TARP Warrants Valuation Methods 

Robert A. Jarrow 

September 22, 2009 

Background and Summary 

I was engaged as a contractor by the U. S. Treasury from July 15, 2009 to August 15, 2009 to assess the 

U.S. Treasury’s TARP warrants valuation methodology. This document details my understanding of the 

Treasury’s approach for valuing TARP warrants, gained from direct dialogue with Treasury staff 

members. 

Under the Capital Purchase Program (“CPP”), the U.S. Department of the Treasury (“Treasury”) received 

warrants in connection with each of its preferred stock investments in a Qualified Financial Institution 

(“QFI”). For investments in publicly traded institutions, Treasury received warrants to purchase common 

shares.1

 When a publicly-traded QFI repays Treasury’s CPP preferred stock investment, the QFI is 

contractually entitled to repurchase the CPP warrants at fair market value. 

The Treasury uses a number of different valuation approaches to help estimate fair market value. These 

approaches include indicative valuations from market participants, independent valuations from external 

asset managers, and modeled valuations using methodologies further described in this paper. The range 

of values provided in these approaches is analyzed by the Treasury to determine the adequacy of a QFI’s 

assessments of fair market value. 

Overview Warrant Repurchase Process under the CPP Contract 

The warrant repurchase process is a multi-step procedure, starting with a QFI who wishes to repurchase 

the warrants submitting a determination of fair market value to Treasury. The Treasury can accept the 

fair market value or not. If the Treasury and the QFI cannot reach an agreement, either party may invoke 

an appraisal procedure. In this appraisal procedure, the bank and Treasury select independent appraisers. 

If these appraisers fail to agree, a third appraiser is hired, and subject to some limitations, a composite 

valuation of the three appraisals is used to establish the fair market value. If Treasury and the QFI cannot 

reach agreement regarding fair market value and neither party invokes the appraisal procedure, the 

Treasury intends to sell the warrants through an auction. 

The Treasury has developed a robust set of procedures for evaluating initial QFI determinations based on 

three inputs: market prices (where available) and quotes from various market participants, financial 

models, and outside consultants/financial agents. The details of this repurchase process can be found at 

http://www.financialstability.gov/docs/CPP/Warrant-Statement.pdf. 

Financial Modeling 

The U.S. Treasury performs an in-depth model valuation as input to its assessment of a warrant’s fair 

market value. The remainder of this report provides an in-depth description of the Treasury’s valuation 

model. 

To value its warrants, the Treasury uses a modified Black-Scholes model. For computations, the Treasury 

employs a binomial approximation to the Black-Scholes model. It is well known that the binomial model 

 

1

 In the case of institutions that are not publicly-traded, Treasury received warrants to purchase preferred stock or 

debt and these warrants were exercised immediately upon closing the initial investment. As such, these warrants are 

no longer outstanding. 
converges to the Black-Scholes model as the number of “steps” in the binomial’s tree approaches infinity. 

The Black-Scholes model and its binomial approximation are well-accepted methods for pricing options 

by both academics and market participants (see Cox and Rubinstein [1985], Hull [2007], Jarrow and 

Turnbull [2000]). 

An unadjusted (or not modified) Black-Scholes model for pricing equity options is based on the following 

simplifying assumptions: (1) no dividends, (2) constant interest rates, (3) the underlying stock’s volatility 

is constant across time, and (4) frictionless markets (liquid markets and no funding costs). The U.S. 

Treasury uses a modified Black-Scholes model to incorporate the relaxation of these simplifying 

assumptions. The modifications employed are discussed below. 

In addition, the unadjusted Black-Scholes model is formulated to price equity options and not warrants. 

Warrants differ from equity options in that when warrants are exercised, to fulfill the conditions of the 

warrant contract, the bank issues new shares. This is not the case with equity options. The U.S. 

Treasury’s valuation method explicitly recognizes this distinction. This potential dilution effect of 

warrants is also discussed below. 

The Standard Inputs 

The standard inputs to the modified Black-Scholes warrant valuation model include the maturity date of 

the warrant, the warrant’s strike price and the underlying stock price. The warrant’s maturity date and 

strike price are as given in the CPP contract. For the current stock price, the Treasury uses a 20-day 

moving average of past stock prices to smooth any aberrations in the stock’s price movements. However, 

the current stock price is also considered to include any recent shifts that may impact valuation. 

