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European Contracts It's a really important question and as @noob2 commented, the FTAP is normally applied to European-style derivatives, even if they are (strongly) path-dependent, including barrier options and Asian options. The idea is always the same, $V_t=B_t\mathbb{E}^\mathbb{Q}\left[\frac{\xi_T}{B_T}\Big|\mathcal{F}_t\right]$, that is the derivative's ...


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Have a look at page 311 in the original paper from Carr, Geman, Madan and Yor (2002). The paramters are for the names of the authors. They explain the role of each parameter there. Note that $C>0$, $G\geq0$, $M\geq0$ and $Y<2$. These parameters play an important role in capturing various aspects of the stochastic process under study. The parameter $C$ ...


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Is there a faster way to calculate the option price? With a recombining binomial tree, the terminal asset price has a binomial distribution -- as you might have expected. For a tree with $n$ steps, the probability of reaching price $S_{n,k}$ where $k$ is the number of up moves is $$P_{n,k} = \frac{n!}{k!(n-k)!}q^k(1-q)^{n-k}$$ The option price is the ...


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I would like to point out some things that might be helpful. I’m trying to figure out the discretization of the Heston model. In the choice of dt ... There exists many discretization schemes that could be implemented to simulate a system of stochastic differential equations (SDEs), such as the Heston model. One of them is the EulerMaruyama scheme, which ...


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It's the interest rate component. That is more meaningful in the formula. Note that the call becomes more expensive. Think about it this way. You could buy the call and sell the put instead of being long the stock. This gives you a synthetic long position. You need to pay the market the cost of borrow (r). That makes the calls more expensive and the ...


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First of all, option contracts normally specify 100 (or in some cases, 1000 contracts of the underlying instrument) to the one option contract, so you are unlikely to encounter the 1:1 scenario you mention. Let's ignore the ratios for now, and look at the risk profile portfolio you described (1 long underlying and 1 short call). The portfolio you have is a ...


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I think you have a small misunderstanding. The hedge for a call (or put) is rarely 1:1 with stock. When you are selling this option you are actually selling the future movements of the stock and specifically the future price jumps that will happen. To hedge an option you would enter into a position that will offset the movements in price of the option ...


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Under Martingale framework you can admit ,without loss of generality, to be under an arbitrage free market. By the way the martingale process is the discounted spot, you then need to use $$\exp^{-3*0.25} E[S_3]=S_0 $$. Finally, remember that under Up event $$S_{t+1} = S_t * u$$. You'll be able to solve your tree recursively. I may have made a mistake but ...


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Perhaps it might help if we define the difference between Brownian Motion (BM) and Geometric Brownian Motion (GBM). BM has independent, identically distributed increments while GBM has independent, identically distributed ratios between successive factors. The definition is inherited from that of arithmetic random walks, which are modelled as sums of random ...


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Does this matlab function help? The expression for $\Upsilon$ (Y) is split up in three parts. The infinite sum is truncated after 10 iterations (bound). We already compared our results for PFunction, QFunction and IFunction. function Y = Upsilon(x,T,mu,sigma,lambda,etaplus,etaminus,p,q) bound = 10; pi0 = exp(-lambda*T); pin = exp(-lambda*T) ...


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The answer to \begin{align} dS = \mu dt + \sigma dW_t \end{align} is simply \begin{align} S(t) - S(0) = \mu t + \sigma W_t \end{align} (as discussed here in the first page, for example)


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Yes, a stochastic volatility SDE can be coupled with any underlying SDE (GBM, diffusion, mean reverting, LMM, etc.). Once stochastic volatility is present, the model earns the right to be labeled 'SV model'. In its name, one may want to specify the names of both SDE's, like in the SABR LMM example found here, or just call it LMM with SV extension. Similarly, ...


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An ATM short call combined with an OTM long call is a bearish call spread not a bullish call spread. Option prices vary as the price of the underlying changes, as time passes and option premium decays as well as due to changes in implied volatility. A combination of the three of these is responsible for the spread widening.


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