*4.1. Pricing Option*

The section prices European call option on the asset *X*1 based on our 4/2 generalized factor model. It explores the implied volatility surface in a three-dimensional plot with strike prices as the x-axis, time to maturity as the y-axis, and corresponding implied volatility as z-axis. We take the strike prices *K* to be 15, 16.4, 17.8, 19.2, 20.6, and 22 and the expiry dates *T* are 0.2, 0.36, 0.52, 0.68, 0.84 and 1.0. By choosing these strike prices, we account for the in-the-money, at-the-money, and out-of-the-money options, given the initial asset price 18. Subsequently, for each strike price and expiry date, we can obtain a simulated call price as follows

$$\mathcal{c}(T,\mathcal{K}) = e^{-rT} \mathbb{E}^Q[(X\_1(T) - \mathcal{K})^+]\_\star$$

where *<sup>X</sup>*1(*T*) is approximated using the Euler method.

We extract the implied volatility by matching the Black–Scholes option price formula with simulated call prices and solve for the volatility parameter. Hence, we can treat the dynamics of *Y*(*t*) as an O-U process such that:

$$d\mathcal{Y}(t) = (L\_1 - 0.5\sigma^2 - \beta \mathcal{Y}(t))dt + \sigma d\mathcal{W}^\*(t).$$

Next, we consider the two cases described above. The first one studies the impact of *b*, which represents the size of the 3/2 component on the covariance; and the second examines the impact of *a*, the size of the commonality.

In the case of *b*, we first extract the implied volatility surface by matching the standard BS formula for changes on *b* and ˜ *b* respectively (see Figures 1 and 2).

(**a**) *b*1=0, *b*1 between (0, 0.008)

(**b**) ˜ *b*1=0.008, *b*1 between (0, 0.008)

**Figure 1.** Impact of *b*1 (common factor, 3/2 component) on implied volatility, Scenario **A**.

(**a**) ˜ *b*1=0, *b*1 between (0, 0.008)

˜

(**b**) ˜ *b*1=0.008, *b*1 between (0, 0.008)

For Scenario **A**, Figure 1a,b illustrates that even small changes in (*b*1) the common factor 3/2 component (from 0 to 0.008) can lead to a 7% difference in implied volatility (from 0.275 to 0.295, or 0.285 to 0.305). The joint effect of the common and intrinsic 3/2 components (*b*1 and ˜ *b*1) can be obtained by combining those two figures leading to a 11% change (from 0.275 to 0.305) in the presence of relatively small values of *b*.

For Scenario **B**, we observe that the impact of intrinsic factor on volatility surface is more significant than in Scenario **A** through a comparison of Figure 2a,b. The effect of *b*1 on implied volatility increase by approximately 31% (0.145 to 0.19), as shown in Figure 2a, when only the common factor is present. In Figure 2b, we observe a volatility "smile" with the difference of approximately 12.2% (0.245 to 0.275). The joint effect of the common and intrinsic 3/2 components in this case is 100% (0.145 to 0.29).

Figures 1 and 2 jointly demonstrate that, given different underlying process for common and intrinsic factors, the impact of the 3/2 component can be crucial.

Next, we study *a*, the weight of the common factor (commonality). We again extract the implied volatility surface from matching the standard BS formula for changes on *a*.

Figure 3a,b displays the significant increase in implied volatility due to the commonality of the asset with the market (*a*1). The change in implied volatility can increase up to 12.5% (from 0.28 to 0.315) in Scenario **A** and up to 30% (from 0.22 to 0.32) in Scenario **B**.

(**a**) *a*1 between (0, 1). Scenario **A**

(**b**) *a*1 between (0, 1). Scenario **B**

**Figure 3.** Impact of commonality (*a*1) on implied volatility.
