概要：本研究では有限状態の資産価格モデルを投資戦略と限界効用に関する双対平坦多様体と考えて，その幾何学的性質のファイナンス的解釈を考察する．具体的には (1) 効用関数の prudence は双対接続に関する接続の係数と対応する；(2) 一部の資産に投資できない非完備市場における均衡は，投資戦略に関する多様体と限界効用（価格）に関する部分多様体の交点として表現される；(3) リスク中立確率の Hansen--Jaganathan 距離は双対平坦多様体における Bregman divergence の特殊例と解釈できる．さらに，Riemann 計量を用いた均衡価格と最適投資戦略の数値計算例も紹介する． 

abstract: Optimal stopping is a fundamental concept in dynamic optimization, with applications in a wide range of fields including economics, statistics, and finance. Some common examples include determining the optimal exercise timing of options, deciding when to stop searching, and solving the quickest detection problem. In the context of Markovian frameworks, one approach to solving optimal stopping problems is to identify the stopping region, which allows us to determine optimal stopping times as the first hitting times of the stopping region. This talk will provide an overview of optimal stopping problems and their applications, with a focus on recent investigations that offer new perspectives on decision-making processes. We will explore optimal retirement planning with incomplete information, consumption habit under ambiguity, and optimal business expansion under uncertainty. Our first two topics illustrate how the challenges of identifying optimal stopping regions in complex market environments, such as incomplete observations and modeling ambiguity, can be overcome using novel duality approaches. The third topic proposes a novel type optimal stopping problem for deciding when to release constraints in optimization, with opportunity cost. By considering these diverse perspectives, we can gain a deeper understanding of how to optimize decision-making processes in various contexts.

abstract: Double barrier backward doubly stochastic differential equations (DB-BDSDEs, for short) are equations with two different directions of stochastic integrals, i.e., the equations involve both a standard “forward” stochastic integral and a “backward” stochastic integral with two mutually independent standard Brownian motions, and with two reflection barriers. This kind of equations is a joint version of backward doubly stochastic differential equations (BDSDEs, for short) and double barrier backward stochastic differential equations (DB-BSDEs, for short). The former has been introduced by Pardoux and Peng. They ave proved the connection with a class of systems of quasilinear SPDEs and the existence and uniqueness result of such PDEs. The latter has been tackled by Hamadene et al. In this talk, we try to show the outline of the proof for the existence and uniqueness of a solution to DB-BDSDEs by using the “penalization method”, so-called under appropriate conditions. At the end of this talk, we introduce our next some studies that we are tackling now.  

abstract: In this talk we discuss methodologies for fair pricing of variable annuity guarantees such as Guaranteed Minimum Withdrawal Benefit (GMWB). These products offer protection from market downturns and gain from market upside, and may include additional death benefit guarantees. Valuation of these products should simultaneously deal with financial risk, mortality risk and human behavior. We consider valuation under the “optimal” and pre-defined policyholder behaviors, extensions to stochastic mortality and stochastic interest rate models, and valuation in a presence of management fees charged for the management of underlying investment account.

The talk is based on several recent papers:

• J. Sun, P.V. Shevchenko, M.C. Fung (2017). A Note on the Impact of Management Fees on the Pricing of Variable Annuity Guarantees. Preprint, http://ssrn.com/abstract=2967045

• P.V. Shevchenko and X. Luo (2017). Valuation of Variable Annuities with Guaranteed Minimum Withdrawal Benefit under Stochastic Interest Rate. To appear in Insurance: Mathematics and Economics. Preprint http://arxiv.org/abs/1602.03238.

• P.V. Shevchenko and X. Luo (2016). A unified pricing of variable annuity guarantees under the optimal stochastic control framework. Risks 4(3), 22:1-22:31, doi:10.3390/risks4030022.  Preprint, http://arxiv.org/abs/1605.00339. 

• X. Luo and P.V. Shevchenko (2015). Fast Numerical Method for Pricing of Variable Annuities with Guaranteed Minimum Withdrawal Benefit under Optimal Withdrawal Strategy. International Journal of Financial Engineering 2(3), 1550024 (26 pages). http://ssrn.com/abstract=2517094 .

