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Consider any option, vanilla or exotic. In between fixing dates it satisfies the Black & Scholes PDE (for simplicity zero interest rate and dividends) $$\frac{1}{2} \sigma^2 S^2 \frac{\partial^2 U}{\partial S^2}(S,t)+\frac{\partial U}{\partial t}(S,t)=0$$ Let ${\cal V}(S,t) = \frac{\partial U}{\partial \sigma}(S,t)$ be the option vega. Differentiating ...

10

It's complicated. Assuming there is no CTD switches, then yes, the theoretical modified duration should be unchanged and the DV01 will be lower. For simplicity, imagine that there is only one bond eligible for delivery into the contract. We'll also ignore all the other complications (e.g., variation margins), then the theoretical futures price is simply the ...

8

The proof is relatively long, so I focus on displaying the reasoning and major steps. We work on a Black-Scholes model. Without loss of generality, we focus on an option with strike $P$ to buy at $t_e$ a European call option expiring at $T$, written on a stock $S$. Expectations are always taken with respect to the risk-neutral measure $Q$ unless otherwise ...

7

It depends a little bit what you're trying to do. If you can statically replicate the payoff of a position at $t=0$, then putting on that hedge will insulate you from all risk coming from the contract. Payoff doesn't need to be linear - for example, you can perfectly replicate a call option using a put option and a futures contract If you want to use only ...

7

They do not calculate it, they set it at a market clearing level based on supply and demand. It is similar to the way equity market makers set the price of a stock: a lot of buyers => raise the stock price (or the IV), a lot of sellers => lower the stock price (the IV). For a new option, not previously traded, they might look at the IV of "...

7

There are two parts to your question which I try to answer separately. The first one is about what calibration actually is whereas the second question deals with risk-neutral pricing. As an example, we can use any model. I continuously refer to the stochastic volatility model from Heston (1993) as an example for equity options. Any thoughts equally apply to ...

