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Black Scholes Calculator

Asset Price

Exercise Price

Time to Expiration

Standard Deviation

Risk Free Rate

	Call	Put
Price
Delta
Gamma
Theta
Vega
Rho

Black Scholes Concept

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].

Black Scholes Formula

Let

$S$ = Asset Price
$K$ = Exercise Price
$r$ = Risk Free Rate
$T$ = Time to expiration
$q$ = Standard deviation
$N$ = The cumulative distribution function of the standard normal distribution
$N^{\prime}$ = The standard normal probability density function

$P_c$ = Price Call
$P_p$ = Price Put

$D_c$ = Delta Call
$D_p$ = Delta Put

$G_c$ = Gamma Call
$G_p$ = Gamma Put

$T_c$ = Theta Call
$T_p$ = Theta Put

$V_c$ = Vega Call
$V_p$ = Vega Put

$R_c$ = Rho Call
$R_p$ = Rho Put

We have

$d_1 = \dfrac{ln(\frac{S}{K}) + (r+\frac{q^2}{2}) \times T}{q\sqrt{T}}$

$d_2 = d_1 - q \sqrt{T}$

$P_c = N(d_1) \times S - N(d_2) \times Ke^{-rT}$

$P_p = Ke^{-rT} - S + P_c$

$D_c = N(d_1)$

$D_p = N(d_1) - 1$

$G_c = \dfrac{N^{\prime}(d_1)}{Sq\sqrt{T}} = G_p$

$V_c = SN^{\prime}(d_1)\sqrt{T} = V_p$

$T_c = -\dfrac{SN^{\prime}(d_1)q}{2\sqrt{T}} - rKe^{-rT}N(d_2)$

$T_p = -\dfrac{SN^{\prime}(d_1)q}{2\sqrt{T}} + rKe^{-rT}N(-d_2)$

$R_c = TKe^{-rT}N(d_2)$

$R_p = TKe^{-rT}N(-d_2)$

Examples

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

References

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) Calculator

Capital Asset Pricing Model (CAPM) Concept

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].

Capital Asset Pricing Model (CAPM) Formula

Let
$E(R_s)$ = Expected Stock Return
$E(R_m)$ = Expected Market Return
$r_f$ = Risk Free Rate
$\beta$ = Beta

We have

$E(R_s) = r_f+\beta \times (E(R_m)-r_f)$

$E(R_m) = \dfrac{E(R_s)-r_f + \beta \times r_f}{\beta}$

$I = \dfrac{E(R_s) - \beta \times E(R_m)}{1-\beta}$

$\beta = \dfrac{E(R_s)-r_f}{E(R_m)-r_f}$

Examples

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%

References

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

Constant Growth Stock Calculator

Constant Growth Concept

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].

Constant Growth Model Formula

Let

$D$ = Dividend
$R_g$ = Growth Rate
$R_r$ = Required Rate of Return

$P$ = Price

If the given dividend is the current dividend, then

$P = D \times \dfrac{1+R_g}{R_r-R_g}$

If the given dividend is the next dividend, then

$P = \dfrac{D}{R_r-R_g}$

Examples

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

References

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 Calculator

Year	Required Rate	Growth Rate
1
2
3
4

Dividend

Period

Results

Price of Stock:

Nonconstant Growth Concept

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].

Nonconstant Growth Model Formula

Let
$P$ = Price of Stock
$P_i$ = Value of Stock at time i
$D_i$ = Expected Dividend at time i
$N$ = Number of Periods
$g_i$ = Growth rate at time i
$r_i$ = Required return on Stock at time i.
$R_i$ = Required return until time i

First, define

$R_0$ = 1

$D_i = D_{i-1} \times (1+g_i)$

If i < N, then

$R_i = R_{i-1} \times (1 + r_i)$

$P_i = \dfrac{D_i}{R_i}$

If i = N, then

$R_i = R_{i-1}$

$P_i = \dfrac{\dfrac{D_i}{r_i-g_i}}{R_i}$

Hence, we have

$P = \sum\limits_{i = 1}^{N}P_i$

Examples

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

References

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

Weighted Average Cost of Capital (WACC) Calculator

Weighted Average Cost of Capital (WACC) Concept

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].

Weighted Average Cost of Capital (WACC) Formula

Let
$E$ = Equity
$E_c$ = Cost of Equity
$D$ = Debt
$D_c$ = Cost of Debt
$R$ = Corporate Tax Rate

$W$ = Weighted Average Cost of Capital

We have

$W = \dfrac{E \times E_c}{E+D} + \dfrac {D \times D_c \times (1-R)} {D+E}$

Examples

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%

References

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) Calculator

Holding Period Return (HPR) Concept

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].

Holding Period Return (HPR) Formula - How to Calculate HPR

Let

$P$ = Initial Value
$E$ = Ending Value
$I$ = Income
$HPR$ = Holding Period Return

We have

$HPR = \dfrac{I+E-P}{P}$

Examples

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%

References

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

Expected Return Calculator

State	Probabilty	Stock A	Stock B
1
2
3
4

Period

	Stock A	Stock B
Expected Return
Standard Deviation

Expected Return Concept

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].

Expected Return Formula - How to Calculate Expected Return

Let
$E$ = Expected Return
$D$ = Standard Deviation
$R_i$ = Expected Return in state i
$P_i$ = Probability of state i
$N$ = The Number of States

We have the following formulas:

$E = \sum\limits_{i=1}^{N} P_i \times R_i$

$D = \sqrt{\sum\limits_{i=1}^{N} P_i \times (R_i - E)}$

Examples

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%

References

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