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Table of Contents Page Conceptual Framework 3 Illiquidity Discounts: No Dividends 8 Illiquidity Discounts: Dividend-Paying Stocks 9 References 36
Conceptual Framework
Conceptual Framework Liquidity Represents the Right to Sell an Asset Put Option: Financial contract which gives the owner the right, but not the obligation, to sell a security by a predetermined date. Liquidity as an Option: Find a derivative D whose payoff, when combined with the cash flows from the illiquid security R, results in a cash flow pattern which is greater than or equal to the cash flows of a freely-traded stock S: $80 $70 $60 $50 $40 $30 Wish to Sell Here (t = U) R + D S S U S T D S R S: Free to sell at any point in time and chooses to sell at time t = U for proceeds S U R: Restricted from selling until time t = T at which point is able to sell for proceeds S T $0 $10 But Forced to Sell Here (t = T) $0 Nov-10 May-11 Nov-11 May-1 Nov-1 May-13 Nov-13 May-14 Nov-14 May-15 Nov-15 D: What derivative has a payoff MMM S U S T, 0? NasdaqGS:GPOR - Share Pricing
Conceptual Framework (cont.) Technical Notes The Unrestricted Stock follows Geometric Brownian motion: GBM equation is dd t = r(dd) + σ(dd) where r = risk-free rate, σ = volatility. S t As a starting point, assume that the stock pays no dividends (q = 0) 1. For simplicity, assume that the risk-free rate is zero (r = 0). The Restricted Stock is subject to investor-specific restrictions (rather than securityspecific ones) which are absolute in nature: Certain investors cannot trade at all (for a fixed time T) while everybody else can. The Put Option is written on the underlying unrestricted stock price: Standard hedging arguments allow risk-neutral pricing of the derivative. The option price is expressed in terms of the underlying restricted stock price. 1 The extension to the dividend case will be reviewed subsequently. For illustrative purposes and ease of presentation only. This assumption allows us to work with stock prices rather than forward prices without changing the analytical framework or the resulting conclusions.
Conceptual Framework (cont.) Restricted Stock The investor cannot sell their shares until the end of the restriction period at time T After the restriction period is over, the stock is fully liquid again The investor sells for proceeds S T $80 $70 $60 $50 $40 $30 $0 $10 Restricted Stock The investor sells for proceeds S T $0 Mar-13 Sep-13 Mar-14 Sep-14 Mar-15 Sep-15 Unrestricted Stock Identical to restricted stock in all aspects other than trading restriction Freely-traded by all other investors What would the investor do in the absence of trading restrictions? $80 $70 $60 $50 $40 $30 $0 $10 Unrestricted Stock What would the investor do in the absence of trading restrictions? $0 Mar-13 Sep-13 Mar-14 Sep-14 Mar-15 Sep-15
Conceptual Framework (cont.) Desirable Model Features The predicted discounts should be bounded in the range from 0% to 100% The predicted discounts should be monotonically increasing from 0% to 100% as the input parameters are varied (e.g. no local maximum or minimum discount) The predicted discounts should vary with respect to the underlying input parameters in a manner consistent with the table below: Input Parameter Input Variation Predicted Discount Increase Increase Restriction Period (T) Decrease Decrease Volatility (σ) Dividend Yield (q) Increase Decrease Increase Decrease Increase Decrease Decrease Increase
Illiquidity Discounts: No Dividends
The Protective Put Option Theoretical Setup What would the investor do in the absence of trading restrictions? The investor will sell immediately at time t = 0 for proceeds S 0. What would the investor do if they are restricted from trading for a time period T? The investor will wait until the restriction period lapses and sell at time t = T for proceeds S T. Under the set of assumptions described above, what is the maximum opportunity loss to an investor who is restricted from trading for a time period T? MMMMMMM OOOOOOOOOOO LLLL = MMM S 0 S T, 0
The Protective Put Option Numerical Example $130 $10 Underprotected Overprotected $114 $110 $14 $100 $90 $1 $80 $70 $60 $79 S 0 = $100 S T = $85 Option Payoff = $15 Jan-13 Jul-13 Jan-14 Jul-14 Jan-15 Jul-15 What if: Want to sell at S t1 = $79? Want to sell at S t = $114?
