I have come across two of these questions: Chapter 13 Q1 and 2023 Q9 in which you have a black scholes question where the payoff is peicewise defined. I am finding these very tricky. In the chapter 13 question 1 solution, i cannot understand how the method in the solutions replicates the payoff diagram. I guess that xt of underlying shares would start the position at xt so that is ok, but they say to then use a long put. the picture in the diagram is a slope up, but a long put is a slope down so this makes no sense to me. it then says to use a short call which is a downwards slope? 2023Q9 does make slightly more sense to me, with the cash being flat and the long call sloping up. I dont really understand why shorting a call is then flat, but i have seen the pdf "cm2 position diagram tips" which suggests a long then a short means flat for calls. Also in that pdf, why are puts from the right? - Thank you
Try some numbers in Excel - this will be good practice in any case. Get a pen and place it on a piece of paper on the left. Get a friend to shout "buy" and "sell". Start moving your pen to the right. If your friend shouts "buy" move up 45 degrees, if your friend shouts "sell" move down 45 degrees. Tell your friend not to be smart arse and shout "buy buy" when you're trying to learn about position diagrams. The diagram you have created has all been done using call options. If you need to start with a slope add a share to your portfolio. If the slope is going down, (short) sell a share in your portfolio. When you've finished this fun exercise, get a new piece of paper and place it on the piece of paper on the right. Get a friend to shout "buy" and "sell". (you could even use the same friend) Start moving your pen to the left. If your friend shouts "buy" move up 45 degrees, if your friend shouts "sell" move down 45 degrees. Again, tell your friend not to be smart arse and shout "buy buy" when you're trying to learn about position diagrams. The diagram you have created has all been done using put options. If you need to start with a slope add a share to your portfolio. Apologies, I've got the slope adding bit the wrong way round with the put options. Try all of this using different strike prices in Excel John
Thank you John, I am shocked at how much the excel graphs have cleared this up so that was a great tip thank you. (i even got tempted to look at what would happen in the buy, buy, and sell, sell case !! ). I am able to recreate the 2023 Q9 graph in excel now, and have also been able to do the paper B questions on the hub so all of that confusion from my question is gone. Chapter 13 q1 does seem as though something is different, i can replicate the graph on page 501 by using cash amount of 0.8, long call with K=0.8 and short call with 1.3 but can just not understand the shares, long put, and short call suggested in the solutions. is it different because the question says to protect against losses? For example if i use S_0=1 and x=0.8 the diagram of returns is @ y=1.3 ------------- / @y=0.8 -----/ but a put with strike price K=0.9 and then a short call with K=1.3 looks like @ y=0.8 \ \ @ y=0 ---------- \ @y=-1.3 \ I really cant make any sense of how the solutions have suggested that this is the right portfolio? Following this, on the paper b approximating questions i assume there is not just one right answer for these types of questions, if i could come up with another reasonable looking graph will i get full marks? I can imagine its quite easy to get bogged down trying to fit it perfectly though... Really appreciate your help
All of this sounds extremely good. Particularly pleased to hear about the proactiveness in your investigative approach - it's how we learn maths, of course. Chapter 13 q1 sounds like a guaranteed equity bond (or a bull spread diagram), lots of different ways of doing it... You can use just calls - buy at 0.8, sell at 1.3 You can use just puts - sell at 1.3, buy at 0.8 You can start with a share (upward slope) and then correct it. If we buy a put at 0.8 then we will stop losing money on the share if S<0.8. If we sell a call at 1.3 then we will stop winning money on the share if S>1.3. The only thing is that you have to get the height of the diagram right. You can do this by setting S to be any value and working out the correct height. You simply add cash to adjust the height. If just using calls, pick a low S so that all the calls expire worthless If just using puts, pick a high S so that all the puts expire worthless If using the third strategy, pick a low S, S=0 should work, where the call is worthless, the share is worthless and the short put is worth 0 - k = - K. So, adjust the height from - K to whatever you need. I'd get some Black-Scholes calcs going in Excel too so you can verify that all of your portfolios have the same value once you've corrected the heights. If you do this, you'll get loads better at CM2 in Excel and you'll understand Black-Scholes stuff, position diagram stuff all much better (2 useful achievements surely?) John
John, you are a real life saver, thanks so much for this. Makes complete sense about the heights. For the value part at the end of your response, do you mean they should be exact, or similar? I am convinced that my right portfolios and heights, have triple checked my black scholes calculations. my heights to the combinations you have mentioned are just calls - buy at 0.8, sell at 1.3, height 0.8 just puts - sell at 1.3, buy at 0.8, height 1.3 the mix of shares, calls put option - no extra height needed let S_0=1.1, volatility 0.2, r=5%, T=1 Long call, K=0.8: value at t=0 0.341354348 Short call , K=1.3: value at t=0 -0.040187878 Cash (0.8): 0.8 Total 1.10116647 Short put K=0.8: value -0.002337888 Long put K=1.3: value 0.17678613 Cash (1.3): 1.3 Total 1.474448242 Share S_0: 1.1 Long put K=0.8: value 0.002337888 Short call K=1.3: value -0.040187878 (no cash needed to get height right) Total 1.06215001 I promise, this is my last question on this and then I will leave you alone!!
Should be exact. I just checked in Excel and all 3 portfolios have a value of 1.06215001. You need to discount the cash and you've got the signs the wrong way round on the puts. Long call, K=0.8: value at t=0 0.341354348 Short call , K=1.3: value at t=0 -0.040187878 Cash (0.8): 0.8*exp(-0.05) Total 1.06215001 Long put K=0.8: value 0.002337888 Short put K=1.3: value -0.17678613 Cash (1.3): 1.3*exp(-0.05) Total 1.06215001 Please don't feel as if you need to "leave me alone". It's a pleasure to help, especially those students who are doing everything they can to help themselves. At the end of all of this you now have a much better understanding of options, Black-Scholes and you're better at CM2 in Excel
Ahh the discounting! I hadn't even thought about that but that makes sense we would need to discount the cash given that we have found c_0,p_0,s_0 etc. I've added the discounting to my excel calculations too now as you have suggested and i also get the 1.06215001 across the board - Phew Thank you so much!! I really appreciate your help on this!!!!