• X. Luo and P.V. Shevchenko (2015). Valuation of Variable Annuities with Guaranteed Minimum Withdrawal and Death Benefits via Stochastic Control Optimization. Insurance: Mathematics and Economics 62, 5-15. http://ssrn.com/abstract=2528355 

• X. Luo, P.V. Shevchenko (2015). Variable Annuity with GMWB: surrender or not, that is the question. In T. Weber, M. J. McPhee, and R. S. Anderssen (Eds.), MODSIM2015, pp. 959-965, http://www.mssanz.org.au/modsim2015/E1/luo.pdf 

講演概要: http://www2.gssm.otsuka.tsukuba.ac.jp/staff/yuji/workshop/Abstract_Japan.pdf

本研究会は，科研費セミナー (基盤研究(A) 課題番号16H01833, 研究代表者: 山田 雄二) との共催です。

(http://www2.gssm.otsuka.tsukuba.ac.jp/staff/yuji/workshop/)

講演概要:Optimal prediction of the ultimate maximum is a non-standard optimal stopping problem in the sense that the pay-off function depends on a process which is not adapted to the filtration at hand.  Our aim is to approximate by stopping times as close as possible the (random) time of the ultimate maximum. For a finite time horizon, this problem has been studied in various papers, including Du Toit, J. and Peskir, G. (AAP 2009) and Bernyk, V., Dalang, R.C. and Peskir, G. (Ann. Probab. 2011) for a Brownian motion and one-sided stable process, respectively. In this talk we will discuss the infinite horizon problem for two classes of processes. On the one hand we consider a general Lévy process and we find an optimal stopping time as a first passage time of the process reflected at its supremum. On the other hand, we will see that for positive self-similar Markov processes the Lamperti transform allows us to consider it as an optimal stopping problem in a Lévy setting. This talk is based on joint work with Dr. Kees van Schaik and with Andreas Kyprianou and Curdin Ott in the Lévy and positive self-similar case, respectively.

講演概要:The classical Reflection Principle allows one to express the joint distribution of a Brownian motion and its running maximum through the distribution of the process itself. It relies on the specific symmetry and continuity properties of a Brownian motion and, therefore, cannot be directly applied to an arbitrary Markov process. I will show that, in fact, there exists a weak formulation of this method that allows to obtain similar results for Markov processes that do not posses any strong symmetry. I will describe the general Weak Reflection Principle and will prove its validity for diffusions and L´evy processes. I will also demonstrate the applications of this technique in Finance, Computational Methods, and Inverse Problems.

講演概要：In this presentation we present an extension of the one factor blended Cheyette model for pricing single currency exotics, allowing for a more adequate fit to the swaption volatility smile. We first present a general framework based on the HJM model and then make a separability assumption on the instantaneous forward rate volatility, thus enabling a representation of the discount curve in a finite number of Markovian state variables. We show a practical application of this family of models by analyzing calibration and pricing in the case of a quadratic volatility function. By doing so, we provide a novel and parsimonious specification of the Cheyette model. Then for calibration purposes, we develop fast and accurate approximations for European swaptions, based on standard projection and averaging techniques. We also improve the usual naïve mean state estimation by the use of Gaussian approximations. Last we present an efficient large step Monte-Carlo simulation for strongly path dependent exotics.

講演概要：To solve utility maximization problems under proportional transaction costs, one can sometimes replace the original market with transaction costs by a frictionless shadow market that gives the same optimal strategy and utility. In this talk we provide sufficient conditions, when this is possible, as well as counter-examples in the case these conditions are not satisfied. For the geometric Ornstein-Uhlenbeck process we explicitly construct such a shadow price to determine the growth-optimal portfolio under transaction costs and explain the new effects resulting from the stochastic investment opportunity set in this construction. The talk is based on joint work(s) with Walter Schachermayer and several other collaborators.

講演概要：We develop statistical methods to detect informed trading in options markets. We apply

these methods to 31 companies from various sectors over 14 years analyzing approximately 9.6

million option prices. We find that option informed trading tends to cluster prior to certain

events, takes place more in put than call options, generates easily large gains exceeding millions, is not contemporaneously reflected in the underlying stock price, involves around the money options during calm times and out-of-the-money options during turbulent times. These findings are not driven by false discoveries in informed trades which are controlled using multiple

hypothesis testing techniques.