The Protective Put Option Numerical Solution What is the price of a derivative with payoff MMM S 0 S T, 0? The exact price of a European put option can be found using the Black-Scholes formula. Protective Put Formula (exact) % DDDD = e rr N d N d 1 d 1 = r + σ T σ T d = d 1 σ T σ T r N stock price volatility restriction period risk-free rate cumulative standard normal
The Protective Put Option Practical Considerations Conceptual Issues Investors do not enter a position only to liquidate it and cash out right away. Liquidity does not provide investors with a downside protection against future losses. Liquidity = Right to Sell at Any Price Protective Put = Right to Sell at Today s Price Numerical Issues The price of a protective put option eventually decreases as its term to maturity gets larger, implying that the theoretical illiquidity discount declines for longer restriction periods.
The Lookback Put Option Theoretical Setup What would the investor do in the absence of trading restrictions? The investor will sell at the maximum achievable price over the restriction period for proceeds S MMM = MMM S 0,, S T. What would the investor do if they are restricted from trading for a time period T? The investor will wait until the restriction period lapses and sell at time t = T for proceeds S T. Under the set of assumptions described above, what is the maximum opportunity loss to an investor who is restricted from trading for a time period T? MMMMMMM OOOOOOOOOOO LLLL = MMM S MMM S T, 0 = S MMM S T
The Lookback Put Option Numerical Example $130 Underprotected $10 Overprotected $114 $5 $110 $100 $40 $90 $80 $70 $60 $79 S MMM = $119 S T = $85 Option Payoff = $34 Jan-13 Jul-13 Jan-14 Jul-14 Jan-15 Jul-15 What if: Want to sell at S t1 = $79? Want to sell at S t = $114?
The Lookback Put Option Numerical Solution What is the price of a derivative with payoff S MMM S T? This is a lookback put option for which an exact closed-form pricing formula exists. Lookback Put Formula (exact) % DDDD = + σ T N σ T + σ T π e σ T 8 1 σ T N stock price volatility restriction period cumulative standard normal
The Lookback Put Option Practical Considerations Conceptual Issues Investors don t have a perfect market-timing ability. Liquidity does not provide investors with a guaranteed price appreciation. Liquidity = Right to Sell at Any Price Lookback Put = Right to Sell at the Highest Price Numerical Issues The price of a lookback put option increases without an upper bound, resulting in theoretical illiquidity discounts exceeding 100%.
The Average-Strike Put Option Theoretical Setup What would the investor do in the absence of trading restrictions? The investor is equally likely to sell on any given day over the restriction period. The average proceeds the investor can realize are S AAA = AAAAAAA S 0,, S T What would the investor do if they are restricted from trading for a time period T? The investor will wait until the restriction period lapses and sell at time t = T for proceeds S T. Under the set of assumptions described above, what is the maximum opportunity loss to an investor who is restricted from trading for a time period T? MMMMMMM OOOOOOOOOOO LLLL = MMM S AAA S T, 0
The Average-Strike Put Option Numerical Example $130 $10 Underprotected Overprotected $114 $110 $17 $100 $90 $18 $80 $70 $60 $79 S AAA = $97 S T = $85 Option Payoff = $1 Jan-13 Jul-13 Jan-14 Jul-14 Jan-15 Jul-15 What if: Want to sell at S t1 = $79? Want to sell at S t = $114?
The Average-Strike Put Option Numerical Solution What is the price of a derivative with payoff MMM S AAA S T, 0? The exact price of an arithmetic average-strike put option can only be found using a numerical solution. Average-Strike Put Formula (Approximations) % DDDD = N v T N v T = N v T 1 1 v T = σ T + ln e σt σ T 1 ln e σt 1 OR v T = ln e σt σ T 1 ln σ T 3 1. N x = 1 N x for all x.. Finnerty [01]; 3. Ghaidarov [009]. σ T N stock price volatility restriction period cumulative standard normal
The Average-Strike Put Option Practical Considerations Conceptual Issues Investors decisions are not random they are more likely to sell on certain days depending on a wide variety of factors. The probability that an investor will sell on a given day in the absence of trading restrictions cannot depend on the length of the restriction period. Liquidity does not provide investors with a tool to dampen volatility or ability to smooth losses. Liquidity = Right to Sell at Any Price Average-Strike Put = Right to Sell at the Average Price
The Average-Strike Put Option Practical Considerations (cont.) Numerical Issues The approximation formula provided by Finnerty [01] results in a maximum discount of 3% because it breaks down for larger volatilities or restriction periods. The size of the error term (difference between exact price and approximation formula) increases as the total variance σ T gets larger. Holding Period: Average-Strike Put Average-Strike Put Difference (Exact Price Monte Carlo) (Finnerty Approximation) (Error Term) Volatility Volatility Volatility 30.0% 60.0% 100.0% 30.0% 60.0% 100.0% 30.0% 60.0% 100.0% 1.0 year 7% 14% 3% 7% 13% 1% 0% 0% % 3.0 years 1% 3% 37% 1% 1% 9% 0% % 8% 5.0 years 15% 30% 46% 15% 6% 3% 1% 4% 15% 10.0 years 1% 40% 60% 0% 30% 3% 1% 10% 7%
The Forward-Starting Put Option Theoretical Setup What would the investor do in the absence of trading restrictions? The investor will sell at a time of their choice t = u for proceeds S u. The exact time at which the investor will sell, and the proceeds they will receive, are not known in advance. What would the investor do if they are restricted from trading for a time period T? The investor will wait until the restriction period lapses and sell at time t = T for proceeds S T. Under the set of assumptions described above, what is the maximum opportunity loss to an investor who is restricted from trading for a time period T? MMMMMMM OOOOOOOOOOO LLLL = MMM S u S T, 0 with 0 u T, where the time of sale u is not known in advance.
