Parameter estimates for the model in Eq. (1) fitted on daily observations from January 2015 to June 2017.
\r\n\tIn this book, the different factors of liquefaction, the field methods and laboratory tests to identify a potentially liquefiable soil aim to be reviewed; in addition with history cases (ground behavior during the occurrence of an earthquake, state of stress, deformation, shear strength, flow, etc.).
\r\n\tA very important aspect of this topic is the presentation of the different constructive techniques used to ground improvement (vibrocompaction, dynamic compaction, jet grouting, chemical injection, replacement, etc.), placing special emphasis on those constructive methods used to solve problems on structures already located in areas of low relative density with liquefaction potential, where the installation of monitoring and control equipment is also required (tiltmeters, piezometers, topographic points, seismographs, pressure cells, etc.).
The placenta is an organ that has evolved to accomplish one thing, promote fetal growth during gestation. This is not a new concept, but the concept has evolved with our understanding of the molecular and cellular components required. The fetal allograft is maintained by several mechanisms including, but not limited to physical sequestration, immune modulation by hormones, cells, and metabolites. In fact, the function of T cells and dendritic cells changes during gestation [1]. In mid gestation, there were reduced antigen-specific T cell responses, upregulation of inhibitory molecules, and reduced myeloid maturation toward dendritic cells suggesting an immune tolerance state was achieved. These changes were reversed in the third trimester toward immune activation that participated in completing pregnancy. In fact, the observation of microchimerism in mothers postpartum indicated that there was a very strong tolerance inductive mechanism produced by the placenta and its physiology. Maternal-fetal chimerism has been recently reviewed [2]. A highlight from this review discussed pregnancy-associated progenitor cells that were proposed to be tissue resident stem cells with long-term survival that led to microchimerism that was detected for decades in some individuals. We propose that these types of tissue resident stem cells can be isolated and cultured from full-term postpartum placenta. After culture, we hypothesize that the immune modulatory properties that these cells exhibit in the placenta will be translated to in vitro and in vivo experiments that form the basis of many clinical development programs.
\nAn example of the cells that can be isolated from the human placenta is the human placenta-derived adherent cells (PDAC). PDAC are a culture expanded, plastic-adherent undifferentiated MSC-like population derived from normal full-term postpartum placental tissue. PDAC exhibit a phenotype of CD34-, CD10+, CD105+, and CD200+. PDAC constitutively express low to moderate levels of HLA Class I and undetectable level of HLA Class II, and these cells do not express the costimulatory molecules CD80 and CD86. PDAC were isolated by mechanical and enzymatic digestion of human placental tissue obtained from a normal, full-term birth as described [3]. PDAC are currently in clinical development as two separate formulations, PDA-001 (infusion product) and PDA-002 (locally injected product). This chapter will focus on developing the connection between placental immune biology (observed microchimerism) and the phenotypic and functional properties of the cells isolated from placenta tissues. These cells regardless of placental tissue source (amnion, cord, and cord blood) can mediate at least some of the immune tolerance properties of the placenta.
\nPlacenta-derived mesenchymal stromal-like cells (pMSC), including PDAC, have been isolated from various anatomical sites of the placenta, including the umbilical cord, chorion, decidua, amniotic membrane, and amniotic fluid. Like their bone marrow counterparts, these cells possess potent immune suppressive properties and exert their effects on T lymphocytes through a multitude of mechanisms that include both cell contact mediated interactions and through the modulation of secreted soluble factors. pMSC have been shown to inhibit both the proliferation and cytokine production of T lymphocytes, as well as, modulate T cell differentiation [3, 4, 5].
\nMany investigators have demonstrated the immune suppressive effects of pMSC in vitro via co-cultures of pMSC with CFSE-labeled, mitogen stimulated, or allogeneic T cells, known as mixed lymphocyte reactions (MLR), or with CFSE-labeled T cells stimulated by anti-CD3/ anti-CD28 monoclonal antibodies/beads, known as bead T cell reactions (BTR). In all cases, in reactions where CFSE-labeled cells were co-cultured with pMSC, significant reductions in both CD4+ and CD8+ T cell proliferation were observed. Reduced levels of proliferation were accompanied by decreased levels of Th1 cytokines (IL-2, IL-12, TNF-α, and IFN-γ) and increased levels of Th2 cytokines (IL-4 and IL-10) [3, 4, 6, 7]. Furthermore, when pMSC were co-cultured with naïve T cells under Th1 or Th17 inducing conditions, inhibition of Th1 and Th17 differentiation was observed [3].
