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11

The PCA analysis does not really tell you what the bonds do but it tells you how the rates move together. The variations of $n$ rates (i.e. 1 y, 2y, ...) are split up in (at first) abstract factors like $$ \Delta R_i = \sum_{j=1}^n e_{i,j} f_j $$ where $\Delta R_i$ is the change in the rate $i$ and $f_j$ is factor $j$ and $e_{i,j}$ is the (factor loading=) ...


9

The majority of the movement in currencies is in the spot rates, rather than in the term structure. A 3-month rolling hedge would always be protecting against movements in the spot rates, no matter when they happen. Using your example, if the current EUR/USD rate is 1.3333, you might be able to get a 3-month forward at 1.3339. (Forgive me if I have the ...


5

Yes. Although sometimes people mean the Euro/Dollar currency pair which can cause confusion. Besides the daily mark-to-market, the counter-party risk is also removed through the clearing house for the futures. No. Eurodollar and FRA are not the same as swaps. A Eurodollar fixes an interest rate for a three month period in the future whereas a swap represents ...


4

Note that $\frac{F(0,s,T)}{F(0,t,T)} = \frac{T-t}{T-s}\frac{B(0,s)-B(0,T)}{B(0,t)-B(0,T)}$ and $\frac{F(s,s,T)}{F(s,t,T)} = \frac{T-t}{T-s}\frac{B(s,s)-B(s,T)}{B(s,t)-B(s,T)}$. Multiplying the numerator and denominator of the last expression with $B(0,s)$ and noting that $B(0,s)B(s,u)=B(0,u)$ (investing one Dollar for $s$ years and then for another $u-s$ ...


4

Suppose that the given condition is true. You want to construct an arbitrage portfolio to take advantage of this. Now, $d$ is an interest rate, and the condition suggests that $d$ is too high. So you will want to receive $d$ in order to profit. If you could, you would borrow money at $r$ and lend it to the stock broker or exchange to collect the interest ...


4

This part of your post In addition, on expiry day the holder (...) is wrong. [Short Story] Due to the daily variation margins calculated by the clearing house on each market close, you have already received/coughed up what you should upon expiry. If the contract is cash-settled, the story thus ends here. In case of physical delivery however, ...


3

If you're asking what the FX Outright for 1M EUR/PLN is, given that table, then yes the answer is just outright = spot + fwd points, which is 3.4550 + 0.0079 = 3.4629 (you had the wrong column for your 1M value). Usually fwd points are quoted directly (i.e. not as an outright), using a divisor set by market convention. I expect EUR/PLN divisor to be 10,000, ...


3

for Japan, act/365 for the domestic market, and act/360 for the euroyen market. For swaps, fixed leg convention is 6m libor act/365, floating leg, if based on libor, is the 6m rate act/360, if tibor, then the 3m rate act/365.


3

The two are not equivalent, because of the cross-currency basis spread (CCBS), which became a risk factor in itself sice 2007, and does depend on term. This practically leeds to a difference in your constantly-assumed notionals (the notional is not constant anymore). What it happens is that you assume having a constant notional cross-currency swap that ...


3

For the last question. We assume that \begin{align*} S_t = S_0 e^{(r-q-\frac{1}{2}\sigma^2)t + \sigma W_t}, \end{align*} where $W$ is a standard Brownian motion, $r$ is the interest rate, $q$ is the dividend yield, and $\sigma$ is the volatility. Then, \begin{align*} X_{u+a}-X_a &= (r-q-\frac{1}{2}\sigma^2)a + \sigma(W_{u+a}-W_u)\\ &\sim (r-q-\frac{...


3

it's easiest to see in terms of replication. The pay-off of a forward contract is $$ S_T - K. $$ We can replicate this precisely and statically by buying one unit of stock, $S_0,$ and $Ke^{-rT}$ riskless bonds growing at rate $r.$ So its value today is $$ S_0 - Ke^{-rT}. $$ This has zero value if and only if $K= S_0 e^{rT}.$ This value is then called ...


3

As a trader I used Black model (amongst others) to value swaptions, where the forward swap rate is the key observable underlying rate. Any market where the forward is the traded instrument would lend itself to Black.


3

Put-call parity says that the difference between a call and a put is equivalent to the difference in the current stock price (adjusted down for dividends) and the strike price discounted at the risk-free rate. $$Call - Put = S_0*e^{-div} - K*e^{-rt}$$ So, if you want to have 120 dollars in the future, you would need to need to have $120 worth of "K" or 2....


3

I think you're arriving at a value for a swap using 2 different expressions of the same thing because FX forward prices are calculated using spot rates and adding or subtracting forward points. The forward points for a currency pair express interest rate differentials between the 2 currencies in the pair. I think your question then moves from arriving at a ...


2

In my mind you are simply right: you arrive at $$ f(t,S) = S(t) - K e^{-r(T-t)}. $$ Assume that $t=0$, so we are at the inception of the contract, then $$ f(0,S) = S(0) - Ke^{-r T}. $$ If you choose $K = S(0) e^{r T}$ then the contract value at inception is zero. This simply means that the fair price for the forward is given by $K= S(0) e^{r T}$ which is ...


