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American options pricing (swaption is just a kind of option) is a bit tricky due to the early exercise. Here is a page listing possible approaches, including some numeric methods, and some close form approximation formula. As I understand, lattice methods (tree, PDE discretization such as forward shooting) are fine to price American options. There're ...


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Anyone can give us an example with Interest Rates Derivative?


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In general, there cannot be a closed-form solution of a random coefficients VG model. The reason is the drift-restriction that needs to be imposed to ensure that the discounted price process is a martingale under the risk-neutral measure. Using the bank account as numeraire, the restriction is $$ \frac{1}{\beta} > \theta + \frac{\sigma^2}{2} $$ where ...


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As I see it the question does not enforce that the market is free of arbitrage. This is why you can get to contradicting prices. Thus you can't actually apply a risk-neutral argument here without making additional assumptions. You yourself provide the example of such an arbitrage. If the underlying process had a B&S dynamics you could just borrow money ...


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The dynamics of the underlying stock process are obviously crucial to the derivative's price. Thus if you don't necessarily assume $S_t$ to be log normally distributed (B&S-Model) you won't get the same price even if the market is arbitrage free. Example: Assume $S_t=C$ $ \forall t \in \mathbb{R}^+$ and $r=0$. Thus $S_t$ is constant and the interest ...


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Generally speaking, if you have two or three sources of noise, you are still going to be much better off pricing American options on a lattice than via LSMC. Too often, LSMC becomes the refuge of academics lacking patience to learn proper lattice techniques. Now, you can frequently reduce the difficulty of pricing American options by considering the ...


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Regarding your second question: one possible approach is to reduce the instrument you are trying to value to something simpler, for which an analytical solution are an alternative methodology does exist. You can then vary parameters and check that the valuation is behaving as expected. If you are using simulations because your price process is more ...


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American Options are a tricky subject and pricing them is almost never easy. A lot depends on the model etc. Assuming you that you a familiar with monte carlo and that you know the risk neutral dynamics of your porcess - you could use monte carlo least squares. The paper on the topic is easily accesible and not too technical. Suggested reading: Valuing ...


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I believe your example describes the payoff of a simple spread option. Some may argue that in reality this spread option has zero strike: $$ (S_T(\omega) - K_T(\omega)-0)^+ $$ Which leads us to the question: What exactly strike is anyway? Is it uniquely identifiable term in each payoff function? No It isn't.


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Asian options: strike is average of underlying over tenor. Underlying is stochastic. Options with kock-ins/knock-outs: Underlying is stochastic and may cross the kock threshold as it evolves. Option value depends on this cross or lack thereof (boolean). Options on Options, too. Motivations for Asian options you can google. Kock-ins and knock-outs ...


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There are several ways to choose a particular EMM. I believe that the most popular approach is to use a "distance" between $\mathbb{P}$ and $\mathbb{Q}$. Most papers use a minimal entropy approach(for example, Fujiwara and Miyahara, Esche and Schweizer, or Hubalek and Sgarra) or a relative q-entropy approach (for example, Jeanblanc, Klöppel, & Miyahara) ...


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An agent with utility function $U$ values a final position $X_T$ by $E\left[U(X_T)\right]$. You can think of this as a function mapping random variables to $\mathbb{R}$, $X_T \mapsto E \left[U(X_T)\right]$. A risk-neutral mapping should be a linear mapping of the kind above. In other words, $f$ should map some space of random variables to $\mathbb{R}$, ...


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A stochastic volatility model for a single risky asset can't be complete because you have two sources of randomness. But you can easily make it complete by adding a derivative whose value depends on the volatility. For example, if you add a variance swap in the Heston model then it becomes complete. This allows you to calibrate the model. But your ...


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Mersenne Twister is currently the most used PRNG in the quant world. It was even incorporated in C++11 so it can be considered standard nowadays. Any PRNG with reasonable statistical quality shall perform well (equivalently) for pricing, so that differences relate more to convenience (speed, parallelizability etc..). If the statistical quality is poor then ...



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