\nSeveral factors secreted from pMSC have been implicated in the immune suppressive activities observed and will be summarized below. Indolamine 2, 3 dioxygenase (IDO) has been identified as a key mediator of pMSC anti-proliferative effects in MLRs/BTRs. IDO is a heme-containing enzyme that catabolizes the essential amino acid tryptophan into L-kynurenine. IDO-induced tryptophan degradation results in T cell cycle arrest in the G1 phase and serves as an instrumental mechanism for maintaining immune cell homeostasis and peripheral tolerance [8]. IDO gene expression and activity were induced in co-cultures of pMSC with MLRs/BTRs [6, 7, 9] and replenishment of tryptophan or treatment with IDO blocking compounds were shown to impair the antiproliferative abilities of pMSC [9, 10]. In addition, IDO was induced and subsequent suppression of T cell proliferation was intensified following stimulation of pMSC with IFN-γ [9, 11].
\nIn addition to IDO, increased production of prostaglandin E2 (PGE2) and transforming growth factor-β (TGF-β) by pMSC, and increased secretion of IL-10 by T lymphocytes have also been implicated as key soluble factors underlying pMSC’s immunosuppressive mechanism. PGE2, a bioactive lipid that is synthesized from arachidonic acid by the COX-1 and COX-2 enzymes, inhibits T cell proliferation and regulates the maturation and antigen presentation function of dendritic cells [3, 12]. TGF-β is a potent immunoregulatory protein that controls the differentiation, proliferation, and activation of various immune cells [13]. IL-10 is a well-known anti-inflammatory cytokine that controls the growth and activation of regulatory and/or anti-inflammatory cells [7]. All three secreted factors have been shown to be significantly increased in pMSC co-cultured with MLRs/BTRs [6, 7, 14]. The addition of blocking or neutralizing agents against PGE2, TGF-β, or IL-10 partially reversed and impaired the inhibitory effects of pMSC on T cell proliferation [5, 10]. Stimulation of pMSC with IFN-γ significantly upregulated the release of the tolerogenic cytokines TGF-β and IL-10 [5, 11].
\nMoreover, the increased levels of PGE2, TGF-β, and IL-10 can also affect T cell differentiation and lead to selective induction of Tregs [15, 16]. Tregs are CD4 + CD25 + FoxP3+ T cells that specialize in inhibiting T cell responses, allergic reactions, autoimmune disease, and graft rejection, while maintaining immune homeostasis [4]. Numerous studies have described an increase in the frequency of Tregs from co-cultures of pMSC with MLRs/BTRs [5, 6, 10, 14].
\nThe immune modulatory effects of pMSC on T cell proliferation and differentiation have also been shown in several animal models. We reported that PDAC suppressed T cell proliferation in an OT-II T cell adoptive transfer model [3]. OT-II transgenic mice, expressing the T cell receptor specific for ovalbumin, were used to evaluate the effects of PDAC on antigen-specific CD4+ T cell proliferation. PDAC at three different doses or vehicle were administered along with the adoptive transfer of CD4+ T cells isolated from OT-II mice into wild-type recipient mice following ovalbumin peptide immunization. PDAC treatment showed a dose-dependent decrease in the ovalbumin-specific CD4+ T cell proliferation in the spleen as compared with vehicle-treated mice. In addition, PDAC treatment resulted in an increase in the percentage of IL-10-producing splenic CD4+ T cells in a dose-dependent manner. In a rat sciatic nerve neuritis model, we showed that PDAC enhanced IL-10, but suppressed IFN-γ and IL-17 gene expression in draining lymph node, indicating that PDAC suppresses Th1 and Th17 cell differentiation [17]. We postulated that immune modulation of T cells in the draining lymph node is the mechanism underlying PDAC mediated neuropathic pain relief. To test this hypothesis, we performed two draining lymph node adoptive transfer studies in rat sciatic nerve neuritis model. Neuritis was induced by surgery and application of 1% carrageenan around sciatic nerve [17]. As shown in Figure 1, donor rats were treated with PDAC or vehicle 3 days after neuritis induction. One day after treatment, donor rats were sacrificed, and the draining lymph node was isolated into a single cell suspension and subsequently administered intravenously into the recipient rats with sciatic nerve neuritis.
\nSchematic chart of draining lymph node cell adoptive transfer study.
Mechanical hyperalgesia measurement in the recipient rats showed that the draining lymph node cells from PDAC-treated animals reduced neuropathic pain in a dose-dependent manner compared with the draining lymph node cells from vehicle-treated animals (Figure 2).
\nThe effect of draining lymph node from PDAC-treated rats on mechanical hyperalgesia measured by 26 g force of von Frey fiber.
