The adjustments represent a manufacturing cost. At expiration, the portfolio and the call option have the same payoff. The amount of money borrowed, and the amount of stock in the portfolio, must be continuously adjusted as the stock price changes. If the current stock price of S is $200, and if it fluctuates randomly, this portfolio will behave precisely like an option in the future, with one year to expiration and a strike of $250. The equation then also shows that this one-dollar call option can be replicated by investing one dollar, borrowing 15 dollars, and then using their sum to purchase 16 dollars’ worth of S. 2 Suppose the solution to the Black–Scholes equation yields a value today of one dollar for the call option on S described above. In 1973, Fischer Black, Myron Scholes, and Robert Merton demonstrated that if stock prices fluctuated randomly as described by the theory of Brownian motion, the future payoff of a stock option could be replicated by a portfolio comprising borrowed money and a fractional share of the stock. Different people would have different expectations about that price. What is the rational price of a one-year call option on S? Before 1973, it seemed obvious that the current value of a call option should reflect the expected future price of its stock. A put option is the corresponding right to sell the stock on a future date for a specified price. If S trades below $250, A need do nothing. If S trades in the market at $300, A can exercise the option, purchasing one share at $250 from B and selling it for $300, obtaining a payoff of $50. If one share of a stock, S, is selling for $200 today, a call option with a strike of $250 and an expiration of one year, gives A the right to buy S one year from today for $250. Stocks and OptionsĪ call option on a stock is a contract between a buyer A and a seller, or counterparty, B, that gives A the right, but not the obligation, to purchase the stock for a specified price on some future date. Markets and prices are social phenomena, not physical ones, and it is unlikely that there is an accurate predictive theory of human behavior. This failure has triggered further ingenious extensions of the model. Although the Black–Scholes model is genuinely useful, it fails to capture the reality of the market’s behavior.
The result has been a transformation in mathematical finance and an explosion of new financial products. An impalpable property of a financial system-volatility-has become concrete enough to be bought and sold. The Black–Scholes equation allows traders to treat volatility as an asset and trade it by buying or selling portfolios of options. 1 The model has become the foundation for valuing options of all kinds. More than 100 years ago, the French mathematician Louis Bachelier developed the theory of Brownian stock-price fluctuations to analyze stock options. The model suggests an analogy between the random fluctuations of a stock price and a concept from physics, the Brownian motion of randomly diffusing particles.
T he Black–Scholes model describes the value of a stock option as a function of the underlying stock price and its volatility.
0 kommentar(er)
