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You introduce a discretized auxiliary variable which represents $S_t$ to solve $N$ PDEs on $[t, t+\tau]$ using finite differences which will give you the present value of the option at time $t$ conditional on $S_t$. Then you solve one PDE using finite differences on $[0, t]$ to obtain the the present value at time $0$. This is the same methodology than that ...

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Here's an approach that's easy to code (but FAR from the fastest). Let $f(T,S,v,K)$ denote the price of a European call in the Heston model with time-to-expire $T$, initial price $S$, initial volatility $v$, strike $K$. First, use the tower property to transform the pricing problem: \begin{align*} V_{0} & ...

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There is a problem in your last step. Note that \begin{align*} P_{t, T_2}E_{Q_{T_2}}\left(\frac{1}{P_{T_1, T_2}} \mid \mathcal{F}_t \right) &= P_{t, T_2}E_{Q_{T_2}}\left(\frac{P_{T_1, T_1}}{P_{T_1, T_2}} \mid \mathcal{F}_t \right)\\ &=P_{t, T_2} \times \frac{P_{t, T_1}}{P_{t, T_2}}\\ &=P_{t, T_1}. \end{align*}

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Look at slide 3 of Mod 4: Fixed Income Derivatives: Bond Forwards, that's the relevant equation. $$G_0 = \frac{E_0^\mathbb{Q}[Z^j_t/B_t]}{E_0^\mathbb{Q}[1/B_t]}$$ Where $G_0$ denotes the price of the forward at $t=0$, $E_0^\mathbb{Q}$ is the risk-neutral price at $t=0$, $Z^j_t$ denotes the ex-coupon price of the bond at time $t$ and state $j$, and $B_t$ ...

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You are mostly right, I don't really get what you don't understand. The answer in the book is quite clear, but let me put it that way : Selling a put and buying a call on the same underlying $S$ with same maturity and same stike $K$ is always equivalent to a long position in a forward contract on $S$ with delivery price $K$. The easiest way to see that is ...

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