The Forward-Starting Put Option Forward-Starting Put vs. Plain Vanilla Put $90 $80 $70 Forward-Starting Put: Strike price will be set at a future date (11/11/13) 11/11/13 Forward-Starting Put: Strike price set to $58 Maturity Date Forward-Starting Put: Payoff = MMM $58 $31,0 = $7 $60 $50 Plain Vanilla Put: Fixed strike price K = $40 set at issuance $40 $30 $0 $10 Plain Vanilla Put: Payoff = MMM $40 $31,0 = $9 $0 Nov-10 May-11 Nov-11 May-1 Nov-1 May-13 Nov-13 May-14 Nov-14 May-15 Nov-15 NasdaqGS:GPOR - Share Pricing
The Forward-Starting Put Option Numerical Example $130 $10 Underprotected Overprotected $114 $110 $100 $90 $80 $70 $60 $79 S u = Variable S T = $85 Option Payoff = Variable Jan-13 Jul-13 Jan-14 Jul-14 Jan-15 Jul-15 What if: Want to sell at S t1 = $79? Want to sell at S t = $114?
The Forward-Starting Put Option Numerical Solution What is the price of a derivative with payoff MMM S u S T, 0 where u is some arbitrary future date at the choice of the option holder when the strike price is set? This is a forward-starting put option (with a start date u) for which an exact closedform pricing formula exists. Forward-Starting Put Formula (exact) % DDDD = N σ T N σ T = N σ T 1 σ T N stock price volatility restriction period cumulative standard normal Note that the formula above is identical to the approximation formula for Average-Strike Put where the argument inside the v T σ T Normal distribution is substituted from with
The Forward-Starting Put Option Practical Considerations The forward-starting put option allows the restricted investor to customize their option by setting the strike price at a time of their choosing thereby fully offsetting their opportunity loss from the trading restrictions in all possible scenarios. Applicable to investors with perfect market-timing skills or investors who are equally likely to sell on any given day Can be utilized by any investor regardless of the trading strategy they follow Liquidity = Right to Sell at Any Price Forward-Starting Put = Right to Sell at Your Chosen Price The price of a forward-starting put option is given by an exact, closed-form solution which depends on the total realized variance σ T in a simple and intuitive manner.
The Forward-Starting Put Option Corroborating Research Longstaff [014] Stopping Rule: decision rule which determines the (potentially random) time when an asset is sold Illiquidity: restriction on the stopping rules that an investor can follow Illiquidity Discount: exchange option for swapping the cash flows under different stopping rules No-Dividend Case: Price of Exchange Option = Price of Forward-Starting Put Option Although based on different premises and following different mathematical derivations, the formulas for the exchange option and the forward-starting put option are identical.
