Hedging The Risk In The Continuous Time Option Pricing Model With Stochastic Stock Volatility

In this work, I address the issue of forming riskless hedge in the continuous time option pricing model with stochastic stock volatility. I show that it is essential to verify whether the replicating portfolio is self-financing, in order for the theory to be self-consistent. The replicating methods in existing finance literature are shown to violate the self-financing constraint when the underlying asset has stochastic volatility. Correct self-financing hedge is formed in this article. PACS number: Typeset using REVTEX 1 It has been indicated by empirical observations that stock price volatility is stochastic. Considerable amount of analytical and numerical work has been devoted to pricing derivatives when the underlying stock has stochastic volatility [1–12]. For example, there were early works done by Merton [1], and that by Garman etc [2]. In current existing finance literature, various authors constructed riskless hedging portfolios in different ways [3,10]. With the principle of no-arbitrage, the riskless hedging portfolio should match the return of a riskless loan. This will result in the partial differential equation satisfied by the option prices. In the situation where the underlying asset has stochastic volatility, investors’ preferences, such as the risk preminum on the stock, will get involved explicitly. In this work, I would like to address the issue of hedging the risk for the above system. In the continuous time models, it is necessary to make sure that the replicating portfolio is self-financing as it was assumed to be. This verification is essential for the theory to be selfconsistent. In the following, I show that in spite of the final correctness of the derived PDE for option prices, the hedging strategies in some current finance literature [3,10] turned out to violate the self-financing condition that was assumed to hold. Arguments and corrections are given in this work when the underlying asset has stochastic volatility. Below, we follow the reference [3] and its notations. Let us first define a probability space (Ω, Q, F ). Consider the stock price obeying the stochastic process dP = αPdt+ σPdZ1, (0.1) where the volatility σ is described by another mean-reverting process dσ = β(σ̄ − σ)dt+ γdZ2. (0.2) Here, both Z1 and Z2 are one dimensional Brownian motions. The co-quadratic process of Z1 and Z2 is assumed to be [Z1, Z2] = tδ. For the process σ described above, there is non-zero chance for the volatility to be negative. However, the following argument of ours will remain unchanged when other positive processes for the volatility are used, such as for the process dlnσ = β( ̄ lnσ − lnσ)dt+ γdZ2. 2 To hedge away the risk, a portfolio was constructed to have two call options and one stock [3]. The two options have different maturity dates T1 and T2. Denote τ1 = T1 − t and τ2 = T2 − t. The option price was assumed to be H(P, σ, τ), a function of stock price P , the stock volatility σ, and τ = T − t. In the following, we may use short notations H(τ1) = H(P, σ, τ1), H(τ2) = H(P, σ, τ2). According to the reference [3], a trading strategy was proposed to be φ = (φ1, φ2, φ3) = (1, ω2, ω3) (0.3)