Subordinated Binomial Tree

We develop a unique tree structure with random move time by subordinating asset price changes to random trade arrival. The asset price change is determined by two independent Bernoulli trials on trade arrival/non-arrival and up/down price movement. Convoluting the two leads to a trinomial tree and a corresponding option pricing problem with stochastic volatility. A time change from calendar-time to tradingtime, however, restores the binomial tree and leads to an isomorphic option-pricing problem with constant volatility but random maturity. Utility-dependent valuation results are derived in both time dimensions and isomorphism is demonstrated. The binomial tree now grows with arrival of trades with a speed dictated by the intensity of arrival, irrespective of the passage of calendar-time. At the continuous-time limit, it contains the information-time option-pricing model of Chang, Chang and Lim (1998) as a risk-neutral special case. The trinomial tree is further extended to allow for correlated price revision and trade arrival to address the leverage effect and test for incremental improvements in pricing and hedging against the Black and Scholes and the CEV models. Please send all correspondence to Professor Jack S.K. Chang, School of Business, University of Hong Kong, Pokfulam Road, Hong Kong, PRC, Tel: 852-28592266, e-mail: jchang@business.hku. hk Tree-based discrete-time option pricing models start with the seminal binomial model of Cox, Ross and Rubinstein (1979) (CRR hereafter) who introduced an intuitive, parsimonious, and robust approach to derivatives valuation. A preference-free valuation result emerges from the simple set-up of a dynamically complete market with two states and two traded assets. The CRR model is widely used by practitioners for embedding the Black and Scholes (1973) model in the continuous-time limit as well as for the efficient pricing of American and path-dependent options. There have been various extensions to the original binomial model. Some aim at improving convergence efficiency while others attempt to incorporate more complex stochastic processes. A completely different set of tree-based models has also been developed for the evolution of interest rates. More recently, much attention has been paid to the so-called implied tree models. These models are the result of concerted efforts to deal with volatility smiles and/or sneers commonly observed in prices of traded options. One attractive feature of implied tree models is that they price all traded options consistently thus eliminate the pricing biases present in parametric models, however the tree parameters are likely to be unstable over time and need to be recalibrated as market conditions change. Throughout these developments however, the original CRR assumption that asset price changes occur uniformly over calendar-time continues to exist. The maturity of the option is divided into equal time intervals over which the asset price moves are of fixed magnitude, implying that a trade should always arrive at the end of each fixed calendar-time period. Taking it to the continuous-time limit, this assumption implies continuous trading. Although convenient, this simplification rules out actual trade arrival dynamics that a trade may not arrive at the end of each calendar-time period, an asset may be thinly traded during certain time periods with lower volatility but heavily traded during other time periods with higher volatility, and that asset price changes may be correlated to the pattern of trade arrival. Although it has been recognized in Amin (1993) and Rubinstein (1994) and some tree-based term structure models that faster convergence can be achieved by introducing non-equally spaced steps, none has randomized trade arrivals. In this vein, the purpose of this research is to investigate tree-based option pricing with random move time by randomizing trade arrival. This idea originates with the celebrated empirical evidence in the subordinated process literature that subordinating asset price changes to random trade arrivals significantly reduces asset return leptokurtosis. It is also consistent with the recent advancements in the Autoregressive Conditional Duration (ACD hereafter) and Autoregressive Conditional Multinomial (ACM hereafter) models literature (e.g., Defour and Engel (2000)) that price and trade co-move with random trade inter-arrival time. Specifically, we first let asset price evolution in each calendar-time interval be governed by two independent i.i. d. (independently and identically distributed) Bernoulli trials. The first trial resolves the trade arrival/non-arrival uncertainty. When no trade arrives, the price of the underlying asset remains unchanged at the end of the time interval. When a trade does arrive however, the asset price evolves in a standard binomial tree with its up/down movement determined by the second Bernoulli trial. Because of this trade inter-arrival