Black Scholes Merton Model

A mathematical model of a financial market which contains derivative investment instruments is called as Black Scholes Merton model [1]. This model provides simple formula regarding asset's price and its volatility, time to maturity of the contract and the risk free interest rate [2]. Black Scholes formula gives a theoretical estimate of the price of European-style options [1].

Let

= Asset Price

= Exercise Price

= Risk Free Rate

= Time to expiration

= Standard deviation

= The cumulative distribution function of the standard normal distribution

= The standard normal probability density function

= Price Call

= Price Put

= Delta Call

= Delta Put

= Gamma Call

= Gamma Put

= Theta Call

= Theta Put

= Vega Call

= Vega Put

= Rho Call

= Rho Put

We have

**Example 1**

*Input*

Asset Price = 125.94

Exercise Price = 125

Time to Expiration = 1

Standard Deviation = 83%

Risk Free Rate = 4.46%

*Output*

Price Call: 42.776, Price Put: 36.383

Delta Call: 0.684, Delta Put: -0.316

Gamma Call: 0.003, Gamma Put: 0.003

Theta Call: -20.534, Theta Put: -15.202

Vega Call: 44.824, Vega Put: 44.824

Rho Call: 43.316, Rho Put: -76.232

**Example 2**

*Input*

Asset Price = 460

Exercise Price = 470

Time to Expiration = 0.17

Standard Deviation = 58%

Risk Free Rate = 2%

*Output*

Price Call: 40.105, Price Put: 48.509

Delta Call: 0.517, Delta Put: -0.483

Gamma Call: 0.004, Gamma Put: 0.004

Theta Call: -132.91, Theta Put: -123.541

Vega Call: 75.592, Vega Put: 75.592

Rho Call: 33.65, Rho Put: -45.979

1. Black–Scholes model. (n.d.). Retrieved August 18, 2016, from https://en.wikipedia.org/wiki/ Black–Scholes_model

2. Black-Scholes-Merton approach – merits and shortcomings. (n.d). Retrieved June 13, 2017 from https://www.essex.ac.uk/economics /documents/eesj/matei.pdf

Capital Asset Pricing Model (CAPM)

In finance, determination of a theoretically appropriate required rate of return of an asset is called as capital asset pricing model (CAPM). This method provides to make decisions about adding assets to a portfolio which is well-diversified [1].

Let

= Expected Stock Return

= Expected Market Return

= Risk Free Rate

= Beta

We have

**Example 1**

*Input*

Expected Return on Stock = 14%

Expected Return of the Market = 12.6%

Beta = 1.6

*Output*

Risk Free Rate = 10.267%

**Example 2**

*Input*

Expected Return of the Market = 4%

Risk Free Rate = 2.7%

Beta = 1.7

*Output*

Expected Return on Stock = 4.91%

1. Capital asset pricing model (n.d.). Retrieved August 18, 2016, from https://en.wikipedia.org/wiki/ Capital_asset_pricing_model

Dividend Discount Model (DDM)

The dividend discount model (DDM) is a method which values a stock price of a company based on the future dividends' net present value (npv) [1].

Gordon Model

One of the class of dividend discount model is the Gordon Model which assumes dividends will increase at a constant growth rate [2].

Let

= Dividend

= Growth Rate

= Required Rate of Return

= Price

If the given dividend is the current dividend, then

If the given dividend is the next dividend, then

**Example 1**

*Input*

Dividend Type = Current

Dividend = 4.56

Required Rate of Return = 13.49%

Growth Rate = 5.97%

*Output*

Price of Stock = 64.258

**Example 2**

*Input*

Dividend Type = Next

Dividend = 5.93

Required Rate of Return = 8.16%

Growth Rate = 1.25%

*Output*

Price of Stock = 85.818

1. Dividend discount model (n.d). Retrieved June 13, 2017 from https://en.wikipedia.org/ wiki/Dividend_discount_model

2. Stock valuation. (n.d.). Retrieved August 18, 2016, from https://en.wikipedia.org/ wiki/Stock_valuation

Nonconstant Growth Stock Calculation

We know that Gordon Model assumes that dividends will rise at a constant growth rate. However, companies' growth rate is not always constant. Nonconstant growth model is a more general method than the Gordon Model and it is based on assuming growth rates are nonconstant until a point, then tehy are constant after that point [1].

