Fischer Black and Myron Scholes made famous dynamic hedging. The basic element of this strategy is the creation of a portfolio containing stocks along with written call options for that stock. When the ratio of stocks to written calls is in the proper ratio the value of the portfolio is independent of infinitesimal fluctuations in the price of the stock.
Let S be the current price of the stock and C the price of a call option on that stock with an exercise price of X and with a duration T for the stock and let r be the risk-free interest rate and σ the volatility of the stock price. Furthermore let h be the hedge ratio and V the value of the portfolio. If N is the number of shares of stock in the portfolio and M is the number of written calls then:
When the stock price changes by an amount dS the price of the call changes by an amount dC. The change in the value of the portfolio is
Consider a stock with a price of $100 and a volatility of 0.2. When the risk-free interest rate is 10% (0.1) the price of a one-year call with an exercise price of $100 based upon the Black-Scholes formula is $12.993.
If the stock price were to go to $110.50 the price of the call would go to 13.354 whereas if the stock price fell to $99.50 the call price would fall to $12.636. The difference of these two call price is approximately the delta of the call option at a stock price of $100.00; i.e., δ = 0.718.
Suppose an investor wanted to create a hedged portfolio involving 1000 written call options. The payment the investor would receive would be $12,993. Since the hedge ratio is ).718 the investor would want to buy 718 shares of stock at $100 per share. This would require an outlay of $71,800. Since $12,993 is covered from the payment received for the written calls the investor would have to contribute an additional $58,807 for the portfolio. The value of the portfolio is $58,807 because $12,993 of the $71,800 in stock is offset by the negative value of the written calls.
Consider now what happens to the value of the portfolio if the stock price moves up to $100.50. The negative value of the written calls increases from $12,993 to $13,354. The value of the shares held increases from $71,800 to $72,159, an increase of $359. The increased cost of the written calls is $361, almost exactly offset by the increase $259 in the value of the stock.
If the stock price moves down to $99.50 there is a loss in the value of the stock of $359 but since the call price decrease to $12.636 the cost of the written calls has falled by $357, almost exactly offsetting the decrease in stock value.
Although the portfolio is perfectly hedged against small changes in stock price this is not true for large price changes. For example, suppose the price of the stock falls from $100 to $0. The value of the stock in the portfolio goes to zero. The price of a call goes down to essentially zero also so the portfolio has a value of zero, a drop from $58,807. So a price decrease of $100 produces a loss of $58,807.
On the other hand consider an increase of $100 to stock price of $200 per share. The value of the stock in the portfolio doubles from $71,800 to $143,600 but the price of the call rises to $109.091 and the cost of the written calls is $109,091 which leaves a net value of the portfolio of $34,509, a drop of $24,298 from its original $58,807 value. Thus portfolios that are perfectly hedged against small changes in stock price are vulnerable to losses from large increase or decreases in stock price.
An interest variation in the Black-Scholes hedging can be created by selling stock short and buy call options. The above number can be used to illustrate this strategies. Suppose the stock is sold short at $100 a share and the investor buys 1000 call options. In order to maintain a hedge ratio of 0.718 the investor would sell short 718 shares. The cash in the portfolio would be $71,800 from the short sale which would counterbalance the shares owed from the short sale. The investor would have to contribute $12,993 that the 1000 call options cost. The net value would then be $0. Now consider the consequences of a small increase in the stock price to $100.50. The cost of owed shares is increased by $359, but the value of the owned calls increases by $361 just about exactly offsetting the increased cost of satisfying the short sales. On the other hand if the stock price decreases to $99.50 the cost of the short sales decrease by $359 and the value of the owned calls decreases by (12,993-12,636)=$357, the two changes essentially offsetting one another.
