June  2022, 1(2): 249-286. doi: 10.3934/fmf.2021010

Making no-arbitrage discounting-invariant: A new FTAP version beyond NFLVR and NUPBR

1. 

Department of Mathematics, ETH Zürich, Rämistrasse 101, CH-8092 Zürich, Switzerland

2. 

Swiss Finance Institute, Walchestrasse 9, CH-8006 Zürich, Switzerland

* Corresponding author: Martin Schweizer, martin.schweizer@math.ethz.ch

Received  July 2021 Revised  December 2021 Published  June 2022 Early access  May 2022

What is absence of arbitrage for non-discounted prices? How can one define this so that it does not change meaning if one decides to discount after all?

The answer to both questions is a new discounting-invariant no-arbitrage concept. As in earlier work, we define absence of arbitrage as the zero strategy or some basic strategies being maximal. The key novelty is that maximality of a strategy is defined in terms of share holdings instead of value. This allows us to generalise both NFLVR, by dynamic share efficienc, and NUPBR, by dynamic share viability. These new concepts are the same for discounted or undiscounted prices, and they can be used in general models under minimal assumptions on asset prices. We establish corresponding versions of the FTAP, i.e., dual characterisations in terms of martingale properties. As one expects, "properly anticipated prices fluctuate randomly", but with an endogenous discounting process which cannot be chosen a priori. An example with N geometric Brownian motions illustrates our results.

Citation: Dániel Ágoston Bálint, Martin Schweizer. Making no-arbitrage discounting-invariant: A new FTAP version beyond NFLVR and NUPBR. Frontiers of Mathematical Finance, 2022, 1 (2) : 249-286. doi: 10.3934/fmf.2021010
References:
[1]

B. AcciaioC. Fontana and C. Kardaras, Arbitrage of the first kind and filtration enlargements in semimartingale financial models, Stochastic Process. Appl., 126 (2016), 1761-1784.  doi: 10.1016/j.spa.2015.12.004.

[2]

A. AksamitT. ChoulliJ. Deng and M. Jeanblanc, No-arbitrage up to random horizon for quasi-left-continuous models, Finance Stoch., 21 (2017), 1103-1139.  doi: 10.1007/s00780-017-0337-3.

[3]

J.-P. Ansel and C. Stricker, Couverture des actifs contingents et prix maximum, Ann. Inst. H. Poincaré Probab. Statist., 30 (1994), 303-315. 

[4] K. E. Back, Asset Pricing and Portfolio Choice Theory, Oxford University Press, 2010.  doi: 10.1093/acprof:oso/9780190241148.001.0001.
[5]

D. Á. Bálint and M. Schweizer, Large financial markets, discounting, and no asymptotic arbitrage, Theory Probab. Appl., 65 (2020), 191-223.  doi: 10.4213/tvp5353.

[6]

D. Á. Bálint and M. Schweizer, Properly discounted asset prices are semimartingales, Math. Financ. Econ., 14 (2020), 661-674.  doi: 10.1007/s11579-020-00269-8.

[7]

W. Brannath and W. Schachermayer, A bipolar theorem for $L^0_+ (\Omega, \mathcal{F}, \mathbb{P})$, In: J. Azéma et al. (eds.), Séminaire de Probabilités XXXIII, Lecture Notes in Mathematics, vol. 1709, Springer, Berlin, 1999,349–354. doi: 10.1007/BFb0096525.

[8]

H. N. ChauA. CossoC. Fontana and O. Mostovyi, Optimal investment with intermediate consumption under no unbounded profit with bounded risk, J. Appl. Probab., 54 (2017), 710-719.  doi: 10.1017/jpr.2017.29.

[9]

T. ChoulliJ. Deng and J. Ma, How non-arbitrage, viability and numéraire portfolio are related, Finance Stoch., 19 (2015), 719-741.  doi: 10.1007/s00780-015-0269-8.

[10]

F. Delbaen and W. Schachermayer, A general version of the fundamental theorem of asset pricing, Math. Ann., 300 (1994), 463-520.  doi: 10.1007/BF01450498.

[11]

F. Delbaen and W. Schachermayer, The no-arbitrage property under a change of numéraire, Stochastics Stochastics Rep., 53 (1995), 213-226. 

[12]

F. Delbaen and W. Schachermayer, The Banach space of workable contingent claims in arbitrage theory, Ann. Inst. H. Poincaré Probab. Statist., 33 (1997), 113-144.  doi: 10.1016/S0246-0203(97)80118-5.

