# 13 Best Automotive Performance Ring & Pinion Gears

List Updated May 2020

## Bestselling Automotive Performance Ring & Pinion Gears in 2020

#### Yukon Gear & Axle (YG D44HD-456) High Performance Ring & Pinion Gear Set for Dana 44-HD Differential

- 1 Year Warranty
- Made From 8620 Steel
- Quiet Running Gear Sets
- 10 Ring Gear Bolts
- 26 Spline Count

#### Yukon (YG D30-427) High Performance Ring and Pinion Gear Set for Dana 30

- 1 Year Warranty
- Made From 8620 Steel
- Quiet Running Gear Sets
- 10 Ring Gear Bolts
- 26 Spline Pinion

#### Yukon (YG D60R-411R) High Performance Ring and Pinion Gear Set for Dana 60 Reverse Rotation Differential

- 1 Year Warranty
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- Quiet Running Gear Sets

#### Yukon (YG C9.25R-373R) High Performance Ring and Pinion Gear Set for Chrysler/Dodge 9.25" Front Differential

- 1 Year Warranty
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- 12 Ring Gear Bolts

#### Yukon Gear & Axle (YG D44RS-513RUB) High Performance Ring & Pinion Gear Set for Dana 44 Short Pinion Reverse Rotation Differential

- 1 Year Warranty
- Made From 8620 Steel
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#### Yukon Gear & Axle (YG D44-411) High Performance Ring & Pinion Gear Set for Dana 44 Differential

- 1 Year Warranty
- Made From 8620 Steel
- Quiet Running Gear Sets
- 10 Ring Gear Bolts
- 26 Spline Count

#### Yukon Gear & Axle (YG D30SR-411JK) High Performance Ring & Pinion Gear Set for Jeep JK Dana 30 Short Reverse Pinion Differential

- 1 Year Warranty
- Made From 8620 Steel
- Quiet Running Gear Sets
- 10 Ring Gear Bolts
- 24 Spline Short Pinion

#### Yukon Gear & Axle (YG GM8.5-456) High Performance Ring & Pinion Gear Set for GM 8.5/8.6 Differential

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- Made From 8620 Steel
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#### Yukon Gear & Axle (YG GM14T-342) High Performance Ring & Pinion Gear Set for GM 14-Bolt Truck 10.5 Differential

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#### Yukon Gear & Axle (YG D44JK-513RUB) High Performance Ring & Pinion Gear Set for Jeep JK Dana 44 Rear Differential

- 1 Year Warranty
- Made From 8620 Steel
- Quiet Running Gear Sets
- 10 Ring Gear Bolts
- 24 Spline Count

#### Yukon Gear & Axle (YG F8.8R-456R) High Performance Ring & Pinion Gear Set for Ford 8.8 Reverse Rotation Differential

- 1 Year Warranty
- Made From 8620 Steel
- Quiet Running Gear Sets

#### Yukon Gear & Axle (YG D30R-456R) High Performance Ring & Pinion Gear Set for Dana 30 Reverse Rotation Differential

- 1 Year Warranty
- Made From 8620 Steel
- Quiet Running Gear Sets
- 10 Ring Gear Bolts
- 26 Spline Pinion

#### Yukon Gear & Axle (YG D80-331) High Performance Ring & Pinion Gear Set for Dana 80 Differential

- 1 Year Warranty
- Made From 8620 Steel
- Quiet Running Gear Sets

## The Black-Scholes Partial Differential Equation: Practice Problems and Solutions

Section 49 of The Actuary's Free Study Guide for Exam 3F / Exam MFE presents the Black-Scholes partial differential equation relation option price to the values of the option Greeks. 5 practice problems and solutions are given.

This is Section 49 of the Study Guide. See Section 1 here. See Section 2 here. See Section 3 here. See Section 4 here. See Section 5 here. See Section 6 here. See Section 7 here. See Section 8 here. See Section 9 here. See Section 10 here. See Section 11 here. See Section 12 here. See Section 13 here. See Section 14 here. See Section 15 here. See Section 16 here. See Section 17 here. See Section 18 here. See Section 19 here. See Section 20 here. See Section 21 here. See Section 22 here. See Section 23 here. See Section 24 here. See Section 25 here. See Section 26 here. See Section 27 here. See Section 28 here. See Section 29 here. See Section 30 here. See Section 31 here. See Section 32 here. See Section 33 here. See Section 34 here. See Section 35 here. See Section 36 here. See Section 37 here. See Section 38 here. See Section 39 here. See Section 40 here. See Section 41 here. See Section 42 here. See Section 43 here. See Section 44 here. See Section 45 here. See Section 46 here. See Section 47 here. See Section 48 here.

The Black-Scholes partial differential equation or Black-Scholes equation (as opposed to the Black-Scholes formula) is as follows:

rC(St) = (1/2)Ïƒ2St2Î“t2 + rStâˆ†t + Î¸

This equation holds for American and European calls and puts, but not at times when it is optimal to exercise the options early.