If the stock price fell to zero there would be no cost for satisfying the short sales. Other other hand the value of the owned calls also falls to zero. But the portfolio has cash equal to the proceeds of the short sale of $71,800 so the gain in the value of the portfolio is $71,800. On the other hand if the price of the stock went to $200 per share then the cost of satisfying the short sales rises from the original $71,800 to $143,600. However the value of the owned calls increases to $109,091. Thus the value of the portfolio is $71,800 in cash minus $143,600 for the shares owed plus $109,091 from the owned calls for a net value of $37,291. This is a gain of $37,291 from the original $0 net value. Thus this hedged portfolio of short sales combined with call options is protected against changes in value due to small changes in prices but it functions like a straddle with respect of large increases or decreases in prices. This is an interest contrast with the standard Black-Scholes hedged portfolio that loses money with large price changes.
Consider a portfolio made up of shares and put options. Let N be the number of shares and M the number of put options. The value of a put option is denoted as P. Then the value of the portfolio is:
If dV is to be zero then the hedge ratio h must be equal to to the negative of ∂P/∂S, the delta of the put option. Since the delta of a put option is negative the negative of a negative produces a positive hedge ratio. For the stock consider in the example involving written call options the value of a put for an exercise price of $100 is $3.92 and the delta of the put option at that price is 0.282. The reader will note that this is the complement of the delta for the call option; i.e. δput = 1.0 - δcall. This follows from the put-call parity formula P = C +PVX - S since differentiation with respect to S gives:
Suppose an investor buys 1000 put options. For a hedge ratio of 0.282 the investor would buy 282 shares of stock. The cost of the stock would be $28,200 and the 1000 put options $3902 for a total portfolio value of $32,102.
Suppose now that the stock price increases to $100.50. There would be a gain of $141 in stock value but since the value of a put goes down to $3.763 there would be a loss of $139 in the value of the puts. The two changes just about exactly offset one another. If the stock price falls to $99.50 there would be a loss of $141 in stock value but the value of a put increases to $4.045 and therefore there is a gain in put value of $143 which essentially offsets the loss in stock value.
For a large change in price, say from $100 to $0, the loss in stock value is $28,200 but the gain in put value is from $3,902 to $90,909, a gain of $87,007 for a net gain in portfolio value of 58,807. On the other hand if the share price increases to $200 the value of the stock doubles to $56,400 but the value of the puts goes down to $0. The value of the portfolio has increased from $32,102 to $56,400 for a net increase of $24,298. So large price changes bring substantial increases in the value of the portfolio.
A mirror image portfolio involving the short sale of stock with the sale of written puts would, with the right hedge ratio, be insulated against small price changes but large price changes would produce losses in portfolio value.
Synthetic shares and short sales can be created with combinations of puts, calls and interest-earning bank accounts. The combination of one written put with exercise price X with one call with the same exercise price along with a bank account having a value of X on expiration day is equivalent to a share. Suppose this synthetic share is substituted for a share in a Black-Scholes hedged portfolio. If N is the number of synthetic shares, M the number of written calls then
If the price of the stock were to go to $100.50 the cost of the written puts would go to $2,702 a decrease of $100. The value of the written calls would go to $3,766 an increase of $102, an almost exact offset for the change in the value of the puts. Similarly a decrease in stock price brings a nearly exact offset and no net change in the value of the portfolio.
For a large price increase to $200 per share the effects are that the value of the puts go to zero and the value of the calls goes to $30,764 on the negative side because the calls are written calls. This means the cost of the written calls to the investor increased from $3,664 to $30,764, a loss for the portfolio holder of $27,100. This is only partially offset by the decline in the cost of the written puts from $2,802 to 0. The net loss on the portfolio as a result of the stock price increase is $24,298.
Likewise if the price of the stock fell by $100 to $0 the cost of the written put options would go to $65,273 and there would only be a $3,664 gain when the cost of the written calls went to zero. Thus the loss would be $61,609.
The mirror image portfolio would involve buying calls and puts. In the proper ratio this portfolio would be insulated against small price fluctuations but would gain from large price changes in either directions. This is a form of a straddle. By the put-call parity formula: