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## Friday, July 11, 2014

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)