To further identify the role of PDAC-mediated T cell modulation in the reduction of neuropathic pain, draining lymph node cells were separated into T cells and non-T cells using magnetic Pan-T microbeads, and adoptively transferred the cells to recipient rats with sciatic nerve neuritis. As shown in Figure 3, the whole population of draining lymph node cells as well as, the T cell fraction reduced neuropathic pain at days 4, 6 and 8. Interestingly, the non-T cell fraction also reduced neuropathic pain, with a slight time-delayed effect. These results from adoptive transfer studies demonstrated that immune modulation of the draining lymph node cells is the underlying mechanism of PDAC-mediated neuropathic pain reduction. Additional studies will be needed to sort out the differing contributions of the lymph node cells and how PDAC mediate these effects.
\nThe effects of T cell, non-T cell subsets, and whole population of draining lymph node cells from PDAC-treated rats on mechanical hyperalgesia measured by 26 g force of von Frey fiber.
The migration of the monocytes to the wound site and the following maturation of monocytes to macrophages play key roles in the process of wound healing. Mesenchymal stem cells have been known to benefit the wound healing process since they have immune regulatory properties including an anti-inflammatory activity [18]. Mesenchymal stem cells derived from different placental tissues have been reported to modulate the maturation and differentiation of macrophages in both in vitro and in vivo studies. pMSC were shown to reduce the endotoxin induced activation of a mouse macrophage RAW264.7 cell line [19]. Human amnion mesenchymal cells, when co-cultured with human THP-1 macrophage cells, were shown to inhibit the mRNA expression and secretion of TNF-α and IL-1β by THP-1 cells in vitro [20]. In addition, human placental mesenchymal stem cells shifted macrophage differentiation from inflammatory M1 to anti-inflammatory M2 macrophages presumably mediated by soluble molecules acting partially via glucocorticoid and progesterone receptors [21]. In a murine hind limb ischemia (HLI) model, a standard preclinical model of peripheral arterial disease (PAD), local administration of placenta tissue-derived mesenchymal stem cells to ischemic hind limb significantly reduced the infiltration of neutrophils and macrophages in the injured tissue compared to the sham-treated group [22].
\nIn a separate report of studying PDAC in an HLI model, we showed that intramuscular administration of PDAC improved angiogenesis in the injured limb. Histological analysis revealed that PDAC-treated mice had an increased level of CD68+ arginase1+ M2-like macrophages in ischemic tissue. Moreover, we demonstrated that the effect of PDAC on macrophage differentiation was T cell dependent. The M2-like macrophage skewing was only observed in wild type and T cell reconstituted nude mice, but not in nude mice [23]. This observation is consistent with other reports that placental mesenchymal stem cells from amniotic membranes shifted macrophage differentiation from an inflammatory M1 to an anti-inflammatory M2 macrophage population [21, 24].
\nAs discussed earlier, the human placenta plays a key role in maintaining immune tolerance between mother and fetus during normal pregnancy and is associated with the presence of Treg cells. It is also apparent that dendritic cells (DC) play a critical role in adaptive immunity and tolerance. While the multiple mechanisms of immune tolerance are not fully understood, it was first reported that amniotic mesenchymal tissue cells from human placenta (AMTC) can inhibit dendritic cell differentiation and maturation of monocytes from both peripheral blood and amniotic tissue [25]. When monocytes were cultured under the differentiation inducing condition, the presence of AMTC inhibited the expression of CD1a and reduced the expression of HLA-DR, CD80, and CD83. This finding suggested that placenta tissue mesenchymal stem cells could contribute to immune tolerance during pregnancy.
\nAnother source of placenta-derived mesenchymal stem cells was reported to induce myeloid DC to a tolerogenic phenotype as demonstrated by its reduced migration in response to CCR7 and impaired ability to stimulate IFN-γ secretion from NK cells [26]. Placenta-derived mesenchymal stem cells were also shown to increase the secretion of IL-10 and reduce the secretion of IFN-γ from DC cells [27]. Placenta chorionic villous-derived mesenchymal stem cells were shown to inhibit the maturation of human dendritic cells induced by LPS in co-culture experiments. The DC cells co-cultured with placenta MSC not only expressed lower levels of costimulatory surface molecules, including CD40, CD80, CD83, and CD86 but, also reduced expressed a reduced ability to activate T cells [28].
