Stock futures

Below are links to the following topics:

• Risk Management with Stock Index Futures
• A Short Hedge and Hedge Ratio Calculation
• A Long Hedge
• A Real World Example of Index Arbitrage
• Hedging with Stock Index Futures
• Altering the Beta of a Portfolio
• Hedge Fund Uses of Stock Index Futures
• Portfolio Insurance
• Combining Options with Bonds and Stocks
• Stock Price Movements

Risk Management with Stock Index Futures

Hedging with stock index futures applies directly to the management of stock portfolios. The usefulness of stock index futures in portfolio management stems from the fact that they directly represent the market portfolio. Before stock index futures began trading there was no comparable way of trading an instrument that gave a price performance that was so directly tied to a broad market index. Further, stock index futures have great potential in portfolio management due to their very low transaction costs. In this section, we consider some hedging applications of stock index futures.

A Short Hedge and Hedge Ratio Calculation

As a first case, consider the manager of a well-diversified stock portfolio with a value of $40 million, and assume that the portfolio has a beta of 1.22, measured relative to the S&P 500. This implies that a movement of 1 percent in the S&P 500 index would be expected to induce a change of 1.22 percent in the value of the stock portfolio. The portfolio manager fears that a bear market is imminent and wishes to hedge his portfolio's value against that possibility. One strategy would be to liquidate the portfolio and place the proceeds in short-term debt instrument and then, after the bear market, return the funds to the stock market. Such a plan is infeasible. First, the transaction costs from such a strategy are quite high. Second, if the fund is large, liquidating the portfolio could drive down stock prices. This would prevent the portfolio manager from liquidating the portfolio at the prices currently quoted for the individual stocks.

As an obvious alternative to liquidating the portfolio, the manager could use the S&P 500 stock index futures contract. By selling futures, the manager should be able to offset the effect of the bear market on the portfolio by generating gains in the futures market. One kind of naïve strategy might involve selling one dollar of the value underlying the index futures contract for each dollar of the portfolio's value. Assuming that the S&P index futures contract stand at 1,060.00, the advocated futures position would be give by

V_P/V_F = - $40,000,000 / 1,060 x $250 = -250.94 = -150.00 contracts

where V_P is the value of the portfolio and V_F is the value of the futures contract.

Table 7.10 A short hedge

Date	Stock market	Futures Market
May 19	Hold $40 million in a stock portfolio	Sell 150 S&P 500 December futures contracts at 1,060.00
August 15	Stock portfolio falls by 5.40% to $37,838,160	S&P futures contract falls by 4.43% to $1,013.00
	Loss: -$2,161,840	Gain: 47 basis points x $250 x 150
	Net loss: -$399,340	contracts = $1,762,500

One problem with this approach is that it ignores the higher volatility of the stock portfolio relative to that of the S&P 500 index. As noted previously, the beta of the stock portfolio as measured against the index, was 1.22. Table 7.10 shows the potential results of a hedge consistent with these facts. The portfolio manager initiates the hedge on March 14, selling 150 DEC futures contracts against the $40 million stock portfolio. By August 16, his fear have been realized and the market has fallen. The S&P index, and the futures, have both fallen by 4.43 percent to 1,013. The stock portfolio, with its greater volatility, has fallen 1.22 times as much (1.22 x 4.43 percent), generating a loss of $2,161,840. This leaves a net loss on the hedge of $390,340. The failure to consider the differential volatility between the stock portfolio and the index futures contract leads to suboptimal hedging results.

The manager might be able to avoid this result by weighing the hedge ratio by the beta of the stock portfolio. According to this scenario, the manager could use the following equation to find the number of contracts to trade:

Equation 7.6: -β_P(V_P/V_F) = number of contracts

where β_P is the beta of the portfolio that is being hedged. Using this approach for our example, the manager would sell 185 contracts.

-1.22($40,000,000/1,060 x $250) = -184.15 = ~185 contracts

Had the manager traded 185 contracts, the futures gain reported in Table 7.10 would have been $2,173,750 instead of $1,762,500. This higher gain would have almost exactly offset the loss on the spot position of $2,161,840. Note, however, that these excellent results depend on two crucial assumptions. First, such results could be achieved only if the movement of the stock portfolio during the hedge period exactly corresponded to the volatility implied by its beta. Second, the technique of Equation 7.6 uses the beta of the stock portfolio as measured against the S&P 500 index itself. This assumes that the futures contracts move exactly in tandem with the spot index. This assumption is clearly violated by recent market experience, because the futures contracts for all of the indexes are more volatile than the indexes themselves. This is reflected by the fact that the futures contracts generally have betas above 1.0 when they are measured relative to the stock index itself. The methodology of Equation 7.6 does not take this into account, since it implicitly assumes the index and the futures contracts to have the same price movements, which would imply equal betas. We will consider more sophisticated approaches to this type of hedging problem later.

A Long Hedge

As with other futures contracts, both long and short hedges are possible in stock index futures. Imagine a pension fund manager who is convinced that she stands at the beginning of an extended bull market in Japanese equities. The current exchange rate is $1 = ¥140. She anticipates that ¥6 billion = $42,857,143 = $43,000,000 in new funds will become available in three months for investment. Waiting three months for the funds to invest in the stock market could mean that the bull market would be missed altogether. An alternative to missing the market move would be to use the stock index futures market. The pension manager could simply buy an amount of a stock index futures contract that would be equivalent in dollar commitments to the anticipated inflow of investable funds. On May 19, with the Nikkei index futures contract standing at 14,400, the CME futures contract represents an underlying cash value of $72,000 ($5 per index point). The pension manager can secure her position in the market by buying $43 million worth of futures. Since she expects the funds in three months, the SEP contract is a natural expiration to use, so she buys 600 SEP contracts, as shown in Table 7.11, because $43,000,000/$72,000 = 597.22 ~ 600. By August 15, the Nikkei spot and futures have risen 2.5 percent or 360 points, to 17,760. Therefore, the ¥6 billion could not buy the same shares that would have been possible on May 19. To offset this fact, the pension manager has earned a futures profit of $1,080,000. This gain in the futures market helps offset the new higher prices that would be incurred in the stock purchase. Because the CME Nikkei contract is based on the Nikkei index but quoted in dollars this strategy still leaves the traded exposed to exchange rate risk, which could be offset by an exchange rate hedge.