Option Models: No-Dividend Case Summary Comparison What would the investor do in the absence of trading restrictions? What is the maximum opportunity loss to the restricted investor? What benefit does the option holder get? Does the option holder get excess protection? Does the option holder get inadequate protection? Protective Put The investor will sell immediately at time t = 0 for proceeds S 0. Lookback Put The investor will sell at the maximum achievable price over the restriction period for proceeds S MMM = MMM S 0,, S T. Average-Strike Put The investor is equally likely to sell on any given day over the restriction period. The average proceeds the investor can realize are S AAA = AAA S 0,, S T Forward- Starting Put The investor will sell at a time of their choice t = u for proceeds S u. The exact time at which the investor will sell, and the proceeds they will receive, are not known in advance. MMM S 0 S T, 0 S MMM S T MMM S AAA S T, 0 MMM S u S T, 0 with 0 u T, where the time of sale u is not known in advance Right to Sell at Today s Price: downside protection Right to Sell at the Highest Price: guaranteed return Right to Sell at the Average Price: loss smoothing Right to Sell at Your Chosen Price: customized payoff Yes - sometimes Yes - always Yes - sometimes No Yes - sometimes No Yes - sometimes No Pricing Formula Exact Exact Approximation Exact
Illiquidity Discounts: Dividend-Paying Stocks
Dividend-Paying Stocks & DLOM Dividend-Paying Stocks and Illiquidity How do dividends impact illiquidity discounts? For a given constant volatility and constant restriction period, the discount should decrease as the dividend (dividend yield) increases. How do dividends impact put options? In general, dividends reduce the stock price which causes the put price to increase. What are the dividend-adjusted prices for the common DLOM models? Average-Strike Put Forward-Starting Put No Dividends S 0 N S 0 N v T σ T Dividend-Adjusted 1 S 0 e qq N 1 S 0 e qq N v T σ T 1 1
Dividend-Paying Stocks & DLOM Model-Implied Discount with Dividends 18.0% Average-Strike Put Option Model: Dividend-Adjusted Price (dividend yield = 10%) Model-Implied Discount 16.0% 14.0% 1.0% 10.0% 8.0% 6.0% 4.0%.0% 0.0% 0.0.0 4.0 6.0 8.0 10.0 1.0 14.0 16.0 Restriction (Holding) Period - Years vol = 30% vol = 60%
Dividend-Paying Stocks & DLOM Dividend-Paying Stocks: Price Components Dividend income stream vs. residual value Dividend Income S 0 1 e qq %DLOM =??? S 0 %DLOM Residual Value S 0 e qq σ T = N 1 Dividend-Adjusted option prices: full DLOM applied to the residual stock price component, no DLOM applied to the dividend income steam component. Expected dividend cash flows treated as fully liquid and immediately available
Dividend-Paying Stocks & DLOM Dividend-Paying Stocks: Hypothetical Scenario Non-Dividend Paying Stock Current stock price is $100, expected volatility is 50% Restricted form trading for 3-year period Implied DLOM = 34% (forward-starting put option) OOOOOO PPPPP = $100 N 50% 3.0 1 $33.5, % DDDD = $33.5 $100 34% Dividend Paying Stock Current stock price is $100, expected volatility is 50% Restricted form trading for 3-year period Stock will pay a large dividend equal to 90% of its value in.9 years Implied DLOM = 3% (forward-starting put option) S 0 = $100, PP oo DDDDDDDDD = $90, RRRRRRRR VVVVV = S 0 e qq = $10 OOOOOO PPPPP = $10 N 50% 3.0 1 $3.4, % DDDD = $3.4 $100 3%
Dividend-Paying Stocks & DLOM Dividend-Paying Stocks: Hypothetical Scenario (cont.) Dividend Paying Stock: A Second Look Current stock price is $100, expected volatility is 50% Restricted form trading for 3-year period Stock will pay a large dividend equal to 90% of its value in.9 years Implied DLOM = 33% (forward-starting put option) S 0 = $100, PP oo DDDDDDDDD = $90, RRRRRRRR VVVVV = S 0 e qq = $10 OOOOOO PPPPP 1 = $10 N OOOOOO PPPPP = $90 N % DDDD = $3.4+$9.7 $100 = 33% 50% 3.0 50%.9 1 $3.4 1 $9.7
Dividend-Paying Stocks & DLOM Dividend-Paying Stocks: Step-by-Step Solution Bifurcate current stock price Present value of expected dividend cash flows (under risk-neutral measure) Residual stock value Calculate the expected (weighted average) timeframe to dividend payment The present values of the discrete dividend cash flows are the weights For a continuous dividend yield, the weighted average timeframe is approximately the middle of the full restriction period Calculate the dollar illiquidity discount (option price) applicable to each of the two stock price components The residual value component gets the full DLOM (full restriction period) The dividend component gets a reduced DLOM (weighted average timing of dividends) The sum of the two option prices is divided by the current stock price to get the overall percentage DLOM
References
References Finnerty, J. An Average-Strike Put Option Model of the Marketability Discount. The Journal of Derivatives, Vol 19, No. 4 (01), pp. 53-69. Ghaidarov, S., Analysis and Critique of the Average-Strike Put Option Marketability Discount Model SSRN Working Paper (009). Ghaidarov, S. Analytical Bound on the Cost of Illiquidity for Equity Securities Subject to Sale Restrictions. The Journal of Derivatives, Vol 1, No. 4 (014), pp. 31-48. Longstaff, F., Valuing Thinly-Traded Assets. National Bureau of Economic Research, Working Paper No. 0589 (014).