\nPlacental trophoblasts express a lower level of CD200 in preeclampsia and that is associated with an increase in production of Th1 cytokines, TNF-α, IL-6, IL-8, and IL-10 [29]. This result suggested that in normal placenta with higher CD200 levels on trophoblasts immune tolerance is favored [29]. Since PDAC express CD200, one can postulate that PDAC can also establish immune tolerance. In the rat neuropathic pain model, when PDAC was administrated via tail vein, it was found that PDAC alleviates mechanical hyperalgesia [17]. This anti-neuroinflammatory activity appeared to be mediated by the suppression of dendritic cell recruitment, maturation and differentiation. Rat DC cells isolated from draining lymph nodes of the PDAC-treated animals showed reduced gene expression of CD11c, CD86, and CD80, markers of DC maturation. The relative expression of IL-12, a key pro-inflammatory cytokine secreted by differentiated DCs, was also significantly reduced in PDAC treated rats. Furthermore, the inhibition of DC infiltration and activation was observed at the ipsilateral sciatic nerve. In an in vitro co-culture experiment, PDAC inhibited differentiation of mouse DC [3], providing direct evidence of PDAC-mediated modulation of DC maturation and differentiation in vitro. In this work, mouse bone marrow cells were induced with GM-CSF and LPS to induce DC maturation with or without PDAC. The bone marrow DC exhibited a phenotype of CD86high and MHC I-A/I-E high, which was reduced in a PDAC cell dose-dependent manner. In addition, the expression of the tolerogenic DC marker, PD-L1, was enhanced. PDAC was further shown to affect the differentiation of human immature peripheral blood DC cells in vitro. When exposed to GM-CSF and IL-4, immature DC (CD1a+) differentiated to mature DC (CD86 high). The inhibition of DC differentiation and maturation by PDAC does not require cell–cell interaction since the co-culture was performed using a trans-well system separating mouse and human DC from PDAC. The conditioned medium from PDAC can also modulate DC differentiation and maturation. The DC modulation activity was in part mediated by PGE2 secreted by PDAC [3].
\nIn a recent review of advanced cell therapy clinical trials highlighting perinatal cells, the authors compiled data on the number of clinical trials conducted in different countries, with different cell types and in many different indications [30]. Since 2008, there was a more rapid advancement in the number of trials registered with most of the trials using cord blood or cord tissue. The use of perinatal mesenchymal stem cells has also increased to about 70% of the annual trials registered and many of the trials were in the early phase of development (phase I or II). The cells from the amniotic membrane and placental tissue seem to be in a growth period for clinical trials. It will be important to track the progress of these trials and see which cells work in which indications.
\nIn Figure 4 below, the search terms on
A narrow search on ClinicalTrials.gov with the search terms placenta derived cells.
A narrow search on ClinicalTrials.gov with the search terms placenta derived cells illustrating the diverse diseases investigated.
A search on ClinicalTrials.gov with the search terms mesenchymal stem cells illustrating the much greater numbers of clinical studies.
In this chapter, the observation of fetal-maternal microchimerism that can last for decades indicated that the placenta exhibited highly specific and strong immune tolerance to the host. In an attempt to explain the immune tolerance mechanisms, we highlighted several cell types from Tregs to dendritic cells and their interactions with placental derived cells using in vitro and in vivo models. There were several molecular mechanisms (examples, IDO, PGE2, T cell proliferation, and DC maturation) invoked to explain some of the interactions with T cells and dendritic cells, which included the effects of cytokine secretion on the activation and differentiation status of immune cells. The in vitro and in vivo data describe a broad array of immune modulation functions suggesting that cells of placental origin have immune modulatory and immune tolerance inducing properties that are independent of tissue source.
\nThese immune modulatory properties highlight some of the possible ways in which the physiology of the placental graft is maintained during pregnancy and well after for microchimerism. To put these interesting results into some physiological context, a recent example of the immune system driving preterm labor described a pro-inflammatory environment at the fetal-maternal interface as a prerequisite for preterm labor [31]. In this study, the authors demonstrated that fetal T cells produced INF-γ and TNF-α which preceded myometrial cell contraction required for parturition. In addition, the authors demonstrated that fetal T cells (both CD4 and CD8) specifically proliferated in response to maternal antigen. It is interesting to speculate that the placental MSC present at the fetal-maternal interface could participate in suppressing the pro-inflammatory signals and the T cell proliferation that drive preterm labor.
\nThis is a very active field of clinical and pre-clinical investigation and has generated huge excitement in the field of advanced cell therapy. The near future will bring us clinical results that allow the advancement of cell therapy to FDA approval and ultimately for the benefit of the patients that eagerly await these therapies.
\nAll authors receive compensation from Celularity Incorporated.