Let

= Price of Stock

= Value of Stock at time i

= Expected Dividend at time i

= Number of Periods

= Growth rate at time i

= Required return on Stock at time i.

= Required return until time i

First, define

= 1

If i < N, then

If i = N, then

Hence, we have

**Example 1**

*Input*

Dividend = 2

Period = 4

Required Rate of Return 1 = 12%

Growth Rate 1 = 8%

Required Rate of Return 2 = 12%

Growth Rate 2 = 4%

Required Rate of Return 3 = 12%

Growth Rate 3 = 5%

Required Rate of Return 4 = 12%

Growth Rate 4 = 6%

*Output*

Price of Stock = 35.06

**Example 2**

*Input*

Dividend = 10

Period = 3

Required Rate of Return 1 = 12%

Growth Rate 1 = 8%

Required Rate of Return 2 = 12%

Growth Rate 2 = 4%

Required Rate of Return 3 = 12%

Growth Rate 3 = 5%

*Output*

Price of Stock = 152.91

1. Stock valuation. (n.d.). Retrieved August 18, 2016, from https://en.wikipedia.org/ wiki/ Stock_valuation

Weighted Average Cost of Capital (WACC)

A compony needs to know how to finance its assets to pay to all its security holders, so that weighted average cost of capital (WACC), which is also defined the cost of capital, is the rate that a company is expected to pay on average to all its security holders. WACC can be used by companies to see whether the investment projects are worth to undertake or not [1].

Let

= Equity

= Cost of Equity

= Debt

= Cost of Debt

= Corporate Tax Rate

= Weighted Average Cost of Capital

We have

**Example 1**

*Input*

Cost of Equity = 15%

Equity = 400,000

Cost of Debt = 8%

Debt = 600,000

Corporate Tax Rate = 5%

*Output*

Weighted Avg Cost of Capital = 10.56%

**Example 2**

*Input*

Cost of Equity = 12.5%

Equity = 8000

Cost of Debt = 6%

Debt = 2000

Corporate Tax Rate = 30%

*Output*

Weighted Avg Cost of Capital = 10.84%

1. Weighted average cost of capital (n.d.). Retrieved August 18, 2016, from https://en.wikipedia.org/ wiki/Weighted_average_cost_of_capital

Holding Period Return (HPR)

Holding period return (HPR) is defined as the total return on an asset or portfolio over a time when it was held.

In finance, holding period return (HPR) is the total return on an asset or portfolio over a period during which it was held. It is called as the simplest and most significant measures of investment performance [1].

Let

= Initial Value

= Ending Value

= Income

= Holding Period Return

We have

**Example 1**

*Input*

Initial Value = 50

Ending Value = 60

Income = 5

*Output*

Holding Period Return = 30%

**Example 2**

*Input*

Initial Value = 200

Ending Value = 320

Income = 10

*Output*

Holding Period Return = 65%

1. Holding period return. (n.d.). Retrieved August 18, 2016, from https://en.wikipedia.org/ wiki/Holding_period_return

Expected Return

Expected return can be calculated using the probability states and expected return states. It measures the center of the variable's distribution [1].

Let

= Expected Return

= Standard Deviation

= Expected Return in state i

= Probability of state i

= The Number of States

We have the following formulas:

**Example 1**

*Input*

Period = 4

State 1 - Probability: 20%, Stock A: 5%, Stock B: 10%

State 2 - Probability: 30%, Stock A: 10%, Stock B: 15%

State 3 - Probability: 30%, Stock A: 15%, Stock B: 20%

State 4 - Probability: 20%, Stock A: 20%, Stock B: 25%

*Output*

Expected Return (A): 12.5%

Standard Deviation (A): 5.123%

Expected Return (B): 17.5%

Standard Deviation (B): 5.123%

**Example 2**

*Input*

Period = 4

State 1 - Probability: 15%, Stock A: 5%, Stock B: 10%

State 2 - Probability: 35%, Stock A: 15%, Stock B: 20%

State 3 - Probability: 35%, Stock A: 25%, Stock B: 30%

State 4 - Probability: 15%, Stock A: 35%, Stock B: 40%

*Output*

Expected Return (A): 20%

Standard Deviation (A): 9.22%

Expected Return (B): 25%

Standard Deviation (B): 9.22%

1. Expected return (n.d.). Retrieved August 18, 2016, from https://en.wikipedia.org/ wiki/Expected_return