Table 7.11 A long hedge with stock index futures

Date	Stock market	Futures Market
May 19	A pension fund manager anticipates having ¥6 billion to invest in Japanese equities in three months	Buys 600 SEP Nikkei futures on the CME at 14,400
August 15	¥6 billion becomes available for investment, stock prices have risen, so the ¥6 billion will not buy the same shares that it would have on May 19	The market has risen and the Nikkei futures stand at 14,760
		Futures profit: 360 points x $5 x 600 contracts = $1,080,000

A Real-World Example of Index Arbitrage

The futures price that conforms to the cost-of-carry model is called the fair-value futures price. In this section, we consider an example of determining the fair value of the December 2001 stock index futures contract traded on November 30, 2001.

The December 2001 S&P 500 stock index futures closed at 1,140 index points on November 30. The cash index price this date was 1,139.45. The value of the compounded dividend stream expected to be paid out between November 30 and December 21, the expiration date of the December contract, totaled 0.9 index points. The financing cost prevailing at the time for large, credit-worthy borrowers was approximately 1.90 percent annualized over a 365-day year, or 0.010943 percent over the 21 days between November 30 and the December 21 expiration date for the futures contract. Using this information we can apply the cost-of-carry model to determine the fair-value futures price:

F_0,t = 1,139.45 x (1 + 0.001093) - 0.9 = 1,139.80 index points

This is the estimated fair value of the December 2001 futures contract at the close of trading on November 30, 2001. Given the way in which the S&P 500 futures contract is designed, each index point is worth $250. This means the expected invoice price of the contract is $284,950; that is, $250 per index point times 1,139.80 index points. Since the closing futures price on this day is 1,140, it would appear that an arbitrage opportunity does not exist. The basis error for this contract -that is, the actual futures price minus the fair-value futures price - is only 0.20 index points. The annualized rate of return from a cash-and-carry arbitrage strategy at these prices would only be 0.3054 percent, less than the financing cost of 1.90 percent. The annualized rate of return for this strategy is determined by calculating the rate of return over the 21-day arbitrage period and then annualizing the 21-day rate of return over a 365-day year (1,140/1,139.80)365/21-1).

Suppose that the December 2001 futures price on November 30, 2001 had been 1,143 instead of the actual 1,140. In this case the rate of return on the strategy would be 4.99 percent annualized over a 365-day year (1,143/1,139.80)365/21-1). This rate of return is well above the annualized financing cost of 1.90 percent. In this case, the futures price is above its fair market value as determined by the cost-of-carry model by 3.20 index points. To exploit this apparent arbitrage opportunity, the trader would simultaneously sell the relatively overvalued futures and buy the relatively undervalued cash index. In other words, the trader would buy low and sell high using a cash-and-carry arbitrage strategy. The cash flows for this cash-and-carry strategy are summarized in Table 8.1.

Table 8.1 Cash-and-carry index arbitrage

Date	Cash market	Futures Market
November 30	Borrow $284,862.5 (1,139.45 x $250) for 21 days at 1.9%; buy stocks in the S&P 500 for $284,862.50.	Sell one DEC S&P 500 index futures contract for $1,143.00.
December 21	Receive accumulated proceeds from invested dividends of $225 (0.9 index points x $250); sell stock for $285,000 (1,140 index points x $250); total proceeds are $285,225; repay debt of $285,173.90.	At expiration, the futures price is set equal to the spot index value of 1,140.00, giving a profit of 3.00 index units; in dollar terms, this is 3.00 index points x $250 per index point.
	Gain: $311.40	Gain: $750
	Total profit: $311.40 + $750 = $1,061.40

Now, suppose instead that the December 2001 futures price on November 30, 2001 had been 1,138.00. In this case, the rate of return from this strategy would be 2.78 percent annualized over a 365-day year (1,143/1,138)365/21-1). This rate of return is above the trader's annualized financing cost of 1.90. In this case, the futures price is below its fair market value as determined by the cost-of-carry model by -1.90 index points. To exploit this apparent arbitrage opportunity, the trader would simultaneously buy the relatively undervalued cash futures and sell the relatively overvalued cash index. The stocks would either be sold short or sold out of inventory. In other words, the trader would buy low and sell high using a reverse cash-and-carry arbitrage strategy. The cash flows for this cash-and-carry strategy are summarized in Table 8.2.

Table 8.2 Reverse cash-and-carry index arbitrage

Date	Stock market	Futures Market
November 30	Sell stock in the S&P 500 index for $284,862 (1,139.45 x $250); lend $284,862.50 for 21 days at 1.9%.	Buy one DEC index futures contract for 1,138.00.
December 21	Receive proceeds from investment of $285,173.90; buy stocks in the S&P 500 index for $285,000 (1,140 x $250); return stocks to repay short sale.	At expiration, the futures price is set equal to the spot index value of 1,140.00, giving a profit of 2.00 index points x $250 per index.
	Gain: $173.9	Profit: $500
	Total profit: $173.9 + $500 = $673.9

Identifying an apparent arbitrage opportunities does not depend on the prices that prevail at expiration on December 21 (which happens to be 1,140 index points). Instead, the arbitrage opportunities arise solely from a discrepancy between the current futures price and its fair value. The arbitrage gain is locked in no matter what happens to stock prices between Novem ber 30 and December 21.

The success of the arbitrage depends upon identifying the misalignment between the actual futures price and the fair value futures price. However, at a given moment the fair value futures price depends upon the current price of 500 different stocks.

Hedging with Stock Index Futures

Previously, we consider the basic techniques for hedging with stock index futures. We presented examples of short and long hedges, and discussed a hedging strategy for hedging a portfolio with stock index futures that reflected the beta of the portfolio begin hedged. The hedge position from above is as follows:

-β_P(V_P/V_F) = number of contracts

where V_P is the value of the portfolio, V_F is the value of the futures contract, and β_P is the beta of the portfolio that is being hedged. In this section, we analyze stock index futures hedging. We begin by showing that the hedges above give the futures position to establish a combined stock and futures portfolio with the lowest possible risk. We illustrate this hedging technique with actual market data. It is also possible to use futures to alter the beta of an existing portfolio. For example, if a stock portfolio has a beta of 0.8 and the desired beta is 0.9, it is possible to trade stock index futures to make the combined stock and futures portfolio behave like a stock portfolio with a beta of 0.9

The minimum risk hedge ratio

Previously, we studied the problem of combining a cash market position with futures to minimize risk. There, we took the cash market position as fixed and sought to find the hedge ratio, HR, that world minimize risk. From equation 4.3 we saw that the risk of a combined cash and futures position is as follows:

Equation (4.3): σ²_P = σ²_s + HR²ρ²_F + 2HRρ_SFσ_Sσ_F

where σ²_P is the variance of the portfolio, P_t; σ²_s is the variance of S_t; σ²_F is the variance of F_t; and ρ_SF is the correlation between S_t and F_t. From Equation 4.3, the risk-minimizing hedge ratio, HR is as follows:

Equation (4.4): HR = -ρ_SFσ_Sσ_F / σ²_F = -COV_SF/σ²_F

where COV_SF is the covariance between S and F. As a practical matter, the easiest way to find the risk-minimizing hedge ratio is to estimate the following regression:

Equation (8.1): S_t = ∝ + β_RMF_t + ε_t

where S_t, is the return on the cash market position in period t. F_t is the returns on the futures contract in period t, ∝ is the constant regression parameter, β_RM is the slope regression parameter for the risk-minimizing hedge, and ε is an error term with zero mean and a standard deviation of 1.0. The negative of the estimated beta from this regression is the risk-minimizing hedge ratio, because the estimated β_RM equals the sample covariance between the independent (F) and dependent (S) variables divided by the sample variance of the independent variable. The R² from this regression shows the percentage risk in the cash position that is eliminated by holding the futures position.

At this point, it is important to distinguish the beta in Equation 8.1 (S_t = ∝ + β_RMF_t + ε_t) and the beta of the portfolio in the sense of the CAPM. The CAPM beta is the beta from regressing the returns of a given assets on the returns from the "true" market portfolio. However, the returns on the true market portfolio are unobservable. Therefore, as a practical measure, proxies are used for the market portfolio and the betas of assets are estimated by regressing the returns of a particular asset on the returns from the proxy of the market portfolio. The potential confusion becomes more dangerous because the S&P 500 spot index is one of the best-known proxies for the true market portfolio.

In Equation 7.6, we computed a hedge ratio using the beta for the portfolio. This beta is the estimated CAPM beta, because it is estimated by regressing the returns from a portfolio on the proxy for the market portfolio. By contrast, the beta in Equation 8.1 is the beta for a risk-minimizing hedge ratio and is not the same as the estimated CAPM beta. The beta in Equation 8.1 is found by regressing the returns of the portfolio on the returns from the futures contract. The estimated CAPM beta is found by regressing the returns of the portfolio on the returns of the spot index being used as a proxy for the unobservable true market portfolio. Thus, the hedging position in Equation 7.6 is not a risk-minimizing hedge. Nonetheless, such hedges can be very useful. We might think of the hedge ratio in Equation 7.6 as a rough-and-ready approximation to risk-minimizing hedging.

Having found the risk-minimizing hedge ratio, (Equation 7.6) -β_P(V_P/V_F) = number of contracts

A Minimum-Risk Hedging Example

In this section, we consider an example of a minimum-risk hedge in stock index futures using actual market data. Let us assume that a trader has a portfolio worth $10 million on November 28. The portfolio is invested in the 30 stocks in the Dow Jones Industrial Average (DJIA). The portfolio manager will hedge this cash market portfolio using the S&P 500 JUN futures contracts. We consider each step that the portfolio manager follows to compute the hedge ratio and to implement the hedge.

Organize Data And Compute Returns

The manager plans to hedge according to Equation 7.6. Therefore, she needs to find the beta for the hedge ratio. Accordingly, she collects data for her portfolio value for 101 days from July 6 through yesterday. She also finds the price of the S&P 500 JUN futures for each day. There is nothing magical about using 101 days, but these data are available and she believes that this procedure will provide a sufficient sample t estimate the hedging beta. From the 101 days of prices, she computes the daily percentage change in the value of the cash market portfolio and the futures price. This gives 100 paired observations of daily returns data.

Estimating Hedging Beta

With the data in place, the portfolio manager regresses the cash market returns on the returns from the futures contract as shown in Equation 8.1 (S_t = ∝ + β_RMF_t + ε_t). From this regression the estimated beta is 0.8801, so β_RM = 0.8801. This indicates that each dollar of the cash market position should be hedged with $0.8801 in the futures position. The R² from the regression is 0.9263, and this high R² encourages the belief that the hedge is likely to perform well. Again, for emphasis, the estimated beta from regressing the portfolio's return on the stock index futures returns is not the same as the portfolio's CAPM beta; βP does not equal βRM.

Compute Futures Position

The portfolio manager wants to hedge a $10 million cash portfolio with the S&P JUN futures contract. Having found the risk-minimizing hedge ratio, she needs to translate the hedge ratio into the correct futures position that takes account of the size of the futures contract. On November 27, the S&P futures closed at 354.75. The futures contract value is for the index times $250. Therefore, applying Equation 7.6, she computes the number of contracts as follows:

Equation7.6: -β_P(V_P/V_F) = -0.8801 x ($10,000,000/354.75 x $250) = -99.2361

The estimated risk-minimizing futures position is -99.24 contracts, so the portfolio manager decides to sell 100 contracts.

Expected Hedging Results

The portfolio's ending value is $9,656,090. The settlement price for the futures on February 22 is 330.60. Therefore, the futures profit is 100 x $250 x (354.75 - 330.60) = $603,750. The futures profit results from trading 100 contracts with each index point being worth $250 and the index having fallen 24.15 points. The value of the hedged portfolio consists of the cash market portfolio plus the futures profit, so the hedged portfolio's terminal value is $10,259,840. In this example, the hedge protected the portfolio against a substantial loss.

In the real world, an institution hedging with stock index futures would use the CME's S&P 500 E-mini contract, in combination with the larger open-outcry version of the contract, to refine the hedge construction. The S&P 500 E-mini contract has the same terms as the regular S&P 500 futures contract except that each index point is worth $50 instead of $250. In this example, the hedge construction could be refined by selling 99 regular S&P 500 futures and one S&P 500 E-mini futures contract.

Ex-Ante Versus Ex-Post Hedge Ratios

In our risk-minimizing hedging example, we computed ΒRM = 0.8801 using historical data and applied the hedge ratio to a future period. It is unlikely that the estimated hedge ratio would equal the hedge ratio that we would have used if we had perfect foresight about the behavior of the cash market position and the futures price. This is the difference between an ex-ante and an ex-post hedge ratio. Ex ante, or before the fact, the best hedge ratio we could find was -0.8801. Ex-post, or after the fact, some other hedge ratio would be likely to perform better than the ex-ante hedge ratio of 0.8801. In this section, we consider the difference between ex-ante and ex-post hedge ratios in the context of our example.