AMTC | amniotic mesenchymal tissue cells |
BTR | bead T cell reactions |
DC | dendritic cells |
HLI | hind limb ischemia |
IDO | indolamine 2, 3 dioxygenase |
MLR | mixed lymphocyte reactions |
PAD | peripheral arterial disease |
PDAC | placenta-derived adherent cells |
pMSC | placenta-derived mesenchymal stromal-like cells |
PGE2 | prostaglandin E2 |
TGF-β | transforming growth factor-β |
Bitcoin is a digital currency built on a peer-to-peer network and on the blockchain, a public ledger where all transactions are recorded and made available to all nodes. Opposite to traditional banking transactions, based on trust for counterparty, Bitcoin relies on cryptography and on a consensus protocol for the network. The entire system is founded on an open source software created in 2009 by a computer scientist known under the pseudonym Satoshi Nakamoto, whose identity is still unknown (see [1]). Hence, Bitcoin is an independent digital currency, not subject to the control of central authorities and without inflation; furthermore, transactions in the network are pseudonymous and irreversible.
\nBitcoin and the underlying blockchain technology have gained much attention in the last few years. Research on Bitcoin often deals with cybersecurity and legitimacy issues such as the analysis of double spending possibilities and other cyber-threats; recently, high returns and volatility have attracted research toward the analysis of Bitcoin price efficiency as well as its dynamics (see, among others, [2, 3, 4]). Moreover, many contributions claim that Bitcoin price is driven by attention or sentiment about the Bitcoin system itself; see [5, 6, 7, 8]. Possible driving factors for the sentiment about the Bitcoin system are the volume of Google searches or Wikipedia requests as in [5], or more traditional indicators as the number or volume of transactions, as suggested in [6]. In [9], the author suggests a time series model in order to identify the dynamic relation between speculation activity and price.
\nIn this chapter, after having introduced the basic concepts underlying Bitcoin, we sum up and describe to a broader audience the recent outcomes of the research reported in [10], by avoiding unnecessary technicalities. Some new insights are also given by looking at possible extensions in order to take into account the presence of bubble effects or the special feature of Bitcoin being traded in different online platforms (exchanges) that will be further investigated in our future research.
\nWe recall that Bitcoin was first introduced as an electronic payment system between peers by Satoshi Nakamoto (pseudonym) in [1]. Opposite to traditional transactions, which are based on the trust in financial intermediaries, this system relies on the network, on the fixed rules and on cryptography. Bitcoins can be purchased on appropriate websites that allow to change usual currencies in the cryptocurrency.
\nThe Bitcoin network has several attractive properties for its users:
No central bank authority for money supply and no regulator;
Transactions are 24/7 and without any country border;
Transaction cost are almost negligible with respect to traded amount;
Transaction are anonymous;
The security of each transaction is guaranteed by cryptography and digital signature;
The security of the whole network is guaranteed by construction unless more than 50% of the network nodes agree on a deceptive action.
As a digital payment system, Bitcoins may be used to pay for several online services and goods. Special applications have been designed for smartphones and tablets for transactions in Bitcoins and some ATMs have appeared all over the world (see Coin ATM radar) to change traditional currencies in Bitcoins. Accepting Bitcoins as a payment method is also related to an advertisement opportunity for companies. However, the high returns achieved in the last few years have transformed Bitcoin in a speculative asset affecting its use as a form of payment.
\nThe Bitcoin system has been subject to many cracks but has proven to be very resilient as the value of the cryptocurrency was able to rise again after all the falls. Nevertheless, at the time of writing, Bitcoin was experiencing a fall in its exchange rate with main fiat currencies.
\nTwo of the main crackdowns were China enforcement in December 2013 and Mt. Gox bankruptcy in February 2014.
\nBesides technical and regulation issues, the Bitcoin system also faces reputational concerns.
\nIn fact, the ambiguity of anonymous transactions has blamed the network of allowing several criminal activities such as buying illegal goods, money laundering or the financing of terrorism actions. As a representative example, we recall that The Silk Road was a website that started selling narcotics and illegal drugs in 2011, payable in Bitcoins. The website was finally shutdown by 2013 and the owner was arrested and sentenced to life in prison. Again, anonymous transactions make it possible to use huge quantities of money, exchanged in Bitcoins, without declaring its origin, hence allowing for possible money laundering. However, according to a research performed by the UK government, the highest score related to money laundering is still cash, followed by the bank, accountancy and legal service providers (see
It is worth noticing that while counterparties are represented by secret addresses and are anonymous, all transactions are recorded and might be traced. Investigation is hence favored by this feature of the network.
\nDespite the flaws in the system, Bitcoin has achieved a notwithstanding rise in recent years.
