- 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**

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 ...

- 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**

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...

- 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**

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...

- 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**

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...

- 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**

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...

- 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**

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...

- 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**

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...

- 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**

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...

- 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**

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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