2

A very good and up-to-date question. Whether you use the LIBOR-rate or any other rate for discounting depends on what you decide to be the fundamental rates in the market. Before the crisis LIBOR-rates were mostly seen as the fundamental market rates (or the "risk-neutral" rates). After the crisis it turned out that these rates were not completely free of ...


2

You need to isolate the risk factors that impact your forward contract, which is your spot fx rate, and the two rates of each currency that underlies the forward contract. You therefore need to estimate the VaRs of each of those risk factors. You also need the correlations between the underlying risk factors. For example, a forward to buy USD in exchange ...


2

Broadly, yes. An IMM dated swap is usually just a standard swap starting on an IMM date. However, there are a few closely related instruments which you could be asking about: IMM-anniversary swap - this not only starts on an IMM date, but also keeps to IMM dates for the roll schedule. It is a better match when hedging with IR futures, because an IMM-start ...


2

The process must contain the spot price. The AnalyticEuropeanEngine will take care of calculating the forward price from the data you're passing in the process (in this case spot and risk-free-rate) and the maturity of the option. As implemented in QuantLib, though, The BlackProcess class assumes there's no dividend yield. If you want to model some kind of ...


2

The FRA A FRA is an agreement to exchange cash flows; the FRA in question is: Start 15/9/14 End 15/5/15 which is 242 days. USD Money Market quoting is Actual/360, so the accrual factor here is 242/360 = 0.6722. The FRA cashflows, therefore, are: on 15/9/14, Fix pays $\$1m * (0.6722 * 0.05) = \$33,611.11$, and Float pays $\$1m * (0.6722 * L)$, where L ...


2

Because the day count of your inquired date is 366 days: Hkd daycount is act/365 therefore 366/365 Usd daycount is act/360 therefore 366/360 $$ \frac{7.7487}{7.7587} = \frac{1+r_2(\frac{366}{365})}{1+0.00965×\frac{366}{360}} $$ Solving for $r_2 = 0.8486$.


2

When you buy a forward you don't have to invest any money, so that's to your advantage in a world of positive interest rates. To charge you the same as the spot rate would be unfair, you would be "getting something for nothing", that is why the appropriate price for a forward is higher. It takes the interest rate into account, balancing things out. In ...


2

The forward price $F$ for a forward contract, determined at the contract inception time today, is the price that the holder will pay at maturity $T$ to buy the underlying equity. Then the payoff, at maturity $T$, of the forward contract is given by \begin{align*} S_T-F. \end{align*} The present value of the contract is then \begin{align*} e^{-rT} \mathbb{E}\...


2

Strictly speaking, any risk-free interest rate can be composed into three components: The rate expectations component is the market's "true" expectation for future interest rate. A bond risk premium component: longer maturity bonds have higher duration risk than cash. Accordingly market participants will demand more compensation for taking on duration ...


2

The rate of interest on cash and the cost of borrowing the stock work in opposite directions. Think of the cost of borrowing the stock as a kind of "dividend" that the stock pays off to its holders. As a stock owner you receive this amount [if you lend the shares] while you pay the interest rate if you hold the stock on margin.


2

Let's use a no-arbitrage argument. Assume that the (continuously compounding) dividend yield is $q$ while the interest rate is $r$. For portfolio 1, we long 1 forward with maturity $T$ and delivery price $K$. The payoff at time $T$ is $S_T - K$. For portfolio 2, we long $e^{-qT}$ unit of a stock (while reinvest all dividends) and short $K e^{-rT}$ unit ...


2

You wrote: outright price -80.4318/80.4610 this is the quote in the spot market. With 80.4610 rubles you can buy 1 USD and with 1 USD you can buy 80.4318 rubles Fwd points 3M - 19650/19950 this is for the forward contract (to receive/pay rubles in 3 months time). These are "points", that have to be added or subtracted from the spot rate to get the actual ...


2

Since all futures are linear instruments you can achieve a perfect hedge by going short or long into the same future depending on your position. If however there are no available futures you can use cross-hedging as explained by Hull (2007) i get an error bellow I'm not sure why so I'll put it in code format: > To answer your question the delta of ...


2

Let's call R the riskless security (100 today, 120 at time T). And call S the stock = 50, and either 70 or 30 at time T. One way to look at it is: A] Consider: buy 2 call options (C), short the stock (S), invest 50 (proceeds from S) in R. At time T: S=70: 2C=40, buy back S=-70, proceeds from R=60. net: 30 S=30: 2C=0, buy back S=-30, proceeds from R=60....


2

If the riskless security cost $100$ at time $t=0$ and $120$ at time $T$ then the risk free rate, $r$, is $20\%$. So that, $r=0.2$. Denote the initial stock price as $S_0$ and price of the call option as $c$. Suppose that at time $t=0$ you buy one stock and sell $\Delta$ options. Your portfolio value at time $t=0$ is $$P_0 = -\Delta\times c + S_0$$. At time $...



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