\nIn Figure 1, we report Bitcoin price and returns from January 2012 to December 2017 (source
Bitcoin price (top) and returns (bottom) from January 2012 to December 2017.
The model we suggest in what follows is motivated by findings in [5, 6, 8, 11] where it is showed that Bitcoin price is related to investors’ attention measured by the trading volume and/or the number of searches in engines such as Google and Wikipedia. Bitcoin is treated as a financial stock as suggested in [12] and the suggested model may be applied in principle to other assets that are proven to depend on market attention.
\nConsider a probability space \n
Let us denote the Bitcoin price process as \n
where \n
It is well known that the above dynamics for the attention factor is a geometric Brownian motion, the solution of which is given by \n
We collect in this subsection the properties of the logarithmic returns obtained by the price process defined in Eq. (1).
\nConsider the discrete process \n
Theorem 2.1. The random vector \n
Proof. In order to prove the theorem it suffices to remind that, for i = 1, 2, …, n, the random variable \n
As for the unconditional distribution, it is easy to obtain, for \n
where \n
Proposition 2.2. The joint probability density of the vector \n
where \n
The proof follows from Bayes’ rule and application of Theorem 2.1.
\nIt is worth to remark that the probability density \n
Precisely, we have that \n
We apply the outcomes above in order to estimate model parameters according to the maximum-likelihood method (see for example [14, 15]) where the likelihood is approximated by applying the Levy approximation [13].
\nParameter estimates are obtained as
\nwhere
The first step in our procedure is to identify possible measures of investors’ attention. As already mentioned in the introduction, we consider the total trading volume on Bitcoin available from
The trading volume of exchange is a classical measure of the attractiveness of a traded asset for an investor; besides, in [16], the authors find evidence that the latter captures the attention of retail/uniformed investors.
\nWe consider daily data from January 1, 2015, to June 30, 2017, for the total volume and the SVI Index. As for the daily value of the Bitcoin, we have considered the average mean across main exchanges represented by the Index in
In Table 1, the outcomes for parameter estimates, obtained by maximizing the approximate likelihood given the observed time series, are summed up.
\n\n | \n\n | \n\n\n | \n\n\n | \n\n\n | \n
---|---|---|---|---|
A = Vol | \n0.9571 | \n1.1346 | \n0.0218 | \n0.0829 | \n
A = SVI | \n1.3584 | \n1.0687 | \n0.0743 | \n0.1559 | \n
Parameter estimates for the model in Eq. (1) fitted on daily observations from January 2015 to June 2017.
In this section, we show how to characterize the price of European call options on Bitcoins in the underlying market model. Let us fix a finite time horizon \n
where \n
Lemma 3.1. Every equivalent martingale measure \n
where \n
The proof can be deduced from that of Lemma 1.4 in [10], where they also account for a possible delay between the attention factor and its effect on Bitcoin prices trend. The process \n
is an \n
Equivalently, we can write the discounted Bitcoin price process \n
Clearly, under the minimal martingale measure \n
where \n
Remark 3.2. Note that, under any equivalent martingale measure that keeps the drift of the attention factor dynamics linear in \n
Now, we compute the fair price of a Bitcoin European call option via the risk-neutral evaluation approach, so it can be expressed as expected value of the terminal payoff under the selected pricing measure, that is, the minimal martingale measure. Let CT = (ST − K)+ be the \n
where
\nand \n
Here, \n
The following result provides the risk-neutral price of the option under the minimal martingale measure \n
Proposition 3.3. The risk-neutral price \n
where the function \n
Hence, the resulting risk-neutral pricing formula when evaluated in \n
In order to appreciate the performance of the pricing formula in Eq. (19), we compute model prices for option traded on the online platform
T-K | \nMarket bid | \nMarket ask | \nModel volume | \nModel Google SVI | \nBenchmark BS | \n
---|---|---|---|---|---|
Aug-2200 | \n0.1662 | \n0.2318 | \n0.2029 | \n0.2282 | \n0.1967 | \n
Aug-2300 | \n0.1670 | \n0.2072 | \n0.1737 | \n0.2032 | \n0.1655 | \n
Aug-2400 | \n0.1390 | \n0.1845 | \n0.1469 | \n0.1802 | \n0.1369 | \n
Aug-2500 | \n0.1142 | \n0.1638 | \n0.1228 | \n0.1591 | \n0.1112 | \n
