, We now construct a stochastic process Z that satisfies the assumptions of Step 3. Towards this end, for each i, let Z i denote the unique P-martingale such that dQ i /dP = Z i (T ), such as the one of Corollary 2.2 in Blanchet and Ruf, vol.4, 2003.

, ?1 1 {Z in (?n)>0} Z in

&. 0}, As zero is an absorbing state for S j under ?n is a, {S j (? n?1 )

·. M?{1, N. R. Jeanblanc, M. Rutkowski, and M. , In this case, the proof of (b) is finished. However, under the more general condition in (b)(iv) it cannot be guaranteed that the P-martingale Z is strictly positive as it might jump to zero on n?N, Step 5B, we shall construct a family of strictly positive P-martingales (Y m ) m?{1,··· ,N +1} that satisfy the following two conditions: References Bielecki, pp.27-126, 2004.

J. Blanchet and J. Ruf, A weak convergence criterion for constructing changes of measure, Stoch. Models, vol.32, issue.2, pp.233-252, 2016.

D. Brigo, Market models for CDS options and callable floaters, 2005.

A. Câmara and S. L. Heston, Closed-form option pricing formulas with extreme events, Journal of Futures Markets, vol.28, issue.3, pp.213-230, 2008.

P. Carr, T. Fisher, and J. Ruf, Why are quadratic normal volatility models analytically tractable?, SIAM Journal on Financial Mathematics, vol.4, pp.185-202, 2013.

P. Carr, T. Fisher, and J. Ruf, On the hedging of options on exploding exchange rates, Finance Stoch, vol.18, issue.1, pp.115-144, 2014.

P. Collin-dufresne, R. Goldstein, and J. Hugonnier, A general formula for valuing defaultable securities, Econometrica, vol.72, issue.5, pp.1377-1407, 2004.

A. Cox and D. Hobson, Local martingales, bubbles and option prices, Finance and Stochastics, vol.9, issue.4, pp.477-492, 2005.

F. Delbaen and W. Schachermayer, A general version of the Fundamental Theorem of Asset Pricing, Mathematische Annalen, vol.300, issue.3, pp.463-520, 1994.

H. Föllmer, The exit measure of a supermartingale, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, vol.21, pp.154-166, 1972.

M. Herdegen and M. Schweizer, Semi-efficient valuations and put-call parity, Mathematical Finance, 2017.

S. Heston, M. Loewenstein, W. , and G. , Options and bubbles, Review of Financial Studies, vol.20, issue.2, pp.359-390, 2007.

H. Hulley and E. Platen, Hedging for the long run, Mathematics and Financial Economics, vol.6, issue.2, pp.105-124, 2012.

J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, 2003.

F. Jamshidian, Valuation of credit default swaps and swaptions, Finance and Stochastics, vol.8, issue.3, pp.343-371, 2004.

R. A. Jarrow and F. Yu, Counterparty risk and the pricing of defaultable securities, Journal of Finance, vol.56, issue.5, pp.1765-1799, 2001.

C. Kardaras, Valuation and parities for exchange options, SIAM Journal on Financial Mathematics, vol.6, pp.140-157, 2015.

M. Larsson and J. Ruf, Notes on the stochastic exponential and logarithm, 2000.

D. Madan and M. Yor, Arbitrage-free pricing before and beyond probabilities, Séminaire de Probabilités, XXXIX, vol.1874, pp.157-170, 2006.

N. Perkowski and J. Ruf, Supermartingales as Radon-Nikodym densities and related measure extensions, Ann. Probab, vol.43, issue.6, pp.3133-3176, 2015.

P. Protter, A mathematical theory of financial bubbles, ParisPrinceton Lectures on Mathematical Finance, pp.1-108, 2013.

P. J. Schönbucher, A note on survival measures and the pricing of options on credit default swaps, 2003.

P. J. Schönbucher, A measure of survival, Risk, vol.17, issue.8, pp.79-85, 2004.

C. A. Sin, Complications with stochastic volatility models, Adv. in Appl. Probab, vol.30, issue.1, pp.256-268, 1998.

M. R. Tehranchi, Arbitrage theory without a numéraire, 2014.

J. Ve?e?, Stochastic Finance: A Numéraire Approach, 2011.

J. Yan, A new look at the Fundamental Theorem of Asset Pricing, Journal of the Korean Mathematical Society, vol.35, issue.3, pp.659-673, 1998.