The Modifications 

1. Dividends 

Unlike common stock, warrants are not entitled to dividend payments, and thus dividends reduce the 

value of the warrant by eroding the value of the underlying shares. The modified Black-Scholes model 

includes this dividend erosion by assuming that the underlying stock pays a constant dividend yield. 

To estimate the dividend yield the Treasury analyzes the company’s dividend payment history and 

investigates the company’s implied or explicit dividend policies. The Treasury also examines recent 

dividend actions or market activity that may have changed dividend yields significantly. The effect of 

these changes is estimated and incorporated into the average dividend yield. 

It is well known that with dividends, an American call option’s value may differ from an otherwise 

identical European call option’s value. This value difference is due to the possibility of early exercise. 

The TARP warrants can be exercised early; hence, they are American-type warrants. Early exercise is 

explicitly included within the binomial approximation procedure when valuing TARP warrants. 

Justification 

For a common stock, over a ten-year horizon, dividends will be stochastic and discrete. The Treasury 

approximates these discrete and stochastic dividend payments using a constant dividend yield. Since the 

underlying stock price is stochastic, a constant dividend yield implies that the total dividends paid over 

any quarter are stochastic. Hence, a constant dividend yield approximation incorporates the stochastic 

nature of these discrete dividend payments. This is a well-accepted approach to handling discrete and 
stochastic dividends (see Jarrow and Turnbull [2000, p. 258]). 

2. Stochastic interest rates 

The Treasury uses as the interest rate input the yield on a Treasury bond that matches the maturity of the 

warrant. Because the warrants in the Treasury portfolio are 9 to 10-year dated, the Treasury finds the 

appropriate matched maturity yield by straight-line extrapolating between the 7-year and 10-year constant 

maturity Treasury bonds. 

Justification 

It is well known (see Amin and Jarrow [1992]) that to modify the Black-Scholes formula for stochastic 

interest rates, there are two necessary adjustments. First, the yield on a Treasury bond matching the 

warrant’s maturity should be used as the input to the Black Scholes formula. The Treasury incorporates 

this first adjustment. Second, when using historical volatility estimates, the volatility input should be 

adjusted to reflect the increased randomness due to the interest rate volatility and its correlation to the 

stock’s return. This adjustment to the historic volatility is typically small and can often be excluded. 

However, when using implied volatilities, this second adjustment is unnecessary (see Jarrow and Wiggins 

[1989]). Because the Treasury uses an implied volatility estimation procedure whenever implied 

volatilities are available, the second adjustment is not used in the Treasury’s valuation method. 

3. Stochastic Volatility

There are two methods for estimating volatility: implied and historical. Without modification, both 

methods have limitations when estimating long-dated warrants with stochastic volatility. The Treasury 

uses a modified procedure involving both methods (where available) to construct a 10-year forward 

volatility curve. A forward volatility curve captures the stochastic nature of volatility. An “average” of 

the forward volatilities across this 10-year curve comprises the input to the modified Black-Scholes 

formula. Importantly, the Treasury also considers warrant values for a range of volatilities around this 

“average” forward volatility input. 

For large financial institutions with liquid public equity and long-term options, the detailed procedure is 

as follows. The Treasury uses both observable implied volatility and historical volatility to construct a 

10-year forward volatility curve. The initial segment of the curve consists of the observed implied 

volatilities for traded options. The last segment of the curve consists of a “normalized” 10-year average 

historical volatility. The volatility is normalized by removing any abnormally high recent volatilities 

from the estimate. The middle segment is determined using straight-line interpolation between the initial 

and terminal segments. The estimated forward volatility curve is typically downward-sloping, consistent 

with a reversion in volatilities to a long-run value. 

Justification 

It is well known (see Eisenberg and Jarrow [1994], Fouque, Papanicolaou and Sircar [2000]) that when 

volatilities are stochastic, a call option's value can be written as a weighted average of (constant volatility) 

Black-Scholes values (each with a different volatility input). The weights in this average correspond to 

the martingale probabilities of the different volatility inputs being realized. The volatility inputs are the 

average of the 10-year realized volatilities, i.e. 

volatility _ input = σ s

ds

0

T

∫

T
where time 0 is today, time T is the maturity of the option (10 years), and σ s is a possible realization of 

the random volatility at a future time s. 

