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2

The last edition (10th, 2017) of Hull's book explains it fairly well. Basically, there is indeed a theoretical arbitrage within the dual curve framework: you could borrow at the overnight rate (Fed funds, SONIA, EONIA, etc.), lend at LIBOR and cash-in the spread in all your dynamic derivatives replication trades. However, such arbitrage is only theoretical : ...

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Already answered, but... from scipy.optimize import root def pv(r): return 2000 / (1+r)**2 + 3000 / (1+r)**4 rate = root(lambda x: pv(x) - 4000, 0.)['x'] print(f"Rate is {rate*100:.5f}%") print(f"Present Value is {pv(rate):,.2f}") Rate is 7.30274% Present Value is 4,000.00

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try this : 4000= 2000/(1+r)^2 + 3000/(1+r)^4 solving this equation for r you'll find equals 7.30274083178438%.

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The digital option pays $H$ at time $T$ if $S_T \geq K$ , so its option time at time $t$ is given by $$V_t=E_t\left[e^{-r(T-t)}H 1_{\{S_T \geq K\}}\right]=e^{-r(T-t)}H* P_t(S_T \geq K)$$ The model used is Black-model, that $$dS_t=rS_tdt+\sigma dW_t$$ or $$S_T=S_te^{\left(r-\frac12 \sigma^2\right)(T-t)+\sigma (W_T-W_t)}{}$$ Calculate $P_t(S_T \geq K)$ ...

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Recall that the price of your contract is \begin{align*} V_t = e^{-r(T-t)} \mathbb{E}^\mathbb{Q} [H1_{\{S_T>K\}}|\mathcal{F}_t] \end{align*} because your option always pays $H$ if $S_T>K$. Next, \begin{align*} V_t &=He^{-r(T-t)} \mathbb{E}^\mathbb{Q} [1_{\{S_T>K\}}|\mathcal{F}_t] \\ &= He^{-r(T-t)} \mathbb{Q} [{\{S_T>K\}}|\mathcal{F}_t] \\...

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$N\left(d_2\right)$ is the risk-neutral probability that the spot is greater than the strike at maturity, therefore the RN probability that you get your payoff.

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