1. RISK MANAGEMENT WITH PROTECTIVE PUT AND

2. Stock-Put Insurance Suppose you want to protect a diversified portfolio such as below as an anticipated market downturn. You could protect the portfolio by buying put options or writing covered calls. Three questions arise: How many puts or call options do you need to buy? What is the best way to achieve the protection? What would be the cost of insurance if market prices rise?

3. Protective Put Based on Minimum Insured Value Suppose we own a portfolio consisting of N S shares of stock and want to insure the portfolio against a fall in the value. We can insure the position with buying N P puts. What is the portfolio position? This is a synthetic call option. The purchase of the put option is also called a Floor, because the Portfolio is guaranteed a minimum value. The put limits losses and permits us to gain if the price increases.

4. Protective Put Based on Minimum Insured Value Assume the stock price is S, and the put price is P. The puts are European, and we assume no dividends on the stock. The value of the portfolio is: V = N S S +N P P Letting N S = N P and calling this N, we have

5. Protective Put Based on Minimum Insured Value This equation tells us how many shares of stock and how many puts we can buy. At expiration the portfolio's value is V T = N S T if S T > E V T = N S T + N(E - S T ) = NEif S T < E where S T is the stock price when the put expires.

6. Minimum Insured Value of Portfolio The worst possible outcome is when S T =0. Suppose we define V min as the minimum value of V T, which occurs when S T =0. Then V min = NE, since N is also equal V/(S+P), This establishes the minimum insured value of portfolio at expiration.

7. Example: On Friday November 20, faced with the prospect of falling stock prices in month of December, upcoming Fed policy, end of the year portfolio window dressing, and cash inflows from investors for their IRA accounts, a portfolio manager wants to protect the aggregate value of his portfolio without liquidating the equity shares. The portfolio has accumulated an impressive return, and the manager wants to protect it against any decline in the stock market. Consider the following portfolio:

8. Portfolio Information Industry BetaMarket Value Asset Allocation Energy1.15$3,800,00019% Chemicals0.952,800,00014% Forest Products1.24,400,00022% Aerospace1.44,400,00022% Utilities0.851,600,0008% Technology1.13,000,00015% Total1.1620,000,000100%

9. Option Information Calls StrikeContract NameLast Implied Volatility 920OEX160115C00920000 2216.69% 930OEX160115C00930000 18.514.53% 940OEX160115C00940000 15.314.77% 950OEX160115C00950000 11.8813.98% 960OEX160115C00960000 6.9213.30% 970OEX160115C00970000 6.313.39% 980OEX160115C00980000 2.113.82% Puts 910OEX160115P00910000 14.9518.37% 930OEX160115P00930000 19.916.69% 940OEX160115P00940000 27.915.82%

10. Protective Put The purchase of a put option can hedge the downside risk of underlying portfolio. This strategy is called a protective put (synthetic Call) that is often referred as portfolio insurance. Synthetic Call = Long cash + long put

11. Example On November 20 the S&P100 index is at 933 and the January 940 put options on the index is priced at $27.9 as shown on Table 1. The option expires on January 15, which is 55 days away, so the time to expiration is 55/365=0.15. The risk free rate is 3.00 percent continuously compounded. The standard deviation is 0.9452%. Given the information about the five factors, the put value is estimated by:

12. Example Black-Scholes Option Valuation (No Dividends) SEtr(f)σ 9339400.150.07%15.82% 3-month T 0.070.0007 d1-0.0896N(d1)0.4643 d2-0.1509N(d2)0.4400 C 19.6011 P 26.5024 PV(E)939.9013 C-P+PV(E)933.0000 S

13. Value of the Minimum Variance Portfolio Suppose the portfolio is constructed in such way to replicates the market index. The portfolio is worth $20 million, which is equivalent to 21739 (20,000,000/920) units of the index. The minimum level of the portfolio is: Thus the minimum level at which we can insure the portfolio is $19,564,991. This means that we can own N shares and N puts, where

14. Value of the Minimum Variance Portfolio Suppose we buy 20814 shares and 20814 puts. Suppose also that at expiration the market index is at 960, then the value of the portfolio is: Value of the stock = 20814(960) = $19,981268 + Value of puts= 20814(0) = 0 Total$19,981268 This exceeds the minimum value. $ 19,564,991.15

15. Value of the stock portfolio If at expiration the index is at 920, then: Value of the stock portfolio = 20814(940) =$19,564,991 by exercising the puts.

16. Cost of Portfolio Insurance There are several different costs such as commissions, bid-ask spreads, and so on. But here the cost of portfolio insurance is the difference between the return of the insured portfolio and the return of uninsured portfolio when the market goes up.

17. Cost of Portfolio Insurance For example, when the market index went to 960, the value of the portfolio is equal to $20,578,778 =(21436X960), so the cost of insurance is equal to: Cost of Insurance = $20,578,778,565 –$19,564,991 =$1,013,787. The cost is about 5% of the portfolio’s original value. The difference between 100 percent and the cost of insurance 5%, or 95%, is called the upside capture. It is the percentage of the uninsured return in a bull market that is earned by the insured portfolio.

18. Protective Put Based Hedge Ratio In this method we use both the systematic risk of the portfolio (Beta of the portfolio) and the delta of the put option to determine the number of put we need to buy. As we know the delta of the put option is equal to absolute value of |1- N (d 1 )| because put increases in value as the stock prices go down. In the case of the above put option Delta = |N (d 1 )-1|= 1-N (d 1 )= 0.5357

19. Protective Put Based Hedge Ratio Should buy 461 puts to hedge the portfolio. Need to buy 461 OEX Jan 940 puts to hedge the portfolio. Suppose that after you buy 461 puts, the market dropped by 5%. This means the S&P100 index will decline to $888.

20. Index declining causes puts value to increase to $57. We purchased 461 at 27.9 and we save 100 x 461($57-27.9)= $1,331,276. The gain on put cancells the losses on portfolio.

21. Writing Covered Calls Writing covered calls is an alternative to protective puts. Appropriate when an investor owns the stock, does not want to sell it, and expects a decline in the stock price. However, it is an imperfect form of portfolio protection.

22. Writing Covered Calls The premium for September 940 call is $15. 3 which means that the price has to decline below $924.70 (Index Value –Call Premium) before an actual loss to occur. Of course, if index value advances above 940, the risk of being called is real and we need to close the position. The strategy of writing covered call provides some protection against market decline but limits the profit if the market increases. So it’s better to use put protection.