[13]

F. Delbaen and W. Schachermayer, The fundamental theorem of asset pricing for unbounded stochastic processes, Math. Ann., 312 (1998), 215-250.  doi: 10.1007/s002080050220.

[14]

____, The Mathematics of Arbitrage, Springer, Berlin, 2006.

[15]

D. Filipović and E. Platen, Consistent market extensions under the benchmark approach, Math. Finance, 19 (2009), 41-52.  doi: 10.1111/j.1467-9965.2008.00356.x.

[16]

G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed., Johns Hopkins University Press, Baltimore, 2013.

[17]

M. Herdegen, No-arbitrage in a numéraire-independent modeling framework, Math. Finance, 27 (2017), 568-603.  doi: 10.1111/mafi.12088.

[18]

M. Herdegen and M. Schweizer, Strong bubbles and strict local martingales, Int. J. Theor. Appl. Finance, 19 (2016), 1650022, 44 pp. doi: 10.1142/S0219024916500229.

[19]

J. Jacod, Calcul Stochastique et Problèmes de Martingales, Lecture Notes in Mathematics, vol. 714, Springer, Berlin, 1979.

[20]

J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Second ed., Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-05265-5.

[21]

Y. KabanovC. Kardaras and S. Song, No arbitrage of the first kind and local martingale numéraires, Finance Stoch., 20 (2016), 1097-1108.  doi: 10.1007/s00780-016-0310-6.

[22]

Y. Kabanov and M. Safarian, Markets with Transaction Costs. Mathematical Theory, Springer Finance. Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-68121-2.

[23]

Yu. M. Kabanov, On the FTAP of Kreps–Delbaen–Schachermayer, In: Yu. M. Kabanov et al. (eds.), Statistics and Control of Stochastic Processes: The Liptser Festschrift, World Scientific, River Edge, NJ, 1997,191–203.

[24]

Yu. M. Kabanov and D. O. Kramkov, Large financial markets: Asymptotic arbitrage and contiguity, Theory Probab. Appl., 39 (1995), 182-187.  doi: 10.1137/1139009.

[25]

J. Kallsen, σ-localization and σ-martingales, Teor. Veroyatnost. i Primenen., 48 (2003), 152-163.  doi: 10.4213/tvp309.

[26]

I. Karatzas and R. Fernholz, Stochastic portfolio theory: An overview, In: A. Bensoussan and Q. Zhang (eds.), Handbook of Numerical Analysis, Special Volume: Mathematical Modeling and Numerical Methods in Finance, Elsevier, 2009, 89–167.

[27]

I. Karatzas and C. Kardaras, The numéraire portfolio in semimartingale financial models, Finance Stoch., 11 (2007), 447-493.  doi: 10.1007/s00780-007-0047-3.

[28]

____, Portfolio Theory and Arbitrage: A Course in Mathematical Finance, American Mathematical Society, 2021.

[29]

C. Kardaras, Finitely additive probabilities and the fundamental theorem of asset pricing, In: C. Chiarella and A. Novikov (eds.), Contemporary Quantitative Finance. Essays in Honour of Eckhard Platen, Springer, Berlin, 2010, 19–34. doi: 10.1007/978-3-642-03479-4_2.

[30]

C. Kardaras, Numéraire-invariant preferences in financial modeling, Ann. Appl. Probab., 20 (2010), 1697-1728.  doi: 10.1214/09-AAP669.

[31]

C. Kardaras, Market viability via absence of arbitrage of the first kind, Finance Stoch., 16 (2012), 651-667.  doi: 10.1007/s00780-012-0172-5.

[32]

____, A time before which insiders would not undertake risk, In: Y. Kabanov et al. (eds.), Inspired by Finance. The Musiela Festschrift, Springer, 2014,349–362.

[33]

C. Kardaras and E. Platen, On the semimartingale property of discounted asset price-processes, Stochastic Process. Appl., 121 (2011), 2678-2691.  doi: 10.1016/j.spa.2011.06.010.

[34]

M. Loewenstein and G. A. Willard, Local martingales, arbitrage, and viability. {F}ree snacks and cheap thrills, Economic Theory, 16 (2000), 135-161.  doi: 10.1007/s001990050330.

[35]

E. Platen and D. Heath, A Benchmark Approach to Quantitative Finance, Springer Finance. Springer-Verlag, Berlin, 2006. doi: 10.1007/978-3-540-47856-0.