本セミナーは東京大学金融教育研究センターとの共催です．

https://drive.google.com/folderview?id=12nEsW6Jw8Gm0ARwndAC2fTr0NsuooUOj 

講演概要：

Abstract. In some non-regular statistical estimation problems, the limiting likelihood processes are generated by fractional Brownian motion (fBm) with Hurst's parameter H, 0 < H  < 1. For these cases we present several analytical and numerical results on the moments of Pitman estimators, also known as Bayesian estimators with a constant prior on real line. We also present selected simulation results for the variance of Pitman estimators.

講演概要：

本研究は，Touch optionを使ってBarrier optionの優劣複製とプライシング手法を提案する．

Touch OptionはBarrier Optionの中でも比較的流動性が高く，カリブレーションにも用いられる．

本研究の優劣複製はモデル非依存な手法で，Plain-vanilla optionとTouch optionで複製ポートフォリオを構築する．

プライシング手法は，Plain-vanilla optionとTouch optionの市場価格を再現し，優劣複製手法から導出される価格の上下限値に含まれることを保証する手法である．

講演概要：This presentation reviews a closed-form approximation approach to numerical problems for pricing derivatives. We derive tractable pricing formulas in local and stochastic volatility models and  show the validity of our method through numerical examples. We also show a numerical scheme for evaluation of derivative prices, where the  risky (or substitution) closeout CVA (credit value adjustment) is taken into account.

講演概要：Ｂｌａｃｋ－Ｓｃｈｏｌｅｓモデルにおけるバリアーオプションのスタティックヘッジ公式の導出においてブラウン運動の鏡像対称性が重要である．しかし一般の拡散過程はこの対称性を持たないため，同様にスタティックヘッジ公式を与えることは不可能である．CarrとLee(2009)は，拡散過程の係数がバリアーの値に関してある対称性を有する場合，Arithmetic Put-Call Symmetry（APCS）と呼ばれる対称性を満たすことを示した．ここでAPCSとはある１点（バリアー）に関してのみ，その拡散過程が対称性を持つことに他ならないが，スタティックヘッジ公式にはこのAPCSで十分である．しかし与えられた拡散過程がこのAPCSを満たすことは一般には期待できない．そこで我々は逆に一般の拡散過程に対してAPCSが成り立つような別の拡散過程を構成する方法を考案した．我々はこれをPut-Call対称化と呼んでいる．この構成方法によってもスタティックヘッジ公式を導くことができるが，ここで構成された拡散過程は実際の市場には存在しないため，もはや実務上の意味をもたない．しかし，この公式によって経路依存したペイオフを持つバリアーオプションの価値が経路依存していないペイオフのオプションによって表現されている点は変わらない．本講演は，この式をバリアーオプションの新しい数値計算手法として解釈し，この手法をSVモデルに実装した実験結果を紹介する．

本研究は奥村敏樹氏，今村悠里氏との共同研究である．

講演概要：TBA

講演概要：TBA

講演概要：

本発表では、資産価格過程がある線型確率ファクターの影響を受けるという仮定のもと、投資家のポートフォリオ価値成長率と、ポートフォリオのリスクとを考慮した連続時間期待冪効用最大化問題、すなわちリスク鋭感的ポートフォリオ最適化問題を考える。

リスク追求的と呼ばれる問題について、線形確率ファクターが2次元の場合に最適投資戦略が求められることを示す。既存の研究では線形確率ファクターが1次元について取り扱われていたが、これはその拡張となっている。

また、サブプライムショック、リーマンショック以降、近年の実務においてはダウンサイドリスクを抑制するポートフォリオ運用を模索する動きがみられる。ここでは上記問題の応用として、リスク鋭感的ポートフォリオ最適化にダウンサイドリスク抑制手法（CPPIアプローチ）を適用した最適投資戦略を求めることについても触れる。

講演概要：

20年前と10年前にそれぞれ三浦が考案したα-quantilesとRanksについて、その後の展開について話します。小話をいくつか並べるような形になりますが、気楽に聴いていただけるとありがたいです。

エキゾチックデリバティブを考案する場合と同様ですが、これらをブラウン運動上の軌跡の統計量とみなした場合、それを利用するには、その確率分布論が必要になります。それが分かったものしか論文には出来ません。今回はまだ確率分布が分かっていないものについても説明して、それがどういうところに使えるかなども議論できれば有意義なのではないかと考えています。例えば、αの値を同時に複数個考えるとどうなるか、ストックオプションのデザインに使うとどうか、平均オプションは市場で取引されているのに、同様の効能があり価格も安いメディアンオプションはなぜ取引されていないか、平均はマルコフ性を持たせることが出来るが、クオンタイルはそうは行かない、などです。