The portfolio manager used historical returns from July 7 to November 27 to estimate the hedge ratio of -0.8801. She applied this hedge ratio on November 28, and maintained the hedge position until February 22 of the next year. The ex-post risk minimizing hedge ratio was not available to her when she made her hedging decision on November 28. What would have been the ideal risk-minimizing hedge ratio have been, had she had complete knowledge about how prices would move from November 28 to February 22? To find this ex-post hedge ratio, we estimated Equation 8.1 (S_t = ∝ + Β_RMF_t + Ε_t) using data from November to February and found this ex-post hedge ratio of -0.9154. This implies a futures position of 51.61 contracts. We round this to 52 contracts. Figure 8.3 shows the results of hedging with the ex-ante and ex-post hedge ratios.

Figure 8.2 Hedged and unhedged portfolio value

In a world with perfect foresight, the ex-post hedge ratio is the risk-minimizing hedge ratio that we would like to use. However, the ex-ante hedge ratio is the best estimate we can make at the time the decision must be implemented. As figure 8.3 shows, the ex-ante hedge ratio performs quite well. The terminal value of the hedge with the ex-ante hedge ratio is $10,259,840. With the ex-post hedge ratio, the terminal value is $10,283,990. While the ex-ante hedge ratio performed well, the ex-post hedge ratio would have been even better. This is exactly the result that we would expect.

Figure 8.3 Ex-ante versus ex-post hedging results

Altering the Beta of a Portfolio.

Portfolio manager often adjust the CAPM betas of their portfolios in anticipation of bull and bear markets. If a manager expects a bull market, she might increase the beta of the portfolio to take advantage of the expected rise in stock prices. Similarly, if a bear market seems imminent, the manager might reduce the beta of a stock portfolio as a defensive maneuver. If the manager trades only in the stock market itself, changing the beta of the portfolio involves selling some stocks and buying others. For example, to reduce the beta of the portfolio, the manager would sell high-beta stocks and use the funds to buy low-beta stocks. With transaction costs in the stock market being relatively low, this procedure can be favorable.

The portfolio manager has an alternative. She can use stock index futures to create a combined stock/futures portfolio with the desired response-to-market condition. In this section we consider techniques for changing the risk of a portfolio using stock index futures.

In the CAPM, all risk is either systematic or unsystematic. Systematic risk is associated with general movements in the market and affects all investment. By contrast, unsystematic risk is particular to a certain investment or a certain range of investments Diversification can almost completely eliminate unsystematic risk from a portfolio. The remaining systematic risk is unavoidable. Studies show that a random selection of 20 stocks will create a portfolio with very little unsystematic risk Therefore, in this section we restrict our attention to portfolios that are well diversified and consequently have no unsystematic risk.

Starting with a stock portfolio that has systematic risk only and combining it with a risk-minimizing short position in stock index futures creates a combined stock/futures portfolio with zero systematic risk. According to the CAPM, a portfolio with zero systematic risk should earn the risk-free rate of interest. Instead of eliminating all systematic risk by hedging, it is possible to hedge only a portion of the systematic risk to reduce, but not eliminate, the systematic risk inherent in the portfolio. Similarly, a portfolio manager can use stock index futures to increase the systematic risk of a portfolio.

A risk-minimizing hedge matches a long position in stock with a short position in stock index futures in an attempt to create a portfolio whose value will not change with fluctuations in the stock market. To reduce, but not eliminate, the systematic risk, a portfolio manger could sell some futures, but fewer than the risk-minimizing amount. For example, to eliminate half of the systematic risk, the portfolio manager could sell half of the number of contracts stipulated by the risk-minimizing hedge. The combined stock/futures position would then have a level of systematic risk equal to half of the stock portfolio's systematic risk.

It is also possible to trade stock index futures to increase the systematic risk of a stock portfolio. If a trader buys a stock index futures, he increases his systematic risk. Therefore, if a portfolio manager holds a stock portfolio and buys stock index futures, the resulting stock/futures position has more systematic risk than the stock portfolio alone. For example, assume a portfolio manager buys, instead of selling, the risk-minimizing number of stock index futures. Instead of eliminating the systematic risk, the resulting stock/long futures position should have twice the systematic risk of the original portfolio.

We can illustrate this principle by considering the same data we used to illustrate the risk-minimizing hedge. In that example, the risk-minimizing futures position was to sell 52 contracts. Selling 52 contracts created a stock/futures position with zero systematic risk. By selling just 26 contracts, the portfolio manager could cut the systematic risk of the original portfolio in half. Similarly, by buying 52 contracts, the resulting stock/futures position would have twice the systematic risk of the original portfolio.

Figures 8.4 shows the price paths of two portfolios over the 60-day hedging period from November 28 to February 22. First, the graph shows the unhedged portfolio. Its value begins at $10 million and terminates at $9,656,090, as we have seen. Over this period, the unhedged portfolio lost about $350,000. The graph also shows the portfolio created by holding the stocks and buying 52 futures contracts. In our analysis of the risk-minimizing hedge, we found that the trader could minimize risk by selling 52 futures contracts. Buying 52 contracts doubles the systematic risk. The new portfolio of stock plus a long position of 52 contracts increases the sensitivity of the portfolio to swings in the stock market. In effect, holding the stock portfolio and buying stock index futures simulates more than 100 percent investment in the stock index. Stock prices in generally fell during this 60-day period. For example, the all stocks portfolio lost 3.44 percent of its value over this period.

Figure 8.4 Price paths for hedged and unhedged portfolio

By buying stock index futures, the portfolio manager would increase the overall sensitivity of the portfolio to changes in the stock market. Not surprisingly, then, the stock/long futures portfolio lost more than the pure stock portfolio. As Figure 8.4 shows, every move of the stock/long futures portfolio exaggerates the movement of the all stock portfolio. For the portfolio of stock plus a long position of 52 index futures, the terminal value is $9,052,340. This portfolio lost 9.48 percent of its value. However, as Figure 8.4 shows, for periods when the stock prices advanced from their initial level, the stock/long futures position rose even more. This is just what we expected, because buying futures increases the systematic risk of the existing stock portfolio.

