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Monte Carlo Simulation and the Black-Scholes Model

Monte Carlo Simulation and the Black-Scholes Model. Monte Carlo Simulation. Monte Carlo Simulation. MCS attempts to manage uncertainty in complex environments.

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Monte Carlo Simulation and the Black-Scholes Model

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  1. Monte Carlo Simulation and the Black-Scholes Model

  2. Monte Carlo Simulation

  3. Monte Carlo Simulation • MCS attempts to manage uncertainty in complex environments. • MCS manages the uncertainty by searching for optimal values of the assigned decision variables through a pre-selected distribution curve.

  4. Monte Carlo Simulation • The desired outcome of the simulation is a range of values in which the probability of those outcomes is known. • For the Black-Scholes Model, the desired outcome was a known probability for which the price of an option on a certain financial asset would fall.

  5. Monte Carlo Simulation • MCS is performed through scenario analysis. • This process determines the possible outcome(s) given specified inputs. • To overcome the limitations of a few simple scenario analyses, MCS allows the user to perform thousands of trials in order to attain an accurate estimate of what will likely happen.

  6. Monte Carlo Simulation • The results of the simulation are summarized on a distribution graph.

  7. Why Use MCS? • MCS allows the user to obtain an advanced evaluation of one’s risk through a range of possible outcomes. • These outcomes are given probability distributions that allow the user to know, with a good degree of certainty, what will occur based on your specific inputs. • The use of MCS allows the user to somewhat control uncertainty, which in turn allow the user to make stronger and hopefully more profitable decisions.

  8. Options

  9. The Basics on Options • What is an Option? • The right, but not the obligation, to buy or sell a security for a specified price on or before a specific date • Two Basic Types of Options: • Call Option • Put Option

  10. Call Options • Call options give the taker (buyer) the right, but not the obligation, to buythe underlying shares at a predetermined price on or before a predetermined date.To acquire this right the taker pays a premium to the writer (seller) of the contract.

  11. Put Options • Put options give the taker (buyer) the right, but not the obligation, to sell the underlying shares at a predetermined price on or before a predetermined date.To acquire this right the taker pays a premium to the writer (seller) of the contract.

  12. Premiums • Premiums are the price of Options. The potential loss to an option buyer can be no greater than the initial premium paid for the contract, regardless of the performance of the underlying security. • This allows the buyer to control risk, as all risk is assumed by seller for the premium.

  13. Exercise Price • The fixed price per share at which a call option conveys the right to purchase the underlying shares • The fixed price per share at which a put option conveys the right to sell the underlying shares.

  14. Expiration Date • The last date on which an option holder can exercise the right conveyed by the option. After that date, the option ceases to exist.

  15. In/Out of the Money (ITM/OTM) • When current market price is greater than the exercise price, the call option is said to be In-The-Money (ITM) • When current market price is less than the exercise price, the call option is said to be Out of-The-Money (OTM)

  16. Black-Scholes Model & Results

  17. The Black-Scholes Pricing Model

  18. Black-Scholes Excel Model • Assumptions: • Current market price $46.63/sh • Risk free rate 3.34% (T-Bill) • Return 1 year volatility 39.20% • Control Variable: • Exercise price • Yr. to expiration (fixed)

  19. Raw Data • Calculate historical return volatility • Compute standard deviation of closing stock price

  20. Distribution Selection • Black-Scholes model is based on normal distribution

  21. Standard Deviations • Calculated stock price standard deviation of $5.02 • One year volatility is relatively stable 39.20% • Use default of 10% of 39.20% • Unlikely the Federal Bank cuts interest by more than 50 basis points

  22. Simulated Variables • Call and Put Option Price Simulation (10,000 trials) • Decision variable: exercise price ranged $45 to $55

  23. Monte Carlo Call Option Result • Reasonable price range $4 - $8 • Deterministic model price: $6.55 • Distribution is positively skewed • Few low exercise prices are below to current market price (few call options “in the money”)

  24. Monte Carlo Put Option Result • Reasonable price range $6 - $9 • Deterministic model price: $8.26 • Distribution is relatively symmetric • Lowest level of exercise price is just below to current market price (most of the call options “in the money”)

  25. Call and Put Conclusion • Put prices range are more expensive (more put options “in the money” than call options) • Black – Scholes model and options market are very dynamic • Options prices always change • Monte Carlo simulation provides reasonable price range for trading

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