# Notation Definition Modeling the Price of a Stock Option Strategies

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## Notation Definition Modeling the Price of a Stock Option Strategies

Baker College of Mount Clemens, US has reference to this Academic Journal, Known dividend so that be paid before option expirationDividend has already been announced or stock pays regular dividendsOption should be priced on S0 ? PV(dividends anticipated before t) = S0 ? Div * exp(-r*time dividend paid)Example: Coca-Cola pays relatively stable quarterly dividend around \$0.31 per share on November 29th each year. Assuming it is November 14, use S0 ? 0.31* exp(-r*15/365) in consideration of current stock price in Black-Scholes model in consideration of January option.Dividend Adjustments so that Black-ScholesContinuous Dividend PaymentsIndex funds on basket of stocks (e.g. S&P 500 index): the many stocks pay out their dividends throughout the yearMerton Model: Assume continuous dividend yield kDividend Adjustments so that Black-ScholesStrategies if anticipate that stock price will rise over period tPurchase callsPurchase stockPurchase stock in addition to put so that insure portfolioWrite putStrategies if anticipate that stock price will decline over period tPurchase putsWrite calls Covered call: purchase stockOption Strategies

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Most financial models of stock prices assume that the stock?s price follows a lognormal distribution. (The logarithm of the stock?s price is normally distributed)This implies the following relationship: Pt = P0 * exp[(?-.5*?2)*t + ?*Z*t.5] Modeling the Price of a StockP0 = Current price of stockt = Number of years in futurePt = Price of stock at time t ?Random Variable!!Z = A standard normal random variable alongside mean 0 in addition to standard deviation 1 ?Random Variable!!? = Mean percentage growth rate of stock per year expressed as a decimal? = Standard deviation of the growth rate of stock per year expressed as a decimal. Also referred so that as the annual volatility.Notation DefinitionOption price is the expected discounted value of the cash flows from an option on a stock having the same volatility as the stock on which the option is written in addition to growing at the risk-free rate of interest.The cash flows are discounted continuously at the risk-free rateThe option price does not depend on the growth rate of the stock!Cox, Ross & Rubenstein (1979) Option Price Theory

Simulate the stock price t years from now assuming that it grows at the risk-free rate rf. This implies the following relationship: Pt = P0 * exp[(rf-.5*?2)*t + ?*Z*t.5] Compute the cash flows from the option at expiration t years from now.Discount the cash flow value back so that time 0 by multiplying by e-rt so that calculate the current value of the option.Select the current value of the option as the output variable so that determine its mean price in addition to other statistics.Option Pricing Simulation LogicWant so that guarantee that t periods from now you will have at least I*zz is a number generally between 0 in addition to 1that guarantees a minimum valueWant so that invest in Stock alongside price S0 in addition to Put in consideration of stock alongside exercise price XA package of share + put costs S0 + P(S0,X)Buy a packages wherea =I/(S0 + P(S0,X))Minimum \$ return = aX which should be set so that I*zPick X so that guarantee that S0 + P(S0,X)=X/zGuaranteeing total \$ return on the total initial investment IVariance-Covariance MethodUsing an assumed distribution in consideration of the asset return (e.g. normally distributed), estimated mean, variances & covariance, compute the associated probability in consideration of the VaRHistorical SimulationUse sorted time series data so that identify the percentile value associated alongside the desired VaRMonte Carlo SimulationSpecify probability distributions & correlations in consideration of relevant market risk factors in addition to build a simulation model that describes the relationship between the market risk factors in addition to the asset return. After performing iterations, identify the return that produces the desired percentile in consideration of the VaR.3 Basic Approaches so that Compute VaR

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Use options so that create other securitiesBull spreads (written in addition to purchased calls)Collars (stock, written call in addition to purchased put)PPUP (Principal-Protected, upside potential: bond plus at-the-money call)ButterflyOptions can be replicated by a long or short position in the underlying stock in addition to a long or short position in the risk-free asset (e.g. bond)Option Strategies

## Johnson, Margaret General Manager

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