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A few years ago I asked a similar question on MO: http://mathoverflow.net/questions/22828/big-picture-concerning-ito-integral-stratonovich-integral-and-standard-results My take today is that you really don't need this heavy mathematical machinery for standard BS but as soon as you move on to more sophisticated (and realistic) models you surely do, so it is ...


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1 - For historical reasons options expire on Saturdays, before noon. This has to do with potential reconciliation issues that are largely gone nowadays because of back-office automation. Practically, you can think of SPX index options as expiring at index settle, which is 4:00 pm EST on Fridays for PM expiring options, or 9:30 am EST on Fridays for serial ...


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It seems that you're looking for data for liquid options worldwide. On the global scale most liquid options are typically index / index futures options on the main local index, not MSCI indexes. Outside of index options, you have only few broad exchange-traded stock options markets - Eurex, Osaka, and ASX are probably top three. I'm surprised that Options ...


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There are "Combo-On-Close" for SPX options but they are OTC. Can you trade OTC? The main IDBs set markets in those. Obviously you would have to hedge your cash somehow.


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Someone asked me similar question last week. No combos never trade at close in spx. Most people trade weekly to Jun {front fut}or various weekly rolls in spx . I looked at btic once, it was trading almost same level as roll mkt. If you traded combo you would hedge with future anyway? Btic seems like cool product to trade efp market. I am not sure if this ...


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This question is extremely interesting and not that straightforward. See answer here. From a financial perspective this is very much like pricing an American call (stopping rule = intrinsic value from exercice (i.e. current cash earned) > continuation value (i.e. what you can expect to gain). Note that you can never win more than 13 nor lose (at worst you ...


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I think the above comment about using ACWI is the only way to trade options on the MSCI World. You can see OTC swaps and futures trade on the index as well but there are no associated options. But just remember how they are priced. The ETF will snap it's close at 4pm like any other stock. The actual index (as well as the ETF NAV) is based on closes from ...


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Consider a payer swaption with maturity $T_0$ and strike $K$. Here the strike $K$ is the fixed rate paid on the fixed leg of the underlying fixed-for-floating swap with reset dates $T_0, \ldots, T_{n-1}$ and payment dates $T_1, \ldots, T_n$, where $0<T_0 < \cdots < T_n$. We assume that the swap exchanges the payments $L(T_{i-1}; T_{i-1}, T_i)\Delta ...


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Let \begin{align*} C(S, K, t) = SN(d_1) - e^{-rt}KN(d_2) \end{align*} denote the Black-Scholes call option price with initial asset value $S$, strike $K$, and maturity $t$. Note that \begin{align*} \frac{\partial C}{\partial S} = N(d_1). \end{align*} For the above barrier option, note that \begin{align*} E_0 &= V_0 N(d_1)-e^{-rt}KN(d_2) -\bigg[V_0 ...


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For this type of question, you basically need only to write the payoff with certain indicator functions. In particular, for the above payoff, we have that \begin{align*} \textrm{Payoff} &= K\, 1_{S_T \le K} + (2K-S_T)\,1_{K < S_T \le 2K}\\ &=K\, 1_{S_T \le K} + (2K-S_T)\big(1_{S_T \le 2K} - 1_{S_T \le K} \big)\\ &=(2K-S_T)\,1_{S_T \le 2K} - ...


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I think this is related to traders jargon. When a dealer quotes the price of a spread between two securities (such as a risk reversal) as "10 cents your choice" or "ten cents around" it means that the bid-ask midpoint is zero and it will cost you 0.10 USD to enter a position long the first security/short the second, and also 0.10 to short the first/long the ...


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I don't agree with the contention that market prices are always used as the benchmark upon which to base model performance. I think this is model dependent. Market prices make sense (for example for modelling an underlying predictor), but for example, for a derivatives model, I would argue the values of those derivatives at expiry (or earlier for path ...


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I don't have a reference for you but I have some experience. Risk management departments at hedge funds and banks would primarily look at the Var in order to capture the risk of an options portfolio. The var indirectly captures all the Greeks in a single measurement , since each Greek generates some exposure. The desk traders would tend to look at all the ...


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$$\begin{array}{rcl} (1) & \partial_KC_t(T,K) & \leq 0 \\ (2) & \partial^2_KKC_t(T,K) & > 0 \\ (3) & \partial_T C_t(T,K) & \geq 0 \\ \end{array}$$ If $(1)$ doesnot hold, it exists $K_1<K_2$ such that $C_t(T,K_1)<C_t(T,K_2)$. Then as barrycarter said in his comment, you sell $C_t(T,K_2)$ and you buy $C_t(T,K_1)$, so your ...


