Published01 Apr Abstract Proactive hedging European option is an exotic option for hedgers in the options market proposed recently by Wang et al.
It extends the classical European option by requiring option holders to continuously trade in underlying assets according to a predesigned trading strategy, to proactively hedge part of the potential risk from underlying asset price changes. To generalize dynamics of options option design for practical application, in this study, a proactive hedging option with discrete trading strategy is developed and its pricing formula is deducted assuming the underlying asset price follows Geometric Fractional Brownian Motion.
Simulation studies show that proactive hedging option with discrete trading strategy still enjoys strong price advantage compared to the classical European option for majority of parameter space.
The observed price advantage is stronger when the underlying asset has more volatility or when the asset price follows closer to Geometric Brownian Motion. Additionally, we found that a higher frequency trading strategy has stronger price advantage if there is no trading cost.
The findings in this research strongly facilitate the practical application of the proactive hedging option, making this lower-cost trading tool more feasible. Introduction Exotic options, such as Asian, lookback, barrier, and passport options, have been a key focus of mathematical finance research since the late s and early s [ 1 — 9 ]. In this paper, we focus on an exotic option that is a proactive hedging strategy bundled into the classical European option, called proactive hedging European option.
This exotic option has a built-in condition that requires option holders to trade the underlying asset and linearly adjust the holding position according to its price fluctuation within the option period. The potential loss of the underlying asset covered by such proactive actions is no longer the responsibility of the option writer.
Therefore, compared to classical European options, this proactive hedging European option can significantly reduce the risk taken by the option writer, thus making it a theoretically less expensive option.
This type of exotic option is particularly suitable to hedgers who seek to cover their risk of exposure at a minimum cost. Although very promising in theory, the continuous linear position strategy makes this option not very practical for trading purposes.
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In this paper, we try to improve the feasibility of this option to make it more adaptable to a real market scenario by making the continuous linear position strategy discrete. We then deduce the theoretical pricing formula for guiding trading practice in the market. Proactive hedging European option with continuous linear position was first introduced by Wang et al. With the addition of a mandatory condition to the classical European option, option holders need to buy in sell out the underlying asset when its price goes up goes down.
Specifically, for a call option in which the prediction is that the future value of the underlying stock will increase, the option dynamics of options holds a certain amount of capital at the beginning of the option period. When the price of the underlying stock goes up tothe option holder spends to buy in the stock.
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The parameter is called the initial capital utilization coefficient and is a constant between 0 and 1. The option holder linearly and continuously adjusts the capital utilization to increase the holding if the price continues to increase, until the price reaches and the total capital spending reacheswhere is a positive number and referred to as the investment strategy index and is the maximum capital utilization coefficient. Figure 1 describes this process. The expected loss resulting from the asset dynamics of options increasing from towhich is supposed to borne by the option writer, is partly retrieved by the dynamic strategy.
The dynamic hedging option works in a similar fashion for a put option in which the prediction is that the future value of the underlying stock will decrease.
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Figure 1 Capital utilization coefficient function with varying stock price under a linear position strategy. Wang et al. Li et al. They compared the price of this exotic option with that of the classical European option using simulations and found that this exotic option almost always has a lower price than the classical European option.
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Although this exotic option enjoys a significant price advantage, it currently remains an unrealistic option choice for hedgers since proactive hedging actions must be taken continuously along a linear function. In this paper, we build on the work of Li et al. Even though making the proactive hedging strategy discrete would likely sacrifice some price advantage, simulations indicate that this discrete strategy still enjoys a strong price advantage compared to the classical European option.
This advantage will be stronger when the underlying asset has more uncertainty, or when the dynamic hedging strategy is more frequent.
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The rest of the paper is organized as follows. In Section 2we describe this exotic option with discrete proactive hedging actions in detail and derive its value function.
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Section 3 gives the theoretical pricing formula binary options strategies on live chart the GFBM assumption and the simplified formula for application to some special cases. In Section 4we use simulations to evaluate the price premium of this exotic option with discrete proactive hedging actions compared to the exotic option with continuous linear proactive hedging actions and the classical European option.
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Since the pricing formula derivations are very similar for call and put options, in this paper, we only present results for call options.
Constraints for the Proactive Hedging European Option The proactive hedging European option is proposed based on the following assumptions: 1 A call option holder holds one piece of contract and an initial capital of amount of at dynamics of options beginning of the option period, where is the number of stock units for the piece of option contract and is the exercising price according to the contract.
Dynamic Discrete Position Strategy The option holder holds a certain amount of capital at the beginning of the option period.
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The dynamic discrete position strategy will be activated when the underlying asset price reaches. The option holder will only buy in when the asset price hits a series of equally spaced pointswhere is a positive constant representing the price distance for two consecutive trading actions and is the total number of trades of the stock in the entire option period. Similar to previous studies, is the maximum capital utilization coefficient, so the maximum amount of capital tradable or available is.
The strategy also assumes the option holder will evenly distribute the capital over the trades, such that each buy-in trade will spend a capital of for pieces of stock.
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Please refer to Figure 2 for an illustration of the discrete position strategy. Figure 2 Graphical illustration of the discrete linear proactive hedging strategy.
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The Value Function for Proactive Hedging European Option with Dynamic Discrete Position Strategy For the classical European option, the option holder will suffer an expected loss as for each piece of the option contract as the stock price rises from tofor.
In an exotic option with proactive hedging strategy, the option holder is required to actively buy in the underlying stock to hedge dynamics of options risk from the fluctuations of the underlying asset. Assume the option holder trades with the discrete position strategy described in Section 2. Then, when the stock price increases from tothe option holder will make a return of by holding the dynamics of options.
Since the discrete position requires the option holder to buy in the underlying stock at every stock price ofeach with a capital ofwhen the stock price reaches withthe option holder will make a cumulative return ofthe expected loss taken by the option writer,should be the expected total loss in 1 minus the cumulative return in 3that is.