Financial Engineering 6

Lecture 26

Introduction to Dynamic Hedging

References: Villalobos

Lecture Topics • Brief Introduction to Dynamic Hedging • Example • Lookup Dynamic Hedging in the Investopedia site

Dynamic Hedging • Clients of derivatives dealers routinely want to buy a call, buy a

put, buy a cap, or buy some exotic derivative.

• Rarely does a client contact a derivatives dealer and ask to sell an option.

• Dealers are left holding massive short options positions. – To hedge those positions, they would like to purchase

offsetting long options, but there is no one to buy them from.

• The solution is to dynamically hedge the short options positions.

Dynamic Hedging • Dynamic hedging is delta hedging of a non-linear position with

linear instruments like spot positions, futures or forwards.

• The deltas of the non-linear position and linear hedge position offset, yielding a zero delta overall.

• However, as the underlying security’s value moves up or down, the delta of the non-linear position changes while that of the linear hedge does not.

• The deltas no longer offset, so the linear hedge must be adjusted, increased or decreased, to restore the delta hedge.

• This continual adjusting of the linear position to maintain a delta hedge is called dynamic hedging.

Example • A derivatives dealer sells a client a put option on STU Corp.

stock. • At the current stock price of $100, the short option position has

a delta of 22,000 shares. • The figure below illustrates the market value of the short option

position as a function of the underlying stock's price. • A tangent line has been fit to that graph, and its positive slope

indicates the positive delta.

Example • To delta hedge the short put, the dealer sells 22,000 shares of

STU stock. • The deltas of the short option and the short stock cancel,

yielding an overall delta of zero. • The market value of the hedged position as a function of the

stock price is shown in the figure below. • A tangent line fit to that graph has zero slope, indicating zero

delta.

Example • With the underlying stock price at $100, the position is delta

hedged, but this doesn't last long. • Soon the stock price rises to $103. • As indicated in figure below, at that stock price, the position

has a slightly negative delta. • It is no longer delta hedged.

Example • At the new stock price, the derivatives dealer adjusts the delta

hedge, buying back some of the underlying stock he had previously shorted.

• The result is a newly delta hedged position at the new stock price of $103.

Example • The continual readjustment of the delta hedge ensures that the

portfolio always has a zero delta. – And, that it loses only a little value each time the underlying

stock price moves. • A delta-hedged negative gamma portfolio loses money

irrespective of whether the underlying stock rises or falls. • We lose money dynamically hedging a negative gamma

position. – We make money dynamically hedging a positive gamma

position. • When Black and Scholes published their option pricing formula,

they asserted that the price of an option should be the discounted value of the cost with dynamically hedging to expiration.

– Black and Scholes' analysis assumed that the underlier's volatility is constant over time.

– Remember, LTCM’s model volatility did not match reality.

Assignments • Lookup Dynamic Hedging in the Investopedia site.