Assume that the investor holds a portfolio that consists of one European call option with strike 20, two European call options with strike \( 60 \), and six European put options with strike \( 90 \). All the options have the same expiration. What is the payoff on this portfolio if the price of the underlying asset at the expiration becomes \( 80 \)?
Assume an one period binomial model in which the initial stock price is \( S=60 \) and in each period the stock price can go either up by a factor of \( u=\frac{7}{3} \) or down by a factor of \( d=\frac{2}{3} \). Assume that the simple interest rate over one time period is \( r=\frac{1}{3} \).
Calculate the price of binary call and put options with strike \(K\). Assume that the stock price follows the binomial model with parameters \(u\), \(d\), \(r\), \(K\). Binary call pays \(1\) if \(S(n)\geq K\) and 0 if \(S(n) < K\). Binary put pays \(1\) if \(S(n) < K\) and \(0\) if \(S(n)\geq K\). What is the price of a portfolio which consists of 1 binary call option and 1 binary put option with the same strike?
Determine the number of sequences \( \left(X_1, X_2, \dots, X_{100}\right) \) with terms in \( \{-1,1\} \) such that the following two conditions are satisfied \begin{eqnarray*}X_1+X_2+\cdots+ X_{100}=0\quad\mbox{ and}&& \newline \left(X_1+X_2+\cdots+X_{k}\right)^2< 30^2 \mbox{ for all }k\in\{1,2,\dots, 100\} .&&\end{eqnarray*}
Assume that instead of binomial model the stock price \( S_n \) evolves in discrete time according to the formula \( S_n=S_0+\sigma W_n \), where \( W_n \) is the standard random walk and \( S_0 \) and \( \sigma \) are given positive real numbers. In other words, the initial price is \( S_0 \) and in each step the stock price increases by \( \sigma \) or decreases by \( \sigma \). Assume that the interest is \(r\) in each step.
Determine the formulas for risk-neutral probabilities for this model and using these risk-neutral probabilities determine the formula for the price of European call option with strike \( K=S_0 \) and expiration 3 under the assumption that \( S_0 > 3\sigma \).