Pricing and Analyzing Equity Derivatives

Introduction

These toolbox functions compute prices, sensitivities, and profits for portfolios of options or other equity derivatives. They use the Black-Scholes model for European options and the binomial model for American options. Such measures are useful for managing portfolios and for executing collars, hedges, and straddles:

• A collar is an interest-rate option that guarantees that the rate on a floating-rate loan will not exceed a certain upper level nor fall below a lower level. It is designed to protect an investor against wide fluctuations in interest rates.

• A hedge is a securities transaction that reduces or offsets the risk on an existing investment position.

• A straddle is a strategy used in trading options or futures. It involves simultaneously purchasing put and call options with the same exercise price and expiration date, and it is most profitable when the price of the underlying security is very volatile.

Sensitivity Measures

There are six basic sensitivity measures associated with option pricing: delta, gamma, lambda, rho, theta, and vega — the “greeks.” The toolbox provides functions for calculating each sensitivity and for implied volatility.

Delta

Delta of a derivative security is the rate of change of its price relative to the price of the underlying asset. It is the first derivative of the curve that relates the price of the derivative to the price of the underlying security. When delta is large, the price of the derivative is sensitive to small changes in the price of the underlying security.

Gamma

Gamma of a derivative security is the rate of change of delta relative to the price of the underlying asset; that is, the second derivative of the option price relative to the security price. When gamma is small, the change in delta is small. This sensitivity measure is important for deciding how much to adjust a hedge position.

Lambda

Lambda, also known as the elasticity of an option, represents the percentage change in the price of an option relative to a 1% change in the price of the underlying security.

Rho

Rho is the rate of change in option price relative to the risk-free interest rate.

Theta

Theta is the rate of change in the price of a derivative security relative to time. Theta is usually small or negative since the value of an option tends to drop as it approaches maturity.

Vega

Vega is the rate of change in the price of a derivative security relative to the volatility of the underlying security. When vega is large the security is sensitive to small changes in volatility. For example, options traders often must decide whether to buy an option to hedge against vega or gamma. The hedge selected usually depends upon how frequently one rebalances a hedge position and also upon the standard deviation of the price of the underlying asset (the volatility). If the standard deviation is changing rapidly, balancing against vega is preferable.

Implied Volatility

The implied volatility of an option is the standard deviation that makes an option price equal to the market price. It helps determine a market estimate for the future volatility of a stock and provides the input volatility (when needed) to the other Black-Scholes functions.

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 Black-Scholes model was the first complete mathematical model for pricing options, developed by Fischer Black and Myron Scholes. It examines market price, strike price, volatility, time to expiration, and interest rates. It is limited to only certain kinds of options.

The binomial model, on the other hand, makes far fewer assumptions about the processes underlying an option. A binomial model is a method of pricing options or other equity derivatives in which 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 higher and one lower) over any short time period. For further explanation, see Options, Futures, and Other Derivatives by John Hull in Bibliography.

Black-Scholes Model

Using the Black-Scholes model entails several assumptions:

• The prices of the underlying asset follow an Ito process. (See Hull, page 222.)

• 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.

• There is no riskless arbitrage (where arbitrage is the purchase of securities on one market for immediate resale on another market to profit from a price or currency discrepancy).

• 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)

OptCall =

   13.6953


OptPut =

    6.3497


CallVal =

    0.6665


PutVal =

   -0.3335


GammaVal =

    0.0145


VegaVal =

   18.1843


LamCall =

    4.8664


LamPut =

   -5.2528

To summarize:

• 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)

Volatility =

    0.5000

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 =

  100.0000  111.2713  123.8732  137.9629  148.6915  166.2807
         0   89.9677  100.0495  111.3211  118.8981  132.9629
         0         0   80.9994   90.0175   95.0744  106.3211
         0         0         0   72.9825   76.0243   85.0175
         0         0         0         0   60.7913   67.9825
         0         0         0         0         0   54.3608

and the tree of option values.

OptionPrice =

   12.1011   19.1708   29.3470   42.9629   54.1653   71.2807
         0    5.3068    9.4081   16.3211   24.3719   37.9629
         0         0    1.3481    2.7402    5.5698   11.3211
         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, and so on. 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.