Hedge Fund Uses of Stock Index Futures [viewable here in Excel]

One particular hedge fund strategy is to establish a portfolio consisting of long positions in some stocks and short positions in other stocks, with no net exposure to market risk. The goal of the strategy is to make money from speculating on individual stocks with the portfolio without worrying about the direction of the overall market. Hedge funds employing this strategy are sometimes called "long/short equity funds," "equity market neutral funds," "hedge equity funds," or "relative value funds." One version of the strategy is referred to as "pairs trading." The manager will position the portfolio to have approximately equal weights assigned to the long and short equity positions. Such a portfolio will be approximately market neutral (i.e., have a beta equal to zero), but typically the portfolio will exhibit some net market exposure. Long/short portfolio managers refer to this net market exposure as "deadweight". The manager may use a stock index futures contract, such as the S&P 500 futures contract, to establish a hedge to account for the remaining market exposure.

As an example, consider a hedge fund that believes that investors are on the verge of reallocating the sector weights within their portfolios. In particular, the hedge fund manager believes that investors will rotate their investments from the pharmaceutical sector to the retail sector. The hedge fund manager believes that as this sector rotation occurs, retail stocks will rise relative to pharmaceutical stocks. In other words, the hedge fund manager believes that stocks in the retail sector are undervalued relative to the pharmaceutical sector. To take advantage of this belief, the hedge fund manager goes long retail stocks while simultaneously going short pharmaceutical stocks.

Table 8.3 displays portfolio characteristics of the hedge fund under the long/short strategy. The weighted average beta of the pharmaceutical sector stocks within the portfolio is 0.39, with an aggregate value of -$42.08 million. The negative sign reflects the fact that these stocks were sold short. The weighted average beta of the retail sector stocks with the portfolio is 1.07, with an aggregate value of $47.22 million. The net cash value of the portfolio (long position plus short position) is $5.14 million. The beta for the net value of the portfolio is 1.07, since the net amount results entirely from the overweighting of stocks in the retail sector, where the weighted beta was 1.07.

To make the portfolio market neutral, the hedge fund manager decides to establish a short position in S&P 500 stock index futures, which are currently trading at 1,154.25 index points. The manager uses Equation 7.6 to determine the number of contracts to trade:

-1.07 x (5,140,000/(1,154.25 x $250) = -19.059 ~ 19 contracts

Portfolio Insurance

As we have seen, traders can tailor the risk of a stock portfolio by trading stock index futures. For a given well-diversified portfolio, selling stock index futures can create a combined stock/futures portfolio with reduced risk. Holding a stock portfolio and buying stock index futures results in a portfolio with greater risk and expected return than the initial portfolio.

Portfolio insurance refers to a collection of techniques for managing the risk of an underlying portfolio. With most portfolio insurance strategies, the goal is to manage the risk of a portfolio to ensure that the value of the portfolio does not drop below a specified level, while at the same time allowing for the portfolio's value to increase. Portfolio insurance strategies are often implemented using options, as we discuss later. However, stock index futures are equally important tools for portfolio insurance. Implementing portfolio insurance strategies using futures is called dynamic hedging. Although the mathematics of dynamic hedging are too complex for full treatment here, we can understand the basis idea behind the portfolio insurance with stock index futures.

A Portfolio Insurance Example

Consider a fully diversified stock portfolio worth $100 million. The value of this portfolio can range from zero to infinity. Many investors would like to put a floor beneath the value of the portfolio. For example, it would be very desirable to ensure that the portfolio's value never falls below $90 million. Portfolio insurance offers a way to control the downside risk of a portfolio. However, in a financial market there is no free lunch, so it is only possible to limit the risk of a large price fall sacrificing some of the potential for a gain. Portfolio insurance, like life insurance, is not free, but it may be desirable for some traders.

We have seen that a risk-minimizing hedge converts a stock portfolio into a synthetic T-bill. By fully hedging our example stock portfolio, we can keep the portfolio's value above $100 million. A fully hedged portfolio will increase in value at the risk-free rate, although full hedging eliminates all of the potential gain in the portfolio beyond the risk-free rate. In dynamic hedging, however, the trader holds the stock portfolio and sells some futures contracts. The more insurance the trader wants, the more futures he or she will sell.

Let us assume that a stock index futures contract has an underlying value of $100 million and a trader sells futures contracts to cover $50 million of the value of the portfolio. Thus in the initial position, the trader is long $100 million in stock and short $50 million in futures, so 50 percent of the portfolio is hedged. Table 8.4 shows this initial position in the time zero row. At t=0, there has been no gain or loss on either the stock or futures. In the first period, we assume that the value of the stock portfolio falls by $2 million. The 50 futures contracts cover half of that loss with a gain of $1 million. Therefore, at t = 1, the combined stock/futures portfolio is worth $99 million. Now, the manger increases the coverage in the futures market by selling five more contracts. This gives a total of 55 short positions and coverage for 56 percent (55/99) of the total portfolio. In the second period, the stock portfolio loses another $2 million, but with 55 futures contracts, the futures gain is (55/99) x $2 million = $1.11 million. This gives a total portfolio value of $98.11 million.

By t = 4, the stock portfolio has fallen $10 million, but the futures profits have been $6.21 million. This gives a total portfolio value of $96.21 million. Also, the manager has increased the futures position in response to each drop in stock price. At t = 4, the trader is short 80 contracts, hedging 83 percent of the stock market portfolio. At t = 5, the stock price drops dramatically, losing $35.86 million. The futures profit covers $30.65 million. This leaves a total portfolio value of $90 million. However, this is the floor amount of the portfolio, so the trader must now move to a fully hedged position. If the stock portfolio is only partially hedged, the next drop in prices can take the value of the entire portfolio below the floor amount of $90 million. At t = 6, the price of the stock drops $10 million, but the futures position fully covers the loss. Therefore, the combined portfolio maintains its floor value of $90 million.

Table 8.4 shows the basic strategy of portfolio insurance with dynamic hedging. Initially, the portfolio is partially hedged. If stock prices fall, the trader increases the portion of the portfolio that is insured. Had the stock portfolio risen in value, the futures position would have lost money. However, the loss on the futures position would have been less than the gain on the stocks, because the portfolio was only partially hedged. As the stock prices rose, the manager would have bought futures, thereby hedging less and less of the portfolio. Less hedging would be needed if the stock rose, because there would be little chance of the portfolio's total value falling below $90 million.

Combining Options with Bonds and Stocks

We now show how to combine options with stocks and bonds to adjust payoff patterns to fit virtually any taste for risk and return combinations. These combinations show us the relationships among the different classes of securities. By combining two types of securities, we can generally imitate the payoff patterns of a third. In addition, this section extends the concepts we have developed earlier. Specifically, we learn more about shapng the risk and return characteristics of portfolios by using options.

