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7

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 ...


5

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".


4

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} ...


4

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 ...


3

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 $S_t$), ...


3

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 ...


3

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


2

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.


2

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


2

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).


1

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


1

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 ...


1

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 ...


1

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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