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Is there a way to run a strap across where is folds and tighten it in the back? You would need to buy some very narrow cotton or nylon webbing and some sort of metal or plastic buckle that you can tighten to the desired tension. If
the strap were narrow enough it would disappear into the fold.
From the picture, it looks like this might be possible, and that it could be disguised.

It doesn't look like the back of the frame goes "upright" enough. Is the problem that the futon keeps slipping forward off the frame? You could try putting one of those rubber, waffle-patterned, non-slip rug pads between the futon and the top part of the frame. Mine actually came with one!

From back in my futon days, I ran a length of nylon rope across the "fold" and then tied it into the frame. Eventually the mattress developed a crease and I removed it.

Get a good rug pad (the kind that is rubbery, not made of foam) and staple it to the futon frame's horizontal half. It should be completely hidden by the futon. You can do just a strip across the front, but the wider you can do, the better and more stable it will be.

A strap that you can get that works well is called "Truck Strapping"

It's the straps that trucks use to hold gear, equipment, ect in place on a truck. It has a metal ratchet as well as hooks that let you connect it together and tighten it up.

I have a futon with the exact same problem (8 inch thick futon mattress) and that has solved the problem. The nice thing is that it pulls it taut enough that you don't end up seeing the strapping from the front, and the back os low enough that you don't notice it unless you are looking for it.

I don't know about this particular frame, but foldable futon frames like yours should have an adequate sized ledge to help keep the futon from sliding when in it's upright postion. Also, your futon is extra thick it appears and the thicker it is, the harder it is to fold them.

I have a black metal frame that's similar to yours and it has 2 large metal lips on one side that serves as the seat when upright and it helps keep the futon in place.

Baring all that, I like the strap idea.

My old futon would slide off the frame all the time. I took the removable cover off and sewed velcro strips to the back of it. I stapled matching velcro strips to the wood frame and the futon never slipped again.

I had this same problem, I have a very heavy mattress which would side off my frame all the time, but I've found 2 great solutions:

You can get futon grips. Google them. You can order them online or get them at a futon store. I got mine at the Futon Shop on La Brea for like $12. It's like that rubber stuff you put under a rug or you can use to line your cabinets. You staple it to at leas the bottom portion (you can do the top too, but I don't know how much more effective that is, just the bottom seems to work for me).

Another thing you can do, to give it crisp lines, is when you fold it up, have two people, one on either side, and as you lift up the back, you want to "punch" the crease with your fist, holding it in place as you sit the futon up. This will give it a sharper fold in the mattress and should help to keep it in place. I learned about this technique while researching futon grips on some futon site.

I have the same problem with my uncooperative futon...in addition to the suggestions about the rug pad, you could try sliding your futon about 8 (or more) inches off-center, with more of the mattress on the back side of the fold line, less on the seat side. When you fold it up and it slides forward (as it wants to do) the mattress should be at the front edge of the frame. After I pull the back of the frame up, I walk to the front of the frame and pull the bask frame up the rest of the way while standing in front to keep the wayward mattress from hanging over the edge.

The truck strapping that Marshall mentioned can be found at Ikea for about $3. They call it Hercules (or some play on the word). It's sold near checkout and used to strap things to the roofs of cars. Also, it looks like you have the futon in it's semi-reclined position. If you place it upright it might help hold the mattress in place better. My old frame used to have that option and it was very similar to yours.