# Tutor profile: Raul G.

## Questions

### Subject: Applied Mathematics

In a contour map, how can you approximate the values of the corresponding partial derivatives?

The contour map gives you the "heights" of the curve in the z-axis (perpendicular to the x-y plane) as contours, where each contour represents a given value of the z-axis. The partial derivatives are defined as: $$\frac{\partial f}{\partial x}=\lim_{\Delta \to 0}\frac{f(x+\Delta,y)-f(x,y)}{\Delta}$$ $$\frac{\partial f}{\partial y}=\lim_{\Delta \to 0}\frac{f(x,y+\Delta)-f(x,y)}{\Delta}$$ Clearly, we don't have the contour lines for every single point in the x-y plane, but we can approximate using the closest contours to the points where the corresponding partial derivatives are to be computed. So, for example, if we were interested in the point (0,0), one could estimate the partial derivative with respect to x as: $$\frac{\partial f}{\partial x} \approx \frac{f(x+\delta,y)-f(x,y)}{\delta}$$, where $$\delta$$ is a small number estimated from the closest two contours to (0,0) in the contour map. Choosing a small $$\delta$$ is essential for this estimate to be reasonably accurate.

### Subject: Econometrics

The EGARCH(1,1) model (Nelson, 1991) is given by: $$y_t=\mu+\epsilon_t,$$ $$\epsilon_t=\sigma_tz_t,$$ $$log(\sigma_t^2)=\omega+[\alpha z_{t-1}+\gamma(|z_{t-1}|-(\frac{2}\pi)^{\frac{1}{2}})]+\beta log(\sigma_{t-1}^2)$$, where $$z_t$$ is a white noise, and $$\sigma_t^2$$ is the conditional variance of $$\epsilon_t$$ Indicate which properties observed in financial returns not captured by the usual GARCH model are captured by the EGARCH model.

The important thing to notice here is that the EGARCH model allows an extra degree of flexibility in modeling financial returns, as negative shocks will be treated differently than positive shocks. This allows for a more accurate modeling of the empirically observed asymmetry between positive and negative returns.

### Subject: Finance

Suppose ABC stock has a current price of 80.00 and the current 1-year interest rate is 5%. The prices for a number of options on the stock are as follows: a) One year, 70-strike put. Price: 3.06 b) One year, 80-strike put. Price: 6.52 c) One year, 90-strike put. Price: 11.50 Question 1: Is it possible that the prices just given are mistaken and the premium prices are in fact for calls instead of puts? Are the prices given above consistent with call price behavior? Question 2: At what strike level will the price of a short put equal the cost of a long call?

Question 1: By put-call parity, $$C+\frac{K}{1+0.05}=P+S$$, where C is the call price, the following term is the discounted strike, P is the Put price, and S is the current stock price. Substituting the corresponding values for the problem you'll see that you would get call prices that seem reasonable, as their values decrease as the strike increases. Question 2: Using the put-call parity relationship specified above, you can set C=P and solve for the strike. I hope this helps. If you have any further questions, please let me know.

## Contact tutor

needs and Raul will reply soon.