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## Introduction

We present the formulas for the Valuation and Risk Models segment.

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## Put Option’s Value

To find a put option’s value, we may utilize the following formula:

Put Value = { [ Strike Price X e^{( – RFR * T )} ] X [ 1 – N ( d_{2 }) ] } – [ Stock Price X ( 1 – N ( d_{1 }) ]

## Call Option’s Value

To find the value of the call option, we may utilize the following formula:

Call Value = [ Stock Price X N ( d_{1 }) ] – [ Strike Price X e ^{( – r X days/365)} ] X [ N ( d_{2 }) ]

## Put^{up}

Put^{up} = Max ( 0, Strike Price – Stock^{Up} )

## Put^{down}

Put^{down} = Max ( 0, Strike Price – Stock^{Down} )

## The Put-Call Parity relationship

Call Price – Put Price = Stock Price – [ Strike Price / ( 1 + RFR )^{T }]

Rearranging:

Call Price = Stock Price – [ Strike Price / ( 1 + RFR )^{T} ] + Put Price

or

Put-call parity relationship:

P_{0} + S_{0} = C_{0} + X / ( 1 + R_{f })^{T }

Where:

C = Call Premium

P = Put Premium

X = Strike Price of Call & Put

r = Annual Interest Rate

t = Time in Years

S_{o} = Initial Price of Underlying Security

## Value at Risk

To calculate the Value at Risk, we may utilize the following formula:

VaR = [ Expected Change in Value ] – [ Z-Score X Price Change Standard Deviation ]

## Annual Standard Deviation

Annual Standard Deviation = Daily Standard Deviation X ( 365 ) ^{½}

## Convexity Adjustment

Convexity Adjustment = Convexity X 100 X ( Δy )^{2}

*Where: Δy is the change in interest rates in decimal form*

## Synthetic Call Price

Formula to determine the synthetic call price:

c_{0} = ( p_{0 }+ S_{0} ) − [ X / (1 + r)^{T} ]

## Covered Call

Consider the following as a quick reminder:

‘ Covered Call = Stock + Short Call ‘

## The Unexpected Loss formula

Unexpected Loss, UL = EA X ( PD X σ^{2} _{LR} + LR^{2} X σ^{2} _{PD }) ^{½}

Where:

LR = loss rate = ( 1 – recovery rate)

EA = exposure amount

PD = probability of default

## Daily Standard Deviation

*Please note: If information is based on a sample, ‘ N – 1 ‘ will be used in the denominator *

Daily Standard Deviation = [ ( Sum of Squared Deviations from Mean / ( N – 1 ) ) ] ^{½}

## Implied 1‐Yr Forward Rate

*From the Pure Expectations Theory: The expected 1-Yr spot rate, one (1) year from now will be equivalent to finding the forward rate for the second year, thus:*

The Implied 1‐Yr Forward Rate is given:

= [ ( 1 + Two-Yr Spot Rate )^{2} / ( 1 + One-Yr Spot Rate ) ] – 1

## Risk Neutral Probability

We may calculate the risk neutral probability from the given formula:

Π = ( 1 + RFR ) – ( 1 – % ^{Down} ) ) / ( ( 1 + % ^{Up} ) – ( 1 – % ^{Down} ) )

## Risk Neutral Probability (single step stock)

The risk neutral probability of a stock going up in a single step may be calculated as follows:

p_{up} = ( e ^{rΔt} – d ) / ( u – d )

## Dirty Price

Dirty Price = Quoted Price + Interest Accrued from last Coupon Date

## Daily Delta Normal VaR

Daily Delta Normal VaR formula is as given:

= [ R – ( z )( sigma ) ] X ( Portfolio Value )

Where:

R is our portfolio’s expected one‐day return

z is our z‐value with respect to the significance level

Sigma is the standard deviation ( one‐day returns )

## Mean Loss Rate

Mean Loss Rate = Probability of Default X ( 1 – Recovery Rate )

## Portfolio’s Beta

Beta, β = Correlation Coefficient X ( Portfolio Standard Dev / Instrument’s Standard Dev )

## Sample-Mean

Sample-Mean may be calculated as:

= ( Summation of change in each day’s yield ) / ( The Number of Observations – 1 )

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