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7

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: $$\frac{2}{T}\ln\frac{F_{0}}{S^{*}} = \frac{2}{T}\ln\left(1+\left(\frac{F_{0}}{S^{*}}-1\right)\right) \approx \frac{2}{T}\left(\left(\frac{F_{0}}{S^{*}}-1\... 7 The CMS represents the value of a swap rate for any point in time, i.e. we are interested in extrapolating the density of the swap rate in a similar way as the IBOR rate. Let us start with the fair value of a swaption under the annuity measure \mathcal{A} with tenor at time \tau:$$\mathcal{A}(t)\mathbb{E}^\mathcal{A}_t[(\mathcal{S}(\tau)-k)^+]Instead ... 7 Futures are in "zero net supply", or "for every long there is a short", which means that at any time there are investors who are long a certain number of contracts and other investors who are short an (exactly matching!) number of contracts. This number is called the Open Interest. It starts at zero when the exchange introduces a new contract (like Sep 2019 ... 6 If you plot the function f, you see that you have a bear spread. You can build such vertical spreads either with call or put options. For example consider a portfolio selling one put option with strike price K_1=30 and purchasing one European-style put option with strike price K_2=35. Then, you obtain the payoff \begin{align*} \max\{35-S_T,0\}-\max\{30-... 5 A forward rate agreement is an agreement to exchange a fixed for a floating rate over one period, with the payment being made at the start of the period. A zero coupon swap (with both legs paid at maturity) is an agreement to exchange a fixed for floating rate over one or more periods, with the payments being made at the end of the final period. So the two ... 5 It would be much easier to start by writing the payoff using indicator functions. For example, \begin{align*} f(S_T) &= 3 \mathbb{I}_{S_T \le 30} + (33-S_T) \mathbb{I}_{30<S_T < 35} -2 \mathbb{I}_{S_T \ge 35}\\ &=3\big(1-\mathbb{I}_{S_T > 30}\big) + (33-S_T) \big(\mathbb{I}_{S_T > 30} - \mathbb{I}_{S_T \ge 35}\big) -2 \mathbb{I}_{S_T \ge ... 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 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 $$which is where the factor 2 \sigma comes from in the ... 3 @Gordon has already given the answer but here is a little more notes to it... At time time T_2 the holder receives X=(S_{T_1}-K)^+. According to Risk Neutral Valuation the value at time t (t<T_1<T_2) is$$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 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 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 Here's another way to do it, that I think is useful if you don't recognize/have knowledge of specific option spreads/techniques. This might help you on exams or other problems, although recognizing the different option plays is probably easier. First you start from the left of the payoff graph, and split the graph into segments, just like how the payoff ... 3 The ATM is an outright position (long 50 delta put and 50 delta call) so the main exposure is vega. It is the riskiest of the three, and demands a higher bid-offer spread from market makers to compensate them for the additional risk. The RR is a spread position (long 25 delta call, short 25 delta put) with zero vega, the main exposure is skew. Because the ... 3 It’s not entirely risk-free. Nothing in life is. The comet could hit etc. The difference is suppose I had a 100m OIS swap line open with Lehman, margined overnight. OIS settles at 1.81% vs 1.80%; and I’m paying. I’m owed 1bp on 100m that I’m not going to get, equals 10 grand. No tears required. If I had lent 100m to them, my lawyers and ops people would ... 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 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) \...

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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, ...

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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 ...

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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.

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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.

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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 ...

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I think this is the old accrual methodology, historically used for the banking book. I believe it is not market standard anymore and regulators require an MTM (mark-to-market) valuation. Here is an article that explains the difference between the two. And someone wrote a more mathematical paper, which should help you better understand the accrual valuation (...

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Let $r(s)$ be the process of a short rate. Then, by risk neutral pricing, $$P(t,T) = \mathbb{E}^\mathbb{Q}\left[ \exp\left( -\int_t^T r(s)\mathrm{d}s\right) \Bigg| \mathcal{F}_t\right].$$ Thus, the zero-coupon bond is determined completely by the short rate process. Here, $P(t,T)$ denotes the time $t$ price of a zero-coupon bond maturing at time $T$. You ...

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Recall that $|x|=\max\{x,-x\}=2\max\{x,0\}-x$. Thus, \begin{align*} f(x)&=|5-x|+|10-x| \\ &= 2\max\{5-x,0\} +x-5 + 2\max\{10-x,0\} +x-10 \\ &= 2x-15+ 2\max\{5-x,0\} + 2\max\{10-x,0\} \\ \end{align*} Thus, by no-arbitrage, the time $t$ price of $f(S_T)$ is given by $$V(t,S_t)= 2S_te^{-q(T-t)}-15e^{-r(T-t)} + 2P(S_t,5,T) +2C(S_t,10,T).$$ If all ...

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Let's say the company was bankrupt (ie, stock price is 0). A put option effectively becomes a bond with face value equal to the strike and maturity equal to the expiration. With positive interest rates, zero coupon bonds generally become more valuable as time passes. In this extreme case, an American option is worth more because you could early ...

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A standard book in the volatility literature is Gatheral (2006). The book begins with stochastic volatility, llocal volatility and the Heston model. Then he adds jumps and default risks. He concludes with barrier options, exotic options and volatility derivatives. He includes many tables and graphs and writes rather well. The only downside is that he does ...

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Liquid market instruments tend to be "building blocks", i.e. the OTC instruments need be "calibrated" to them, since futures/options etc are used in hedging OTC. OTC instruments are valued in risk neutral or equivalent martingale measures, but liquid markets are in physical measure. One can build some models to tackle the liquid markets, but not in the ...

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As Alex C says in the comments, Longstaff and Schwarz did consider multiple factors and mention it as one of the advantages (page 114 in the journal): By its nature, simulation is a promising alternative to traditional finite difference and binomial techniques and has many advantages as a framework for valuing, risk managing, and optimally exercising ...

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