tradable/non-tradable underlier in a drv contract

I guess in many, many entry-level quant questions, we are often given a task to find the Risk Neutral [2] dynamics of some variable X. Simple examples include Xa(t)=S(t)^2 or Xb=600/S, Xc=sqrt(S), Xd=exp(S), Xe=S*logS … where S is the IBM price following a GBM. In many simple cases the variable X is also GBM under the Risk Neutral measure. We use Ito’s rule…

Then we are asked to price a contract that guarantees to pay X(T) at maturity.

At this point, it’s easy to forget the X itself is not tradeable i.e. the X process is not the price process of a tradeable asset. When interest rate goes from 200 to 201, the mid-quote (of any security) doesn’t go from $200 to $201, even though the implied vol or implied yield could go from 200 to 201.Another Eg – suppose I were to maintain tight bid/ask quotes around current value of 600/ S_IBM. If IBM is trading at $30 then I quote $20. If IBM trades at $40 then I quote $15. This market-maker would induce arbitrage (– intuitive to the practitioners but not the uninitiated). A contract paying 600/S_T on maturity has a fair price today X_0 that’s very, very different from 600/S_0  [1].

Given X(t) process isn’t a tradeable (not a price process), X doesn’t have drift equal to risk-free rate “r” 😦

However, don’t lose heart — noting this Contract is a tradable , the contract’s price process C(t) is tradeable and C(t) has (exponential) drift = r 🙂

Q: Basic question – Given X(t) isn’t a price process, does it make sense to apply Ito’s on X = 600/S ?
A: Yes because 1) Ito lets us (fully) characterize the dynamics of the X(t) process, albeit NOT a price process. In turn, 2) the SDE (+ terminal condition) reveals the distro of X(T). From the distro, we could find the 3) expectation of X(T) and the 4) pre-expiry price. Note every step requires a probability measure, since dW, BM, distro, expectation are all probabilistic concepts.

[1] Try to develop intuition — By Jensen’s inequality, it should be above 600/S_0, provided S process has non-zero volatility.
[2] (i.e. using money market account probability measure)

quiz: add 0 to return value: gotcha # lval

Given calcTax(), I sometimes convert the returned value — simply add 0. I used to think this is completely harmless and has no effect on the program.

Now I don’t feel that way. calcTax() could return by reference, so we can take the address of this “expression” like


In other words, calcTax() is an lvalue expression. If you add 0, you get a rvalue expression! Basically a temporary object. You can’t take its address.