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13

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=) ...


12

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 ...


11

We assume that, under the probability measure $Q$, \begin{align*} dS_t &= S_t\big(r_t dt + \sigma dW_s(t)\big),\\ dr_t &= -k\, r_t dt + \alpha dW_r(t),\tag{1} \end{align*} where $d\langle W_s(t), W_r(t)\rangle_t = \rho dt$. From $(1)$, for $s\ge t$, \begin{align*} r_s = e^{-k(s-t)}r_t + \alpha\int_t^s e^{-k(s-u)} dW_r(u). \end{align*} Then, for $T\ge ...


10

Amazingly, there are several different methods for computing bond forward price – the underlying ideas are the same (forward price = spot price - carry), but the computational details differ a bit based on market convention. Let's start with the basics. Assume between now ($t_0$) and the forward settlement date $t_2$, the bond makes a coupon payment at time ...


7

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, although ...


6

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 $r_2 = 0.8486$.


6

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 ...


5

I would do as follows: A) First do PCA on an arbitrage-free monthly curve (assuming the most granular contract you will use is individual months). To ensure no arbitrages, you will need to drop out certain contracts, I would drop the most illiquid ones. To give you an example, if you are in Dec, you might see Jan, Feb and Mar quoted, but also Q1. In this ...


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

Another way to obtain this result is, as I mentioned in the comment, to think about how you would replicate the forward contract. It has the following cash-flow structure: type | t | t1 | t2 ---------------------------------------------------------------------------- forward | 0 | +P(t1,...


4

The forward price $K$, determined at time $t$, is the amount such that the payoff at time $T$ is $S_T-K$, while the value at time $t$ is zero. That is, \begin{align*} B_t E\left(\frac{S_T-K}{B_T}\mid \mathcal{F}_t \right)= 0, \end{align*} Where $E$ is the risk-neutral expectation operator. Then, \begin{align*} K&=\frac{E\left(\frac{S_T}{B_T}\mid \...


4

Assuming zero dividend and a constant interest rate $r$, the 1y forward price is then \begin{align*} 120 = K = S_0 e^r = 100\, e^r. \end{align*} Consequently, $e^r = 1.2$. The fair value of the forward contract, at 6M, is given by \begin{align*} e^{0.5 r} E\left(\frac{S_{1Y}-120}{e^{r}} \mid \mathcal{F}_{6M} \right) &= e^{0.5 r}\left(\frac{S_{6M}}{e^{0....


4

At the heart of the (relative) pricing theory is the concept of no arbitrage and replication. I'll focus on equities here because as stated in the comments it may be more complicated for commodities. Forwards deliver a payout linear in the future value of the underlying asset. Hence they can be replicated statically by a simple cash & carry replication ...


4

Interest rate derivative trading relies on curves. The LIBOR rate, be it 1month, 3month, 6month etc is published and determined every day but derivative contracts continue to speculate on what futures day's LIBOR publications will be. A 6M Libor curve does one thing and one thing only. It estimates what 6M Libor will be on any future date. I.e you can '...


4

There are two types of contract (a) a forward contract and (b) a futures contract. In (a) there is no payment of margin on a daily basis. Its value is $(F_1-F_0)e^{-r(T-t)}$ as you describe. In (b) there is a direct payment of $F_1-F_0$ on each day and the value of the contract is always zero at the close of business. Does that answer your question?


4

This has been posted a few times now, so I will invest the time on a full response. FRA / Futures convexity has nothing to do with profits/losses being immediately recognised on the future through margin settlement, whilst deferred on the FRA. Although this seems to be a very common belief amongst many practitioners it is not correct. Let me ...


3

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 ...


3

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 ...


3

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 ...


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

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 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

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

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.


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

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 $...


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

It is a very badly worded question in my humble opinion. There are three "prices" to contend with. (1) If you want to buy a stock and pay for it now, you pay the current stock price S. (2) If you want to buy a stock and not have to pay for it until a future delivery date T, then you enter into a "forward" or (in the United States) a "futures contract" ...


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