only psychological phenomena, but also have physiological and neurological correlates. Some researchers have used electroencephalography (EEG) and functional magnetic resonance imaging (fMRI) to measure the brain activity of traders and investors in response to market fluctuations.

They found that greed and fear are associated with different brain regions and networks, such as the amygdala, the insula, the anterior cingulate cortex, and the orbitofrontal cortex. These brain regions are involved in emotion processing, reward anticipation, risk perception, and decision making.

Furthermore, some studies have also measured the hormonal levels of traders and investors, such as cortisol and testosterone. They found that these hormones can modulate the levels of greed and fear, as well as the risk-taking behavior of market participants.

Conclusion

I had just give you the formula to reduce your risk in options trading and giving you the reminder that behind all of those trades are human. So combine both and voila!