### The Black-Scholes equation entails the following assumptions:

### 1. The underlying asset does not pay any dividends.

### 2. The option does not pay any dividends.

### 3. The interest rate and volatility are constant.

### 4. The stock moves one standard deviation over a small time interval.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2020, Ch. 13, pp. 429-430.

### Meaning of variables:

### St = stock price at time t.

### C = call option price.

## âˆ† = option delta.

## Î“ = option gamma.

## Î¸ = option theta

### r = annual continuously compounded risk-free interest rate

### Ïƒ = annual standard deviation of the stock price movement.

Original Practice Problems and Solutions from the Actuary's Free Study Guide:

Problem BSPDF1. The stock of Company Îž has a current price of $56 per share, and the standard deviation of its price is 0.34. A certain call option on this stock has a delta of 0.42, a gamma of 0.001, and a theta of -0.05. The annual continuously compounded risk-free interest rate is 0.06. What is the price of this call option, as found using the Black-Scholes equation?

Solution BSPDF1. We use the formula rC(St) = (1/2)Ïƒ2St2Î“t2 + rStâˆ†t + Î¸, which we rearrange thus: C(St) = [(1/2)Ïƒ2St2Î“t2 + rStâˆ†t + Î¸]/r = [(1/2)0.3425620.0012 + 0.06*56*0.42 - 0.05]/0.06 = C(56) = $22.68968768

Problem BSPDF2. Assume that the Black-Scholes framework holds. The stock of Vicious Co. has a price of $93 per share, and the price has a standard deviation of 0.53. A certain call option on Vicious Co. stock has a price of $4, a delta of 0.53, and a gamma of 0.01. The annual continuously compounded risk-free interest rate is 0.02. What is the theta for this call option?

Solution BSPDF2. We use the formula rC(St) = (1/2)Ïƒ2St2Î“t2 + rStâˆ†t + Î¸, which we rearrange thus: Î¸ = rC(St) - (1/2)Ïƒ2St2Î“t2 - rStâˆ†t. Thus, Î¸ = 0.02*4 - (1/2)0.5329320.012 - 0.02*93*0.53 =

## Î¸ = -1.027275205

Problem BSPDF3. Assume that the Black-Scholes framework holds. The price of the stock of Devious Co. has a standard deviation of 0.74. A certain call option on this stock has a price of $25, a delta of 0.29, a gamma of 0.02, and a theta of -0.04. The annual continuously compounded risk-free interest rate is 0.08. What is the current price of one share of Devious Co. stock?

Solution BSPDF3. We use the formula rC(St) = (1/2)Ïƒ2St2Î“t2 + rStâˆ†t + Î¸, which we rearrange thus: 0 = (1/2)Ïƒ2St2Î“t2 + rStâˆ†t - (rC(St) - Î¸). We find that

rC(St) - Î¸ = 0.08*25 + 0.04 = 2.04, (1/2)Ïƒ2Î“t2 = (1/2)0.7420.022 = 0.00010952, and râˆ†t = 0.08*0.29 = 0.0232. We thus have the following equation:

0.00010952St2 + 0.0232St - 2.04 = 0. By the quadratic formula, the relevant positive value of

### St is St = $66.84058205.

Problem BSPDF4. Assume that the Black-Scholes framework holds. The price of the stock of Imperious LLC is $516 per share and has a standard deviation of 0.22. A certain call option on this stock has a price of $67, a delta of 0.03, and a theta of -0.05. The annual continuously compounded risk-free interest rate is 0.1. What is the gamma of this option?

Solution BSPDF4. We use the formula rC(St) = (1/2)Ïƒ2St2Î“t2 + rStâˆ†t + Î¸, which we rearrange thus: rC(St) - Î¸ - rStâˆ†t = (1/2)Ïƒ2St2Î“t2 , so Î“t = âˆš[(rC(St) - Î¸ - rStâˆ†t)/((1/2)Ïƒ2St2)] =

âˆš[(0.1*67 + 0.05 - 0.1*516*0.03)/((1/2)0.2225162)] = 5.202/6443.3952 = Î“t = 0.0008073383424

Problem BSPDF5. The stock of Obsequious Co. has a price of $40, and this price has a standard deviation of 0.43. A certain call option in this stock has a price of $5, a gamma of 0.02, and a theta of -0.02. The annual continuously compounded risk-free interest rate is 0.056. Find the delta of this call option.

Solution BSPDF5. We use the formula rC(St) = (1/2)Ïƒ2St2Î“t2 + rStâˆ†t + Î¸, which we rearrange thus: (rC(St) - Î¸ - (1/2)Ïƒ2St2Î“t2 )/rSt = âˆ†t

Thus, (rC(St) - Î¸ - (1/2)Ïƒ2St2Î“t2 )/(rSt) = (0.056*5 + 0.02 - (1/2)0.4324020.022 )/(0.056*40) =

## âˆ†t = 0.1075142857.

See other sections of The Actuary's Free Study Guide for Exam 3F / Exam MFE.