This paper develops a new option pricing formula that pushes the underlying source of risk back to the risk of individual assets of the firm. The formula simultaneously encompasses differential riskiness of the assets of the firm, their relative weights in determining the value of the firm, the effects of firm debt, and the effects of a dividend policy with both constant and random components. Although this setting considerably generalizes the Black‐Scholes  analysis, it nonetheless produces a formula via riskless arbitrage arguments that, given estimated inputs, is as easy to use as the Black‐Scholes formula.
"Other works improved the log-normal approximation allowing for improved skewness and kurtosis calibration. The displaced diffusion introduced by Rubinstein (1981) considers the shifted basket value as being log-normally distributed. Borovkova et al. (2007), henceforth BPW, proposed a generalized log-normal approach that is superior to the model in Rubinstein (1981) because it allows distributions of a basket to cover negative values and negative skewness . "
[Show abstract][Hide abstract] ABSTRACT: Theoretical models applied to option pricing should take into account the
empirical characteristics of the underlying financial time series. In this
paper, we show how to price basket options when assets follow a shifted
log-normal process with jumps capable of accommodating negative skewness. Our
technique is based on the Hermite polynomial expansion that can match exactly
the first m moments of the model implied-probability distribution. This method
is shown to provide superior results for basket options not only with respect
to pricing but also for hedging.
"An example is the CEV process proposed by Cox  and Cox and Ross . A different example is the displace diffusion model by Rubinstein . A general class of problems was presented by Carr, Tari and Zariphopoulou . "
[Show abstract][Hide abstract] ABSTRACT: We introduce a multivariate diffusion model that is able to price derivative
securities featuring multiple underlying assets. Each underlying shows a
volatility smile and is modeled according to a density-mixture dynamical model
while the same property holds for the multivariate process of all assets, whose
density is a mixture of multivariate basic densities. This allows to reconcile
single name and index/basket volatility smiles in a consistent framework.
Rather than simply correlating one-dimensional local volatility models for each
asset, our approach could be dubbed a multidimensional local volatility
approach with state dependent diffusion matrix. The model is quite tractable,
leading to a complete market and not requiring Fourier techniques, contrary to
multivariate stochastic volatility models such as Wishart. We provide a
semi-analytic formula for the price of European options on a basket/index of
securities. A comparison with the standard approach consisting in using Monte
Carlo simulation that samples simply-correlated suitably discretized
one-dimensional paths is made. Our results show that our approach is promising
in terms of basket option pricing. We also introduce a multivariate uncertain
volatility model of which our multivariate local volatilities model is a
multivariate markovian projection and analyze the dependence structure induced
by our multivariate dynamics in detail. A few numerical examples on simple
contracts conclude the paper.
"Jäckel (2010a) prices quanto derivatives using the displaced diffusion model introduced by Rubinstein (1983). Unfortunately, this model allows for the underlying asset to attain negative values and this circumstance can affect the pricing results. "
[Show abstract][Hide abstract] ABSTRACT: European quanto derivatives are usually priced using the well-known quanto adjustment corresponding to the forward of the quantoed asset under the assumptions of the Black–Scholes model. In this article, I present the quanto adjustment corresponding to the local volatility model that allows pricing quanto derivatives consistently with the observed market equity skew and exchange rate smile. I then examine the model risk arising in the standard quanto adjustment by fitting the local volatility model to market data and then comparing the prices of European quanto euro derivatives on the Nikkei 225 index with those generated by the standard quanto adjustment. The results show that the standard quanto adjustment can be subject to significant pricing errors when compared with the local volatility model.
I also compare the pricing performance of the local volatility model with a multivariate stochastic volatility model. The results show that when the correlation between the instantaneous variances associated with the underlying asset and the exchange rate is close to one, as it is the case when we consider historical data, there is little evidence of model risk for the local volatility model in the pricing of European quanto euro derivatives on the Nikkei 225 index.
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