The risk-neutral probability measure is defined in terms of its numeraire. For the usual risk-neutral probability measure the numeraire is the bank account, $\beta(t)$. If we have a tradeable asset $X(... View answer Accepted answer 5 votes I think you can make the exact same argument for the cash-settled futures. The only difference is that you have to sell the underlying at expiry and delivery the cash instead of delivering the actual ... View answer Accepted answer 5 votes I think you are absolutely correct if the hazard rate is deterministic, although I think you are forgetting a discounting factor in your example. But sometimes the hazard rate cannot be assumed to be ... View answer Accepted answer 4 votes Let us say we have a yearly interest rate of$r$that compounds over$n$periods. With annual compounding that means$n=1$, with semi-annual compounding that means$n=2$and with daily compounding ... View answer Accepted answer 3 votes You forget the$e^{-rt}$term in the diffusion: $$dS^*=-rS^*dt+e^{-rt}dS=(\mu e^{-rt}-rS^*)dt+e^{-rt}\sigma B^\mathbb{P}(t)$$ Using Girsanov we can write$B^\mathbb{P}(t)=B^\mathbb{Q}(t)+\phi(t)dt$, ... View answer 2 votes The data stored in the object is adjusted such that compounding is Continuous and frequency is NoFrequency. The C++ source code is available here: zerocurve.hpp. I think that the reason for this is ... View answer Accepted answer 2 votes Your intuition is correct for a symmetric distribution. However, the log-normal distribution, as is assumed in the Black-Scholes model, is an asymmetric distribution. I have illustrated the effect of ... View answer Accepted answer 2 votes On page 119 in Björk (3rd edition) we have the replicating portfolio (equations 8.20 and 8.21): Hold$\frac{\partial C}{\partial s}$of the stock and$\frac{X_{t}-S_{t}\frac{\partial C}{\partial s}}{...

To find the $S$-dynamics under $\mathbb{Q}$ we have to use Girsanov's theorem: $$dW_t^P=\varphi_t dt+dW_t^Q$$ Dynamics under $\mathbb{Q}$ is thus $$dS_t=a(b-S_t)dt+\sigma S_t(\varphi_t dt+dW_t^Q)=abdt-... View answer 1 votes I do not recall having seen that formula before and I cannot make much sense of it. I will try to go through the steps necessary to price the option. This is by no means the fastest way to calculate ... View answer 1 votes For a plain vanilla swap we can define the annuity$$A(t)\equiv\sum_{i=0}^{N-1} \tau_i P(t,T_{i+1})$$and write the swap rate as$$S(t)\equiv\frac{P(t,T_0)-P(t,T_N)}{A(t)}$$such that the swap price ... View answer 1 votes Edit: This is probably incorrect. The Quadratic Exponential scheme is the best one I have seen as it converges in distribution and is pretty fast, so nice choice there! When \eta is constant you can ... View answer 1 votes In [Andersen & Piterbarg (2010)] on pages 415-417 the so-called General One-Factor Gaussian Short Rate Model is described, where the short rate is assumed to follow SDE$$ dr(t)=\kappa(t)(\theta(t)...

The Cholesky decomposition works in the following way: Suppose you have a vector of $n$ independent normal realisations: $Z$. Set the elements of $\Sigma$ to be $\rho_{i,j}$ where $i$ indicates the ...

I will assume that the interest rate is 0. The price of a binary option is then the same as the risk-neutral probability that the event will occur \mathbb{E}^{\mathbb{Q}}\left[\mathbb{1}_{S(T)\geq K}...