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Reading this paper, I'm struggling to understand what the author is saying with paragraphs below (see pages 39-42):

We define Implied Gamma ($\Gamma_{\operatorname{implied}}$) as the value of the parameter $\Gamma$ that is to be used as input to our model so that the price calculated from the model matches the market price, or price from simulation if market price is not obtainable. Implied volatility is defined as the volatility to be substituted into the Black-Scholes formula to get the market price. The observation of implied volatility smile shows that for different strike prices, the volatility to be input to the pricing model is different. However, volatility is a characteristic of the asset returns that should not be affected by the strike price. In our approach, $\Gamma_{\operatorname{implied}}$ is a measure of investor's risk appetite. When the strike price is far away from the current stock price, the investor would be more risk averse, thus he wants to use a higher $\Gamma$ and wants the price of the option to reflect this fact.

$\Gamma$ vs $\frac{K}{S_0}$- quadratic dependence

We propose and verify empirically that a quadratic variation of $\Gamma_{\operatorname{implied}}$ with $\frac{K}{S_0}$ would be adequate to characterize the risk aversion of investor towards different strike prices. We use the following function to describe the relationship: $$\Gamma_{\operatorname{implied}}=a_0+a_1\frac{K}{S_0}+a_2(\frac{K}{S_0})^2,a_2\geq0$$

$\frac{K}{S_0}$ captures the distance between the strike and the spot price. We use a quadratic regression model to compute the coefficients $a_0,a_1,a_2$ so that the prices given using these $\Gamma$'s match the price from the model we want to compare. We can then use this quadratic model $\Gamma(\frac{K}{S_0})$ to calculate $\Gamma$ and input it to yield the price for options with other strike prices.

Sampling

After we obtain the various strike prices $K$ which are observed in the market, we divide them into 2 groups: in-sample and out-of-sample. If it is in-sample, we find $\Gamma_{\operatorname{implied}}$ that matches our model price to the market price for this specific $K$. We then input these $K$'s and $\Gamma_{\operatorname{implied}}$'s to a regression model to find the coefficients of equation above. After that we use $\Gamma(\frac{K}{S_0})$ to find the prices for out-of-sample options.

After all this, author says that using $\Gamma_{\operatorname{implied}}$ in the problem of page 17 the model price is closest to the market price...

This is what I know:

  • Implied volatility is the only value for volatility that satisfies $c_0^{\operatorname{BS}}(\sigma)=c_0^*$, where $c_0^{\operatorname{BS}}(\sigma)$ is the theoretical price of an European Call option evaluated with Black-Scholes and $c_0^*$ is the market price of the same option evaluated through rate-spreads (i.e. the mean between bid price and ask price). It can be calculated using $$c_t(T,K)=e^{-r(T-t)}\int_{0}^{+\infty}(S_T-K)^+\mathbb{Q}(S_T\equiv S)\operatorname{d}s$$ so if integrand function is regularly enough $$\frac{\partial^2c}{\partial K^2}=-e^{-r(T-t)}\mathbb{Q}(S_T\equiv S)$$ that represents $\operatorname{IV}$ (i.e. the probability that option can be exercised by the expiry date). In practice: $$\widetilde{\sigma}\doteq \operatorname{IV}(K,T,S_0)=\mathbb{E}^{\mathbb{Q}}[(\int_t^T \frac{\operatorname{d}S_u}{S_u})^2]=$$ $$2\int_0^{+\infty} \frac{c_0(T,Ke^{r(T-t)})-c_0(t,Ke^{r(T-t)})}{K^2}\operatorname{d}K$$

  • in B&S volatility is constant, but smiles with moneyness $\frac{K}{S_0}$ on x-axis and $\sigma$ on y-axis show that we don't have a line but a curve initially descending and then increasing. So $\sigma$ is not a constant. (From here Heston, Dupire and all that).

  • $\Gamma \in \mathbb{R}^+$ is a risk-aversion parameter that can be selected by the modeler consistent with his/her preferences.

  • $\widetilde{R}_t^S$, that represents random cumulative returns, is bounded by $\begin{bmatrix} \underline{R}_t^S,\overline{R}_t^S \end{bmatrix}$, i.e. $U=\begin{Bmatrix} \widetilde{R}_t^S|\underline{R}_t^S\leq \widetilde{R}_t^S \leq \overline{R}_t^S \end{Bmatrix}$ with $U \doteq \begin{Bmatrix} \widetilde{R}_t^S:|\frac{\frac{1}{t}\operatorname{log}\widetilde{R}_t^S-\mu_{\operatorname{log}}}{\sigma_{log}\sqrt{t}}|\leq \Gamma \end{Bmatrix},\forall t$ is the uncertainty set of log cumulative returns. It is applied CLT Theorem for the assumptions of normality and independence of returns (second order stationarity of time series). Thus $$U=\begin{Bmatrix} \widetilde{R}_t^S|e^{t\mu_{\operatorname{log}}-\Gamma\sqrt{t}\sigma_{\operatorname{log}}}\leq \widetilde{R}_t^S \leq e^{t\mu_{\operatorname{log}}+\Gamma\sqrt{t}\sigma_{\operatorname{log}}} \end{Bmatrix},\forall t \space \space \space (1)$$ with $\mu_{\operatorname{log}}$ and $\sigma_{\operatorname{log}}$ sample mean and standard deviation, respectively. You can see the notation used at page 37-38.

  • $\widetilde{r}_t^S$, that represents random singe-period returns, is bounded by $\begin{bmatrix} \underline{r}_t^S,\overline{r}_t^S \end{bmatrix}$, i.e. $$U=\begin{Bmatrix} \widetilde{r}_t^S|\underline{r}_t^S\leq \widetilde{r}_t^S \leq \overline{r}_t^S \end{Bmatrix}\doteq \begin{Bmatrix} \widetilde{r}_t^S|\mu-\Gamma_t \sigma\leq \widetilde{r}_t^S \leq \mu+\Gamma_t \sigma \end{Bmatrix},\forall t \space \space \space (2)$$

  • EDIT - Author says that "The volatility smile is a long-observed pattern in which at-the-money options tend to have lower implied volatilities than in- or out-of-the-money options. The traditional way the research and practice community deals with this phenomenon is to assume that the volatility of the underlying security varies with the strike $K$, which is not a sound concept. Our proposal is to assume that the parameter $\Gamma$ to define the uncertainty set $(1)$-$(2)$ depends on $\frac{K}{S_0}$. The intuition here is that we want to select a larger $\Gamma$, that is to make our approach more robust, when $\frac{K}{S_0}$ is away of one (that is in-the-money or out-of-the-money). In other words, the parameter $\Gamma$ attempts to capture the risk-aversion of the modeler to define the uncertainty set."

Could you please explain me what the author's doing? It's really unclear (at least for me). Thanks in advance just for reading.

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  • $\begingroup$ Good question. I don’t know offhand, but am researching. Interesting find. $\endgroup$ Commented Dec 15, 2020 at 7:35
  • $\begingroup$ @MonteCarloSims Thanks for your answer, I appreciate your concern. In short, I am trying to use theory described before to replicate the procedure that you can see in Computational Results - Chapter 7. I don't have a significant knowledge about programming, and if I understood how author had got ECO-LP problem in page 17 I don't know how to use it in practice to obtain Figure 7.1 and 7.2. He says "We solve ECO-LP with different $\Gamma$ that ranges $0$ to with a step size of $0.01$, and find the $\Gamma_{\operatorname{implied}}$ so that the model price is the closest to the market prices"... $\endgroup$
    – user853717
    Commented Dec 15, 2020 at 8:31

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