7

The CMS represents the value of a swap rate for any point in time, i.e. we are interested in extrapolating the density of the swap rate in a similar way as the IBOR rate. Let us start with the fair value of a swaption under the annuity measure $\mathcal{A}$ with tenor at time $\tau$: $$\mathcal{A}(t)\mathbb{E}^\mathcal{A}_t[(\mathcal{S}(\tau)-k)^+]$$ Instead ...


5

The piece you are missing is an approximation via the Taylor formula of the logarithm: $$\ln(1+x) \approx x-\frac{x^2}{2} \; .$$ Apply this to the first term in the final formula of the technical paper: $$\frac{2}{T}\ln\frac{F_{0}}{S^{*}} = \frac{2}{T}\ln\left(1+\left(\frac{F_{0}}{S^{*}}-1\right)\right) \approx \frac{2}{T}\left(\left(\frac{F_{0}}{S^{*}}-1\...


4

No, it can be negative. The price of risk is what you agree to receive on average in exchange for positive returns when the risk measure is high, and determined by the covariance of the risk measure with your marginal utility of consumption. That said, stochastic volatility risk is negatively priced: you happily agree to a negative return on average in ...


4

In an Asian-style swap, instead of using the last price quote of the underlying (such as commodity price), they take an average, such as the average closing price over the last month. This is fairly common in commodity swaps. As for pricing, a good start is this paper on pricing Asian-style interest rate swaps (where the floating leg uses the average of an ...


4

The market is in backwardation, so there is a positive roll yield. December settled yesterday at 5083.0 vs November at 5098.5 That's a difference of 15.5 points (we could also compare the bid ask midpoints, that would give a difference of 14.5 points. I trust this number less because the bid ask spread for dec is very wide, december is not trading very ...


3

Current USDTRY is about 5.7. The jargon means this: The strike of a 2 month USD Call/ TRY put corresponding to a delta of 0.05 is 8, and it costs 0.6% of USD notional. TRY to plunge is what might be implied!


3

Just my 2cts' worth: With commodity swaps exchanging typically a daily spot price (i,e, immediate delivery price) vs a fixed rate payable in regular intervals, the only difference to a truly Asian product is that discount factors are not perfectly equal to unity. So while rates are not high or tenors very long, regular commodity swaps would be pretty close ...


3

This is a well known issue. There are three possible tricks: I am surprised that none of the answers so far mention the work of Lord and Kahl Optimal Fourier Inversion in Semi-Analytical Option Pricing. They study this oscillation problem and propose an optimal contour for the integration. The challenge is to write a small algorithm to obtain the optimal $\...


3

The price of a swap is the rate on the fixed leg that puts the market value to zero. Swaps are typically quoted this way, so in your example it would be the fixed rate of a swap with the same dates and floating leg as your previous one. However, if you wish to sell the swap you could also ask for a quote in absolute currency terms (could be negative or ...


3

@Gordon has already given the answer but here is a little more notes to it... At time time $T_2$ the holder receives $X=(S_{T_1}-K)^+$. According to Risk Neutral Valuation the value at time $t$ $(t<T_1<T_2)$ is $$V_t = e^{-r(T_2-t)}E_t[(S_{T_1}-K)^+] = \\ e^{-r(T_2-t+T_1-T_1)}E_t[(S_{T_1}-K)^+]=\\ e^{-r(T_2-T_1)}e^{-r(T_1-t)}E_t[(S_{T_1}-K)^+] $$ $e^...


3

Look at the infinitesimal version of the change in variance: $$ d\sigma^2 = 2\sigma d\sigma + (d \sigma)^2 $$ The Ito term $(d\sigma)^2$ is non-zero for stochastic processes, and is of order $dt$, but if we ignore that then we get the approximate relation $$ d\sigma^2 \approx 2 \sigma d\sigma $$ which is where the factor $2 \sigma$ comes from in the ...


2

Actually, all investments, retirement accounts, mutual fund accounts, utility bills, supermarket price listings are reported or stated in the Constant Numeraire, which may also be called Dollar-kept-under-the-mattress Numeraire It is the most widely (indeed the only) Numeraire used in real life. How nice it would be if my retirement account or mutual fund ...


2

If you already have the zero rates, you can construct the zero curve using the set of maturities (dates) and zero rates values, in addition to a day count convention in this way: import QuantLib as ql ql.Settings.instance().evaluationDate = ql.Date(26, 7, 2018) dates = [ql.Date(26, 7, 2019), ql.Date(26, 7, 2020), ql.Date(26, 7, 2030)] zero_rates = [0.03, 0....


2

There is a well known identity for the Black Scholes model: $S_0 n(d_1)-X e^{-rT} n(d_2) = 0$ (proof). Using this allows you to combine these two terms: $$S_0 n(d_1)\frac{\partial d_1}{\partial t} - Xe^{-rT}n(d_2) \frac{\partial d_2}{\partial t}$$ into $$S_0 n(d1) (\frac{\partial d_1}{\partial t}-\frac{\partial d_2}{\partial t})$$ or $$S_0 n(d1) \...


