Mathematical Modeling in Finance with Stochastic Processes
7.3. PUT-CALL PARITY 241 Figure 7.4: Value of the put option at maturity ...
242 CHAPTER 7. THE BLACK-SCHOLES MODEL Figure 7.5: Value of the put option at various times ...
7.3. PUT-CALL PARITY 243 Figure 7.6: Value surface from the put-call parity formula ...
244 CHAPTER 7. THE BLACK-SCHOLES MODEL Sources This section is adapted from: Financial Derivatives by Robert W. Kolb, New York I ...
7.4. DERIVATION OF THE BLACK-SCHOLES EQUATION 245 Key Concepts The derivation of the Black-Scholes equation uses (a) tools fro ...
246 CHAPTER 7. THE BLACK-SCHOLES MODEL purchases and sales can be made in any amounts, that is, the stock and bond are divisibl ...
7.4. DERIVATION OF THE BLACK-SCHOLES EQUATION 247 The second assumption is a reasonable assumption for a modeling attempt althou ...
248 CHAPTER 7. THE BLACK-SCHOLES MODEL In a short time interval, we can take the changes in the portfolio to be δΠ =φ(t)δS+ψ(t)r ...
7.4. DERIVATION OF THE BLACK-SCHOLES EQUATION 249 portfolio. Either way, the existence of risk-free profits to be made in the ma ...
250 CHAPTER 7. THE BLACK-SCHOLES MODEL Black-Scholes equation derived by more advanced probabilistic methods. In this equivalent ...
7.4. DERIVATION OF THE BLACK-SCHOLES EQUATION 251 Sources This derivation of the Black-Scholes equation is drawn from “Financial ...
252 CHAPTER 7. THE BLACK-SCHOLES MODEL Outside Readings and Links: Bradley University, School of Business Administration, Finan ...
7.5 Implied Volatility Mathematical Ideas Historical volatility Estimates ofhistorical volatilityof security prices use statisti ...
254 CHAPTER 7. THE BLACK-SCHOLES MODEL where ̄uis the mean of theui’s. Sometimes it is more convenient to use the equivalent for ...
7.5. IMPLIED VOLATILITY 255 C = 1.7647, which is too low. SinceC is a increasing function ofσ, this suggests we try a value ofσ= ...
256 CHAPTER 7. THE BLACK-SCHOLES MODEL Problems to Work for Understanding Suppose that the observations on a security price (in ...
7.6 Sensitivity, Hedging and the “Greeks” consequences for hedging investments. The sensitivity of the Black-Scholes formula (o ...
258 CHAPTER 7. THE BLACK-SCHOLES MODEL Figure 7.7: Value of the call option at various times Mathematical Ideas To start the exa ...
7.6. SENSITIVITY, HEDGING AND THE “GREEKS” 259 Delta TheDeltaof a European call option is the rate of change of its value with r ...
260 CHAPTER 7. THE BLACK-SCHOLES MODEL we first considered options, see Options. The increase in security value inS is visible i ...
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