In this section, we consider five combinations of options with bonds or stocks. First, we consider the popular strategy of the covered call - a long position in the underlying stock and a short position in a call option. Second, we explore portfolio insurance. During the 1980's, portfolio insurance became one of the most discussed techniques for managing the risk of a stock portfolio. We illustrate some of the basic ides of portfolio insurance by showing how to insure a stock portfolio. Third, we show how to use options to mimic the profit-and-loss patterns of the stock itself. For investors who do not want to invest the full purchase price of the stock, it is possible to create an option position that gives a profit-and-loss pattern much like the stock itself. Fourth, by combining options with the risk-free bond, we can synthesize the underlying stock. In this situation, the option and bond position gives the same profit-and-loss pattern as the stock, and it has the same value as the stock as well. Finally, we show how to combine a call, a bond, and a share of stock to create a synthetic put option.

The Covered Call: Stock Plus a Short Call

In a covered call transaction, a trader is generally assumed to already own a stock, and writes a call option on the underlying stock. (The strategy is "covered" because the trader owns the underlying stock, and this stock covers the obligation inherit in writing the call.) This strategy is generally undertaken as an income enhancement technique. For example, assume a trader owns a share currently priced at $100. She might write a call option on this share with an exercise price of $110 and an assumed price of $4. The option premium will be hers to keep. In exchange for accepting the $4 premium, our trader realizes that the underlying stock might be called away from her if the stock price exceeds $110. If the stock price fails to increase by $10, the option she has written will expire worthless, and she will be able to keep the income from selling the option without any further obligation. As this example indicates, the strategy turns on selling an option with a strike price far removed from the current value of the stock, because the intention is to keep the premium without surrendering the stock through exercise.

Although writing covered calls can often serve the purpose of enhancing income, it must be remembered that there is no free lunch in the options market. The writer of the covered call is actually exchanging the chance of large gains on the stock position in favor of income from selling the option. For example, if the stock price were to rise to $120, the trader would not receive this benefit, because the stock would be called away from her.

Figure 11.21 graphs the profits and losses at expiration for the example we been considering. The solid line shows the profits and losses for the stock itself, while the dashed line shows the profits and losses for the covered call (the stock plus short call). For any stock price less than or equal to $110, the written call cannot be exercised against our trader, and she receives whatever profits or losses the stock earns plus the $4 option premium. Thus, she is $4 better off with the covered call than she would be with the stock alone for any stock price of $110 or less. If the stock price exceeds $110, the option will be exercised against her, and she must surrender the stock. This potential exercise places an upper limit on her profit at $14. If the stock price had risen to $120 and the trader had not written the call, her profit would have been $20 on the stock investment alone. In the covered call position, she would have made only $14, because the stock would have been called away from her. The desirability of writing a covered call to enhance income depends upon the chance that the stock price will exceed the exercise price at which the trader writes the call.

Portfolio Insurance: Stock Plus a Long Put

Along with program trading, portfolio insurance was a dominant investing technique developed in the 1980's. Portfolio insurance is an investment management technique designed to protect a stock portfolio from severe drops in value. Investment managers can implement portfolio insurance strategies in various ways. Some use options while other use futures, and still others use combinations of other instruments. We analyze a simple strategy for implementing portfolio insurance with options. Portfolio insurance applies only to portfolios, not individual stocks. Therefore, for our discussion we assume that the underlying good is a well-diversified portfolio of common stocks. We may think of the portfolio as consisting of the S&P 100. This is convenient, because a popular stock index option is based on the S&P 100. Therefore, the portfolio insurance problem we consider is protecting the value of this stock portfolio from large drops in value.

In essence, portfolio insurance with options involves holding a stock portfolio and buying a put option on the portfolio. If we have a long position in the stock portfolio, the profits and losses from holding the portfolio consist of the profits and losses from the individual stocks. Therefore, the profits and losses for the portfolio resemble the typical stock's profits and losses. Let S_t be the cost of the stock portfolio at time t, and let P_t be a put option on the portfolio. The cost of an insured portfolio is, therefore

S_t + P_t

Because the price of a put is always positive, it is clear that an insured portfolio costs more than the uninsured stock portfolio alone. At expiration, the value of the insured portfolio is:

S_T + P_T = S_T / MAX{0,X-S_T}

As the profit on an uninsured portfolio is S_T - S_t, the insured portfolio has a superior performance when S_T<X-P_t.

As an example of an insured portfolio, consider an investment in the stock index at a value of 96.00. Figure 11.22 shows the profit-and-loss profiles for an investment in the index at 96 and for a put option on the index. The figure assumes that the put has a strike price of 96.00 and costs 4.00 (we are expressing all values in terms of the index). Figure 11.23 shows the effects of combining an investment in the index stocks and buying a put on the index. For comparison, Figure 11.23 also shows the profits and losses from a long position in the index itself.

Figure 11.22 Profits and losses at expiration for a stock index and a long put

The insured portfolio, the index plus a long put, cost 100.00, 96.00 for the stock index and 4.00 for the put. This insured portfolio offers protection against large drops in value. If the stock index suddenly falls from 96.00 to 80.00, for example, the insured portfolio loses only 4.00 and is still worth 96.99. No matter how low the market index goes, the insured portfolio can lose only 4.00 points. However, this insurance has a cost. Investment in the index itself shows a profit for any index value over 96.00. By contract, the insured portfolio has a profit only if the index climbs above 100.00. In the insured portfolio, the index must climb high enough to offset the price of buying the insuring put option. Because the put option will expire, keeping the portfolio insured requires that the investor buy a series of put options to keep the insurance in force. In Figure 11.23 notice that the combined position of a long index and a long put gives a payoff shape that matches a long position in a call. Like a call, the insured portfolio protects against extremely unfavorable outcomes as the stock price falls. This similarity between the insured portfolio and a call position suggest that a trader might buy a call and invest the extra proceeds in a bond in order to replicate a position in an insured portfolio.

Figure 11.23 Profits and losses at expiration for an insured portfolio

As a further comparison, we consider the likely returns profit from holding the stock portfolio and insured portfolio. Assume that the option expires in one year. The stock portfolio returns are normally distributed with a 10 percent return and a standard deviation of returns of 15 percent, as shown in Figure 11.24. There is an approximately 68 percent chance that the returns will lie between -5 percent and +25 percent, because in a normal distribution about 68 percent of all observations lie within one standard deviation of the mean.

