## Quoting Prices With Volatility Measures In Fixed Income Options Markets

Market participants often use yield-to-maturity to quote bond prices because interest rates are in many ways more intuitive than bond prices. Similarly, market participants often use volatility to quote option prices because volatility is in many ways more intuitive than option prices. Chapter 3 defined the widely accepted relationship between yield-to-maturity and price. This section discusses the use of market conventions to quote the relationship between volatility and option prices.

Many options trading desks have their own proprietary term structure models to value fixed income options. If customers want to know the volatility at which they are buying or selling options, these trading desks have a problem. Quoting the volatility inputs to their proprietary models does not really help customers because they do not know the model and have no means of generating prices given these volatility inputs. Furthermore, the trading desk may not want to reveal the workings of its models. Therefore, markets have settled on various canonical models with which to relate price and volatility.

In the bond options market, Black's model, a close relative of the Black-Scholes stock option model, is used for this purpose. As discussed in Chapter 9, direct applications of stock option models to bonds may be reasonable if the time to option expiry is relatively short. Further details are not presented here other than to note that Black's model assumes that the price of a bond on the option expiration date is lognormally distributed with a mean equal to the bond's forward price.5

Figure 19.7 reproduces a Bloomberg screen used for valuing options using Black's model. The darkened rectangles indicate trader input values. The header under "option valuation" indicates that the option is on the U.S. Treasury 5s of February 15, 2011. As of the trade date January 15, 2002, this bond was the double-old 10-year. The option expires in six months, on July 15, 2002. The current price of the bond is 101-81/4 corresponding to a yield of 4.827%. The strike price of the option is 99-181/4 corresponding to a yield of 5.063%. At the bottom right of the screen, the repo rate is 1.58% which, given the bond price, gives a forward price of 99-181/4. The option is,

<HELP> for explanation.

Change values, or for quick use, see DV <Help> for help ori

OPTION VALUATION

Option T 5 02/15/11

SETTLE DELAY Option Bond

N1'59 Govt tails.

I exercise

PRICE

101-08'4

Yield

Yield [Jorst to

Option T 5 02/15/11

SETTLE DELAY Option Bond

I exercise

PRICE

101-08'4

Yield

Yield [Jorst to

7s16/ 2 exercise settle |
S100-00 | ||||

Save ? E |
i fl/E EUROPEAN | ||||

[ffijMrisk free |
CALL |
PUT |
Model H | ||

0ft-32nds,B-YldBP |
dec. |
32nds |
dec. |
32nds |
B - Black-Derman-Toy |

Option Price |
H.ÏJB |
■MEM |
UM&iU I |
fcBMfll |
L - Lognormal |

Price I. Vol |
kkee |
HEmfi |
N - Normal Mean rev | ||

Yield Vol(X) S |
■JiWlIlIl] |
»¡Willi!] |
P - Price based | ||

dPdY |
6.875 |
Model properties | |||

Delta (Price) |
.512 |
-.487 |
Price based | ||

Gamma (Price) |
,063 |
.063 |
1 factor: bond price | ||

Vega (Price) |
.277 |
0-09 |
.277 |
0-09 |
mean reversion! no |

B-day decay |
.048 |
0-01+ |
.048 |
0-01+ |
bond price lognormal |

Time Value |
2.521 |
2-1634 |
2.521 |
2-1634 |
vol; const, relative |

Expi rati on furd px|

Expi rati on furd px|

Australia 61 2 9777 8600 Brazil 5511 3048 4500 Europe 44 20 7330 7500 Germany 49 69 920410

Hong Kong 852 2977 6000 Japan 81 3 3201 8900 Singapore 65 212 1000 U.S. 1 212 318 2000 Copyright 2002 Bloomberg L.P.

1984-583-1 15-Jan-02 17'39'26

■Bloomberg

■ professional9

FIGURE 19.7 Bloomberg's Option Valuation Screen for Options on the 5s of

February 15, 2001

Source: Copyright 2002 Bloomberg L.P.