アブストラクト:

  In this article, I present a closed-form and exact solution for pricing futures of Nikkei Stock Average Volatility Index (Nikkei 225 VI) in a stochastic volatility model with simultaneous jumps in both the asset price and volatility processes. The pricing formula proposed in this article is derived in [ZL11a]  for someone to price VIX futures. VIX and Nikkei 225 VI have been developed under the same policies that the volatility index (i.e., VIX or Nikkei 225 VI) is inferred in the volatility of the stock index expected in one month and is computed from the option prices (premiums) in the stock option markets.  For this reason, the pricing formula in [ZL11a] can be directly applicable to pricing Nikkei 225 VI futures.

To price Nikkei 225 VI futures with the newly derived formula in [ZL11a], I am required to estimate all parameters of the model which I assume as the dynamics of stock index price. In this research, I program the algorithm of the Markov Chain Monte Carlo (MCMC) method in R and R-package, and I estimate all parameters with MCMC method from the daily-returns of Nikkei 225.

  I prove the newly derived formula with same technique in [ZL11a], estimate all parameters with MCMC method from daily-returns of Nikkei 225, and price the Nikkei 225 VI futures empirically.

[ZL11a]:S., P., Zhu, G., H., Lian, An Analytical Formula for VIX Futures and Its Applications, The Journal of Futures Markets, Vol. 00, No. 00, 1-25 (2011).

アブストラクト: 金融機関の信用リスク管理実務では, 自らの与信ポートフォリオが抱えるリスク量をバリュー・アット・リスク(VaR)によって定量的に把握している場合が多い. このVaRの計算はモンテカルロ法を用いることが一般的であるが, 最近ではより計算負荷の低い極限損失分布による方法やグラニュラリティ調整法など近似的な解析解を用いる方法が提案されている. しかし, これらの解析的手法は, 債務者が少ない場合や与信が集中している場合に近似精度が悪化するという点や, 信用力を表すモデルとしてMulti-Factor Merton Modelを用いると計算負荷が大きくなるといった問題点がある. そのため本発表では解決策として, Wavelet変換を用いた解析的なVaR計算方法を提案し, Multi-Factor Merton Modelの枠組みで与信ポートフォリオのVaRを短時間・高精度に計算できることを示す.

Abstract: 市場性金融商品の時価評価において、汎用的手法の必要性が高まる中、計算機の進歩と相俟ってモンテカルロ法の利用価値が高まっている。一方でリスクコントロールの観点からはグリークスの安定性が問題になる。今回の発表ではマリアバン解析を用いた解決策を中心に、他の手法との比較をしながら実用上のメリットや課題を紹介する。

Pricing Kernel 概説

The theory of Lévy models for asset pricing simplifies considerably if we take a pricing kernel approach, which enables one to bypass market incompleteness issues. The special case

of a geometric Lévy model (GLM) with constant parameters can be regarded as a natural generalisation of the standard geometric Brownian motion model used in the Black-Scholes theory. In the one-dimensional situation, for any choice of the underlying Lévy process the associated GLM model is characterised by four parameters: the initial asset price, the interest rate, the volatility, and a risk aversion factor. The pricing kernel is given by the product of a discount factor and the Esscher martingale associated with the risk aversion parameter. The model is fixed by the requirement that for each asset the product of the asset price and the pricing kernel should be a martingale. In the GBM case, the risk aversion factor is the so-called market price of risk. In the GLM case, this interpretation is no longer valid as such, but instead one finds that the excess rate of return is given by a non-linear function of the the volatility and the risk aversion factor. We show that for positive

values of the volatility and the risk aversion factor the excess rate of return above the interest rate is positive, and is monotonically increasing in the volatility and in the risk aversion factor. In the

case of foreign exchange, we know from Siegel's paradox that it should be possible to construct FX models for which the excess rate of return (above the interest rate differential) is positive both for the

exchange rate and the inverse exchange rate. We show that this condition holds for any GLM for which the volatility exceeds the risk aversion factor. Similar results are shown to hold for multiple-asset markets driven by vectorial Lévy processes, and for market models based on certain more general classes of Lévy martingales. (Work with D. Brody, E. Mackie, F. Mina, and M. Pistorius.)