The conditions mentioned in II and III are necessary for an asset price to have a lognormal distribution. The lognormal distribution assumes that the rate of return on the asset is normally distributed, which implies that the asset price changes smoothly with no jumps. This is because the lognormal distribution is a continuous distribution, and any jumps in the asset price would disrupt this continuity. Furthermore, the lognormal distribution assumes that the volatility of the asset is constant. This is because the standard deviation (which is a measure of volatility) of the normal distribution (which the lognormal distribution is based on) is a constant. Therefore, any variation in the volatility of the asset would violate this assumption and prevent the asset price from having a lognormal distribution.

Choice A is incorrect. While condition I may be necessary for an asset price to exhibit a lognormal distribution, condition II is not. The lognormal distribution assumes that returns are continuously compounded, which implies that the return on an asset can take any value from negative infinity to positive infinity. However, this does not necessarily mean that the asset price itself must also take any value from negative infinity to positive infinity.

Choice B is incorrect. Although condition I might be necessary for an asset price to follow a lognormal distribution, condition III is not necessarily required. The assumption of constant volatility in financial markets is often unrealistic as volatility tends to change over time due to various factors such as changes in market sentiment or economic conditions.

Choice D is incorrect. As explained above, all three conditions are not necessary for an asset price to exhibit a lognormal distribution.

Things to Remember

• Lognormal distribution is commonly used in finance to model the distribution of asset prices, as it allows for a wide range of possible values and is skewed to the right.
• Continuous compounding is a key assumption in the lognormal distribution, where returns are assumed to be continuously compounded over time.
• Volatility plays a crucial role in determining the shape of the lognormal distribution curve, with higher volatility leading to a wider distribution.
• Lognormal distribution is often used in option pricing models, such as the Black-Scholes model, to estimate the probability distribution of future asset prices.