This article will introduce the idea of Binomial Options Pricing Model (BOPM) and the principle of Risk-Neutral Valuation.
Option/option (option) pricing is more complicated than stock, bond, futures, swap/swap/exchange (swap), the famous Black-Hughes-Merton model (Black-Scholes-Merton model, BSM) because of its He was awarded the 1997 Nobel Prize in Economics for his contribution to option pricing. This article will introduce a simpler option/option pricing method than BSM – the Binomial Options Pricing Model (BOPM) (also known as the binary tree option pricing model). Essentially, BOPM and BSM are consistent, and taking the limit of BOPM will get the result of BSM.
Binomial Options Pricing Model (BOPM)
The idea of BOPM is to assume that the price of the underlying asset of the option has only two outcomes after a period of time T. By constructing a combination of the underlying asset and the European option (European option), there is no uncertainty in the return of the combination, which is transformed into a one-to-one Price the assets that generate deterministic cash flows, that is, use the risk-free interest rate to discount the cash flows, so as to obtain the price of the portfolio at the beginning of the investment period.
For example, suppose the price of a non-dividend stock today (t=0) is 10 (S0=10), and the price after 3 months is either 12 or 8, then the expiration price will be K in 3 months. What is the price (C0) of a European call at 10?
First, we know what the call price should be 3 months from now (when the call expires). When the stock price rises to 12, the value of the call option is 2, and when the stock price falls to 8, the value of the call option is 0.
Knowing the prices of the stock and the option in the two situations, it is possible to construct a risk-free (return without uncertainty) portfolio. The combination of 1 long stock and 2 short calls can make the cash flow after 3 months without any uncertainty. (Verification: When the stock price rises to 12, the income of the combination is 12-(2×2)=8; when the stock price falls to 8, the income of the combination is 8-(2×0)=8)
(The essence of constructing a combination is to solve linear equations (groups). It is recommended that the author refer to “Bond Duration/Duration and Convexity” to use duration/duration and convexity. The principles behind hedging a bond portfolio are similar to the options in this example for hedging stocks.)
Knowing that a portfolio will infallibly provide a return of 8 after 3 months, then according to the arbitrage-free equilibrium principle, the cost of this portfolio (at today’s prices) should be the discounted value of the risk-free rate of 8.
(For the principle of no-arbitrage equilibrium, readers are advised to refer to “A Brief Talk on Forward Pricing: Cost-Benefit Analysis of No-Arbitrage Equilibrium”
Regarding the risk-free interest rate, readers are advised to refer to “A Brief Talk on the Most Basic Questions About Interest Rates: Treasury Bond Yield Curve, OIS Zero Interest Rate Curve”)
Assuming today’s 3-month risk-free interest rate is 4% (annualized), the discounted value of 8 is about 7.92 (8/1.01) (3-month annualized interest rate is 4%, that is, the quarterly interest rate is 1%, the discount factor is 1.01), this is the theoretical price of 1 long stock and 2 short call options today, that is, S0-2×C0=7.92 (S0=10 is today’s stock price, C0 is today’s call option price), therefore, the call price should be 1.04.
ignore all mathtransforming all values in this example into mathematical symbols yields the general form of the binomial option pricing formula:
Note: In the above formula, f is today’s option price (in this example, the call option price is 1.04), e^-rT is the continuous risk-free interest rate discount factor (in this example, the risk-free interest rate discount factor is 1/1.01), fu is the option value if the underlying asset rises when the option expires (2 in this case), fd is the option value if the underlying asset falls when the option expires (0 in this example), p and (1-p ) are the probabilities of the underlying asset going up and down, respectively, in the risk-neutral world. u is the ratio of the rising price of the underlying asset to the initial price (12/10=1.2 in this example), and d is the ratio of the falling price of the underlying asset to the initial price (8/10=0.8 in this example).
Why can it be written in this form? Because using risk-free interest rate pricing for the constructed asset portfolio pricing is equivalent to using risk-free interest rate pricing for underlying assets and options in a risk-neutral world, the above formula is the risk-free interest rate pricing for options in a risk-neutral world formula.
What is risk-neutral?In simple terms, investors believe that certain cash flows are equal to the value of uncertain cash flows.
For example, if there are two investments, the first investment has a certainty of getting $100, the second investment has a 50% probability of getting $120, and the second investment has a 50% probability of getting $80 (the expected return of the second investment is $100), how would you value these two investments?
Risk aversion (risk aversion) people think that the first investment is better than the second investment;
Risk seeking people think that the first investment is worse than the second investment;
Risk neutral (risk neutral) that there is no difference between the two.
In a risk-neutral world, the expected return (in this case,[ p×fu+(1-p)×fd ]) is priced at the risk-free rate (e^rT in this example) (similarly, expected return in a risk-averse world is priced at a discount rate higher than the risk-free rate, since risk-averse people need more compensation to accept risk; risk-loving world medium expected return priced at a discount rate lower than the risk-free rate)
What is the relationship between the risk-neutral world and the real world? The author believes that transforming the complex option pricing problem in the real world into the simpler option pricing problem in the risk-neutral world is a transformational idea and a methodology that can stand scientific testing.(This article omits all the mathematical proof process)
The relationship between the binomial distribution and the real world
In the real world, it is reasonable to think that the price of the underlying asset or its rate of return obeys a normal/normal distribution within a certain period of time. In mathematics, the limit of the binomial distribution is the normal/normal distribution. Therefore, the real world can be approximated by increasing the degree of the binomial distribution (the number of steps of the binary tree). (In the real world, there is a minimum unit for asset quotations, which is not a continuous distribution in the true sense.) By taking the limit of the binomial option pricing formula, the BSM option pricing formula can be obtained.
The core idea of the Binomial Option Pricing Model (BOPM) is to transform the complex real world into a simple risk-neutral world, and in this world, transform the complex normal/normal distribution into a simple binomial Distribution (binomial distribution). When the problem is simplified, then use the hedging/hedging/avoiding (hedge) method and the principle of no-arbitrage equilibrium (no-arbitrage equilibrium) to price options.
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