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24

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 ...


12

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 ...


8

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 ...


8

An FX swap exposes the user to a risk that is intrinsic to the interest rate differentials and supply and demand factors of one currency relative to another, but fundamentally there is negligible exposure to the spot FX rate, since one essentially agrees to a buy price and a sell price separated by a fixed amount. A forward FX contract is an agreement to ...


8

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, and potential reinvestment, whilst deferred on the FRA. Although the opposite seems to be a very common belief amongst many practitioners (...


7

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 $r_2 = 0.8486$.


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

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 ...


6

You are not giving the constructor a discountCurve. The constructor is: ql.ForwardRateAgreement(valueDate, maturityDate, position, strikeForward, notional, iborIndex, discountCurve=ql.YieldTermStructureHandle()) So you should add a the spotCurveHandle as the last parameter: fra = ql.ForwardRateAgreement(ql.Date(7, 5, 2018), ql.Date(15,12,2020), ql.Position....


5

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 ...


5

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,...


5

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 \...


5

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 ...


5

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?


5

Decompose the first formula as $F_0=(S_0 - S_0(1-e^{-dT}))e^{rT}$ then let $PV_{I} = S_0(1-e^{-dT})$ which represents the present value of dividends (dividend rate = $d$) paid on the security during the life of the contract, and you obtain the second formula.


5

When you look at EUR/USD Cross Currency basis historical chart, you will notice that it was very similar in magnitude before 2008 to what it is now: in other words, there has always been some cross-currency basis, it hasn't really gotten more pronounced post 2008. Many people (even seasoned industry professionals) believe that the Cross-Currency basis has to ...


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

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 ...


4

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 go long 1 forward contract with maturity $T$ and delivery price $K$. The payoff at time $T$ is $S_T - K$. For portfolio 2, we go long $e^{-qT}$ unit of a stock (while reinvest all dividends) and short $K ...


4

Your method assumes you can borrow or lend at OIS in both currencies, but in practice you cannot. That's why there is a current basis swap market , where you lend at OIS in one currency versus borrowing at OIS + X in the other currency , where X is not zero. That is the missing piece of your calculation. Why, you may ask , is X not zero , as many ...


4

I think your statement has a typo. I can't find the statement you made in the article you cite. The forward measure is the measure induced by using a bond as the numeraire instead of the risk free asset. Letting $H(X_T)$ be the payoff function for an asset $X_t$, $$ \tilde{\mathbb{E}}\left[\frac{B(t)H(X_T)}{B(T)}\right]=P(t, T)\tilde{\mathbb{E}}\left[\...


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

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

If EURUSD Spot is currently trading at 1.21, then trading today for the usual settlement of EURUSD on today+2 would be at 1.21. If you wanted to trade for immediate exchange of EURUSD, what would be the right rate? 1.21? Not really; if EURUSD was expected to go up or down between today and the spot date, I could arbitrarily profit from taking one side or the ...


4

Interest rate parity is not sufficient now There is a cross currency basis between the IRP result and observed market prices, because essentially Libor does not represent the cost of funding; in particular the implied USD funding rate deviates significantly from the USD Libor. Cross currency basis (as a swap) is a traded quantity which covers that ...


4

I guess the author's argument is that, because of the frequent settlements, one needs to invest the mark-to-market gains and fund the losses. As the exchange traded futures contract is negatively correlated to interest rates, the mark-to-market gain happens when interest rates are low, so not a great time to invest, while the loss happens when interest rates ...


4

There could be any number of explanations for copper to be backwardated and aluminum to be in contango right now. The simplest (and most correct) explanation is the most vague: that these futures curves represent time-varying expectations of supply and demand over the next half year or so. More helpful is to know that copper and aluminum took a sharp dip ...


4

Based on notations in this question, assuming the market value recovery mechanism, the pre-default value at time $T_1$ of a zero-coupon bond with maturity $T_2$, where $T_1 < T_2$, is given by \begin{align*} P(T_1, T_2) = E\Big(e^{-\int_{T_1}^{T_2}(r_s +(1-R)h_s)ds}\,\big|\, \mathscr{F}_{T_1}\Big). \end{align*} Let $B_t=e^{\int_0^t r_s ds}$ be the credit ...


4

Assume today is $t$, and the 1st coupon pays at time $T_1$, the 2nd one at $T_2$, etc. Then your term structure of spot rates would be $R_1 = R(T_1) = f(t,T_1)$ for the 1st maturity, and $R_2 = R(T_2)$ for the 2nd maturity, and so on... Note that by no arbitrage $1+R_2 = (1+f(t,T_1)) (1+ f(T_1,T_2))$. Here I denote by $f(x, y)$ today's value of a forward ...


4

Yes. The swap is quoted in fwd points relative to spot (sssuming that what you mean by r_term is the term interest rate of one currency and r_base the term interest rate of the other). Also, best to use market convention for the FX quotes.


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