SERVING THE QUANTITATIVE FINANCE COMMUNITY

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by Rabelais
October 2nd, 2020, 10:05 pm
Forum: Numerical Methods Forum
Topic: What are the boundary conditions for the Forward contract PDE?
Replies: 45
Views: 2766

Re: What are the boundary conditions for the Forward contract PDE?

I was trying to answer to your point above : " At the end of the finite difference scheme, to obtain the value of the option at time 0 we compute  [ltr] V ( S 0 , 0 ) = ( F ( S 0 , 0 ) − K ) e − r T V(S0,0)=(F(S0,0)−K)e−rT   which in general is not 0, in contrast with what I said above about  F F  ...
by Rabelais
October 2nd, 2020, 10:00 pm
Forum: Numerical Methods Forum
Topic: What are the boundary conditions for the Forward contract PDE?
Replies: 45
Views: 2766

Re: What are the boundary conditions for the Forward contract PDE?

This is the plot with spot prices at time T (vector  ST  in the code) on x-axis and option prices at time 0 ( (F-K)*exp(-r*T)  in the code) on y-axis. As you can see the there is a problem when the option price approaches the biggest value of the spot price, since the curve goes from linear to expo...
by Rabelais
October 2nd, 2020, 9:56 pm
Forum: Numerical Methods Forum
Topic: What are the boundary conditions for the Forward contract PDE?
Replies: 45
Views: 2766

Re: What are the boundary conditions for the Forward contract PDE?

 whose solution is, letting \(\tau=T-t\) \[ \tag2 F(S_t,\tau) = \mathbb E[S_t] = \exp\bigg(e^{-\alpha\tau}\log S_t +\Big(\mu-\frac{\sigma^2}{2\alpha}-\lambda\Big)(1-e^{-\alpha\tau})+\frac{\sigma^2}{4\alpha}(1-e^{-2\alpha\tau})\bigg)  \] I know it's cheating but can you use the exact solution at Sma...
by Rabelais
September 30th, 2020, 4:56 pm
Forum: Numerical Methods Forum
Topic: What are the boundary conditions for the Forward contract PDE?
Replies: 45
Views: 2766

Re: What are the boundary conditions for the Forward contract PDE?

This is the plot with spot prices at time T (vector  ST  in the code) on x-axis and option prices at time 0 ( (F-K)*exp(-r*T)  in the code) on y-axis. As you can see the there is a problem when the option price approaches the biggest value of the spot price, since the curve goes from linear to expo...
by Rabelais
September 30th, 2020, 4:52 pm
Forum: Numerical Methods Forum
Topic: What are the boundary conditions for the Forward contract PDE?
Replies: 45
Views: 2766

Re: What are the boundary conditions for the Forward contract PDE?

I am not getting your point: suppose that we enter today a contract where I will pay to you 1 euro in one year. This contract worth the zero-coupon today, hence is not null, even if we do not exchange money today.. Maybe you are confusing payoff and fair-value ? I don't fully understand your argume...
by Rabelais
September 30th, 2020, 4:45 pm
Forum: Numerical Methods Forum
Topic: What are the boundary conditions for the Forward contract PDE?
Replies: 45
Views: 2766

Re: What are the boundary conditions for the Forward contract PDE?

I see your predicament. It is based on heuristical reasoning. What I do is to use domain transformation [$]y = S/(1+S)[$] to get a PDE in [$]y[$]. Then use Fichera theory (and/or integration by parts) to get 'numerical BCs'. There might  a Feller-style condition because of mean-reversion. You *migh...
by Rabelais
September 29th, 2020, 11:34 pm
Forum: Numerical Methods Forum
Topic: What are the boundary conditions for the Forward contract PDE?
Replies: 45
Views: 2766

What are the boundary conditions for the Forward contract PDE?

European call When solving the PDE for the value \(V\) of a European call option under the Black-Scholes model using a finite difference scheme, we have that Initial/terminal condition . \(V(S_T,T) = \text{payoff}(S_T) = \max(S_T-K,0)\) (initial since the scheme is solved backward, terminal since i...
by Rabelais
September 24th, 2020, 9:33 pm
Forum: Numerical Methods Forum
Topic: Accuracy of Explicit Euler method (finite difference) decreases as Δx decreases, shouldn't it increase?
Replies: 8
Views: 947

Accuracy of Explicit Euler method (finite difference) decreases as Δx decreases, shouldn't it increase?

The price of a commodity can be described by the Schwartz mean reverting SDE \[ dS = \alpha(\mu-\log S)Sdt + \sigma S dW \] where W is the standard Brownian motion and alpha is the strength of mean reversion. From it is possible to derive the PDE for the price of the forward contract having the com...
by Rabelais
September 22nd, 2020, 8:17 pm
Forum: Numerical Methods Forum
Topic: How to perform Monte Carlo simulations to price a Forward contract under the Schwartz mean reverting model?
Replies: 0
Views: 764

How to perform Monte Carlo simulations to price a Forward contract under the Schwartz mean reverting model?

I have 200 monthly spot prices of a commodity (oil) and I have to model the prices using the Schwartz mean reverting SDE: https://i.imgur.com/6kmNkyX.png where W is the standard Brownian motion, mu is the drift, sigma the volatility, alpha the strength of mean reversion (these parameters are estima...
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