2

There needs to be a 4th transaction: the cashflow at t=1y needs to be invested at Libor until t=2y. You will then find that all the cashflows cancel. This means that the hedge is not quite static, but the convexity adjustment is still zero, because we assume money can be invested (or borrowed) at Libor for free at any time. (Note: this is not quite ...


2

My impression is that cash is a more generic term: deposits are a cash security, but so are US Treasury bills, commercial paper, repos, and so on, and each security has different yields. When you construct a yield curve, the short end is calibrated to a certain cash security.


2

IV for a put and call is the same so it doesn’t matter (in theory). In practice you use puts for low strikes and calls for high strikes, since the OTM are more liquid. Low/high is relative to the forward price.


2

Repo is essentially collateralised lending/borrowing, but it is executed via sale and repurchase. The repo rate works the same way as the deposit rate, so would be annualised. General collateral means that the seller has a choice regarding which particular security to provide,e.g., any US treasury as opposed to a specific issue. Triparty means that the ...


2

The Greeks chapter in Hull’s Options, Futures, and other derivatives book would be a good start if you have not read Hull’s. The Greeks and Hedging Explained by Peter Leoni is also very accessible and provides good coverage of the concepts. And if you prefer the traders style then you might like Taleb’s Dynamic Hedging. The title makes it sound technical ...


1

I am assuming you are asking this question as a programmer, not as a trader. From a traders perspective, due to put/call parity, we look at puts and calls at the same expiry and strike price as the same thing. Each option has an intrinsic value and a time value. After accounting for the cost of carry, the time value for the put and call are equal as is their ...


1

Questions: 1=> Does anyone have a suggestion to determine a trend correctly. My answer is in general and an opinion. Hong Kong Stock Exchange is third largest market behind Tokyo and Shanghai and most volatile market in the world. It is related to Singapore, Shanghai and Shenzhen, Korea, Taiwan and other famous Asian markets. For rough overall trends, ...


1

I think part of the problem may be a lack of a formal definition for what constitutes the trend. At least it lacks a definition of some kind of statistical property. (I was thinking autocorrelation but then realised that a random walk could also look as if it were trending) I am not convinced that the MACD indicator is the best way to do this. I would ...


1

The closest contract to this is gap risk which does trade, either as OTC swap (client looking for a hedge) or embedded inside a structured note (bank looking to recycle risk). Basic starting point is daily close-to- close observations against a 80-90% putspread, cancel upon payment, equity payer receives a spread for being short the risk (major equity ...


1

Indeed, if the collateral is not segregated, it could be lost if the counterpart defaults. So, the credit exposure should be computed taking into account the collateral balance $C(t)$: $$ Exposure(t) = \max \left( MtM(t) - C(t) , 0 \right) $$ This means that, even if $MtM(t) < 0$, if $C(t) < MtM(t) < 0$, then you will have a strictly positive ...


1

Only just saw this. Commodity swaps are very simple products. For each averaging period, they will pay off the notional for the period multiplied with the difference between the strike and the average of some reference contract. The thing that's a little hard is knowing what the conventions are for how the reference price is determined. For example, for WTI ...


1

The labelling is indeed a bit confusing- this guy is normally very smooth! Let’s focus on the weekly rebalancing column, here are the detailed steps. 1) Simulate the path of the stock price as per weekly frequency (20 time steps here as the option maturity is 20 weeks). 2) Calculate the delta of the option at each time step. 3) At the initial time, ...


1

Black-Scholes does not really require a constant interest rate. For a european option with maturity $T$ the only rate involved is the zero coupon rate for maturity $T$. The theory behind this comes from working under the $T$-forward measure (the risk neutral measure associated with the zero coupon bond as numeraire). The only subtelty is that the model ...


1

This is explained in Hull. Alternatively you can check this link https://web.ma.utexas.edu/users/mcudina/m339d-lecture-ten-forwards-pricing.pdf Essentially the seller of the forward contract earns the income associated to the stock lending activity so it needs to be discounted from the forward price to ensure absence of arbitrage opportunity. Absence of ...


1

Let's skip calling it volatility and variance. Let us deal with variance and standard deviation. For normally distributed variables, it is very important to distinguish between the true variance and the estimator of the variance and the estimator of the standard deviation. Variance in its raw form is important in defining the statistical law that governs ...


1

First of all, it is part of the Ito formula theorem that if $S_t$ is an Ito process then $V(t,S_t)$ is also an Ito process. This includes the regularity property you mention for the integral you mention and only assumes $V$ is continuous in time and $C^2$ in space. See Oksendal theorem 4.1.2 http://th.if.uj.edu.pl/~gudowska/dydaktyka/Oksendal.pdf In order ...


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