Assignment 5 - Modelling the Volatility Smile.

The Black Scholes Option Pricing Model can be modified to value options on securities paying a continuous dividend. This modification is known as the Garman Kohlhagen model and is frequently applied to options on currencies and stock market indices.

Whereand.

Required:

1.Build a function to value options using the Garman Kohlhagen model above.

2.Build a function to calculate implied volatility.

3.Use the implied volatility function to compute the market volatility values associated with the strike prices and option prices supplied in the spreadsheets Option Data A, Option Data B and Option Data C depending on the letter assigned to you.

4.Sketch the volatility smile, write a short explanation of the volatility smile.

Price data forcalloptions onthe S&P 500

Pricing Date 7th June 2021

Expiration Date 17th June 2022

Valuation Date 7 June 2021 Spot Price 4218.54 Interest Rate 0.15% Dividend Yield 1.50% Ticker Strike Bid Ask SPX 6/17/22 C4000 4000 433.3 435.4 SPX 6/17/22 C4025 4025 416 418.1! SPX 6/17/22 C4050 4050 398.8 400.9 SPX 6/17/22 C4075 4075 381.9 384 SPX 6/17/22 C4100 4100 365.3 367.3 SPX 6/17/22 C4125 4125 348.9 350.9 SPX 6/17/22 C4150 4150 332.7998 334.8 SPX 6/17/22 C4175 4175 316.8999 319 SPX 6/17/22 C4200 4200 301.3999 303.5 SPX 6/17/22 C4225 4225 286.0999 288.3! SPX 6/17/22 C4250 4250 271.2 273.2998 SPX 6/17/22 C4275 4275 256.7 258.6001 SPX 6/17/22 C4300 4300 242.4 244.4 SPX 6/17/22 C4325 4325 228.6 230.6 SPX 6/17/22 C4350 4350 215.2 217.1 SPX 6/17/22 C4375 4375 202.2 204.1 SPX 6/17/22 C4400 4400 189.7 191.6 SPX 6/17/22 C4425 4425 177.6 179.5 SPX 6/17/22 C4450 4450 166 167.9 SPX 6/17/22 C4475 4475 155 156.8