Estimated reading time: 3 minutes
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 ( d2 ) ] } – [ Stock Price X ( 1 – N ( d1 ) ]
Call Option’s Value
To find the value of the call option, we may utilize the following formula:
Call Value = [ Stock Price X N ( d1 ) ] – [ Strike Price X e ( – r X days/365) ] X [ N ( d2 ) ]
Putup
Putup = Max ( 0, Strike Price – StockUp )
Putdown
Putdown = Max ( 0, Strike Price – StockDown )
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:
P0 + S0 = C0 + X / ( 1 + Rf )T
Where:
C = Call Premium
P = Put Premium
X = Strike Price of Call & Put
r = Annual Interest Rate
t = Time in Years
So = 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:
c0 = ( p0 + S0 ) − [ 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 σ2 LR + LR2 X σ2 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:
pup = ( 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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