This has been taken from a real life example but all data has been modified.

Jason Barrick (not his real first name) was trading US equity options in May 2005 and was in fact making the market in a certain auto company's calls and puts. In the second quarter of 2005 the stocks of the US auto companies were experiencing big swings and the volatility was significantly up. Also, there was great turmoil in the credit default swaps (CDS) and the CDO markets during that time due to hedge fund activities.

A customer calls him up one day in that month and asks him for the price of a 3 month 8.50 call on *SpecialMotors *. The *SpecialMotors * stock was trading at $10.00. Jason's bank was using state of the art volatility surface models and all he needed to do was to pluck a value for the local volatility and price the 3 month in the money option. However, he was convinced that there was something missing in the whole model. And he believed that the missing piece was some kind of a measure of jump that the stock was experiencing.

His view was that the unprecedented turmoil in the credit derivatives market was spilling over to the equity market and driving the volatility of the auto stocks up. The 5 year CDS rate on *SpecialMotors * was close to 7% and all hell was breaking loose there. This was a clear indication - and in fact it was very much a part of public knowledge and discussion - that the possibility of a default on major auto companies is very real. Therefore, the probability of default should somehow need to be accounted for in the volatility of the stock.

Here is what Jason did, somewhat to the dismay of his superiors and other fellow traders. He used a Jump-to-Ruin model to derive the local volatility of the stock, *SpecialMotors *. Actually, Merton's Jump to Ruin model is a very elegant and theoretically appealing model which is used by quite a few traders to price options on stocks which have high default probability. The key is the definition of a *hazard rate *or what we call the jump to ruin probability. It simply says that there is some probability, per unit time of a stock price jumping to zero. If this is the case then Dupire's local volatility formula reduces to:

For an excellent theoretical treatment of the above, Jason recommends all the readers to go through Jim Gatheral's NYU Stern's lecture notes or the text, "Volatility Surface" by him. How did Jason proceed to make the above estimation of the local volatility in the presence of a jump:

**Step 1 **

He observed the 5 year credit default swap rate and substituted that for a 3 month spread to calculate the hazard rate. This was quite a bit of a stretch to take the 5 year rate and use it as a 3 month rate, but he was willing to run with it. The 5 year rate was 7% and the recovery rate (market's estimate) on *SpecialMotors *bonds was 0.3. Then using the simple formula he calculated the probability of default for the bonds of *SpecialMotors *.

The probability of default, turned out to be 2.48%. This was an exceptionally high number (please remember that the example is modified from the actual one) for a 3 month probability of default, and placed the company in the lowest spectrum of junk status. But once again he was willing to run with it. His rationale was that this probability of default, or the hazard rate, is the correct jump to ruin probability of the stock. In the event that the company defaults on its bonds the stock will be worthless. So this was his estimate of .

**Step 2 **

The following market data for 3 month option was observed on Jason's screen.

Using the above market data he estimated the implied volatility of all the above strikes. The implied volatilities for all the strikes were actually available on his screen as well. Then using the implied vols and the modified local variance formula given above he calculated the local variance and the volatilities.

**Step 3**

The next step was to assume a quadratic (polynomial) variation between the log strike, and the local volatility. A plot of the volatility surface is shown below. A regression equation is fit to the data in the graph and the R-squared shows a very good fit.

From the fitted regression equation we get:

**Step 4 **

Finally, Jason, took the strike price of the customer, i.e. 8.50 and the current spot price of the stock, i.e. 10 and estimated the local volatility from the above equation. The estimate for the local volatility turned out to be 72%. Thus he priced the 3 month 8.50 call on the stock using a risk free rate of 5% and a volatility measure of 72%.

**Questions : **

- How valid was Jason's reasoning and the approach to incorporating jumps in the local vol model?

- Will the approach to pricing of the puts on the stock - much more relevant for stocks with high vol and default probabilities - be any different that the above?

- What are the major loopholes or flaws in the above method?

- Given a choice would you rather be Jason Barrick or Jason Bourne?

Please send your answers to team@risklatte.com .

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