The authors introduce forward curve models for Energy in Chapter 8 of their book, Energy Derivatives, Pricing and Risk Management. For those interested:
I'm trying to solve and implement (excel first) the equation 8.1 shown in the attachment "clelow_pages.png". As a first attempt, I thought of solving a non-stochastic equation by removing the dz(t) term; this attempt is shown in the attachment clewlo_soln. My first question is - is this even correct? I think I have an error as the solution doesn't produce the expected graph of a decay (see the first attachment, figure 8.3). My excel file is also attached....pls refer to ColP (shaded green)...instead of a decay, I get an exponential growth....so I have to be doing something terribly wrong; would deeply appreciate some advice!!!
Thanks!
Chet
Serving the Quantitative Finance Community
Clewlow and Strickland - one factor energy model, solution steps
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Re: Clewlow and Strickland - one factor energy model, solution steps
Looks like my link above (after the words...."for those interested....") didn't make it. so posting it here.
Clewlow book
Clewlow book
Re: Clewlow and Strickland - one factor energy model, solution steps
Figured out one mistake I made.....the factor inside the exponential term is not the one that creates the shape of the curve, it only impacts the magnitude of the shock and decays it across time. The original forward curve itself has a decay OR structure imposed on it. The shock is then applied on TOP of an existing forward curve (with an existing shape). I was mistakenly thinking that the exponential decay term itself creates the convex shape. Going to continue researching.
Re: Clewlow and Strickland - one factor energy model, solution steps
cool! keep us informed as you make progress.
Re: Clewlow and Strickland - one factor energy model, solution steps
@tags - appreciate the encouragement!! More coming soon as I try to wade through this stuff.
Re: Clewlow and Strickland - one factor energy model, solution steps
Here's the first attempt at creating the declining forward curve as shown in the screenshots from the book. At this time, I'm only focused on the non-stochastic part for now, the stochastic part to follow later.
Some things are not quite clear.....pls see the attached sheet and the comments below:
Some things are not quite clear.....pls see the attached sheet and the comments below:
- Col S: F0 is the "original" fwd curve. I made this up using a simple decay. It's not a real forward curve of course, but it's the starting point.
- Col T: dF is the shock based on the equation I wrote out in my screenshot above. I used a fixed arbitrary one time shock of 20 (cell T7), enough to show a clear separation b/w the original and the new forward curve.
- Col U: F1 is the new shocked forward curve (F0 and F1 in the first cart at the top in COl V-AE)
- Col Q: "non_stoch", also represents F1, but as a direct solution (refer to hand written solution screenshot above). In this case, I don't need an explicit shock to create F1 (bottom chart in Col V-AE).
- Pls ignore all other charts and the columns I-N, greyed out font
- Things look fine so far...but one thing bothers me. As you can see, I'm using Col G (time) instead of H (ttm; T-t) to create F1. Clewlow's equation uses T-t, not just "t"......and if I switch to Col H, I get a weird chart....where the fwd curves separate "at the end". Pls do try it...just change the formula in Col Q and T to point to Col H instead of G.
- Using time to maturity in the calculation feels more natural....but I get the right results when I use the absolute time index "t"; not quite sure what to make of that right now - if you folks have a good explanation, pls do enlighten!
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Re: Clewlow and Strickland - one factor energy model, solution steps
It helps to have the limits of integration correct. My original solution has a mistake; the solution should read:
F(T) = F0 * sigma/alpha * [1 - exp(-alpha * (T-t))]
F(T) = F0 * sigma/alpha * [1 - exp(-alpha * (T-t))]
- This fixes the issue with the interpretation of the T-t; i.e. time to maturity "flavor" I mentioned in #6 above. To be clear, the curve derived in Col Q; using the direct solution works fine if I use Col H now.
- However, when I use the dF formulation (numerical) - I need to use the index t instead of T-t to get the right shape of the fwd curve.
Re: Clewlow and Strickland - one factor energy model, solution steps
Finally - a eurkea moment....at-least it seems to be!
I found another error in Col S - the original/starting Forward Curve I had created, labeled F0. The mistake was that I used the same alpha to decay the curve as I was using to create the exponential shocks - and that created the strange results. There's no specific reason to use an alpha decay factor here....except that it creates a clean and smooth forward curve. It's not part of the model per se....and I could have used any other scheme to correct it. But I can't apply the same decay factor (alpha) to create the forward curve and also apply it to the exponential shock (i.e. the exp(-alpha * ttm). This is corrected in the new file attached and now I see the right curvature in the shocked curve, using either method:
I found another error in Col S - the original/starting Forward Curve I had created, labeled F0. The mistake was that I used the same alpha to decay the curve as I was using to create the exponential shocks - and that created the strange results. There's no specific reason to use an alpha decay factor here....except that it creates a clean and smooth forward curve. It's not part of the model per se....and I could have used any other scheme to correct it. But I can't apply the same decay factor (alpha) to create the forward curve and also apply it to the exponential shock (i.e. the exp(-alpha * ttm). This is corrected in the new file attached and now I see the right curvature in the shocked curve, using either method:
- The numerical solution (Col T and U), Chart 1
- The full solution (Col Q); Chart 2
- The scale of the two charts is not the same....but that is not an issue. For the #2, I just took some arbitrary fixed shock (cell T7) and applied the decay factor to that shock for the entire term of the curve.
- I'm now consistently able to use the T-t column (Col H) for both curves to create the two forward curves (Charts 1 and 2).
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Re: Clewlow and Strickland - one factor energy model, solution steps
Ok folks
Here's my attempt at the full solution - i.e. including the stochastic; dW term.
The screenshot shows the algebra, the attached sheet , tab clew_full shows the implementation:
Thanks!
Here's my attempt at the full solution - i.e. including the stochastic; dW term.
The screenshot shows the algebra, the attached sheet , tab clew_full shows the implementation:
- Col I shows the original forward curve (Fo) from the clew_nonstoch tab.
- Cells D11 and columns J-M build out the components of the SDE as solved in the paper solution (see screenshot)
- Col J, labeled as A is the implementation of the portion of the SDE labeled as A in the screenshot
- Col K is the brownian motion
- Col L is the the implementation of the portion of the SDE labeled as B in the screenshot
- Col M adds the components together, the constant in D11 is multiplied by the cumulative integral in col L
- Col N is then just the exponential of the entire term
Thanks!
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