JOHNSON, H. and D. SHANNO, "Option Pricing when the Variance is Changing," Journal of Financial and Quantitative Analysis, Vol. 22, 1987, pp. 143-151.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for both the asset returns and interest rate dynamics. This model nests the Amin Ng (1993) model as a special case when # 2 =0. Following are three alternative model specifications: # Submodel 1: No stochastic interest rates, i.e. interest rate is constant, r t = r,asintheHull White (1987) Johnson Shanno (1987) and Wiggins (1987) models; # Submodel 2: Constant asset return volatility but stochastic interest rate, # s;t = #,asinthe Merton (1973) Turnbull Milne (1991) and Amin Jarrow (1992) models; # Submodel 3: Constant asset return volatility and constant interest rate, # s;t = #;r t = r,asin ....

....payoffs under a model that incorporates the 8 In this case the individual t values are all about the same. This is a consequence of the fact that the individual t values are asymptotically equal with probability one in case of only one degree of freedom in the test. 9 Hull White (1987) Johnson Shanno (1987), Bailey Stulz (1989) Stein Stein (1991) and Heston (1993) 10 See e.g. Scott (1987) Wiggins (1987) Chesney Scott (1989) Melino Turnbull (1990) and Bakshi et al. 1997) 15 appropriate compensation for systematic asset, volatility, interest rate, or jump risks. Similarly, the ....

Johnson, H. & Shanno, D. (1987), `Option pricing when the variance is changing', Journal of Financial and Quantitative Analysis 22, 143--152.

.... suggested for option pricing bias such as the inconstant volatility of the underlying stock, the observed changes in interest rates or the difference between the observed and the assumed stock price path (like the jump processes) Merton [1976] Hull and White [1987] Scott [1987] Wiggins [1987] Johnson and Shanno [1987], Stein and Stein [1991] Bates [1996] Bakshi and Chen [1997] Cox [1996] Corrado and Su [1996] Madan, Carr and Chan [1998] are examples of models where the basic Black Scholes assumptions are dropped. Some other authors impute the smile to the behaviour of traders or to their risk aversion, ....

JOHNSON H. and D. SHANNO (1987), "Option Pricing When Variance is Changing", Journal of Financial and Quantitative Analysis, 22 (2), June, pp. 143-151.

....proposed by Hull and White (1988) and Heston (1993) In this model the volatility is related to the square root process of Cox, Ingersoll and Ross (1985) and oe can be interpreted as the radial distance from the origin of a multidimensional OU process. Two other models of note were proposed by Johnson and Shanno (1987) who modelled both the price and volatility as CEV processes, and Melino and Turnbull (1990) who took the price to be a CEV process and the logarithm of the volatility to be an OU process. 2.4 Transition densities Consider the model dP t P t = oe t dB t dt (13) doe t = fl(oe t )dW t ....

....analysis. However there is strong empirical evidence that ae is non zero. A negative value of ae provides one method of capturing the observed negative correlation between volatility and price. Hence it is worthwhile to pursue the general case and to resort to numerical methods if necessary (see Johnson and Shanno (1987) and Wiggins (1987) Heston (1993) has devised an efficient method for calculating options using characteristic functions. It is possible to recover the level dependent volatility models of, for example, Cox and Ross and Geske by taking jaej = 1 and choosing an appropriate, though potentially ....

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JOHNSON, H. and SHANNO, D. (1987): Option pricing when the variance is changing. Journal of Financial and Quantitative Analysis, 22, 143-151.

.... shows that the real volatility is time varying, random and correlated with the stock prices (see e.g. Black and Scholes (1972) Derman et al. 1996) Mayhew (1995) For modified models, a number of deterministic and stochastic equations for volatility was proposed (see e.g. Christie (1982) Johnson Shanno (1987), Hauser Lauterbach (1997) Hull White (1987) Masi et al. 1994) For these modifications, the corresponding Black and Scholes and Merton hedging strategies require information about the future values of volatility. In another approach, a special temporal scale is used to find times when the ....

Johnson, H., and Shanno, D. (1987): Option pricing when the variance is changing. Journal of Financial and Quantitative Analysis, 22, 143-151.

....Bensoussan et al. (1994) describes the stock price as a diffusion with level dependent volatility. This may either be a modelling assumption or follow from more fundamental properties, for example by relating stock price to the value and the debt of a firm. The second approach, exemplified by Johnson and Shanno (1987), Scott (1987) Hull and White (1987, 1988) and Wiggins (1987) defines the volatility as an autonomous diffusion driven by a second Brownian motion. The asset price process is driven by the first Brownian motion. Further details of these two approaches, which we label as level dependent ....

....differential equation for an option price which they solve numerically. 2.2 Stochastic Volatility via an SDE Several authors have proposed models of volatility in which the volatility is defined via a stochastic equation. The following model is due to Hull and White (1987) For related papers see Johnson and Shanno (1987), Scott (1987) Wiggins (1987) and Hull and White (1988) Hofmann et al. (1993) consider a more general Markovian model which contains the model of Hull and White as a special case. Let the stock price P t and the volatility oe t be defined via the pair of stochastic differential equations dP t = ....

JOHNSON, H. and SHANNO, D. (1987): Option pricing when the variance is changing. Journal of Financial and Quantitative Analysis, 22, 143-151.

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JOHNSON, H. and D. SHANNO, "Option Pricing when the Variance is Changing," Journal of Financial and Quantitative Analysis, Vol. 22, 1987, pp. 143-151.

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Johnson, H. and D. Shanno (1987). Option pricing when the variance is changing. Journal of Financial and Quantitative Analysis 22, 143--152. 22

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Johnson, H. and D. Shanno (1987). Option pricing when the variance is changing. Journal of Financial and Quantitative Analysis 22, 143--152.

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Johnson, H. and D. Shanno, 1987, Option Pricing when the Variance is Changing, Journal of Financial and Quantitative Analysis 22, 143-151. 14

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Johnson H. and D. Shanno (1987), Option pricing when the variance is changing, Journal of Financial and Quantitative Analysis, 22, 143-151.

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