New answers tagged options
-1
It's hard to help without knowing how you tried to solve the problem.
However, here's an idea:
try to make the model simpler (simpler parameters) and price a simpler claim. Then look at the results. Are they correct now? If yes, then you can look at your code and figure out what might be going wrong when you change the parameters.
If your result is ...
4
In a practical manner, here is how you get to the PDE of your option:
Use Girsanov theorem to go from the real-world measure to the risk-neutral measure (basically subtract the market price of risk $\mathrm dW^Q_t = \mathrm d W^P_t - \frac{\mu -r}{\sigma} \mathrm dt$). This will change your SDE.
Discounted option price $e ^{-rt} v(t, S_t)$ has to be a ...
2
Let $c_t$ be the price of an European call with maturity $T$ and $D_{t,T}$ the discount factor from $T$ to $t$. We assume deterministic rates. Then note that for $s<t\leq T$:
$$\begin{align}
E^Q_s\left(c_t\right)&=E^Q_s\left(E^Q_t\left(D_{t,T}(S_T-K)^+\right)\right)
\\[3pt]
&=E^Q_s\left(D_{t,T}(S_T-K)^+\right)
\\[3pt]
&=E^Q_s\left(\frac{D_{s,t}...
4
To compute the price of an American option or a callable instrument in general, at each potential exercise date, one is required to compare its continuation value (discounted risk-neutral expectation of what the option would pay off if it was not exercised) to the relevant exercise value/early redemption price.
By construction, lattice and finite difference ...
1
Exactly what @Alex C said. It's the time homogeneous diffusion proprety. You can't state such an argument in models where volatility is no longer time homogeneous ( that's being time independant and depending only on the underlyings).
5
Joshi is correct. The no arbitrage argument implies that the stock price instantaneous return under the risk neutral measure is equal to the short rate, and the girsanov theorem implies that the instantaneous volatility $\sigma$ is the same under the historical measure and under the risk neutral measure, so under the risk neutral measure the stock price is a ...
1
I assume we are talking about swaptions here? Then your formula looks correct, under a couple of assumptions; first, from the context of the question, you are assuming that you delta hedge once a day at the close of business. Second, you have implicitly assumed that the gamma is constant over the region of the daily move, which is ok as long as the option ...
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