[36]

P. A. Samuelson, Proof that properly anticipated prices fluctuate randomly, Industrial Management Review, 6 (2) (1965), 41–49.

[37]

A. N. Shiryaev and A. S. Chernyi, Vector stochastic integrals and the fundamental theorems of asset pricing, Proc. Steklov Inst. Math., 237 (2002), 6-49. 

[38]

C. A. Sin, Strictly Local Martingales and Hedge Ratios on Stochastic Volatility Models, PhD thesis, Cornell University (1996), available online at https://ecommons.cornell.edu/bitstream/handle/1813/9055/TR001171.pdf.

[39]

K. Takaoka and M. Schweizer, A note on the condition of no unbounded profit with bounded risk, Finance Stoch., 18 (2014), 393-405.  doi: 10.1007/s00780-014-0229-8.

[40]

M. R. Tehranchi, Arbitrage theory without a numéraire, preprint, arXiv: 1410.2976v2, 2015.

[41]

J.-A. Yan, A new look at the fundamental theorem of asset pricing, J. Korean Math. Soc., 35 (1998), 659-673. 

show all references

References:
[1]

B. AcciaioC. Fontana and C. Kardaras, Arbitrage of the first kind and filtration enlargements in semimartingale financial models, Stochastic Process. Appl., 126 (2016), 1761-1784.  doi: 10.1016/j.spa.2015.12.004.

[2]

A. AksamitT. ChoulliJ. Deng and M. Jeanblanc, No-arbitrage up to random horizon for quasi-left-continuous models, Finance Stoch., 21 (2017), 1103-1139.  doi: 10.1007/s00780-017-0337-3.

[3]

J.-P. Ansel and C. Stricker, Couverture des actifs contingents et prix maximum, Ann. Inst. H. Poincaré Probab. Statist., 30 (1994), 303-315. 

[4] K. E. Back, Asset Pricing and Portfolio Choice Theory, Oxford University Press, 2010.  doi: 10.1093/acprof:oso/9780190241148.001.0001.
[5]

D. Á. Bálint and M. Schweizer, Large financial markets, discounting, and no asymptotic arbitrage, Theory Probab. Appl., 65 (2020), 191-223.  doi: 10.4213/tvp5353.

[6]

D. Á. Bálint and M. Schweizer, Properly discounted asset prices are semimartingales, Math. Financ. Econ., 14 (2020), 661-674.  doi: 10.1007/s11579-020-00269-8.

[7]

W. Brannath and W. Schachermayer, A bipolar theorem for $L^0_+ (\Omega, \mathcal{F}, \mathbb{P})$, In: J. Azéma et al. (eds.), Séminaire de Probabilités XXXIII, Lecture Notes in Mathematics, vol. 1709, Springer, Berlin, 1999,349–354. doi: 10.1007/BFb0096525.

[8]

H. N. ChauA. CossoC. Fontana and O. Mostovyi, Optimal investment with intermediate consumption under no unbounded profit with bounded risk, J. Appl. Probab., 54 (2017), 710-719.  doi: 10.1017/jpr.2017.29.

[9]

T. ChoulliJ. Deng and J. Ma, How non-arbitrage, viability and numéraire portfolio are related, Finance Stoch., 19 (2015), 719-741.  doi: 10.1007/s00780-015-0269-8.

[10]

F. Delbaen and W. Schachermayer, A general version of the fundamental theorem of asset pricing, Math. Ann., 300 (1994), 463-520.  doi: 10.1007/BF01450498.

[11]

F. Delbaen and W. Schachermayer, The no-arbitrage property under a change of numéraire, Stochastics Stochastics Rep., 53 (1995), 213-226. 

[12]

F. Delbaen and W. Schachermayer, The Banach space of workable contingent claims in arbitrage theory, Ann. Inst. H. Poincaré Probab. Statist., 33 (1997), 113-144.  doi: 10.1016/S0246-0203(97)80118-5.

[13]

F. Delbaen and W. Schachermayer, The fundamental theorem of asset pricing for unbounded stochastic processes, Math. Ann., 312 (1998), 215-250.  doi: 10.1007/s002080050220.

[14]

____, The Mathematics of Arbitrage, Springer, Berlin, 2006.

[15]

D. Filipović and E. Platen, Consistent market extensions under the benchmark approach, Math. Finance, 19 (2009), 41-52.  doi: 10.1111/j.1467-9965.2008.00356.x.