As discussed previously, the Treasury provides a range of Black-Scholes for various volatility inputs 

around the average of the 10-year forward volatility curve. These inputs can be interpreted as various 

possible averages of the future realized volatilities. The midpoint of this range is, therefore, an estimate 

for the option's value under a stochastic volatility model.2

3. Market Imperfections 

The Treasury considers a number of market imperfections that could potentially cause the fair market 

value of a warrant to deviate from the model value (such as illiquidity of the warrant instrument or the 

bank’s underlying equity). Judgment is used on a case-by-case basis to determine which adjustments, if 

any, for these market imperfections are appropriate. 

Justification 

Directly capturing market imperfections - market illiquidity and funding costs - in an option model is 

difficult (see Jarrow and Protter [2008], Broadie, Cvitanic and Soner [1998], Naik and Uppal [1994], and 

Cuoco and Liu [2000]). Each of these market imperfections can be considered as a type of transaction 

cost. It is well known that transaction costs make a market incomplete. In an incomplete market, there is 

a range of arbitrage free prices determined by the buying price (highest part of range) and a selling price 

(lowest part of range). The standard Black-Scholes value (without market imperfections) can be shown to 

lie between these two prices. 

The buying and selling prices are determined by a trader's cost of replicating the identical cash flows to 

the warrant synthetically (for a long position and for a short position), including all market imperfections, 

via delta hedging. Note that with these market imperfections, there will be a buying premium and a 

selling discount reflecting the additional costs of obtaining the required cash flows. 

Specific considerations with respect to market illiquidity and funding costs follow. 

a. Illiquidity 

The level of the stock price input into the Black-Scholes value captures the general impact of a depressed 

and/or illiquid market - making the warrants less valuable. This is distinct, however, from market 

liquidity considered as an endogenous transaction cost, i.e. a quantity impact on the price. A quantity 

impact on the price captures the notion that if you buy many warrants in an illiquid market, you need to 

pay more per share than if you buy only a single unit. Similarly, if you sell many warrants in an illiquid 

market, you will receive less per share than if you sell only a single unit.

The quantity impact on the market price is market-wide, and not trader-specific. For quoted prices 

(options prices for one round lot) or transaction prices (actual trading volume) the liquidity impact of the 

trade is captured by using an implied volatility (see Jarrow and Wiggins [1989]). 

For large market trades of warrants, this component has to be separately included (as a discount to the 

model price) after the model's value is determined based on the Black Scholes formula using implied 

 

2 Of course, the midpoint assumes that each estimate of the realized future volatility provided is equally 

likely under the martingale probabilities.
volatilities. The magnitude of the potential discount is difficult to estimate. It depends on the size of the 

warrant position and the liquidity of the equity underlying the warrants. To access the magnitude of the 

liquidity impact, one can compute the number of shares an investor would short to “delta-hedge” the 

warrant position and compare that number to the average daily trading volume of the stock. The number 

of days of daily trading volume needed to delta hedge the position, across different banks, provides 

information on the relative liquidity of the warrant market. 

It is important to emphasize that the price to the buyer (the bank) will exceed the price to the seller (the 

Treasury). The bank would pay a liquidity premium in buying (i.e. analogous to the “ask” in a bid-ask 

spread). The Treasury would incur a liquidity discount in selling (i.e. analogous to the “bid” in a bid-ask 

spread). Since the buyer and seller are meeting in a negotiated transaction, the "fair market" price should 

be in between the two prices.3

b. Funding Costs 

Each bank has its own unique funding costs. These costs are determined by its existing balance sheet and 

credit worthiness. These unique funding costs determine the bank's internal cost of constructing a 

warrants cash flows synthetically (trading in the stock and borrowing/lending) via delta hedging. 

In contrast, the fair market value is determined by the "marginal trader's" funding costs. The marginal 

trader is often not constrained, e.g. they can sell stock from inventory rather than shorting and incurring 

short sale fees. The marginal trader is the lowest cost transactor and their trades determine the fair market 

price. A market price that differs from the marginal trader’s cost of construction - their buying/selling 

costs - will generate arbitrage opportunities for them. The marginal trader taking advantage of any such 

arbitrage opportunities will force the market price to reflect their funding costs. 

