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4

This link presents several discretization schemes for Heston: https://www.degruyter.com/view/journals/math/15/1/article-p679.xml For example, Milstein is a popular one. An alternative to the Euler discretization scheme for the Heston model is the second-order discretization method. The system of SDE under the risk-neutral measure \begin{eqnarray*} dS_t &=...


4

Given a initial discount bond $P^M(0, T)$ curve, the expression for $\theta(t)$ in the Hull White Short Rate model is a know result given by: $$ \theta(t) = \frac{1}{\kappa} \cdot f'(0, t) + f(0, t) + \frac{1}{2} \cdot \left( \frac{\sigma}{\kappa} \right)^2 \cdot \left( 1 - e^{-2 \kappa t} \right). $$ I have used a notation where the spot rate dynamics is ...


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A call option on zero coupon bonds $P(t, T)$ imply that its price, at time $t$, is given by $$ V(t) = E_t^Q \left[ D(t, T) \cdot \max \left(P(\tau, T) - K, 0 \right) \right], $$ with $t \leq \tau \leq T$. $P(t, T)$ is the discount bond or zero coupon bond and it is given by: $$ P(t, T) = E_t^Q \left[ \exp \left( - \int_t^T r(s) ds \right) \right]. $$ ...


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It sounds to me that they just mean that each bound can be seen as a function of the parameter(s) in the parametrization and this function is Lipschitz continuous. An example: Consider the XY-plane. Let $Y(x)$ be a function of $x$. This function can be seen as describing the upper bound of the area below the graph. This function can then have the Lipschitz ...


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Some of the assumptions here are wrong. The issue here is that $$S_0 \neq e^{-rT} E[S],$$ but $$F = E[S].$$ And thus Z should be Z=V-theta*(VC-exp(-rT)*F). If you output mean(VC) it's very clear. It suggests that the choice of parameters for the Schwartz model are not consistent with the interest rate r, unless a non-zero convenience yield is expected.


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Let's see. You have the following SDE for the stock price under the measure $P$: $$ dS(t) = \alpha \cdot (\mu - \log S(t)) \cdot S(t) \cdot dt + \sigma \cdot S \cdot dW^P(t), $$ with initial condition $S(0) = S_0$. Moreover, defining $X(t) = \log S(t)$, assuming a constant market price of risk $\lambda = \mu - r$ and performing a change of measure, you get ...


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