For the insured portfolio we noted that the maximum loss is 4.00, so the terminal value must be at least 96.00, a loss of -4.00/100.00 = -4.00 percent. Figure 11.23 and 11.24 imply that there is a good chance that the insured portfolio will actually lose this 4.00 percent. Any return of zero or less on the stock portfolio give a -4.00 percent return for the insured portfolio. The chance that the stock portfolio will have a zero or lower return equals the chance that the stock portfolio will have a return that is two-thirds of a standard deviation below its mean return of 10 percent. As a feature of the normal distribution, this probability is 25.25 percent. Therefore, there is a 25.25 percent probability that the insured portfolio will have a return of -4.00 percent.

Although the insured portfolio protects against large losses, it has a lower chance of larger returns. For example, there is a 2.2750 percent chance that the stock portfolio will have a return of 40 percent or higher, which is two standard deviations above the mean return of 10 percent. For the insured portfolio to have a return of 40 percent or higher, the terminal value of the insured portfolio would have to be at least 140.00 = 100 x 1.40. This result for the insured portfolio implies that the stock portfolio itself would have to be worth 140.00, which is a return of 45.833 percent on the stock portfolio, or 2.38887 standard deviations above the mean return of 10 percent (0.10 + 2.3887 x 0.15 = 0.458333). The chance of a return 2.3887 standard deviations above the mean is 0.8450 percent. Thus, there is a 2.2750 percent chance that the stock portfolio will have a return of 40 percent or higher, but only a 0.8450 percent chance that the insured portfolio will do so. This numerical example shows that there is a smaller chance of high returns in the insured portfolio compared to the stock portfolio, as the insured portfolio provides insurance against the large loses at the expense of sacrificing large gains. Said another way, the insured portfolio has a truncated returns distribution at exactly -4.00 percent, which is purchased at the cost of sacrificing the chance of high returns.

Figure 11.25 illustrates the truncation of the returns distribution by comparing the cumulative distribution of returns for both portfolios, where the smooth line and kinked line pertain to the stock and insured portfolios, respectively. For the insured portfolio, the vertical line at -4.00 reflects the zero probability of a return below -4.00 percent and the 25.25 percent probability that the return on the insured portfolio will be exactly -4.00 percent. For the stock portfolio to have a return of -4.00 percent or worse, the return must be at least 14.00 percent below the expected return, which is 0.93333 standard deviation or more below the mean return. The probability of this occurring is 17.53 percent. Similarly, compared to the insured portfolio, there is a better chance that the stock portfolio will achieve any given return above -4.00 percent. For example, we have already seen that the chance of a return of 40 percent or better for the stock portfolio is 2.2750 percent, but only 0.8450 percent for the insured portfolio. This means that there is an 82.47 percent chance that the stock portfolio will outperform the insured portfolio, because there is an 82.47 percent chance that the stock portfolio will return more than -4.00 percent.

Figure 11.24 The probability distribution for a stock index's returns

The cumulative distributions of the insured portfolio and the stock portfolio in Figure 11.25 dramatizes the truncation of the returns distribution and the reduced potential for high returns for the insured portfolio. As such, Figure 11.25 echoes Figure 11.23, where the insured portfolio has potential losses truncated at -4.00 and reduced profit potential compared to the stock index alone. In Figure 11.25, the insured portfolio has truncated distribution on the loss side at -4.00 percent, reflected by the vertical line at -4.00 percent. However, the insured portfolio also has a reduced chance for large positive gains. Shown by the fact that the cumulative distribution line for the insured portfolio lies above that for the stock returns larger than -4.00 percent. As these cumulative distributions show, if stocks do well, the stock portfolio will outperform the insured portfolio. Thus, Figures 11.23 and 11.25 present two views of the same phenomenon.

Figure 11.25 A comparison of return distributions for an insured and an uninsured stock portfolio

This portfolio insurance example emphasizes the role of options in adjusting the returns distribution for the underlying investment. With options, we can adjust the distribution to fit our tastes - subject to the risk and return tradeoff governing the entire market. Finally, we note that the insured portfolio has the same profits and losses as a call option with a strike price of 96.00 and a premium of 4.00. This does not mean, however, that the insured portfolio and such a call option would have the same value. At expiration, the call will have no residual value beyond its profit and loss at that moment. By contrast, the insured portfolio will include the value of the investment in the stock index. Therefore, for a particular time horizon, two different investments can have the same profit-and-loss patterns without having the same value.

Stock Price Movements

In actual markets, stock prices change to reflect new information. During a single day, a stock price may change many times. From the ticker, we can observe stock prices when transactions occur. When trading ceases overnight, however, we cannot observe the stock price for hours at a time. Where the price wanders during the night, no one can know. Our observations are also limited because stock prices are quoted in eights of a dollar (or much less). The true stock price need not jump from one eighth to the next, but the observed stock price does. Sometimes the observed price remains the same from one transaction to another. But just because we observe the same price twice in succession does not mean it remained at that price between the two observations. From these reflections, we see that we can never know exactly how stock prices change, because we cannot observe the true stock price at every instant. Therefore, any model of stock price behavior deviates from an exact description of how stock prices move. Nonetheless, it is possible to develop a realistic model of stock price movements. In this section, we review a particular model that has been very successful in a wide range of finance applications.

Let us consider the random information that affects the price of a stock. We assume that the information arrives continuously and that each bit of information is small in importance. Under this scenario, we consider a stock price that arises or falls a small proportion in response to each bit of information. We know that finance depends conceptually on the twin ideas of expected return and risk. Thus, we might also think of a stock as having a positive expected rate of return. In the absence of special events, we expect the stock price to grow along the path of its expected rate of return. However, the world is risky. Information about the stock is sometimes favorable and sometimes unfavorable. As this random information becomes known, it pushes the stock away from its expected growth path. When the information is better than expected, the stock price jumps above its growth path. Negative information has the opposite effect; it pushes the stock price below its expected growth path. Thus, we might imagine the stock price growing along its expected growth path just as a drunk walks across a field. We expect the drunk to reach the other side of the field, but we also think he will wander and stumble in unpredictable short-term deviations from the straight path. Similarly, we expect a stock price to rise, because it has a positive expected return, but we also expect it to wander above and below its growth path, due to new information.