6

I'll give a heuristic "proof" for general European claims which will cause mathematicians to feel sick, but which physicists / practitioners would probably be quite happy work with: Write the Black-Scholes PDE as $$\frac{\partial F}{\partial\tau}(\tau) = \mathcal{A} F(\tau)$$ with $\tau = T- t$, and the operator $\mathcal A$ is defined as \... 6 Let \begin{align*} \mathrm{d}S_t&=\mu S_t\mathrm{d}t+\sqrt{v_t}S_t\mathrm{d}B_{S,t}, \\ \mathrm{d}v_t&=\kappa(\bar{v}-v_t)\mathrm{d}t+\xi\sqrt{v_t}\mathrm{d}B_{v,t}, \end{align*} where \mathrm{d}B_{S,t}\mathrm{d}B_{v,t}=\rho\mathrm{d}t. The market price of risk (or Girsanov kernel or Sharpe ratio) is {\varphi}_t=\left(\frac{\mu-r}{\sqrt{v_t}},\... 6 I am not sure if you can classify it like that. Mind you, I never wrote a book. I'll write what I know below and you can decide if the classification makes sense or not. 1 ) STIR: as the term indicates - short term like Eurodollar frequently modelled with Black or Bachelier (normal) model. HW1F is also a short rate model. 2 ) HJM is a framework (M is not ... 5 I think you are referring to a compound option. It's valuation under Black-Scholes assumptions is given in the link. The option was first derived by Geske (1978), see here for the original paper . 5 This is a slightly deeper question that it appears at first. Depending on your treatment of rates (deterministic vs. stochastic), it can indeed be model-dependent. Let's first think about a forward contract F_T(t) locks you in to a transaction at price F_T(t) on an underlying S(t) at time T. The well-known price of this contract at time t is \begin{... 4 At time T_0, the strike price becomes known and the option turns into a normal'' put option, i.e. \begin{align*} V(T_0,S_{T_0}) &= S_{T_0}e^{-r(T-T_0)}\Phi(-d_2)-S_{T_0}e^{-q(T-T_0)}\Phi(-d_1) \\ &= S_{T_0}\underbrace{\left(e^{-r(T-T_0)}\Phi(-d_2)-e^{-q(T-T_0)}\Phi(-d_1)\right),}_{=:p} \end{align*} where p is indeed independent of the stock ... 4 You're in luck. There's an excellent book about Brazil markets Marcos C. S. Carreira, Richard J. Brostowicz. Brazilian Derivatives and Securities: Pricing and Risk Management of FX and Interest-Rate Portfolios for Local and Global Markets. Palgrave Macmillan 2016. Also if you can find Credit Suisse Guide to Brazil Local Markets (2014), it may help. To answer ... 4 I believe there's a Bloomberg manual in the help function which explicitly shows these calculations in a spreadsheet, very useful: 4 Just an addendum to the above answers and comments: The main decision is whether to use single or multiple factor dynamics. LMM models term forward rates. HJM models instantaneous forward rates. The main disadvantage of HJM, high-dimensional stochastic process as underlying, was overcome by Cheyette, back in 1994, by restricting the general HJM model to a ... 4 I may risk some negative reactions with that post. However, I prefer to write this rather than not doing it. After all, it should be fine to disagree and express concern. From what I can tell, no one mentioned you cannot trade barrier options. Your question was about exotic options (whatever they are). You mentioned triple barriers. In a now deleted comment ... 3 So, a future is basically like a forward. F_0(T) = S_0e^{T(r_{f,T}-r_{d,T}+x_T)} The longer dated you go, the more you have exposure to the stuff in the exponential (rates in the two currencies, and the xccy basis x_T). That's a trading choice: do you want to trade pure spot FX (or close to it), or the forward (for which maturity?) The answer of ... 3 Interesting question. To answer this, we need to think about what it means to 'liquidate' Bob, and how this is achieved. Exchanges don't take principle positions against their clients, they only match client orders. So to liquidate Bob's short position, they need to find another client to step in to the position - as long as the market is functioning ... 3 You can get Options data from algoseek.com. They provide historical and live OPRA feed from 2012 to the present, and you are not required to install any specific software. AlgoSeek provides flexible data aggregations from TAQ (Trade+Quotes), TANQ (Trade+NBBO Quote), to various minute bars and analytics such as Greeks. Live data services are currently in beta ... 3 The present value of a Vanilla Swap (the word Vanilla is used since I am considering the simplest swap, i.e., notional equal to one, contiguous time intervals, constant rate, etc) is given by: \begin{align} V_s(t) &= \mathbb{E}_t^Q \left[ \sum_{i=1}^N D(t, T_{i+1}) \cdot \tau_i \cdot (L(T_i, T_i, T_{i+1}) - k) \right] \end{align} where T describes the ... 3 In FX swaps and FX forwards, the following formula holds:S_{AUD/USD}(1+r_{USD})=(1+r_{AUD}+r_{basis})F_{AUD/USD}$$The Spot S_{AUD/USD} and the Forward F_{AUD/USD} are traded and their prices are observed in the market. If you take the USD OIS rate for r_{USD} and also the AUD OIS rate for r_{AUD}, you will be able to extract the term r_{basis} ... 3 See 'Collateral Posting and Choice of Collateral Currency' (Theorem 1, in particular) and 'A Note on Construction of Multiple Swap Curves with and without collateral' by Fujii et. al., and also 'Cooking with Collateral' by Piterbarg (the basics of unsecured and collateralized cases for forward contracts in general are covered in Piterbarg's paper 'Funding ... 3 The CVA on a cross currency swap comes mostly from the final exchange (being the biggest flow). If you as an end user are paying the EUR, then the bank is receiving the EUR a d paying the USD. They will see that this position is a net receivable, because EUR is more valuable on a forward basis than spot. Receivables attract a higher CVA than payables. (... 3 Simple example: euro based investor wants to buy a USTreasury, currency hedged back into Euro. Investor executes the following 2 trades at t=0: purchase Treasuries for next day settle. Assume usd12mm purchase price. execute fx swap with cashflows at t=0 : receive usd12mm/pay €10mm and cashflow at t=1yr : pay usd12.0mm/ Rec €9.9mm. (I used spot =1.20 and ... 3 Adding some details: CNY is problematic because it is a nonconvertible currency (that is why user42108 suggests using the offshore yuan CNH instead, or a nondeliverable forward on CNY). See this post Spot/Next and Tom/Next FX forward swaps for more detail about T/N Swaps and how they can be used to postpone delivery of a spot transaction by 1 day (... 3 Assume that the payoff is L(T1,T1,T2)=:X paid at T_1. This is equivalent to paying off X(1+X) at time T_2. You can do this because in the risk neutral setting, a certain payment known at time T_1 can be paid later at T_2 if the beneficiary were compensated with exactly the fair rate of growth present at T1, for the period between T_1 and ... 3 For r=q=0 and t\leq T'\leq T:$$ C_t(T)=E_{t}[(S_T -K)^+] = E_{t}[E_{T'}[(S_T -K)^+] \geq E_t[(S_{T'} -K)^+]=C_t(T'), where we used the tower property of conditional expectation and the sub-martingality of $(S_{T'}-K)^+$ they mentioned (which is a consequence of Jensen inequality for conditional expectation). A calendar spread (one long call with ...

3

This is not an answer but a comment which is way too long for the comments section. What you can trade will be very much restricted by where you live. In Europe, you have Priips, which changes the game a lot - even US-ETFs are generally not easily available on most platforms (within Europe). How much money do we actually talk about?. No need to answer this ...

2

Discretisation schemes If you want to simulate the path, then common practice is to sample from the exact distribution, as for the CIR process this is known. The distribution can be found from the original CIR process (1985). However, this requires sampling from a non-central $\chi^2$-distribution, which can be very expensive, and a bit more difficult to ...

2

Let $\sigma_J$ be the volatility of the index $J$. Assume that $J(0)\leq B$. Consider the following 2 extreme cases: $\sigma_J=0 \Rightarrow \forall x\in[0,T],J(x)=J(0)\leq B$: hence you will always be paid $I(T)$ at expiry. $\sigma_J=\infty \Rightarrow \exists\epsilon>0, J(\epsilon)>B$: hence you will almost immediately be paid \$I(0)\approx I(\...

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