Price European basket options using Nengjiu Ju approximation model

## Syntax

```Price = basketbyju(RateSpec,BasketStockSpec,OptSpec,Strike,Settle,Maturity) ```

## Description

`Price = basketbyju(RateSpec,BasketStockSpec,OptSpec,Strike,Settle,Maturity)` prices European basket options using the Nengjiu Ju approximation model.

## Input Arguments

 `RateSpec ` Annualized, continuously compounded rate term structure. For more information on the interest rate specification, see `intenvset`. `BasketStockSpec ` `BasketStock` specification. For information on the basket of stocks specification, see `basketstockspec`. `OptSpec` Character vector or `2`-by-`1` cell array of character vectors with values of `'call'` or `'put'`. `Strike` Scalar for the option strike price. `Settle ` Scalar of the settlement or trade date specified as a character vector or serial date number. `Maturity` Maturity date specified as a character vector or serial date number.

## Output Arguments

 `Price` Price of the basket option.

## Examples

Find a European call basket option of two stocks. Assume that the stocks are currently trading at \$10 and \$11.50 with annual volatilities of 20% and 25%, respectively. The basket contains one unit of the first stock and one unit of the second stock. The correlation between the assets is 30%. On January 1, 2009, an investor wants to buy a 1-year call option with a strike price of \$21.50. The current annualized, continuously compounded interest rate is 5%. Use this data to compute the price of the call basket option with the Ju approximation model.

```Settle = 'Jan-1-2009'; Maturity = 'Jan-1-2010'; % Define RateSpec Rate = 0.05; Compounding = -1; RateSpec = intenvset('ValuationDate', Settle, 'StartDates', ... Settle, 'EndDates', Maturity, 'Rates', Rate, 'Compounding', Compounding); % Define the Correlation matrix. Correlation matrices are symmetric, and % have ones along the main diagonal. Corr = [1 0.30; 0.30 1]; % Define BasketStockSpec AssetPrice = [10;11.50]; Volatility = [0.2;0.25]; Quantity = [1;1]; BasketStockSpec = basketstockspec(Volatility, AssetPrice, Quantity, Corr); % Compute the price of the call basket option OptSpec = {'call'}; Strike = 21.5; PriceCorr30 = basketbyju(RateSpec, BasketStockSpec, OptSpec, Strike, Settle, Maturity)```

This returns:

```PriceCorr30 = 2.12214```

Compute the price of the basket instrument for these two stocks with a correlation of 60%. Then compare this cost to the total cost of buying two individual call options:

```Corr = [1 0.60; 0.60 1]; % Define the new BasketStockSpec BasketStockSpec = basketstockspec(Volatility, AssetPrice, Quantity, Corr); % Compute the price of the call basket option with Correlation = -0.60 PriceCorr60 = basketbyju(RateSpec, BasketStockSpec, OptSpec, Strike, Settle, Maturity)```

This returns:

```PriceCorr60 = 2.27566```

The following table summarizes the sensitivity of the option to correlation changes. In general, the premium of the basket option decreases with lower correlation and increases with higher correlation.

 Correlation -0.6 -0.3 0 0.3 0.6 Premium 1.5283 1.76006 1.9527 2.1221 2.2756

Compute the cost of two vanilla 1-year call options using the Black-Scholes (BLS) model on the individual assets:

```StockSpec = stockspec(Volatility, AssetPrice); StrikeVanilla= [10;11.5]; PriceVanillaOption = optstockbybls(RateSpec, StockSpec, Settle, Maturity,... OptSpec, StrikeVanilla) ```

This returns:

```PriceVanillaOption = 1.0451 1.4186```

Find the total cost of buying two individual call options:

`sum(PriceVanillaOption) `

This returns:

`ans=2.4637`

The total cost of purchasing two individual call options is \$2.4637, compared to the maximum cost of the basket option of \$2.27 with a correlation of 60%.

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A basket option is an option on a portfolio of several underlying equity assets.

Payout for a basket option depends on the cumulative performance of the collection of the individual assets. A basket option tends to be cheaper than the corresponding portfolio of plain vanilla options for these reasons:

• If the basket components correlate negatively, movements in the value of one component neutralize opposite movements of another component. Unless all the components correlate perfectly, the basket option is cheaper than a series of individual options on each of the assets in the basket.

• A basket option minimizes transaction costs because an investor has to purchase only one option instead of several individual options.

## References

Nengjiu Ju. “Pricing Asian and Basket Options Via Taylor Expansion.” Journal of Computational Finance. Vol. 5, 2002.