Implied Volatility - Monte-Carlo Pricing - Performance of Delta Hedging - Accounting and Finance Assignment Help

Assignment Task:

1. Implied volatility

Sub bisecIVO extracts the implied volatility with a bisection convergence algorithm. Sub NewtonRaph son() does the same job but using a supposedly more efficient convergence algorithm. 
• Explain how both codes function by commenting in the VBA 
• Modify Newton-Raphson() so that it incorporates the tolerance parameter as an argument (input) instead of defining it in the code. - reports in the spreadsheet the number of iterations needed for convergence. 
• Compare both algorithms.

2. Monte-Carlo pricing 

Function MCVanilla() returns the Monte-Carlo estimate of a vanilla option. Modify this code such that it returns the x% interval of confidence for the option price as well as the running time. 
Optional: If you feel like having fun with barrier option you can also do the following Function cuo() has been coded so that it returns the Rubinstein and Reiner (1991) closed-form solution for the value of an up-and-out call option in a continuous-time when the underlying follows a geometric Brownian motion. 
Program function MCcuo() that returns the Monte Carlo estimate of up and outcall options that pay off at expiration max(ST - K;0) x 1s<h 

2.1 Monte Carlo simulation of the performance of the strategies for vanilla options:

We shall now investigate how the dynamic strategy performs in a Monte Carlo simulation framework. The objective is to programme subroutines in VBasic that return the mean gain or loss with standard deviation to the writer of the call following a delta-neutral hedging scheme for a number of simulations nbSimul.Your VBA code should be organized as follows:
• Step 1: Extract the relevant information from the Excel spreadsheet, send it to VBA and initialize the hedging portfolio composition.
• Step 2: Programma first loop (inner loop) on the number of rebalancing steps that draws a random return for the period - calculates the hedge ratio updates the portfolio composition and value.
• Step 3: After the end of the loop, store the terminal gain or loss for this sample-path.
• Step 4: Programma second loop (outer loop) on the number of simulations that re-runs the whole hedging strategy for another sample-path.
• Step 5: Return in the Excel spreadsheet the relevant performance measures (mean, standard deviation, min, max, ...)

2.2 Analysis:

Briefly analyze your results by explaining the impact of a change in the rebalancing frequency.

2.3 Style requirements and VBA code:

• Marks will be deducted for untidy, badly designed spreadsheets and badly presented code 
• Here are some tips
- Distinguish inputs and outputs in your spreadsheet 
- Format numbers to the appropriate number of decimal places 
- Please use comments in your VBA code where appropriate 
- In VBA use a variable naming style similar to the one in the notes (for ex. Option Type) 
- Don't forget Option Explicit 
– Indent IF ... THEN ... ELSE ... ENDIF statements and also FOR... NEXT loops 
where 1sch = 1 if St <H for any t < T and = 0 otherwise 
You can build your code by modifying the MCVanilla() function (you will need to simulate the whole trajectory instead of only the final price) or build a completely different code. Note that: 
• Your code need not store the whole trajectory. 
• Your code should also return the x% interval of confidence for the option price as well as the running time.

3. The performance of delta-hedging:

The objective of this exercise is to measure the relative performance of dynamic delta-hedging as a hedging tool for a short option position. To do so, you will have to programme a Monte-Carlo simulation in Excel VBA.

3.1 Delta hedging a short option position for a single trajectory:

Assume the situation is the one described in Hull (2012): you sold a European call option for $300,000. The call is written on 100,000 shares of a non-dividend paying stock with the following parameters:
 
• Current stock price = So = $ 49 
• Strike price = K = $ 50 
• Stock volatility = 0 = 20% 
• Risk-free interest rate = s = 5% 
• Option time to maturity = T = 20 weeks 
• Stock expected return = u = 13% 

In a spreadsheet named weekly_reb, simulate the trajectory of the stock price (assuming the distribution is lognormal with mean u and standard deviation o) and hedge the short position with weekly rebalancing with a delta-hedging strategy.
 
You need to:
 
• write VBasic functions that return the Black-Scholes price and the delta for both European call and put options on an underlying that pays dividends at a continuous rate q,
• simulate a random lognormal path for the stock price over the 20 weeks
• program the delta-hedging scheme and compute the total net gain or loss to the writer in each case.

In addition, you can add a button allowing to re-run the weekly rebalancing for a new trajectory.

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