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2

You have set up the wrong formula for monthly compounded interest. It should be $$ 2P = P\left(1 + \frac{0.08}{12}\right)^{t} $$ where $t$ is in months. Note that the answer you have is for $t$ in years, which is about 108 months, so its not as wrong as it might first look (it's still incorrect though).


2

The objective of the investor is to receive a return. The expected return is indicated by the yield rate. If the bond were not callable, the investor will receive back the face value of $100$ in $15$ years along with $5\%$ coupon payments on earlier dates. If the bond is callable for $ 10 \leqslant t \leqslant 15$ at $80$, then the issuer has the option ...


2

Notice that $M_n$ is $\mathcal F_n$ measurable hence so is $S_n$, and $M_n$ is integrable since it is bounded. It remains to check that $\mathbb E(S_{n+1} \mid \mathcal F_n)=S_n$. To this aim, we have to compute $\mathbb E(e^{bM_{n+1}} \mid \mathcal F_n)$. As $$e^{bM_{n+1}}=\underbrace{e^{bM_{n+1}-bM_n}}_{\mbox{independent of }\mathcal F_n}\cdot ...


2

Everything you need is in the graphic per the instructions. As mentioned in the comments and shown in the graphic, you could also solve for $t$ using numerical methods like Newton's Method. The equation becomes: $$10000e^{.072 t}+10000(1+0.084)^t = 35000 \implies e^{.072 t}+(1.084)^t = 3.5$$ The result is $t = 7.32505 ~~\mbox{years}$ (see update below ...


1

You shouldn't start with a PDE. The Black-Scholes-Merton equation is a PDE that lists the relationship between the greeks. You will need to differentiate with the solution of the PDE . You have listed the solution to the PDE for a call option in your first part of the question. You will need to do partial differentiation for the solution. $\frac{\partial^2 ...


1

If we assume that $ud=1$(Pretty standard approach) We find that $$ u^2 -Ru +1 =0 $$ Where $$ R = \mathrm{e}^{-r\Delta t}+\mathrm{e}^{r\Delta t}+\sigma^2\Delta t \mathrm{e}^{-r\Delta t} $$ We know we can't taylor expand now since the volatility will disappear straight away! So solutions to the quadratic for $u$ is $$ u =\frac{R\pm\sqrt{R^2 -4}}{2} $$ ...


1

The yield to maturity $y$ satisfies $$111.98 = 100(1+y)^{-3} + 10\sum_{k=1}^{3} (1+y)^{-k} = 10\frac{(1+y)^{-1}[1-(1+y)^{-3}]}{1 - (1+y)^{-1}}+100(1+y)^{-3}\\=10\frac{1-(1+y)^{-3}}{y}+100(1+y)^{-3}.$$ Solve numerically using, for example, bisection to find $y = 0.0556$.


1

The book's answer makes sense if the interest is not 8% per month, but 8% "per annum", or more precisely $\frac8{12}$% per month. (I'm getting 104.32 rather than 104.28 under that assumption, but that may just be a rounding issue).


1

I like the book Brownian Motion - An Introduction to Stochastic Processes by René Schilling & Lothar Partzsch pretty much. It does not only cover stochastic differential equations (in particular, several possibilites are presented how to solve SDEs, e.g. by transforming them into linear SDEs), but also contains a very interesting and detailed exposition ...


1

I like "Brownian Motion Calculus" by Ubbo F. Wiersema (2008). I found it more approachable than other books I've seen. But it depends on your level of mathematical sophistication (I'm a mathematical hick).


1

It may not be possible to determine VaR analytically and you will need to run a Monte Carlo simulation. As you indicate, for a univariate distribution $F$, VaR is specified as the $\alpha$-percentile: $$P(X \leq \text{VaR}) = \alpha \implies \text{VaR}=F^{-1}(\alpha).$$ In the multivariate case, with $X = X_1 + X_2$, you need the distribution ...


1

The strategy is to borrow by selling the 6-month bond $B_{0.5}$ and invest the proceeds in the 1-year bond $B_1$. Then after 6 months, when the loan is due ($B_{0.5}$ matures), sell $B_1$ to repay the loan. There will be a net profit as long as the 6-month spot rate in 6 months is less than the forward rate. Suppose the face value of a bond is $100$. Let ...


1

In practical terms, you will usually want to find $i$, given balances and cash flows at certain times, so the second formula is useful. However, once you are given values for $A,B,C_k,$ and $t_k$, it usually isn't very difficult to solve for $i$, so you could get by just knowing the first. It can be time-consuming, though, so consider learning both. In ...


1

By the premium/discount formula, which is $ P = C + (Fr - iC)a_{n|i}$ you can see that, if $ Fr -iC >0 $, the bond is selling at premium so the earliest redemption date is the most favorable for the issuer. (Because he would like to stop repaying the premium via the coupon payments as soon as possible.) For $ i^{(2)} = 0.08 $, the bond is ...


1

As stated, $P_1$ and $P_2$ are bond prices per $100$ of face amount. The amount invested in each bond, given face amounts $F_1$ and $F_2$, is $$V_1 = F_1\frac{P_1}{100}, \\V_2 = F_2\frac{P_2}{100}.$$ The portfolio value is the total amount invested: $$V = V_1 + V_2 = F_1\frac{P_1}{100} + F_2\frac{P_2}{100}.$$ In order to find the portfolio value, we ...



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