5For more details, see Hull (2000), pp. 533-537.

therefore, an at-the-money forward (ATMF) option, meaning that the strike price equals the forward price. The risk-free rate equals 1.58%, used in Black's model to discount the payoffs of the option under the assumed lognormal distribution. Because the double-old 10-year was not particularly special on January 15, 2002, the repo rate and the risk-free rate are equal. If the bond were trading special, the repo rate used to calculate the bond's forward price would be less than the risk-free rate.

As can be seen above the words "call" and "put," the option is a European option. To the right is the model code "P" used to indicate the price-based or Black's model. Below this code is a brief description of the model's properties. It is a one-factor model with the bond price itself as the factor. There is no mean reversion in the process, the bond price is lognormal, and the volatility is constant. The description also indicates that the volatility is relative, that is, measured as a percentage of the bond's forward price.

The main part of the option valuation screen shows that at a percentage price volatility of 9.087% put and call prices equal 2.521.6 This means, for example, that an option on $100,000,000 of the 5s of February 15, 2011, on July 15, 2002, at 99-181/, costs

2.521

The price volatility is labeled "Price I. Vol" for "Price Implied Volatility" because the pricing screen may be used in one of two ways. First, one may input the volatility and the screen calculates the option price using Black's model. Second, one may input the option price and the screen calculates the implied volatility—the volatility that, when used in Black's model, produces the input option price.

While Black's model is widely used to relate option price and volatility, percentage price volatility (or, simply, price volatility) is not so intuitive as volatility based on interest rates. Writing the percentage change in the forward price as APfwd /Pfwd, the percentage change may be rewritten as

6At-the-money forward put and call prices must be equal by put-call parity.

fwd fwd J fwd J fwd fwd J fwd

Letting op denote price volatility and <7y denote yield volatility, it follows from equation (19.20) that o p = f 10,000 x DV01fwd o y (19.21)

Pfwd

In the example of Figure 19.7,

Note that all of these inputs are on the Bloomberg screen. Since the strike is equal to the forward price, the yield corresponding to the strike is the forward yield. Also, the forward DV01 is computed next to the symbol "dPdY" (i.e., the derivative of price with respect to yield). Solving equation (19.22), o y = 26% (19.23)

as reported in the row labeled "Yield Vol (%)." The input "F," by the way, indicates that volatility should be computed using a forward rate, as done here.

Many market participants find yield volatility more intuitive than price volatility. With yields at 5.063%, for example, a yield volatility of 26% indicates that a one standard deviation move is equal to 26% of 5.063%. This also suggests measuring volatility in basis points: 26% of 5.063% is 131.6 basis points. Letting obp denote basis point volatility, then, as explained in Chapter 12, o bp = yfwdo y (19.24)

It is crucial to note that while volatility can be quoted as yield volatility or as basis point volatility, Black's model takes price volatility as input. In other words, it is price volatility that determines the probability distribution used to calculate option prices. To make this point more clearly, consider three models: Black's model with price volatility equal to 9.087%, a model with a lognormally distributed short rate and yield volatility equal to 26%, and a model with a normally distributed short rate and basis point volatility equal to 131.6 basis points. These three models are different. They will not always produce the same option prices even though the volatility measures are the same in the sense of equations (19.21) and (19.24).

Return now to the trading desk with a proprietary option pricing model. A customer inquires about an at-the-money forward option on the 5s of February 15, 2011, and the desk responds with a price of 2.521 corresponding to a Black's model volatility of 9.087%. The customer knows the price and has some idea what this price means in terms of volatility, whether by thinking about price volatility directly or by converting to yield or basis point volatility. But the customer cannot infer the price the trading desk would attach to a different option on the same bond nor certainly to an option on a different bond. Plugging in a price volatility of 9.087% on a Bloomberg screen to price other options on the 5s of February 15, 2011, will not produce the trading desk's price unless the trading desk itself uses Black's model.

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