IN Tutorial 2: Q15, 16, 18 I am struggling to understand the intuition behind the choices made. I understand that certain of these decisions are made in chapter 10 - and given the open book nature would be available to refer to - but I severely struggle with understanding how to decide on the choice. In April 2019 - Q6 it is suggested (apologies for the long paste): "There is a methodical approach to answering forward pricing questions: 1) Start by assuming the market is arbitrage-free and hence the law of one price holds. 2) At time 0, set up two portfolios that have identical payoffs at time T. 3) By the law of one price, since the payoffs on the two portfolios are equivalent at time T, the values of the two portfolios must be identical at time 0. This equation can then be used to solve for K. The difficult step is determining the composition of the two portfolios. There is no unique way of doing this but one method that works well is to use the same Portfolio A in all forward pricing questions and to manipulate Portfolio B to achieve the desired payoff. For example: • Always set Portfolio A up to include a) one long forward and b) the cash that needs to be invested at time 0, rT Ke , to enable the long party to achieve the forward price, K , to exchange for the underlying asset at time T. • The payoff on Portfolio A at time T is the payoff on the forward, TS K and the payoff on the cash, K. Together, this gives a payoff at time T of T T S KKS . • Then work out the composition of Portfolio B to achieve the same payoff as that on Portfolio A, namely TS , at time T . If no income nor costs are involved we simply need one unit of the underlying asset in Portfolio B. In cases where consideration needs to be given to income and costs, we need to manipulate Portfolio B, either to include less than or more than one unit of the underlying asset or to borrow or invest cash so that the overall payoff stays at TS . The beauty of this approach is that it enables you to get at least one of the portfolios and payoffs each time and provides a systematic way of achieving the other portfolio" But then in Q15 - there is only one portfolio. Then in Q16 - portfolio B I am unsure how it has been gathered. Then in Q6 - April 2022 - we see that the approach is quite different to that suggested in chapter 10 part 4 in the derivation provided. There is a lot of inconsistency and I simply can't understand how to actually go about these questions. Likewise in Q4)iv) in tutorial 3 - what does a " no arbitrage approach based on the construction of a risk-free portfolio." generally entail? It is always going to be minus one derivative + delta shares?
There are ALWAYS two portfolios with these. Portfolio A is always... a) one long forward and b) cash Ke^-rT This is guaranteed to deliver one unit of the security at time T. So you just need to make Portfolio B do the same. Q15 is annoying as we are standing at time t but I'm sure we can cope... Portfolio A is always... a) one long forward and b) cash Ke^-r(T-t) This is guaranteed to deliver one unit of the security at time T. So you just need to make Portfolio B do the same. Portfolio B is e^-delta(T-t) units of the asset. Value = S e^-delta(T-t) As the asset pays dividends at rate delta, we use them to buy more units of the asset ending up with one unit at time T. This is guaranteed to deliver one unit of the security at time T. Hence, their values must be the same... Ke^-r(T-t) = S e^-delta(T-t) and we rearrange for K = S e^(r-delta)(T-t) Have a go at the others! John
