New answers tagged discounting
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 : ...
2
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'][0]
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
2
try this :
4000= 2000/(1+r)^2 + 3000/(1+r)^4 solving this equation for r you'll find equals 7.30274083178438%.
2
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)$
...
2
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] \\...
1
$N\left(d_2\right)$ 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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