Financial Toolbox

Analysis Models

Toolbox functions for analyzing equity derivatives use the Black-Scholes model for European options and the binomial model for American options. The Black-Scholes model makes several assumptions about the underlying securities and their behavior. The binomial model, on the other hand, makes far fewer assumptions about the processes underlying an option. For further explanation, please see the book by John Hull listed in the Bibliography.

Black-Scholes Model

Using the Black-Scholes model entails several assumptions:

• The prices of the underlying asset follow an Ito process. (See Hull, 196 - 198)
• The option can be exercised only on its expiration date (European option).
• Short selling is permitted.
• There are no transaction costs.
• All securities are divisible and pay no dividends.
• There is no riskless arbitrage.
• Trading is a continuous process.
• The risk-free interest rate is constant and remains the same for all maturities.

If any of these assumptions is untrue, Black-Scholes may not be an appropriate model.

To illustrate toolbox Black-Scholes functions, this example computes the call and put prices of a European option and its delta, gamma, lambda, and implied volatility. The asset price is $100.00, the exercise price is $95.00, the risk-free interest rate is 10%, the time to maturity is 0.25 years, the volatility is 0.50, and the dividend rate is 0. Simply executing the toolbox functions

• [OptCall, OptPut] = blsprice(100, 95, 0.10, 0.25, 0.50, 0);
  [CallVal, PutVal] = blsdelta(100, 95, 0.10, 0.25, 0.50, 0);
  GammaVal = blsgamma(100, 95, 0.10, 0.25, 0.50, 0);
  VegaVal = blsvega(100, 95, 0.10, 0.25, 0.50, 0);
  [LamCall, LamPut] = blslambda(100, 95, 0.10, 0.25, 0.50, 0);
  

yields:

• The option call price OptCall = $13.70
• The option put price OptPut = $6.35
• delta for a call CallVal = 0.6665 and delta for a put PutVal = -0.3335
• gamma GammaVal = 0.0145
• vega VegaVal = 18.1843
• lambda for a call LamCall = 4.8664 and lambda for a put LamPut = -5.2528

Now as a computation check, find the implied volatility of the option using the call option price from blsprice.

• Volatility = blsimpv(100, 95, 0.10, 0.25, OptCall);
  

The function returns an implied volatility of 0.500, the original blsprice input.

Binomial Model

The binomial model for pricing options or other equity derivatives assumes that the probability over time of each possible price follows a binomial distribution. The basic assumption is that prices can move to only two values, one up and one down, over any short time period. Plotting the two values, and then the subsequent two values each, and then the subsequent two values each, and so on over time, is known as "building a binomial tree." This model applies to American options, which can be exercised any time up to and including their expiration date.

This example prices an American call option using a binomial model. Again, the asset price is $100.00, the exercise price is $95.00, the risk-free interest rate is 10%, and the time to maturity is 0.25 years. It computes the tree in increments of 0.05 years, so there are 0.25/0.05 = 5 periods in the example. The volatility is 0.50, this is a call (flag = 1), the dividend rate is 0, and it pays a dividend of $5.00 after three periods (an ex-dividend date). Executing the toolbox function

• [StockPrice, OptionPrice] = binprice(100, 95, 0.10, 0.25,... 
  0.05,  0.50, 1, 0, 5.0, 3); 
  

returns the tree of prices of the underlying asset

• StockPrice =
  
  100.00     111.27     123.87     137.96     148.69     166.28
       0      89.97     100.05     111.32     118.90     132.96
       0          0      81.00      90.02      95.07     106.32
       0          0          0      72.98      76.02      85.02
       0          0          0          0      60.79      67.98
       0          0          0          0          0      54.36
  

and the tree of option values.

• OptionPrice =
  
  12.10      19.17      29.35      42.96      54.17      71.28
      0       5.31       9.41      16.32      24.37      37.96
      0          0       1.35       2.74       5.57      11.32
      0          0          0          0          0          0
      0          0          0          0          0          0
      0          0          0          0          0          0
  

The output from the binomial function is a binary tree. Read the StockPrice matrix this way: column 1 shows the price for period 0, column 2 shows the up and down prices for period 1, column 3 shows the up-up, up-down, and down-down prices for period 2, etc. Ignore the zeros. The OptionPrice matrix gives the associated option value for each node in the price tree. Ignore the zeros that correspond to a zero in the price tree.

Sensitivity Measures

Analyzing Portfolios