The proposed model modifies option pricing formulas for the basic case of log-normal probability distribution providing correspondence to formulated criteria of efficiency and completeness. The model is self-calibrating by historic volatility data; it maintains the constant expected value at maturity of the hedged instantaneously self-financing portfolio. The payoff variance dependent on random stock price at maturity obtained under an equivalent martingale measure is taken as a condition for introduced "mirror-time" derivative diffusion discount process. Introduced ksi-return distribution, correspondent to the found general solution of backward drift-diffusion equation and normalized by theoretical diffusion coefficient, does not contain so-called "long tails" and unbiased for considered 2004-2007 S&P 100 index data. The model theoretically yields skews correspondent to practical term structure for interest rate derivatives. The method allows increasing the number of asset price probability distribution parameters.
Deep Dive into Mirror-time diffusion discount model of options pricing.
The proposed model modifies option pricing formulas for the basic case of log-normal probability distribution providing correspondence to formulated criteria of efficiency and completeness. The model is self-calibrating by historic volatility data; it maintains the constant expected value at maturity of the hedged instantaneously self-financing portfolio. The payoff variance dependent on random stock price at maturity obtained under an equivalent martingale measure is taken as a condition for introduced “mirror-time” derivative diffusion discount process. Introduced ksi-return distribution, correspondent to the found general solution of backward drift-diffusion equation and normalized by theoretical diffusion coefficient, does not contain so-called “long tails” and unbiased for considered 2004-2007 S&P 100 index data. The model theoretically yields skews correspondent to practical term structure for interest rate derivatives. The method allows increasing the number of asset price proba
In the standard Black-Scholes-Merton option pricing model [1,2], the delta-hedged portfolio growth determines the diffusive partial differential equation in the underlying price-time coordinates (BSM PDE). The famous BSM formula can be derived as the PDE particular solution with a terminal condition represented by a payoff function, or as a discounted expected derivative value at maturity obtained under martingale measure Q, equivalent, according to Girsanov [3], to the real-world log-normal probability measure P (see also Wilmott e.a. [4]). The BSM PDE asserts upholding constant hedged portfolio value for given payoff function, at that the condition at maturity is denoted by a real variable. Using the real variable in condition expression is inconsistent with a formulation of underlying stochastic process with a random variable at arbitrary time. On the other hand, employing random-variable condition at maturity would contradict to requirement of upholding the constant portfolio value at arbitrary time. The widely-used exponential discount from expected payoff implicitly assumes that a derivative value forwarded to maturity is a martingale under both measures P and Q, which is true for an underlying security (or forward) but not for an option with asymmetric payoff function.
As can be shown, discounting the derivative value back to the current time under the realworld measure P rather than under appropriate log-normal-world martingale measure Q causes the bias between implied and historical volatilities; that undermines the model efficiency. In our opinion, for corresponding the efficient market hypothesis, the BSM model requires modification applying Girsanov’s equivalent martingale measure to derivation of both expected derivative value at maturity and its value discounted to the current time.
The delta-hedging is practically feasible for local volatility models of Dupire [5],
Rubinstein [6], describing the diffusion process with variable coefficients, which can be recovered from conditional probability density (“volatility smile” level, slope, and curvature). However, theoretically, according to Ait-Sahalia e.a. [7], the differences between the stock and option implied risk-neutral densities within the framework of BSM diffusion with exponential discount ultimately would lead to the pricing inefficiency.
The determination of diffusion coefficients is complicated by well-known fact that assumed log-normality is violated in stock return distribution time-series. Besides an empirical phenomenon called “volatility smile” in option markets, the leptokurtic feature takes place. The return distribution of assets may have a higher peak and asymmetric tails, heavier than those of the normal distribution; this led many authors to consider jump-diffusion models with Levy flights (first proposed by Merton [8]). For example, Kou [9] assumes a double-exponential conditional distribution for the jump size; such many-parameter model is sufficient for description of the volatility smile parameters. The models with increased number of parameters price options across strikes and maturities more accurately; however, the issue of parameter stability arises. The mentioned long tails can be eliminated by using normalized distributions introduced below.
Analogously to equity derivatives, the fixed income options are priced by Black [10] formula as an exponentially discounted to the current time expected payoff value at maturity for the case of log-normally distributed forward price. Within the framework of Heath-Jarrow-Morton [11] term structure of interest rates expressed as functions of their volatilities, the bond and its derivative prices at arbitrary time are determined by exponential discount with integral-average rate for the period to maturity. Brace-Gatarek-Musiela [12] Libor forward rate structure model describes the dynamics of a family of forward rates under a common measure. But, unlike stocks, the interest rate futures are derivatives. The existing differences between implied volatilities of two derivative types -the interest rate futures and options, according to de Jong e.a. [13], should theoretically lead to the possibility of arbitrage. According to Gupta and Subrahmanyam [14], for improving a pricing accuracy of interest rate options, there is a need for introducing a second stochastic factor, mean-reversion coefficient determining the term structure evolution through time. For consistent pricing and hedging, a further increasing of a number of parameters is suggested, however, at expense of model stability and extensive computation resources.
The general stochastic volatility models of Heston [15], Hull and White [16] introduce an additional stochastic process for underlying security’s volatility, governed by its price level. It allows introducing necessary corrections to exponentially discounted expected price value, which is dependent on volatility. However, improving the pricing accuracy is achieve
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