[16]

G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed., Johns Hopkins University Press, Baltimore, 2013.

[17]

M. Herdegen, No-arbitrage in a numéraire-independent modeling framework, Math. Finance, 27 (2017), 568-603.  doi: 10.1111/mafi.12088.

[18]

M. Herdegen and M. Schweizer, Strong bubbles and strict local martingales, Int. J. Theor. Appl. Finance, 19 (2016), 1650022, 44 pp. doi: 10.1142/S0219024916500229.

[19]

J. Jacod, Calcul Stochastique et Problèmes de Martingales, Lecture Notes in Mathematics, vol. 714, Springer, Berlin, 1979.

[20]

J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Second ed., Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-05265-5.

[21]

Y. KabanovC. Kardaras and S. Song, No arbitrage of the first kind and local martingale numéraires, Finance Stoch., 20 (2016), 1097-1108.  doi: 10.1007/s00780-016-0310-6.

[22]

Y. Kabanov and M. Safarian, Markets with Transaction Costs. Mathematical Theory, Springer Finance. Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-68121-2.

[23]

Yu. M. Kabanov, On the FTAP of Kreps–Delbaen–Schachermayer, In: Yu. M. Kabanov et al. (eds.), Statistics and Control of Stochastic Processes: The Liptser Festschrift, World Scientific, River Edge, NJ, 1997,191–203.

[24]

Yu. M. Kabanov and D. O. Kramkov, Large financial markets: Asymptotic arbitrage and contiguity, Theory Probab. Appl., 39 (1995), 182-187.  doi: 10.1137/1139009.

[25]

J. Kallsen, σ-localization and σ-martingales, Teor. Veroyatnost. i Primenen., 48 (2003), 152-163.  doi: 10.4213/tvp309.

[26]

I. Karatzas and R. Fernholz, Stochastic portfolio theory: An overview, In: A. Bensoussan and Q. Zhang (eds.), Handbook of Numerical Analysis, Special Volume: Mathematical Modeling and Numerical Methods in Finance, Elsevier, 2009, 89–167.

[27]

I. Karatzas and C. Kardaras, The numéraire portfolio in semimartingale financial models, Finance Stoch., 11 (2007), 447-493.  doi: 10.1007/s00780-007-0047-3.

[28]

____, Portfolio Theory and Arbitrage: A Course in Mathematical Finance, American Mathematical Society, 2021.

[29]

C. Kardaras, Finitely additive probabilities and the fundamental theorem of asset pricing, In: C. Chiarella and A. Novikov (eds.), Contemporary Quantitative Finance. Essays in Honour of Eckhard Platen, Springer, Berlin, 2010, 19–34. doi: 10.1007/978-3-642-03479-4_2.

[30]

C. Kardaras, Numéraire-invariant preferences in financial modeling, Ann. Appl. Probab., 20 (2010), 1697-1728.  doi: 10.1214/09-AAP669.

[31]

C. Kardaras, Market viability via absence of arbitrage of the first kind, Finance Stoch., 16 (2012), 651-667.  doi: 10.1007/s00780-012-0172-5.

[32]

____, A time before which insiders would not undertake risk, In: Y. Kabanov et al. (eds.), Inspired by Finance. The Musiela Festschrift, Springer, 2014,349–362.

[33]

C. Kardaras and E. Platen, On the semimartingale property of discounted asset price-processes, Stochastic Process. Appl., 121 (2011), 2678-2691.  doi: 10.1016/j.spa.2011.06.010.

[34]

M. Loewenstein and G. A. Willard, Local martingales, arbitrage, and viability. {F}ree snacks and cheap thrills, Economic Theory, 16 (2000), 135-161.  doi: 10.1007/s001990050330.

[35]

E. Platen and D. Heath, A Benchmark Approach to Quantitative Finance, Springer Finance. Springer-Verlag, Berlin, 2006. doi: 10.1007/978-3-540-47856-0.

[36]

P. A. Samuelson, Proof that properly anticipated prices fluctuate randomly, Industrial Management Review, 6 (2) (1965), 41–49.

[37]

A. N. Shiryaev and A. S. Chernyi, Vector stochastic integrals and the fundamental theorems of asset pricing, Proc. Steklov Inst. Math., 237 (2002), 6-49. 

[38]

C. A. Sin, Strictly Local Martingales and Hedge Ratios on Stochastic Volatility Models, PhD thesis, Cornell University (1996), available online at https://ecommons.cornell.edu/bitstream/handle/1813/9055/TR001171.pdf.

