# A forward contract to buy a foreign currency can be handled by a linear model

In Hull's book, he says that: "An example of a derivative that can be handled by the linear model is a forward contract to buy a foreign currency." Then he continues with, "For the purposes of calculating VaR, the forward contract is therefore treated as a long position in the foreign bond combined with a short position in the domestic bond." I know what a linear model, forward contract, bonds, long/short position, and VaR mean by studying Hull's book. But I still cannot understand his reasoning. My intuition says that a forward contract can be handled by a linear model because a change in the price of the underlying asset translates linearly to the value of the contract at maturity.

• Hull is saying that the forward is a derivative that can be thought of as consisting of two more basic assets: a domestic zcb and a foreign zcb both which we already know how to price. So we can use the linear model. Later on when he gets to Options I imagine he will say that the option cannot be statically broken up into two things that we already know how to price, so we have a non-linear problem on our hands. (Of course the option can be broken up into two positions but their size changes dynamically, the option is not the superposition of two fixed things). Jun 21 '20 at 0:35
• Hull is saying some derivatives are nothing new, just existing instruments combined together in a fixed way, but some (options) are more complex and have a time-varying structure. They are made with a dynamic recipe, which makes them nonlinear. Jun 21 '20 at 0:52
• Instead of "linear" Hull might have used "additive": the value of the forward is the value of two bonds (one long, one short) added together. Jun 21 '20 at 0:59
• The part that I'm not getting is the "the value of the forward is the value of two bonds (one long, one short) added together." Why he decided that the domestic bond is short and the foreign bond is the long? I understand the linear vs nonlinear part. Jun 21 '20 at 2:01
• Write down the cash flows now and the cash flows at maturity for "long a foreign currency zero coupon bond and short a domestic currency zcb" and compare to the cash flows (now and at maturity) for fx forward. Make a nice 2 x 2 table. Are the two rows in this table the same or different ? Jun 21 '20 at 14:18