Aug-2600 | \n0.0922 | \n0.1376 | \n0.1014 | \n0.1399 | \n0.0887 | \n
Aug-2700 | \n0.0749 | \n0.1202 | \n0.0828 | \n0.1226 | \n0.0695 | \n
Aug-2800 | \n0.0572 | \n0.1047 | \n0.0684 | \n0.107 | \n0.0535 | \n
Aug-2900 | \n0.0442 | \n0.0983 | \n0.0549 | \n0.0931 | \n0.0405 | \n
Sept-2200 | \n0.1991 | \n0.2648 | \n0.2546 | \n0.3204 | \n0.2173 | \n
Sept-2300 | \n0.1766 | \n0.2432 | \n0.2321 | \n0.3019 | \n0.1906 | \n
Sept-2400 | \n0.1890 | \n0.2230 | \n0.2113 | \n0.2844 | \n0.1662 | \n
Sept-2500 | \n0.1375 | \n0.2042 | \n0.1919 | \n0.2679 | \n0.1439 | \n
Sept-2600 | \n0.1207 | \n0.1828 | \n0.1741 | \n0.2523 | \n0.1239 | \n
Sept-2700 | \n0.1120 | \n0.1668 | \n0.1576 | \n0.2377 | \n0.1060 | \n
Sept-2800 | \n0.0953 | \n0.1504 | \n0.1463 | \n0.2239 | \n0.0903 | \n
Sept-2900 | \n0.0848 | \n0.1422 | \n0.1325 | \n0.2109 | \n0.0764 | \n
Motivated by empirical evidences (see for example [21, 22]), we discuss a generalization of the model introduced in Section 3.1, which is capable to describe speculative bubbles in Bitcoin markets.
\nPrecisely, we fix a finite time horizon \n
Without loss of generality, we assume that the interest rate is fixed and equal to zero. In this setting, the discounted Bitcoin price trend and the market attention factor dynamics are described by
\nwhere we have set \n
By simulating trajectories for the asset price \n
Simulated trajectories with n = 250 daily observations for the attention process (red) and the corresponding Bitcoin price dynamics for \n\nρ\n=\n0\n\n (black), \n\nρ\n=\n0.5\n\n(green), and \n\nρ\n=\n1\n\n (blue).
Indeed, we will show formally that the possibility of Bitcoin speculative bubbles is related to the sign of the correlation parameter \n
The mathematical theory of financial bubbles is developed, among others, in [23, 24, 25]. Precisely, we introduce the following definition from [23].
\nDefinition 4.1. The Bitcoin price process \n
The term strict \n
Recall that the absence of arbitrage opportunities is “essentially” equivalent to the existence of a probability measure \n
Remark 4.2. Note that stock bubbles arise if \n
Then, to exclude arbitrage opportunities from the market, we define the process \n
where \n
To ensure that \n
and we can consider the corresponding family of equivalent (local) martingale measures \n
where the \n
\n\n
Now, suppose that the risk perception process is zero, that is, \n
is a true \n
Proposition 4.3. In the model outlined in Eq. (24), the Bitcoin price process \n
The proof is based on the application of some of Sin’s results given in [27], where the existence of risk-neutral measures for the Hull-White stochastic volatility model [19] and for similar frameworks is determined by the possibility of explosion in finite time for solutions of certain auxiliary stochastic differential equations. Precisely, it is possible to show that the martingale property of the discounted stock price \n
Let us generalize the model introduced in Eq. (1) by assuming a possible delay \n
where \n
Analogous results as those in Section 2 can be derived by similar computations, and model parameters, for a fixed delay, can be estimated by means of the maximum likelihood method. In order to estimate the delay parameter, we maximize the profile likelihood as defined in [15]. Details of this procedure can be found in [10]. The estimation results of model in Eq. (27) on the same daily data considered in Section 2 are summed up in Table 3.
\n\n | \n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n
---|---|---|---|---|---|
A = Vol | \n1 day | \n0.4881 | \n1.0459 | \n0.0282 | \n0.0924 | \n
A = SVI | \n7 days | \n1.0964 | \n0.9946 | \n0.1005 | \n0.1885 | \n
Parameter estimates for model in Eq. (27) fitted on daily observations from January 2015 to June 2017.
In Figure 3, we plot simulated trajectories of the price process in Eq. (27) by letting the delay parameter vary.
\nSimulated trajectories of n = 250 daily observations of the attention factor (red) and the Bitcoin price according to model in Eq. (27) when the delay parameter is \n\nτ\n=\n1\n\n day (black) and \n\nτ\n=\n10\n\n days (blue).
The different delays result in a shift to the south-east between the faster and slower reacting trajectories; in the picture, this behavior is sharp since the other model parameters are kept constant. By looking at the picture, the idea to model the price of Bitcoin in different exchanges by the same model in Eq. (27) but allowing different parameters naturally arises.
