The Black-Scholes-Merton model for stock pricing in discrete time Let So be the initial stock price...
2. Let Bt denote a Brownian motion. Consider the Black-Scholes model for the price of stock St, 2 So-1 and the savings account is given by β,-ea (a) Solve the equation for the price of the stock St and show that it is not a (b) Explain what is meant by an Equivalent Martingale Measure (EMM) martingale. State the Girsanov theorem. Give the expression for Bt under the EMM Q, hence derive the expression for St under the EMM, and...
Question 1 Consider the derivation of the Black-Scholes model of option pricing. Let S=S(t) be the underlying stock price at time t and let f=f(S, t) be the option price at time t. a) Write down the value P of the portfolio defined in the Black-Scholes model. [2 marks] b) Use Itô’s lemma to find an expression for the change Δf in the discrete time Δt. [5 marks] c) Use the expression you have found in point b) to find...
3. Some computations related to a stock S(t) following the Merton-Black-Scholes Model. (a) Let S(t) = S(0) exp((u - 02/2)t +oW(t)), where W(t) is a standard Brownian motion. Compute that u is the expected annual return rate, i.e., E[S(T)] = S(O)eMT, where T > 0. Is o2 the variance of S(T)/S(O)? (b) Let X be the continuously compounded annual rate of return between 0 and T, i.e., S(T) = S(0) exp(XT). Compute E(X) and Var(X) (find the distribution of X...
In this question we assume the Black-Scholes model. We denote interest rate by r, drift rate pi and volatility by o. A European power put option is an option with the payoff function below, Ka – rº, ha if x <K, 0, if x > K, for some a > 0. In particular, it will be a standard European put option when a = 1. (a) Derive the pricing formula for the time t, 0 <t< T, price of a...
14. Note that the Black-Scholes formula gives the price of European call c given the time to expiration T, the strike price K, the stock’s spot price S0, the stock’s volatility σ, and the risk-free rate of return r : c = c(T, K, S0, σ, r). All the variables but one are “observable,” because an investor can quickly observe T, K, S0, r. The stock volatility, however, is not observable. Rather it relies on the choice of models the...
Black-Scholes
1. C8: Provide a formula for the forward price based on the
stock price S, the risk-free rate r and the time to expiration
T.
2. Columns N, O: Provide formulas for the future value (at
expiration) value of the option premiums using the BlackScholes
option prices C(K,T) and P(K,T), the risk free rate r and the time
to expiration T.
Black-Scholes 2.45-Y 100% Q- Search in Sheet Home Layout Tables Charts SmartArt Formulas Data Review Edit Font Number...
SOME DRAWBACKS OF BLACK-SCHOLES To provide one motivation for the development of ARCH models (next handout), we briefly discuss here some difficulties associated with the Black Scholes formula, which is widely used to calculate the price of an option. For example, consider a European call option for a stock. This is the right to buy a specific number of shares of a specific stock on a specific date in the future, at a specific price (the exercise price, also called...
1. Consider the following discrete time one-period market model. The savings account is given by Bo 1 and B1 1.1. The stock price is given by So 1 and S,-ξ where ξ is a random variable taking two possible values u 1.2 and d = 0.9. Consider a put option whose payoff at time l is P = (1-S)+. (a) Find a replicating strategy for this option. By considering the value of the replicating strategy, find the time 0 price...
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Budgetary Policy and Economic Growth Errol D'Souza The share of capital expenditures in government expenditures has been slipping and the tax reforms have not yet improved the income...