# Tag Info

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$$\Delta_1(H)=\frac{V_2(HH)-V_2(HT)}{S_2(HH)-S_2(HT)}=-\frac{1}{12}$$ and $$\Delta_1(T)=\frac{V_2(TH)-V_2(TT)}{S_2(TH)-S_2(TT)}=-1$$ and $$\Delta_0=\frac{V_1(H)-V_1(T)}{S_1(H)-S_1(T)}=-0.433$$ The optimal exercise time is $$\tau(HH)=\infty$$ $$\tau(HT)=2$$ $$\tau(TH)=1$$ $$\tau(TT)=1$$ AS a result, you should borrows $1.36$ at time zero and buys the put ...

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I think you might be looking for "in-the-money" for the calls with strikes below spot and the puts with strikes above spot. And then the options that are close to spot will be called "at-the-money". And the remaining options would be "out-of-the-money".

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Yes it's arbitrary, when you agree on the condition you define the how many shares and the Strike price if the contract is not standard

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Call options are usually standardized product: in the contract you can (but are not obliged to) buy a certain amount, which is specified. The most common quantity is 100 shares (see for example the description by J. Hull, Options, Futures, and Other Derivatives).

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You are trying to price an option through Monte Carlo simulations. Here is how it should work, assuming the Black-Scholes diffusion framework. Under the Black-Scholes model's assumptions, the value of a risky asset $S$ at the time $t=T$ is a random variable which reads $$S_T = S_0 e^{\left(\mu-\frac{\sigma^2}{2}\right)T + \sigma \sqrt{T} Z}\tag{1}$$ with ...

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I provided an answer, based on an elementary approach, to an exactly same question yesterday. However, that question has disappeared, even though I like to keep a record for what I wrote. I would suggest that people do not delete their questions as they may be helpful for others. Here, I re-post that answer. We assume that, under the risk-neutral ...

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See this excellent paper by @MarkJoshi which defines/discusses the use of power numeraires. Starting from a dynamics specified under the risk-neutral measure $\mathbb{Q}$ \begin{align} &\frac{dS_t}{S_t} = (r-q) dt + \sigma dW_t^{\mathbb{Q}}\\ \iff& S_T\ \vert\ \mathcal{F}_t = S_t e^{(r-q-\frac{\sigma^2}{2})(T-t) + \sigma(W_T-W_t)} \tag{EQ.0} ...

3

I do not have any reference, but I think $H$ is for hitting.

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I assume this is a plot of option value versus price of the underlying. The only case where it ought to be symmetric is if the pdf of the underlying is symmetric eg normally distributed. I'm guessing your chart assumes a lognormal underlying, which is a non symmetric pdf, so the graph is non symmetric.

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This formula is used for replication of certain payoffs, for example, the log-payoff in Variance replication using options. The value of $\kappa$ can be set to any number, for example, $\kappa=E(S_T)$. This is a decomposition of the payoff, which is not a valuation of the payoff itself, and then further valuation is still needed. For example, based on the ...

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Time $T$ boundary condition is correct $u(T,x)=(x-K_1)^+-(x-K_2)^+$. Time $x\to 0$ boundary condition is known and is equal to $0$. Time $x\to\infty$ boundary condition is also known and is correct $\lim_{x\to\infty}u(t,x)=(K_2-K_1)e^{-r(T-t)}.$ You need to be precise if you want your boundary be "absorbing" or "reflecting".

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Yes, your broker could have used one or combination of many factors: estimated volatility surface from historical returns of your target index, historical returns of similar indexes, implied volatility of similar indexes, existing inventory,etc. Check out these two approaches to deriving surfaces from returns starting slide 14

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Generalizing, some people who write options trading software are not aware of a few small, but important details, resulting in some pricing idiosyncrasies. That is often the case with retail trading platforms, and you often read statements like "implied vol blows up in days before expiration", "greeks become unreliable before expiration", or suggestion of ...

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Not sure what they use in Thinkorswim but I would assume some variation of a Cubic Spline, taking points along the skew and interpolating between them.

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$$s\partial_{s}\Phi = S_T I_{S_T>K}.$$ so no. (I am not absolutely sure whether you want to differentiate wrt S_T or S_0 however.)

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I don't have much experience with database designing so I am not sure how to proceed. Read an introductory database book! Which database will be best to use? MSSQL, MySQL, NoSQL, etc.? MySQL and Microsoft SQL Server are database management systems that follow the relational database paradigm. NoSQL is not a database management system, it is a ...

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You are on the right track IMO, except that there are no exceptions to "the greater the expected value, the higher the option value" rule, since an option premium's is precisely the discounted expectation of its payout by absence of arbitrage - at least under the risk-neutral measure. As far as the influence of volatility on the option value is concerned: ...

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The answer is 'no', no single database for all exchange traded instruments in the world. You can try to use Interactive Brokers symbol search service. For example, just now I've tried to search Brazil instruments. There are another services like this, for example IQ Feed symbol lookup. I can't insert a link to it because of low reputation, but just google ...

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The significant open position at some strike might be treated as a hope of those, who opened it that at expiration market will be there and further, so options will be in the money.

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The answer by @HenriK is certainly correct. However, for justification, technique such as the Jensen inequality is needed. For example, since $x^+$ is a convex function, assuming zero interest and zero divdiend, \begin{align*} E\big((S_{T_{2}}-K)^+ \mid \mathcal{F}_{T_1} \big) &\ge \big(E(S_{T_{2}} \mid \mathcal{F}_{T_1})-K\big)^+\\ &=(S_{T_1}-K)^+. ...

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IMHO the 'definition' you mention is not a mathematical definition per se, but rather an approximation used by some practitioners. Mathematically, it is $N(d_2)$ in the BS formula which figures the conditional probability that the terminal asset price $S_T$ will finish above the strike level $X$ given the information we possess today (represented by $S_t$), ...

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