Taiga Saito

Self-financing strategy expression in general shape limit order book with market impacts in continuous time

In this paper, we extend the self-financing strategy expression in the linear supply curve model with market impacts by Roch (2011) and Cetin et al. (2004) to a non-linear case. Option hedging with liquidity costs and market impacts has been a key issue since the financial crises. We generalize the continuous time expression of a self-financing strategy in the linear supply curve model with market impacts proposed by Roch (2011) and Cetin et al. (2004), which is a useful result when one considers an option hedging strategy under an illiquid market, to a non-linear case. After showing an expression of the maximum price to when a hedger buys a certain amount under the non-linear supply curve, we define a non-linear market impact and show the self-financing expression under the non-linear supply curve model with market impacts. We also show examples of the strategy in non-linear supply curves observed in practice.

Trading and Ordering Patterns of Market Participants in High Frequency Trading Environment - Empirical Study in the Japanese Stock Market

In this study, we investigate ordering patterns of dii¬€erent types of market participants in Tokyo Stock Exchange (TSE) by examining order records of the listed stocks. Firstly, we categorize the virtual servers in the trading system of TSE, each of which is linked to a single trading participant, by the ratio of cancellation and execution in the order placement as well as the number of executions at the opening of the afternoon session. Then, we analyze ordering patterns of the servers in the categories in short intervals for the top 10 highest trading volume stocks. By classifying the intervals into four cases by returns, we observe how dii¬€erent types of market participants submit or execute orders in the market situations. Moreover, we investigate the shares of the executed volumes for the dii¬€erent types of servers in the swings and roundabouts of the Nikkei 225 index, which were observed in September in 2015. The main findings of this study are as follows: Server type A, which supposedly includes non-market making proprietary traders with high-speed algorithmic strategies, executes and places orders along with the direction of the market. The shares of the execution and order volumes along with the market direction increase when the stock price moves sharply. Server type B, which presumably includes servers employing a market making strategy with high cancellation and low execution ratio, shifts its market making price ranges in the rapid price movements. We observe that passive servers in Server type B have a large share and buy at low levels in the price falls. Also, Server type B, as well as Server type A, makes profit in the price falling days and particularly, the aggressive servers in the server type make most of the profit. Server type C, which is assumed to include servers receiving orders from small investors, constantly has a large share of execution and order volume.

Hedging and pricing illiquid options with market impacts

In this paper, we consider hedging and pricing of illiquid options on an untradable underlying asset, where an alternative asset is used as a hedging instrument. Particularly, we consider the situation where the trade price of the hedging instrument is subject to market impacts caused by the hedger and the liquidity costs paid as a spread from the mid price. Pricing illiquid options, which often appears in trading of structured products, is a critical issue in practice because of its difficulties in hedging mainly due to untradablity of the underlying asset as well as the liquidity costs and market impacts of the hedging instrument. Firstly, by setting the problem under a discrete time model, where the optimal hedging strategy is defined by the local risk-minimization, we present algorithms to obtain the option price along with the hedging strategy by an asymptotic expansion. Moreover, we provide numerical examples. This model enables the estimation of the effect of both the market impacts and the liquidity costs on option prices, which is important in practice.

Derivatives pricing with market impact and limit order book

This paper investigates derivatives pricing under existence of liquidity costs and market impacts for the underlying asset in continuous time. Firstly, we formulate the charge for the liquidity cost and the market impact on the derivatives prices through a stochastic control problem that aims to maximize the mark-to-market value of the portfolio less the quadratic hedging error during the hedging period and the liquidation cost at maturity. Then, we obtain the derivatives price by reduction of this charge from the premium in the Bachelier model. Next, we solve a second order semilinear PDE of parabolic type reduced from the HJB equation for the control problem, which is analytically solved or approximated by an asymptotic expansion around a solution to an explicitly solvable nonlinear PDE.We also present numerical examples of the pricing for a quadratic payoff and a European call payoff in different settlement types, and show comparative static analyses.

Hedging and Pricing Illiquid Options with Market Impacts

In this paper, we consider hedging and pricing of illiquid options on an untradable underlying asset, where an alternative asset is used as a hedging instrument. Particularly, we consider the situation where the trade price of the hedging instrument is subject to market impacts caused by the hedger and the liquidity costs paid as a spread from the mid price. Pricing illiquid options, which often appears in trading of structured products, is a critical issue in practice because of its difficulties in hedging mainly due to untradability of the underlying asset as well as the liquidity costs and market impacts of the hedging instrument. First, by setting the problem under a discrete time model, where the optimal hedging strategy is defined by the local risk-minimization, we present algorithms to obtain the option price along with the hedging strategy by an asymptotic expansion. Moreover, we provide numerical examples. This model enables the estimation of the effect of both the market impacts and the liquidity costs on option prices, which is important in practice.

Self-Financing Strategy Expression in Extended Roch-CJP Supply Curve Model

In this paper, we show a self-financing strategy expression in an extended Roch-CJP supply curve model. Liquidity has been a key issue in options hedging especially since the financial crises. The supply curve model developed by Roch and Cetin et al. is one of the widely recognized approaches incorporating liquidity in derivatives pricing. We extend the Roch-CJP model to include a general form of the supply curve, giving the expression of the self-financing strategy in the extended model. We also show results on the relation between the supply curve and the limit order book density.

Optimal Room Charge Under Choice Models for Hotels with Limited Capacity

In this paper, we investigate an optimal room charge problem for a hotel that considers maximization of the expected sales. We introduce a waterfall model and a multinomial logit model for hotels in the same area and the same price range which offer limited number of rooms. The models reflect choice behaviors of customers in a transparent on-line hotel booking system. We show how to estimate the parameters of the models from on-line hotel booking data which are accessible for anyone through internet. Moreover, we give an algorithm to solve the maximization problem and examples of the optimal room charge for hypothetical hotel booking data. The examples show a potential of the sales increase for each hotel by solving the maximization problem in the waterfall model. With these models, it is expected that hotel managers as well as hotel investors, such as hotel REITs and hotel funds, are able to know the potential of the sales increase of hotels from on-line booking data and use the result as a tool for making investment decisions.

Pricing Foreign Exchange Options under Intervention

We develop a new framework of foreign exchange (FX) options pricing under intervention and incorporate the effect of intervention by assuming a FX rate process which is stopped by a hitting time of an absorption boundary. More importantly, the proposed model satisfies the arbitrage-free condition. We derive closed-form pricing formulas for European and digital put options, and Greeks of a European put option. We show numerical examples for the case of the EUR/CHF options in the market, where the Swiss National Bank has been intervening since September 2011. We investigate how the present values of the options differ under intervention by comparing the models prices with those of the market. We observe that the model indicates significantly lower present values than the market prices for the options which have a payoff in a low strike area.