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6

$$\begin{array}{rcl} (1) & \partial_KC_t(T,K) & \leq 0 \\ (2) & \partial^2_KKC_t(T,K) & > 0 \\ (3) & \partial_T C_t(T,K) & \geq 0 \\ \end{array}$$ 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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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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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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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 ...


3

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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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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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 ...


2

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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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.


1

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 ...


1

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


1

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 ...


1

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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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 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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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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