**Case Study: Volatility Trading**

(a) The understanding in this case study deal with Convertible as well as Reverse-Convertible bonds. These are interesting instruments by themselves however this Case Study begins with some volatility models since these types of structured equity products are essentially instruments of volatility.

Let's the volatility at time *t *of a stock be signifies by *σ*_{t }and let this be a random process. Then the denote reverting model for this stochastic process will be given by

*dσ*_{t}* *= *λ*(*μ **- **σ**t*)*d*_{t} + *γσ*_{t}*dW*_{t}

At this point the *λ, μ, γ *are a variety of parameters of this model. The *γ *is the volatility of volatility and the *μ *is average long-run volatility and the *λ *is the rate of convergence of the spot volatility towards this long-run average.

The first reading deals with an instance of how this type of model can be used in position taking.

(b) These are conversed in the text. The putrefaction of a convertible bond is shown in diagram.

(c) Here is a recent instance of a convertible bond as mentioned in the IFR November 2004.

*Thus far another Asian convertible bond (CB) was at the centre of controversy last week as a US$125m CB for CMC Magnetics began life amid a haze of revised terms and a convoluted post-launch repricing.*

*The buy deal was launched by lead manager JP Morgan late in the afternoon of October 29 as a fixed-priced *zero-yield *deal at a *conversion premium *15% above the stock's close that day. At the end of the evening the deal had priced with a 0.8% yield a 20% conversion premium and an October 28 reference price.*

*Subsequent to launching on the initial terms JP Morgan realised that the conversion premium fell outside terms filed with the Securities and Futures Bureau (SFB) which had promised a 20%-50% premium.*

*The bank was therefore obliged to return to investors with revised terms, lifting the conversion premium to 20% while compensating investors with the 0.8% yield to put/maturity. The new deal as well used a lower reference share price. The original term sheet utilized the closing price on October 29 (NT$14.9) the final one referenced to the close on October 28 (NT$14).*

The conversion premium is the sum either in percentage terms or in dollar amount by which the conversion price of a convertible security exceeds the current market value of the underlying common stock. If the bond is converted after that the issuer needs to issue new stocks. This will have a dilution result on the existing positions.

(d) A convertible bond encloses a call option. The investor has in a sense brought an embedded call. If the price of the equity goes beyond the conversion price then the investor will call the stocks. In a reverse-convertible bond it is the issuer who has bought an option actually this is a put option. The issuer determines if as well as when the to convert the bond into stock. The investor alternatively is short an embedded put. The investor will believe the delivery of a bond or a stock at a pre-determined price if the issuer chooses to convert. For this additional risk the investor will receive a higher coupon.

(e) When volatility rises it gives the following opportunities to dealers. Elevated volatility implies high option prices. Therefore reverse convertibles can be structured with higher coupons. This attracts investors together the issuing company will be long an option. By hedging this company is able to isolate the gamma. Therefore if the option is purchased at a reasonable price from the investor which is quite likely in such cases then the gamma gains are able to very well exceed the premium paid for the option. The structures gain two ways. From higher volatility as well as from selling new instruments.

(f) Regulators may perhaps worry that such instruments are making investors *sell *options. Many investors mayn't realize how to price options given a certain volatility structure. Under such circumstances they may sell options below the fair price.