ECONOMICS DEPARTMENT
Thayer Watkins
and Its Variations
Fischer Black and Myron Scholes made famous dynamic hedging. The basicelement of this strategy is the creation of a portfolio containing stocksalong with written call options for that stock. When the ratio of stocks to written calls is inthe proper ratio the value of the portfolio is independent of infinitesimalfluctuations in the price of the stock.
Let S be the current price of the stock and C the price of a call optionon that stock with an exercise price of X and with aduration T for the stock and let r be the risk-free interestrate and σ the volatility of the stock price. Furthermore let h bethe hedge ratio and V the value of the portfolio. If N is the number ofshares of stock in the portfolio and M is the number of written calls then:
V = NS - MC
V = MhS - MC
V/M = hS - C.
When the stock price changes by an amount dS the price of the callchanges by an amount dC. The change in the value of the portfolio is
dV = MhdS - MdC
if dV is to be zero then h must be such that
hdS - dC =0
h = ∂C/∂S,
this is called the delta of the call option.
Thus the proper hedge ratio for the portfolio is the delta of theoption.Consider a stock with a price of $100 and a volatility of 0.2. When the risk-free interest rateis 10% (0.1) the price of a one-year call with an exercise price of $100 basedupon 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 pricefell to $99.50 the call price would fall to $12.636. The difference of thesetwo call price is approximately the delta of the call option at a stockprice of $100.00; i.e., δ = 0.718.
Suppose an investor wanted to create a hedged portfolio involving1000 written call options. The payment the investor would receive wouldbe $12,993. Since the hedge ratio is ).718 the investor would want to buy718 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 thewritten calls the investor would have to contribute an additional $58,807for the portfolio. The value of the portfolio is $58,807 because $12,993of the $71,800 in stock is offset by the negative value of the writtencalls.
Consider now what happens to the value of the portfolio if the stockprice moves up to $100.50. The negative value of the written calls increasesfrom $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 thewritten calls is $361, almost exactly offset by the increase $259 in the valueof the stock.
If the stock price moves down to $99.50 there is a loss in the value of thestock of $359 but since the call price decrease to $12.636 the cost of thewritten calls has falled by $357, almost exactly offsetting the decrease instock value.
Although the portfolio is perfectly hedged against small changes instock 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 thestock in the portfolio goes to zero. The price of a call goes down toessentially zero also so the portfolio has a value of zero, a dropfrom $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 andthe cost of the written calls is $109,091 which leaves a net value ofthe portfolio of $34,509, a drop of $24,298 from its original $58,807value. Thus portfolios that are perfectly hedged against small changesin stock price are vulnerable to losses from large increase or decreasesin stock price.
An interest variation in the Black-Scholes hedging can be created byselling stock short and buy call options. The above number can be used toillustrate this strategies. Suppose the stock is sold short at $100 ashare and the investor buys 1000 call options. In order to maintaina hedge ratio of 0.718 the investor would sell short 718 shares. The cashin the portfolio would be $71,800 from the short sale which would counterbalancethe shares owed from the short sale. The investor would have to contribute $12,993 thatthe 1000 call options cost. The net value would then be $0. Now considerthe consequences of a small increase in the stock price to $100.50. Thecost of owed shares is increased by $359, but the value of the ownedcalls increases by $361 just about exactly offsetting the increased costof satisfying the short sales. On the other hand if the stock price decreasesto $99.50 the cost of the short sales decrease by $359 and the value of theowned calls decreases by (12,993-12,636)=$357, the two changes essentiallyoffsetting one another.
If the stock price fell to zero there would be no cost for satisfyingthe short sales. Other other hand the value of the owned calls alsofalls to zero. But the portfolio has cash equal to the proceeds of theshort 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 pershare 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 $143,600for 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. Thusthis hedged portfolio of short sales combined with call options isprotected against changes in value due to small changes in prices but itfunctions like a straddle with respect of large increases or decreases inprices. This is an interest contrast with the standard Black-Scholeshedged portfolio that loses money with large price changes.
Put Hedged Portfolio
Consider a portfolio made up of shares and put options. LetN be the number of shares and M the number of put options. The valueof a put option is denoted as P. Then the value of the portfolio is:
V = NS + MP
so a change in stock price of dS results in
dV = NdS + MdP.
If dV is to be zero then the hedge ratio h must be equal to to thenegative of ∂P/∂S, the delta of the put option. Since thedelta of a put option is negative the negative of a negative producesa positive hedge ratio. For the stock consider in the example involvingwritten 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 thecall option; i.e. δput = 1.0 - δcall.This follows from the put-call parity formula P = C +PVX - S sincedifferentiation with respect to S gives:
∂P/∂S = ∂C/∂S - 1
so
- ∂P/∂S = 1 - ∂C/∂S
Suppose an investor buys 1000 put options. For a hedge ratio of0.282 the investor would buy 282 shares of stock. The cost of thestock would be $28,200 and the 1000 put options $3902 for a totalportfolio value of $32,102.
Suppose now that the stock price increases to $100.50. There wouldbe a gain of $141 in stock value but since the value of a put goesdown 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 stockprice falls to $99.50 there would be a loss of $141 in stock value butthe value of a put increases to $4.045 and therefore there is again in put value of $143 which essentially offsets the loss in stockvalue.
For a large change in price, say from $100 to $0, the loss in stockvalue 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 ofthe stock doubles to $56,400 but the value of the puts goes downto $0. The value of the portfolio has increased from $32,102 to $56,400for a net increase of $24,298. So large price changes bring substantialincreases in the value of the portfolio.
A mirror image portfolio involving the short sale of stock with thesale of written puts would, with the right hedge ratio, be insulatedagainst small price changes but large price changes would produce lossesin portfolio value.
A Hedged Portfolio Without Stock Shares
Synthetic shares and short sales can be created with combinations ofputs, calls and interest-earning bank accounts. The combination ofone written put with exercise price X with one call with the same exerciseprice along with a bank account having a value of X on expiration day isequivalent to a share. Suppose this synthetic share is substituted fora share in a Black-Scholes hedged portfolio. If N is the number of syntheticshares, M the number of written calls then
V = N(C - P + (pvX)) - MC = (N-M)C - NP + NpvX)
and
N/M = h = ∂C/∂S.
The portfolio matching the first example would involve 1000 writtencalls with 718 synthetic shares which consist of 718 written puts and718 owned calls. The net number of written calls would be 282 sothe portfolio would consist of a bank account with a value equal tothe exercise price of the put options on expiration day; i.e.,$71,800. The payment received for the 718 written puts would be $2,802 and$3,664 forthe 282 written calls. Therefore the portfolio would be roughly $78 thousandin bank accounts and 718 written puts and 282 written calls for a net valueof $71,800.If the price of the stock were to go to $100.50 the cost of the writtenputs would go to $2,702 a decrease of $100. The value of the written callswould go to $3,766 an increase of $102, an almost exact offset for thechange in the value of the puts. Similarly a decrease in stock price bringsa 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 thevalue of the puts go to zero and the value of the calls goes to $30,764on the negative side because the calls are written calls. This means thecost 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 partiallyoffset 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 thewritten 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 theloss would be $61,609.
The mirror image portfolio would involve buying calls and puts. In theproper ratio this portfolio would be insulated against small pricefluctuations but would gain from large price changes in either directions.This is a form of a straddle. By the put-call parityformula:
S = C - P + PVX.
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