# ReplicatingVarianceSwap

Create ReplicatingVarianceSwap pricer object for VarianceSwap instrument using ratecurve object

## Description

Create and price a VarianceSwap instrument object with a ratecurve object and a ReplicatingVarianceSwap pricing method using this workflow:

1. Use fininstrument to create a VarianceSwap instrument object.

2. Use ratecurve to specify a curve model for the VarianceSwap instrument.

3. Use finpricer to specify a ReplicatingVarianceSwap pricer object for the VarianceSwap instrument.

For more information on this workflow, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.

For more information on the available pricing methods for a VarianceSwap instrument, see Choose Instruments, Models, and Pricers.

## Creation

### Description

example

ReplicatingVarianceSwapPricerObj = finpricer(PricerType,'DiscountCurve',ratecurve_obj,'VolatilitySmile',volatilitysmile_value,'SpotPrce',spotprice_value) creates an ReplicatingVarianceSwap pricer object by specifying PricerType and sets properties using the required name-value pair arguments DiscountCurve, VolatilitySmile, and SpotPrice.

example

ReplicatingVarianceSwapPricerObj = finpricer(___,Name,Value) sets optional properties using additional name-value pairs in addition to the required arguments in the previous syntax. For example, ReplicatingVarianceSwapPricerObj = finpricer("ReplicatingVarianceSwap",'DiscountCurve',ratecurve_obj,'VolatilitySmile',smiletable,'SpotPrice',1000,'CallPutBoundary',"forwardprice",'InterpMethod',"cubic") creates a ReplicatingVarianceSwap pricer object. You can specify multiple name-value pair arguments.

### Input Arguments

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Pricer type, specified as a string with the value "ReplicatingVarianceSwap" or a character vector with the value 'ReplicatingVarianceSwap'.

Data Types: char | string

ReplicatingVarianceSwap Name-Value Pair Arguments

Specify required and optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: ReplicatingVarianceSwapPricerObj = finpricer("ReplicatingVarianceSwap",'DiscountCurve',ratecurve_obj,'VolatilitySmile',smiletable,'SpotPrice',1000,'CallPutBoundary',"forwardprice",'InterpMethod',"cubic")
Required ReplicatingVarianceSwap Name-Value Pair Arguments

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ratecurve object for discounting cash flows, specified as the comma-separated pair consisting of 'DiscountCurve' and the name of a ratecurve object.

Note

• Specify a flat ratecurve object for DiscountCurve. If you use a nonflat ratecurve object, the software uses the rate in the ratecurve object at Maturity and assumes that the value is constant for the life of the equity option.

• The software uses the Basis value of the specified ratecurve object to calculate both the discounting and accrual for the VarianceSwap instrument object.

Data Types: object

Volatility smile table, specified as the comma-separated pair consisting of 'VolatilitySmile' and a table with the columns "Strike" and "Volatility" or a NumVols-by-2 matrix where the first column is the strikes and the second column is the volatilities in decimals.

Data Types: table | double

Spot price of the underlying asset, specified as the comma-separated pair consisting of 'SpotPrice' and a scalar nonnegative numeric.

Note

SpotPrice must be covered by the range of strikes in VolatilitySmile.

Data Types: double

Optional ReplicatingVarianceSwap Name-Value Pair Arguments

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Boundary strike for call and put options, specified as the comma-separated pair consisting of 'CallPutBoundary' and a scalar numeric or one of the following character vectors or strings:

• "spotprice" — The call and put option boundary strike is the spot price.

• "forwardprice" — The call and put option boundary strike is the forward price.

Note

CallPutBoundary must be covered by the range of strikes in VolatilitySmile.

Data Types: double | char | string

Interpolation method for SmileTable, specified as the comma-separated pair consisting of 'InterpMethod' and a scalar string or character vector using a supported value. For more information on interpolation methods, see interp1.

Data Types: char | string

## Properties

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ratecurve object for discounting cash flows, returned as the ratecurve object.

Note

The software uses the Basis value of the specified ratecurve object to calculate both the discounting and accrual for the VarianceSwap instrument object.

Data Types: object

Volatility smile table, returned as a table with the columns "Strike" and "Volatility" or a NumVols-by-2 matrix where the first column is the strikes and the second column is the volatilities in decimals.

Data Types: table | double

Strike price of the underlying asset, returned as a scalar nonnegative numeric.

Data Types: double

Boundary strike for call and put options, returned as a numeric or as a string with the value "spotprice" or "forwardprice".

Data Types: double | char | string

Interpolation method, returned as a scalar string.

Data Types: string

## Object Functions

 price Compute price for equity instrument with ReplicatingVarianceSwap pricer

## Examples

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This example shows the workflow to price a VarianceSwap instrument when you use a ratecurve and a ReplicatingVarianceSwap pricing method.

Create VarianceSwap Instrument Object

Use fininstrument to create a VarianceSwap instrument object.

VarianceSwapInst = fininstrument("VarianceSwap",'Maturity',datetime(2021,5,1),'Notional',150,'StartDate',datetime(2020,5,1),'RealizedVariance',0.05,'Strike',0.1,'Name',"variance_swap_instrument")
VarianceSwapInst =
VarianceSwap with properties:

Notional: 150
RealizedVariance: 0.0500
Strike: 0.1000
StartDate: 01-May-2020
Maturity: 01-May-2021
Name: "variance_swap_instrument"

Create ratecurve Object

Create a flat ratecurve object using ratecurve.

Settle = datetime(2020, 9, 15);
ZeroTimes = [calmonths(6) calyears([1 2 3 4 5 7 10 20 30])];
ZeroRates = [0.0052 0.0055 0.0061 0.0073 0.0094 0.0119 0.0168 0.0222 0.0293 0.0307]';
ZeroDates = Settle + ZeroTimes;
Basis = 1;
ZeroCurve = ratecurve("zero",Settle,ZeroDates,ZeroRates,'Basis',Basis)
ZeroCurve =
ratecurve with properties:

Type: "zero"
Compounding: -1
Basis: 1
Dates: [10x1 datetime]
Rates: [10x1 double]
Settle: 15-Sep-2020
InterpMethod: "linear"
ShortExtrapMethod: "next"
LongExtrapMethod: "previous"

Create ReplicatingVarianceSwap Pricer Object

Use finpricer to create a ReplicatingVarianceSwap pricer object and use the ratecurve object for the 'DiscountCurve' name-value pair argument.

Strike = (50:5:135)';
Volatility = [.49;.45;.42;.38;.34;.31;.28;.25;.23;.21;.2;.21;.21;.22;.23;.24;.25;.26];
VolatilitySmile = table(Strike, Volatility);
SpotPrice = 100;
CallPutBoundary = 100;

outPricer =  finpricer("ReplicatingVarianceSwap",'DiscountCurve', ZeroCurve, 'VolatilitySmile', VolatilitySmile, ...
'SpotPrice', SpotPrice, 'CallPutBoundary', CallPutBoundary)
outPricer =
ReplicatingVarianceSwap with properties:

DiscountCurve: [1x1 ratecurve]
InterpMethod: "linear"
VolatilitySmile: [18x2 table]
SpotPrice: 100
CallPutBoundary: 100

Price VarianceSwap Instrument

Use price to compute the price and fair variance for the VarianceSwap instrument.

[Price, outPR] = price(outPricer,VarianceSwapInst,["all"])
Price = 8.1997
outPR =
priceresult with properties:

Results: [1x2 table]
PricerData: [1x1 struct]

outPR.Results
ans=1×2 table
Price     FairVariance
______    ____________

8.1997      0.21701

outPR.PricerData.ReplicatingPortfolio
ans=19×6 table
CallPut    Strike    Volatility      Weight       Value     Contribution
_______    ______    __________    __________    _______    ____________

"put"        50         0.49        0.0064038    0.39164      0.002508
"put"        55         0.45        0.0052877    0.49353     0.0026097
"put"        60         0.42        0.0044402    0.67329     0.0029895
"put"        65         0.38        0.0037814    0.80343     0.0030381
"put"        70         0.34        0.0032592     0.9419     0.0030698
"put"        75         0.31        0.0028382      1.223     0.0034711
"put"        80         0.28        0.0024938       1.58     0.0039403
"put"        85         0.25        0.0022086     2.0456     0.0045177
"put"        90         0.23        0.0019696     2.9221     0.0057554
"put"        95         0.21        0.0017675     4.1406     0.0073183
"put"       100          0.2       0.00082405     6.1408     0.0050603
"call"      100          0.2       0.00077087     6.4715     0.0049887
"call"      105         0.21        0.0014465     4.7094     0.0068119
"call"      110         0.21        0.0013178     3.1644     0.0041701
"call"      115         0.22        0.0012056      2.307     0.0027814
"call"      120         0.23        0.0011072     1.7127     0.0018962
⋮

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## Algorithms

The fair value of the future variance Kvar is approximated in terms of the following portfolio of options ᴨCP:

$\begin{array}{l}{K}_{\mathrm{var}}=\frac{2}{T}\left\{rT-\left(\frac{{S}_{0}}{{S}_{*}}{e}^{rT}-1\right)-\mathrm{log}\frac{{S}_{*}}{{S}_{0}}+{e}^{rT}{\pi }_{CP}\right\}\\ {\pi }_{CP}=\sum _{i}w\left({K}_{ip}\right)P\left(S,{K}_{ip}\right)+\sum _{i}w\left({K}_{ic}\right)C\left(S,{K}_{ic}\right)\end{array}$

Here:

• Call option strikes — The call option strike are K0 < K1c < K2c < K3c … < Knc.

• Put option strikes — The put option strikes are Kmp < … < K3p < K2p < K1p < K0 = S*.

• Kvar — is the fair value of future variance

• CP — is the portfolio of call and put options

• S0 — is the current asset price

• S* — is the boundary between the call and put option strikes (for example, the spot price S0 or forward price S0erT)

• P(K) — is the current put option price with strike K

• C(K) — is the current call option price with strike K

If the options portfolio ᴨCP has an infinite number of options with continuously varying strikes, it has the following payoff function at maturity:

$f\left({S}_{T}\right)=\frac{2}{T}\left[\frac{{S}_{T}-{S}_{*}}{{S}_{*}}-\mathrm{log}\frac{{S}_{T}}{{S}_{*}}\right]$

Since it is not possible to construct such a portfolio with an infinite number of options and continuously varying strikes, the appropriate weights w(Kip) and w(Kic) for a portfolio with a finite number of options and discretely varying strikes can be computed by approximating the continuous payoff function f(ST) in a piecewise linear fashion. Starting with the strike at K0, the first call option weight can be computed as the slope of the first piecewise linear function:

${w}_{c}\left({K}_{0}\right)=\frac{f\left({K}_{1c}\right)-f\left({K}_{0}\right)}{{K}_{1c}-{K}_{0}}$

The next call option weight with the strike K1c is computed as the slope of the next piece-wise linear function minus the previous weight:

${w}_{c}\left({K}_{1c}\right)=\frac{f\left({K}_{2c}\right)-f\left({K}_{1c}\right)}{{K}_{2c}-{K}_{1c}}-{w}_{c}\left({K}_{0}\right)$

This procedure is continued for the remaining call option strikes:

${w}_{c}\left({K}_{n,c}\right)=\frac{f\left({K}_{n+1,c}\right)-f\left({K}_{nc}\right)}{{K}_{n+1,c}-{K}_{n,c}}-\sum _{i=0}^{n-1}{w}_{c}\left({K}_{i,c}\right)$

To compute the put option weights, a similar procedure can be used in the opposite direction (starting from K0):

${w}_{p}\left({K}_{m,p}\right)=\frac{f\left({K}_{m+1,p}\right)-f\left({K}_{mp}\right)}{{K}_{m,p}-{K}_{m+1,p}}-\sum _{i=0}^{m-1}{w}_{p}\left({K}_{i,p}\right)$

Once the fair variance is computed, the actual price paid in the market at time t for the variance swap with a StartDate at time 0 is computed as follows:

$VarianceSwap\left(t\right)=Notional×Disc\left(t,T\right)×\left[\frac{t}{T}\mathrm{Re}alizedVariance\left(0,t\right)+\frac{T-t}{T}FairVariance\left(t,T\right)-StrikeVariance\right]$

Here:

• t is the time from the start date of the variance swap to the settle date.

• T is the time from the start date to the maturity date of the variance swap.

• Disc(t,T) is the discount factor from settle to the maturity date.

• RealizedVariance(0,t) is the realized variance from start date to the settle date, in basis points.

• FairVariance(t,T) is the fair variance for the remaining life of the contract as of the settle date, in basis points.

• StrikeVariance is the strike variance predetermined at inception (start date), in basis points.

## References

[1] Demeterfi, K., Derman, E., Kamal, M., and J. Zou. “More Than You Ever Wanted To Know About Volatility Swaps.” Quantitative Strategies Research Notes. Goldman Sachs, 1999.

### Functions

Introduced in R2020b