Finance uses a standard mathematical model that is consistent with the story of the preceding paragraph. We assume that the stock grows at an expected rate μ with a standard deviation σ over some period of time Δt:

Equation (13.8) $Delta;S = S_t+1 - S_t = S_tμΔt + S_tN(0,1)&sigma,√Δt

where S_t is the stock price at the beginning of the interval; ΔS is the stock price change during time Δt; S_tμΔt is the expected value of the stock price change during time Δt; N(0,1) is the normally distributed random variable with μ = 0 and σ = 1; and σ is the standard deviation of the stock price. Equation 13.8 says that the stock price change during Δt depends on two factors: the expected growth rate in the price and the variability of the growth. First, the expected growth in the stock price over a given interval depends on the mean growth rate, μ, and the amount of time, Δt. Therefore, if the stock price starts at S_t, the expected stock price increases after an interval of Δt equals S_tμΔt. However, this is only the expected stock price increase after the interval. Due to risk, the actual price change can be greater or lower. Deviations from the expected stock price depends on chance and on the volatility of the stock. The equation captures risk by using the normal distribution. For convenience, the model uses the standard normal distribution, which has a zero mean and a standard deviation of 1. The standard deviation represents the variability of a particular stock. We multiply the standard deviation of the stock by a random drawing from the normal distribution to capture the riskiness of the stock. Also, the equation says that the variability increases with the square root of the interval Δt. In other words, the farther into the future we project the stock price with our model, the less certain we can be about what the stock price will be.

Dividing both side of Equation 13.8 by the original stock price, S_t, give the percentage change in the stock price during Δt:

ΔS / S_t = μΔ_t + N(0,1)σ√Δt

From this equation, it is possible to show that the percentage change in the stock price is normally distributed:

Equation (13.9) ΔS / S_t ~ N[μΔt,σ√Δt]

Figures 13.3 shows two stock price paths over the course of a year, with both stock prices starting at $100. The straight line graphs a stock that grows at 10 percent per year with no risk.

Figure 13.3 Two stock price path

The jagged line shows a stock price path with an expected growth rate of 10 percent and a standard deviation of 0.20 per year. We generated the second price path by taking repeated random samples from a normal distribution to create price changes according to Equation 13.8. As the jagged line in Figure 13.8 shows, a stock might easily wander away from its growth path due to the riskiness represented by its standard deviation.

To construct Figure 13.8, we used 1,000 periods per year and a growth rate of 10 percent. To construct the straight line, we assumed that the stock price increased by 0.10/1000 each period. However, this gives an ending stock price of $110.51, not the $110 we expect if the stock price grows at 10 percent per year. This difference results from using 1,000 compounding intervals during the year.

However, we hypothesize that information arrives continuously, so that the stock price could always change. To avoid worrying about the compounding interval, we now employ continuous compounding. Therefore, we focus on logarithmic stock returns. For example, consider a beginning stock price of $100 and ending price of $110 a year later. The logarithmic stock return over the year is ln(S_t/S₀) = ln($110/$100) = Ln(1.1) = 0.0953. The logarithmic stock return is just the continuous growth rate that takes the stock price from its original value to its ending value. Thus, $100 x exp^ut = $100 x exp^0.0953 x 1 = $110. We now need a continuous growth model of stock prices that is consistent with our model for the percentage change stock price model of Equation 13.9. With some difficult math, it is possible to prove the following result:

Equation (13.10) ln (S_t+Δt/ S_t ~ N[μ - 0.5σ²

This expression asserts that logarithmic stock returns are distributed normally with the given mean and standard deviation. For a latter time, t + Δt, the expected price and variance of the stock price are as follows:

E(S_t+Δt) = S_te^μ+Δt

VAR(S_t+Δt) = S²_te^μ+Δt(e^μ+Δt - 1)

Thus, the expected stock price at t + Δt depends on the original stock price, S_t, the expected growth rate, m, and the amount of time that elapses, Δt. Similarly, the variance of the stock price depends of the original stock price, the expected growth rate, and the elapsed time as well. The longer the time horizon, the larger will be the variance. The increasing variance reflects our greater uncertainty about stock prices far in the future.

As an example, consider a stock with an initial price of $100 and an expected growth rate of 10 percent. If the stock has a standard deviation of 0.2 per year, we can compute the expected stock price and variance for six months into the future. For this example, we have the following values:

S_t = $100, μ = 0.1, σ = 0.2, Δt = 0.5
E(S_t+Δt) = $100 x e^{0.1 x 0.5} = $105.13
VAR(S_{t + Δt}) = $100 x $100 x e^{0.1 x 0.5}(e^{0.2 x 2} x 0.5 - 1) = $223.26
Standard deviation = √$223.26 = $14.94

The standard deviation of the price over period Δt is $14.94. Figure 13.4 (not shown) shows stock price realizations that are consistent with this example. We found these prices by drawing random values from a normal distribution and using our example growth rate and standard deviation. Each dot in the figure represent a possible stock price realization. Notice that the price tends to drift higher over time, consistent with a rising expected value. This is shown by the regression line that is fitted through the points. However, there is considerable uncertainty about what the price will be at any future date. The farther we go into the future, the greater that uncertainty becomes.

Research on actual stock price behavior shows that logarithmic stock returns are approximately normally distributed. So we can say that, as an approximation, stock returns follow a log-normal distribution. Stock returns themselves are not normally distributed. As an example, Figure 13.5 shows a distribution of stock returns with a mean of 1.2 and a standard deviation of 0.6. It is easy to see that this distribution is not normal, because it is skewed to the right. There is a greater chance of larger returns than one would expect with a normal distribution. Figure 13.6 shows the log-normal distribution that corresponds to the values in Figure 13.5. The values graphed in Figure 13.6 are the logarithms of the values used to construct Figure 13.5. The graph in Figure 13.6 shows a normal distribution. We will assume that stock returns are distributed as Figure 13.6 shows, except that the mean and standard deviation differ from stock to stock.

Figure 13.5 A Log-normal distribution

While the log-normal distribution only approximates stock returns, it has two great virtues. First, it is mathematically tractable, so we can obtain solutions for the value of call options if stock returns are log-normally distributed. Second, the resulting call option prices that we compute are very good approximations of actual market prices. In "Interest Rate Options" we treat stock returns as log-normally distributed with a specified mean and variance.

Figure 13.6 The normal distribution

Sources:Damodaran, Hull, Basu, Kolb, Brueggeman, Fisher, Bodie, Merton, White, Case, Pratt, Agee