Computational Finance

Lecture 7

The “Greeks”

Agenda

Sensitivity Analysis Delta and Delta hedging Other Greeks

One Example

A financial institution just sold a European call option on 100,000 share of a non-dividend-paying stock.

Information known: Current stock price: Strike price: Interest rate: Volatility: Time to maturity:

One Example

According to the B-S formula,

Thus, totally it should charge for about

$240,000.

How about When the Stock Price Changes?

Suppose one minute after the financial institution makes the deal, the stock price increases $1 per share to $50.

Now,

Totally, the call option price should be more than $295,000.

Sensitivity Analysis

Option prices are sensitive to the movement of the underlying asset prices.

People care about: How sensitive are their investment

portfolios? How can they avoid such risk? (risk

hedging)

Sensitivity Analysis and Function Derivatives

Mathematically, we can put the first question in the following way:

?

Sensitivity Analysis and Function Derivatives

Recall a basic concept in Calculus: function derivative

In other words, if is very small,

Sensitivity Analysis and Function Derivatives

Illustration of function derivative:

The Example Revisited

Taking derivative on function with respect to ;

Delta:

The Example Revisited

The Delta of the European call option is

In other words, as the underlying stock rises up $1 per share, the European call option is worth $52,160.2 more approximately.

Delta Hedging

Hedging the option risk using Delta:

The financial institution may hedge its position in the call option by buying

shares. When the stock increases $1 per share Option: loss = 295,444.7- 240,046.1=$55,398.6 Stock: gain = $52,160.2 Total change = 52,160.2- 55,398.6 = -$3,238.4

Static vs. Dynamic Delta Hedging

The hedge may be set up initially and never adjusted afterwards. Such hedging scheme is referred to as static-hedging or hedge-and-forget scheme.

Realize that the Delta is also a function of the underlying stock price. As the underlying price changes, the value of Delta changes too.

Dynamic Delta Hedging

Dynamic Delta hedging: Contrasted with the static hedging,

a hedging scheme is known as dynamic delta hedging if changing the holding of stocks periodically according to the updated value of Delta.

The Example Revisited

Take the previous example as an illustration of the operation of dynamic delta hedging.

Suppose that the hedge is adjusted or rebalanced weekly.

Procedure:

Stock Price Delta ( )Option: 1Stock:

The Example Revisited

In week 9, the stock price is $53 per share. According to the B-S formula, the call option value should be $414,500.

The financial institution loses

in its option position; but its stock holding appreciates

The Example Revisited

The total gain in its portfolio is $1,590,555.18-

$172,414.06=$1,418,141.12

The cumulative costs of dynamic hedging is $1,419,925.87.

Therefore, the total position change is

$1,418,141.12-$1,419,925.87=-$1,784.76.

Dynamic Delta Hedging

The risk profile of the financial institution remains unchanged if it does dynamic delta hedging when it trades options.

Other Greeks

Other Greeks used frequently in practice: Gamma: Rho: Vega: Theta:

Gamma

Gamma:

In words, Gamma is the sensitivity of Delta with respect to the underlying stock price. If Gamma is large, the Delta is very liable to

the price change of the stock. We need rebalance the portfolio frequently.

Gamma small. Rebalance infrequently.

Gamma

For European call or put options on the non-dividend-paying stocks:

Vega

Vega:

In words, it is the sensitivity of the option value with respect to the underlying stock’s volatility.

Volatility is usually unobservable in the market. Vega measures how severe the mistake could be if we use wrong volatility to price options.

Vega

For European call or put options on the non-dividend-paying stocks:

Rho

Rho:

In words, it is the sensitivity of the option value with respect to the risk free interest rate.

Prevailing risk free interest rates in an economy change from time to time. Rho gives us an idea how much the change impacts the options values.

Rho

For European call or put options on the non-dividend-paying stocks: Call:

Put:

Theta

Theta:

In words, the theta measures how fast the values of options changes as time passes by.

Reference

John Hull, Options, Futures and Other Derivatives, Sixth Edition, Prentice Hall, Upper Saddle River, NJ, 2006.

Chapter 15, the Greek Letters.