[39]

K. Takaoka and M. Schweizer, A note on the condition of no unbounded profit with bounded risk, Finance Stoch., 18 (2014), 393-405.  doi: 10.1007/s00780-014-0229-8.

[40]

M. R. Tehranchi, Arbitrage theory without a numéraire, preprint, arXiv: 1410.2976v2, 2015.

[41]

J.-A. Yan, A new look at the fundamental theorem of asset pricing, J. Korean Math. Soc., 35 (1998), 659-673. 

Figure 1.  Graphical summary of Theorem 4.1. Assumptions are $ {S}\ge0 $ and that $ \eta $ is a reference strategy (which is assumed to exist). The shorthand sm stands for share maximal, vm stands for value maximal. The equivalences on the left side need in addition that $ \eta $ and $ {{{S}}^{\eta}} $ are bounded (uniformly in $ (\omega,t) $)
Table 1.  Overview of existing FTAP-type results. Note that we have $ {\rm NA1} = {\rm{NUPBR}} $ on $ [0,T] $
price process $ {S} $ time condition dual condition
KK [27] $ (1,X) \in \mathcal{S}^{1+d}_{++} $ $ [0,\infty) $ NUPBR $ \exists $ $ {S} $-tradable SMD $ D>0 $, $ \forall\ H{\cdot} X $ with $ H\in{L_{{\rm{adm}}}}(X) $, with $ D_\infty>0 $
TS [39] $ (1,X) \in \mathcal{S}^{1+d} $ $ [0,T] $ NUPBR $ \exists \sigma \mathrm{MD}\ D>0 \text { for } X $
K [31] $ (1,X) \in \mathcal{S}^{1+1} $ $ [0,T] $ NA1 $ \exists \operatorname{LMD} D>0 \text {, } \forall H \cdot X \text { with } H \in L_{\mathrm{adm}}(X) $
KKS [21] $ (1,X) \in \mathcal{S}^{1+d} $ $ [0,T] $ NA1 $ \exists S\text {-tradable LMD } D>0, \forall H \cdot X \text { with } H \in L_{\mathrm{adm}}(X), \text { in any neighbourhood of }P $
H [17] $ \text { in } \mathcal{S}^{N} $ $ [0,T] $ NINA $ \exists (\text discounter, \text {E} \sigma \mathrm{MM}) \text { pair for }S $
here $ \text { in } \mathcal{S}^{N}_+ $ $ [0,\infty) $ DSV for $ \eta $ $ \exists \sigma \mathrm{MD}=\operatorname{LMD} D>0\text { for }S\text { with }\inf _{t \geq 0}\left(\eta_{t} \cdot\left(S_{t} / D_{t}\right)\right)>0 $
price process $ {S} $ time condition dual condition
KK [27] $ (1,X) \in \mathcal{S}^{1+d}_{++} $ $ [0,\infty) $ NUPBR $ \exists $ $ {S} $-tradable SMD $ D>0 $, $ \forall\ H{\cdot} X $ with $ H\in{L_{{\rm{adm}}}}(X) $, with $ D_\infty>0 $
TS [39] $ (1,X) \in \mathcal{S}^{1+d} $ $ [0,T] $ NUPBR $ \exists \sigma \mathrm{MD}\ D>0 \text { for } X $
K [31] $ (1,X) \in \mathcal{S}^{1+1} $ $ [0,T] $ NA1 $ \exists \operatorname{LMD} D>0 \text {, } \forall H \cdot X \text { with } H \in L_{\mathrm{adm}}(X) $
KKS [21] $ (1,X) \in \mathcal{S}^{1+d} $ $ [0,T] $ NA1 $ \exists S\text {-tradable LMD } D>0, \forall H \cdot X \text { with } H \in L_{\mathrm{adm}}(X), \text { in any neighbourhood of }P $
H [17] $ \text { in } \mathcal{S}^{N} $ $ [0,T] $ NINA $ \exists (\text discounter, \text {E} \sigma \mathrm{MM}) \text { pair for }S $
here $ \text { in } \mathcal{S}^{N}_+ $ $ [0,\infty) $ DSV for $ \eta $ $ \exists \sigma \mathrm{MD}=\operatorname{LMD} D>0\text { for }S\text { with }\inf _{t \geq 0}\left(\eta_{t} \cdot\left(S_{t} / D_{t}\right)\right)>0 $
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