New answers tagged

1

You don't need any assumption about the distributional properties of $S_t$. What matters for the FTAP is the drift only. By definition, the risk neutral measure $Q$ is the measure, equivalent to the natural measure $P$ (*), under which the local rate of return (i.e. the instanteneous drift of the SDE of $S_t$ per unit of $S_t$) of "any" traded asset $S_t$ (...


0

For a few basic option types under Black-Scholes you can also cross-check your results using this free option pricing application. It does vanillas, barriers (continuous and discrete monitoring) and Asians (discrete sampling) with European/Bermudan/American exercise. It has two separate engines, PDE/Finite Differences and Monte Carlo with Sobol sequences. ...


0

You can use this article Probability distribution of returns in the Heston model with stochastic volatility Let $$\begin{align} & d{{S}_{t}}=r{{S}_{t}}dt+\sqrt{{{\nu }_{t}}}\left( \rho dW_{1}^{Q}(t)+\sqrt{1-{{\rho }^{2}}}dW_{2}^{Q}(t) \right) \\ & d{{v}_{t}}=\kappa (\theta -{{v}_{t}}){{d}{t}}+{{\sigma }_{v}}\sqrt{{{\nu }_{t}}}dW_{1}^{Q}(t) \\...


2

The classical and naïve procedure for generating Poisson Hypersphere samples is by acceptance rejection, which has complexity over $O(N^2)$ and is thus unfeasible for most practical usage with on-the-fly generation. This cost could be improved by space partitioning techniques at low dimensions, but at high ones afaik they become useless again with uniform ...


0

I think the point of this approach is to model the firm value $V(t) $ using some appropriate probability distribution, then deduce the dustribution of the CB price. Thus the CB price depends on the firm value, but not vice versa.


2

The proof is fine. For example, $D(t)S(t)$ is a martingale and then \begin{align*} E\big(D(t)S(t)\big) = S(0). \end{align*} Regarding the function $C(1, T-T_0, K)$, it is the value, at time $T_0$, of the option payoff \begin{align*} \left(\frac{S(T)}{S(T_0)} - K \right)^+. \end{align*} Here, you can treat $\frac{S(T)}{S(T_0)}$ as the normalized value or ...


1

you can find from the CBOE paper above mentioned that the value is pretty much that of a strip of vanillas, weighted by 1/K^2. Typically if spx spot goes down, then realized vol increases. Together with the increase of realized vol, implied vol gets "re-evaluated" and typically marked higher with a steeper skew etc. the remark of the implied vol surface ...


0

The Black 76 swaption formula works for all these cases. The expiration time T= 1mo, 2mo or 3mo but the forward rate of the swap is the same in each case. The market will place different implied volatilities on these 3 options, according to the expectations of realized volatility in these 3 time periods.


1

Indeed parameters are selected so that the quoted option prices are as close as possible to the model option prices. Alternatively, quoted and model implied volatilities can be used instead of prices.The first category are those that minimize the error between quoted and model. The second category,are those that minimize the error between quoted and model ...


-1

Sorry if this is late, but this is the bible of Heston (and it has code) https://www.amazon.co.uk/Heston-Model-Extensions-Matlab-Finance-ebook/dp/B00EMADBN2/ref=sr_1_1?ie=UTF8&qid=1468410988&sr=8-1&keywords=Heston+matlab


2

Besides the code's problem, I highly recommend the Brownian Bridge correction method which can compensate the pricing error resulting from discretization of the continuous path.


1

There are many things wrong with your code. I'll leave aside the manner in which it is implemented, but note that it is: (1) not Matlab friendly with all the for loops (you should vectorise), (2) the fact that you have splitted the case j==0 in the main loop is a poor coding practice. for i=1:n I=1; for j = 0:(m-1); Z(j+1)= randn (1 ,1); dW=sqrt (T/...


2

You are trying to write a program which solves the following pricing PDE (Black-Scholes assumed) $$ \frac{\partial V}{\partial t}(t,S) + (r-q)S\frac{\partial V}{\partial S}(t,S) + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}(t,S) - rV(t,S) = 0 $$ where $V_0:=V(0,S_0)$ is the target option premium. The terminal condition is that, at $t=T$ (...


5

No offense but it will be much more complicated than what you think... I'm not even sure that you are familiar with risk-neutral pricing in the first place? I'll try to give you some clues. This security is called a basket option. On top of the multi-asset feature, there are non-trivial mechanisms embedded in the contract you mention: an auto-callable ...


0

"Monte Carlo convergence" means that you've sampled enough individuals to represent (and understand) a general population. If the probability models behind your Monte Carlo simulation are accurate, then your results will match reality as you increase your sampling size. Monte Carlo convergence becomes difficult when you try to study a low-probability sub-...


0

One way to do this would be to try to replicate the VIX calculation, which is calculated as the square root of a 30 day variance swap level. A variance swap can be replicated (in theory) using standard European calls and puts (you would need to convert American style stock option prices to European style prices using option models). The weighting scheme is ...


1

For historical volatility I actually like this article: http://www.todaysgroep.nl/media/236846/measuring_historic_volatility.pdf it provides several of the better known methods for calculating historical vol, which of course could be done manually. Just being aware of the upsides and downsides of each method. As for implied vol, yes as onlyvix has said it'...


4

Under GBM $$ \frac {dS_t}{S_t} = \mu dt + \sigma dW_t $$ we get $$ S_T = S_0 e^{(\mu - \frac{1}{2}\sigma^2)T + \sigma W_T} $$ suggesting that $$ S_T \sim \text{ln}\mathcal {N} ( \tilde {\mu}, \tilde {\sigma}) $$ where \begin{align} \tilde {\mu} &= \ln S_0 + (\mu - \frac{1}{2}\sigma^2)T \\ \tilde {\sigma} &= \sigma \sqrt {T} \end{align} Now if $X \...



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