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$H$ is in general random. The position of a trading firm into a stock is clearly random in terms of it being dependent of the realisation of the stock price. If a firm is not invested in a stock but changes its mind because it keeps increasing, then they may alter their opinion and begin investing in the asset. So, a trading strategy depends on the random ...


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It could depend on the brownian - e.g., could be a function of B, $H(B)$. What it means is you can change your holding over time depending on how the Brownian/randomness evolves, but for Ito's definition, H is supposed to be kinda non-anticipating, roughly speaking H cannot depend on the next move as you cannot predict the next change in the Brownian when ...


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No, the simulation is not exact in general, precisely for the reason you mentioned. By "exact", it is meant that there is no discretization error in time. Of course, there will always be a Monte-Carlo sampling error. For the Black-Scholes model, the simulation is exact if you simulate the log asset, as it is a standard arithmetic Brownian motion, and then ...


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There are several reasons why continuous models are often preferred: 1.) It is more realistic: Trading happens continuously. Especially regarding HFT, trades can and might happen at any time. 2.) It therefore is also more coherent. You don't have to find and deal with artifical time steps. 3.) It is easier to handle. The lack of artificial time steps ...


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You're right. Discrete models are easier to simulate and indeed, if you have a time continuous model, you typically first discritse it before you can implement it (computers live and work in a discrete world) (This, by the way, gives rise to discretisation errors). So, why do we love continuous models then? Because they're much easier. Think about the Black-...


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Sorry I am bit late to the party. Just saw your post while trying to write my own black model. I am going to the mistake is a typo in dplus d_plus = ((math.log(F_0 / y) + 0.5 * vol * vol * expiry)/ vol / math.sqrt(expiry)) Should be: d_plus = ((math.log(F_0 / y) + 0.5 * vol * vol * expiry)/( vol * math.sqrt(expiry))) Warm Regards, Varun


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You know the bond price formula takes this form: $P \left( t, T \right)= A \left( t, T \right) e^{ -r_{t} B \left(t, T \right) }$ Now apply Ito's lemma, so you will get after some manipulation: $\frac{dP}{P}= \left(\frac{1}{A} \frac {\partial A}{\partial t} -r \frac {\partial B}{\partial t} - \kappa \theta B + \kappa r B+ \frac{1}{2} B^2 {\sigma}^2\right)...


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The word "hedge" can be ambiguous because it is not always clear what the risk is that we are trying to eliminate. The "business model" that Shreve has in mind here (which is very common) is that an investment bank sells a derivative to a customer and now is short that derivative. They are exposed to changes in the market value of that derivative which they ...


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Here are a few FX structured product examples: All of these can be notes or swaps, notes will pay back the notional at the end and carry no credit risk (and are normally set so that they are worth 100% at inception - i.e. they'll be worth 99% and the seller will take some profit/hedging costs). Swaps will either be set to be worth 0% (same deal as above, ...


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That means that to determine the price of a security by non arbitrage you can either find the strategy that will hedge your short position ( when you sold an option) or you can find the strategy that will hedge your long position (when you buy an option). Both need to lead to a profit of 0. With the short position of an European call, he is considering here ...


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As explained by @byouness, using Itô's Isometry, we get: $$\begin{align} V(S_T)&=V^{\mathbb{Q}}\left(\int_0^T\sigma e^{r(T-s)} dW^\mathbb{Q}_s\right) \\[9pt] &=E^{\mathbb{Q}}\left(\left(\int_0^T\sigma e^{r(T-s)} dW^\mathbb{Q}_s\right)^2\right)-{\underbrace{E^{\mathbb{Q}}\left(\int_0^T\sigma e^{r(T-s)} dW^\mathbb{Q}_s\right)}_{=\int_0^T\sigma e^{r(T-s)...


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