I'm reading a book that discusses about derivatives pricing and have some doubts about a particular problem and really appreciate your advice:


Assume a non-dividend paying stock follows a geometric Brownian motion. What is the value of a contract that at maturity T pays the inverse of the stock price observed at the maturity?

Here is the solution:

Under risk-neutral measure, $dS = rSdt+σSdW(t)$. Apply Ito's lemma to $V=1/S$, we get $dV = (-r+σ^2)Vdt-σVdW(t)$. So V follows a geometric Brownian motion as well and we can apply Ito's lemma to $lnV$, we can get: $dln(V)=(-r+0.5*σ^2)dt-σdW(t)$. If I understand correctly, we don't need to find out the risk neutral measure for V because V is dependent on stock price S, and we already have the risk neutral measure for S and have $dS = rSdt+σSdW(t)$, due to fundamental theorem of asset pricing, the risk neutral measure is unique in this case and the unique risk neutral measure is derived from stock price S. I'm wondering if my understanding is correct?

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    $\begingroup$ I suggest you check my answer to my own question here. In the context of financial markets modeled with Brownian Motions (BM), the model is complete iff there are more assets than BMs; arbitrage-free iff there are more BMs than assets; and complete/arbitrage-free iff there are exactly the same number of assets and BMs. So in your case there is one BM and one asset (the stock) thus your model is complete and arbitrage-free. $\endgroup$ – Daneel Olivaw Dec 16 '19 at 15:55
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    $\begingroup$ traded assets, of course. $\endgroup$ – Daneel Olivaw Dec 16 '19 at 16:01
  • $\begingroup$ @DaneelOlivaw,thanks a lot:) $\endgroup$ – M00000001 Dec 16 '19 at 16:24

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