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there are a number of ways to do this. You do have to make some modelling assumptions, however. eg continuity, BS model holds, or log stock price process is independent of level. The most common way is to take the pay-off and geometrically reflect in the barrier. (i.e. pass to log coordinates and reflect). i.e. write the function as $f(x)$ where $x= \log ...


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Under the risk neutral measure, the expected present value of the butterfly payoff is: $$V_0 = e^{-rT} * \int_{S_T=K_1}^{K_3}P(T,S_T)f_{S_T}dS_T$$ And if we assume that $f_{S_T}$ is constant from $K_1$ to $K_3$, then: $$V_0 = e^{-rT} * \dfrac{1}{\Delta K} \int_{S_T=K_1}^{K_3}P(T,S_T)dS_T = e^{-rT} *\dfrac{\delta^2}{\Delta K} $$


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Check out this post: http://www.macroption.com/option-greeks-excel/ Let me know if it answers your query. The Excel equations used to calculate Delta, Gamma, Theta and Vega are shown in the above link.


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Peter Jaeckel has written various papers on this. "by implication" and "Let's be rational" are the most recent ones. He also provides code on his website www.jaeckel.org. (Note: the question asked for literature.)


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Look on Google for Asymptotic behavior of Implied Volatility Near Infinity you will find results like : $$I(K) \stackrel{K\to\infty}{=} \sqrt{\frac{2}{T}}\left(\sqrt{\ln \frac{K}{C(K)}}-\sqrt{\ln\frac{1}{C(K)}}\right) +\text{O}_{K\to \infty}\left(\frac{\ln\ln\frac{1}{C(K)}}{\sqrt{\ln\frac{1}{C(K)}}}\right)$$


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We assume that the inequality is given by \begin{align*} B > N C(K-1/N, T) - N C(K, T).\tag{1} \end{align*} The argument for the case with the inequality \begin{align*} B < N C(K, T) - N C(K+1/N, T) \end{align*} is similar. $$$$ For the binary option, \begin{align*} \pmb{1}_{\{S_T \ge K\}} = \begin{cases} 1, & \textrm{if } S_T \ge K,\\ 0, & ...


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The risk free rate is used to get the present value of future payoff, so you should use the rate of a risk-free instrument (e.g. a Treasury note) that has roughly the same maturity of the option you are valuing. If you option expires in a time that does not have an exact Treasury instrument, you can get a rough approximation by interpolating between two ...


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Don't use a nosql DB like Mongo. Use an SQL like MySql or MSSQL. MSSQL: https://www.microsoft.com/en-us/download/details.aspx?id=42299 MYSql: http://dev.mysql.com/downloads/mysql/ Sounds like you have structured columns like a spreadsheet. If you are a fiance guy you, intuitively you see data the way it is stored in SQL. Tables and columns. If your tech ...


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What you are trying to do is fit a volatility surface for a given underlying. Once you have a volatility surface you can price an option for an arbitrary expiration and strike. There are numerous approaches to do this and the linear interpolation methods mentioned in the other examples are okay but be careful in the following situations where there is: a ...


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You haven't written down your equations correctly. Ignoring discounting, the equations should be: C(70)-P(70)= -4 (not 66), from put-call parity. Also, C(70) + P(70)= 27; from these two we get C(70)= 11.5 and P(70)=15.5 Also P(60)-P(50)= 2.5 and P(70)-2P(60)+P(50)=0.2 from which P(70)-P(60)=2.7, hence P(60)=12.8 and P(50)=10.3 so now we know all the ...


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You could check at the methodology for VIX. The VIX itself yields one number - but you might instead return a set of numbers for your skew analysis.


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You use a form of interpolation(start with linear) between the 30 day to maturity IV and the 90 to get the 60,


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[Short Answer] You write $E [S_T]=S_0(1+r)^T $ but you actually compute the RHS as $X (1+r)^T$ in your numerical application. [Long Answer] The stock price is a martingale in an equivalent measure using the risk-free asset as numeraire i.e. $$ E [S(T)] = (S_0 u) q + (S_0 d) (1-q) = S_0 (1 + r ) \Delta t $$ In that case, dividing each member by $S_0$ and ...


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Well stock prices change all the time when markets are open. American options give you the opportunity to exercise it at any time up until maturity, whereas a European option only allows you to exercise it at a specific date and time. A simple example is to compare an American option that matures in 1 day and European option where it matures at the last ...



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