Sample work

PARAMETERS

SIMULATING THE RANDOM VARIABLES

$$ dWt_{i,j} \sim \mathcal{N}(0,dt)$$

THE VOLATILITY FUNCTION

The implied volatility function is

$$ \sigma_{IV}(K,\tau,S) = Be^{\gamma \tau} +\frac{\alpha}{\alpha(K + S) + 1} $$

The parameter $\gamma$ controls the term structure and can be finetuned.

As is evident, this function produced 2 kinds of volatility, pre-specified or implied. This function helps in quickly specifiying the sea of greeks you will see below

OPTION PRICES SET UP

The arguments I have used are

BSM GREEKS

The arguments for the Greeks specificed above are as discussed before. At first glance, you may find it unintuitive to include a model specifiying variable when the function name clearly indicates the greeks are under BSM. However, I retain this variable as the IV-adjusted greeks require evaluating BSM Greeks at $\sigma_{IV}$.

$\text{1}^{st}$ AND $\text{2}^{nd}$ DERIVATIVES OF THE VOLATILITY SURFACE, WITH RESPECT TO K (OR EQUIVALENTLY S)

IV ADJUSTED GREEKS

NOTES

STRIKE SELECTORS

Since the fund resets options at strikes in reference to the stock price at the start of the quarter, the followinh functions help us reset the option strikes when the position is rolled over

GREEKS FOR DEALER PORTFOLIOS

SETTING UP THE PRICE GENERATING PROCESS

Our process is

$$ dS_t = \psi_tM^{\Delta}_tdt + \sigma S_t^{1-\zeta}dW_t$$

if we discretize this, we have

$$ S_{t+1}-S_t = \psi S_t M_t^{\Delta} + \sigma S_t^{1-\zeta}(W_t - W_{t-1}) $$

where

$$M_t^{\Delta} = \Sigma_{u=1}^t e^{-\varphi (t-u)}(\bar\Delta_{u}-\bar\Delta_{u-1})$$

and $\bar\Delta_{u}-\bar\Delta_{u-1}$ is $\Delta_{u}^d-\Delta_{u-1}^d$, ie., the change in market makers $\Delta$ for all times but the reset dates. On the reset dates, it also includes JHEQX's change in portfolio size ($\theta$).

NOTES

GENERATING THE SAMPLE PATHS

ANALYSIS AND VISUALISATION

Let's create a multi-indexed dataframe containing some upward, flat and downward paths. We will use it as a database for the next part of the code.

Let's create two strategies to exploit the above graphs.

STRATEGY 1 - MOTIVATION

This is a stock strategy based on the fact the the drift of the process $d\bar\Delta$ is subject to a discrete jump, namely $\theta_{\frac{j}{4}} - \theta_{\frac{j-1}{4}} $. This is a non trivial amount given the size of JHEQX. The collar/spread derivative strategy they use(and the structure of the question) means that in quarter $j$, the fund's value is capped at $\theta_{\frac{j}{4}} 1.05S_{\frac{j}{4}}$. As such, if this barrier is breached, the fund effectively sustains losses as the short call will be exerised. Combined with high stock prices and sef-financingness, this means when the roll over their portfolio in quarter $j+1$:

$$\theta_{\frac{j}{4}} > \theta_{\frac{j+1}{4}}$$

$$\implies \theta_{\frac{j+1}{4}} - \theta_{\frac{j}{4}} < 0 $$

So, we expect a drop in share price.

Further, as the market makers hold an ITM call, their delta as we approach quarter $j$ expiry is

$$\Delta_{MM,\frac{j}{4}} \ = \theta_{\frac{j}{4}}$$

Now since we know $$(\Delta_{p,\frac{j+1}{4}}^{0.8} - \Delta_{p,\frac{j+1}{4}}^{0.95} + \Delta_{c,\frac{j+1}{4}}^{1.05} )< 1$$

$$\Delta_{MM,\frac{j+1}{4}} \ = \theta_{\frac{j+1}{4}}(\Delta_{p,\frac{j+1}{4}}^{0.8} - \Delta_{p,\frac{j+1}{4}}^{0.95} + \Delta_{c,\frac{j+1}{4}}^{1.05} ) < \theta_{\frac{j+1}{4}} $$

So,

$$\Delta_{MM,\frac{j+1}{4}} - \Delta_{MM,\frac{j}{4}}<0$$

This further bolsters our confidence in the conclusion that the stock price will fall

STRATEGY 1 - OUTLINE

For each path:

If we want to modify this strategy to options (you can try to compare results), we do the following: