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2

This will depend on the nature of your tree. For a re-combining binomial tree, the number of nodes, including the initial one, will be \begin{align*} \sum_{i=1}^n i = \frac{n(n+1)}{2}. \end{align*} For the paths, as at each time $j$, there are two possibilities from each node, the total path number is $2^n$.


0

Translate your forecast of yields into a forecast of bond prices: you believe long term bonds will fall in price rel. to short term bonds. So, what to do? Shorten the duration of your portfolio, i.e. sell long term bonds and/or buy short term bonds. Since you don't like long term bonds (and the fixed payments they make to you), you may also enter into ...


1

Note that, for $K_1 < K < K_2$, \begin{align*} -(K_2-K_1) \le (S_T-K_2)^+ - (S_T-K_1)^+ \le 0. \end{align*} Taking the conditional expectation with respect to information set $\mathcal{F}_t$, \begin{align*} -(K_2-K_1)B_t(T) \le C(T, K_2, S, t) - C(T, K_1, S, t) \le 0. \end{align*} That is, \begin{align*} -B_t(T) \le \frac{C(T, K_2, S, t) - C(T, K_1, ...


3

For the case where $\sum_{i=1}^n \lambda_i =1$, you need only note that the payoff \begin{align*} (x-K)^+ \end{align*} is a convex function in $x$. That is, \begin{align*} \Big(\sum_{i=1}^n \lambda_i S_i -K\Big)^+ \le \sum_{i=1}^n\lambda_i(S_i-K)^+. \end{align*} Then \begin{align*} e^{-rT}E\left(\Big(\sum_{i=1}^n \lambda_i S_i -K\Big)^+\right) \le ...


1

Let $B_t$ be the value of the risk-free asset at time $t$. Then $B_0=1$ and $B_{t+1} = (1+R) B_t$. Moreover, let $\beta_t$ be units invested in the risk-free asset at time $t$. It is clear that $\beta_0 = w_0 - \Delta_0 S_0$. Since the strategy is self-financing, \begin{align*} \Delta_{t-1} S_{t-1} + \beta_{t-1} B_{t-1} = \Delta_t S_{t-1} + \beta_t ...


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Have you looked at the PerformanceAnalytics R package functions? It should allow you to calculate delta normal Var quite easily. I recommend you look at the instructions manual but here is the code for it: VaR(R = NULL, p = 0.95, ..., method = c("modified", "gaussian", "historical", "kernel"), clean = c("none", "boudt", "geltner"), portfolio_method = ...


3

Using the answer from: Chris Taylor, on math stackexchange (link): Let the price of an option at strike $K$ be given by $V(K)$. To say that the price is convex in the strike means that $$V(K-\delta) + V(K+\delta) > 2 V(K)$$ for all $K>0$ and $\delta>0$. Let's assume that the opposite is true, i.e. that there exist tradeable option contracts ...


0

american options are at least as expensive as their european counterparts. So it's enough to argue that european options increase in value as time to expiry prolongs, given other metrics remain the same. This is because of the "time value" of options. On the other hand, longer time give you more opportunity to early exercise, which adds in zero or positive ...


3

For American options, the longer the maturity, the more choices for the optimal exercises time, then the option value is bigger. For example, consider maturities $T_1$ and $T_2$, for the same option except for different maturities. Any optimal exercise time within $[0, T_1]$ is a possible exercise time within $[0, T_2]$, with a better time possibly falls in ...



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