\nIn particular, considering for instance two exchanges, we have
\nwhere \n
Note that within this model, prices for Bitcoin traded in different exchanges are perfectly correlated. Indeed, this is what happens in observed data; considering daily prices from January 2015 to June 2017 for Bitstamp, Kraken, Cex.io, Gdax, and The Rock exchanges we get cross-correlation values larger than 0.999.
\nWe fit model in Eq. (28) for the Bitstamp and Gdax exchanges on daily observations of Bitcoin price from January 2015 to June 2017 obtaining the outcomes reported in Table 4, when the attention is measured by the trading volume, and in Table 5, when attention is measured by the Google SVI index.
\nExchange | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n
---|---|---|---|---|---|
Bitstamp | \n1 | \n0.4994 | \n1.0461 | \n0.0281 | \n0.0896 | \n
Gdax | \n2 | \n0.4997 | \n1.0420 | \n0.0326 | \n0.1036 | \n
Model fitting with delay parameter: outcomes for Bitstamp and Gdax exchanges when attention is measured by the trading volume.
Exchange | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n
---|---|---|---|---|---|
Bitstamp | \n7 days | \n1.0934 | \n0.9946 | \n0.0992 | \n0.1782 | \n
Gdax | \n7 days | \n1.0964 | \n0.9946 | \n0.1160 | \n0.2087 | \n
Model fitting with delay parameter: outcomes for Bitstamp and Gdax exchanges when attention is measured by the SVI index.
It is evident from the outcomes in Table 4 that the model parameters are not significantly different while the delay might be quite different as if the reaction to the attention factor is faster for some exchanges and slower for others. On the contrary, when attention is measured by the Google SVI Index, the delay is unchanged, but the difference between estimated parameters for the price dynamics is nonnegligible.
\nBy analyzing the outcomes and considering the shift effect as depicted in Figure 3, it is tempting to conjecture that the faster reaction determines the leader exchanges and that the slower exchange will then follow. If we could forecast that the next day price of the slower exchange will reach the price today for the faster one, we could obtain a profit by suitably investing in the two exchanges. However, it is worth noticing that the estimation of the delay parameter is obtained by maximizing the likelihood over a whole time series and is a product of averaging so arbitrage cannot be achieved in a direct way.
\nNevertheless, in a multivariate setting as ours, the theory guarantees that arbitrage opportunities are ruled out if the market price of risk in the market is unique. Without entering technical details and assuming \n
It is evident that these values are not equal if we plug parameter estimates in Eq. (30); hence, arbitrage opportunities are not ruled out at least from a theoretical point of view. We will address this issue more precisely in future research.
\nIn this chapter, we have introduced a model in continuous time in order to describe the dynamics of Bitcoin price depending on an exogenous stochastic factor, which represents market attention on the Bitcoin system. Market attention is measured either by the total trading volume in Bitcoins or by means of the Google Search Volume Index, which, as suggested in [16], is a direct measure of the revealed attention for uniformed retail investors. More precisely, the attention factor affects directly the instantaneous mean and volatility of logarithmic returns; in addition, it may be also correlated with the price changes. An estimation procedure to fit the model to observed data is also suggested and, under the assumption of no correlation, a closed formula for standard European option prices on Bitcoin is provided.
\nBy applying outcomes within the mathematical theory of bubbles [23, 24, 25, 27], we are able to show that Bitcoin boosts in a bubble if and only if there is a positive correlation between changes in the price and in the attention factor. This finding is reasonable and claims that a stronger positive dependence between the two processes in Eq. (21) may result in an explosion of the price process.
\nFinally, we allow for a delay on the effect of market attention on the Bitcoin price, and, based on this generalized model, we introduce a multivariate setting for our model (Eq. (28)) in order to take into account the special feature of multiple exchanges where it is possible to trade in Bitcoins. Preliminary results indicate that arbitrage opportunities may arise in two exchanges that are characterized by different delays.
\nWe gratefully acknowledge Marco Patacca for having provided the routines in Matlab® to develop the empirical sections of the paper and for useful suggestions and comments. We also acknowledge funding from Fondazione Cassa di Risparmio di Perugia (Grant 2015:0459013) and Bank of Italy (Grant 407660/16).
\nThe authors declare no conflict of interest.
Gianna Figà-Talamanca gratefully acknowledges interesting discussions on the topic of this chapter with colleagues of the Department of Finance and Risk Engineering where she was visiting professor while preparing this research. The authors also wish to thank Stefano Bistarelli for having introduced them to the intriguing and worth to explore world of cryptocurrencies.
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