There is significant empirical evidence that supports the claim that option models fit market prices well 

without explicitly including funding costs (see for example Pan [2002]). This supports the assertion that 

the fair market price is determined by marginal traders with small funding costs. 

From the bank's perspective, if their funding costs are too high, for a long position, the bank would prefer 

to buy the warrants on the market rather than creating the same cash flows synthetically on their balance 

sheet via delta hedging. Similarly, for a short position, the bank would prefer selling the warrants in the 

market rather than holding the warrants on their balance sheet and shorting the warrants synthetically via 

delta hedging. 

The Treasury's mandate is to obtain the "fair market" value of the warrants. Consequently, the marginal 

trader's funding costs are those that are relevant, not the bank's. Since the modified Black-Scholes model 

captures dividends, stochastic interest rates, and stochastic volatility, the only remaining considerations 

are market liquidity4

 and funding costs. If one uses the Black-Scholes model (as modified above) without 

market liquidity and funding costs, and (for a given future realized volatility) if one computes an option's 

implied volatility, then the implied volatility will incorporate both the historic volatility and the marginal 

trader's liquidity impact and funding costs (see Jarrow and Wiggins [1989]). 

 

3 Note that the Black-Scholes value using implied volatilities is near the midpoint of the range determined 

by the buying premium and selling discount. 

4 The adjustment for market liquidity – a quantity adjustment on the price - is given by a scale adjustment 

after the model's value is computed using the implied volatilities which are based on typically small 

transaction volumes for the traded options.
For this reason, using the implicit volatility is key to including market liquidity and funding costs into the 

valuation procedure. 

Bank's Buying Price

Treasury's Selling Price

Black Scholes Value

Marginal Trader's

Buying Price

Marginal Trader's

Selling Price Fair market value

lies within red

Figure 1: Various Warrant Prices 

To understand the difference between the various prices, consider the following alternatives as reflected 

in Figure 1. 

i. Bank’s buying price: The bank can keep the warrants on its balance sheet, but borrow 

and trade in the underlying stock to remove the economic impact of the warrants. Note 

that the bank is short the warrants to the Treasury. Hence, it has to synthetically create a 

long position in the warrants via delta hedging, i.e. it has to buy the stock and borrow. 

When borrowing, the bank is incurring its higher funding costs (through a higher 

borrowing rate). It can be shown that the cost of synthetic construction - the bank's 

buying price - exceeds the Black Scholes value without the inclusion of these funding 

costs.5

 

ii. Marginal trader’s buying price: The fair market price is determined by the marginal 

trader's funding cost for a long position in the warrant. The marginal trader's funding 

costs are less than those of the bank's (see Figure 1). 

iii. Treasury’s selling price: The Treasury has the analogous decision to the bank’s. It 

can keep the warrants on its balance sheet and remove their economic risk by delta 

hedging. If the Treasury keeps the warrants on its balance sheet, it needs to sell the 

underlying stock and lend cash. The Treasury's funding costs are the relevant 

consideration here. The lending rate is the Treasury rate (zero funding cost). Although 

there are no funding costs, the Treasury's selling price would include short sales fees and 

the liquidity discount. Short sales fees and the liquidity discount make the selling price 

below the Black Scholes value without the inclusion of these costs (see Figure 1). It is 

important to note that the bank's selling price would be approximately the same as the 

 

5 Of course, the inclusion of liquidity costs in the bank's delta hedging will further increase the buying 

price due to the liquidity impact on the price. 
Treasury's. The reason is that the bank’s lending rate is the Treasury rate as well (zero 

funding cost). 6

iv. Marginal trader’s selling price: The fair market price is determined by the marginal 

trader's funding cost for a short position in the warrant. The marginal trader's funding 

costs are less than the bank's (see Figure 1). 

v. Fair market price: The fair market price lies between the marginal trader’s buying and 

selling prices. It is determined by market equilibrium considerations. 

A negotiated transaction between the bank and Treasury has as the range for negotiation all prices 

between the bank's buying price and the Treasury's selling price. The Black-Scholes value without the 

inclusion of funding costs lies between these two. The fair market value is also between the two and 

determined by the marginal trader's funding costs. The Treasury's mandate is to obtain the "fair market" 

value of the warrants. Consequently, if Treasury’s model were the only valuation mechanism is use, 

starting discussions regarding value with the modified Black-Scholes value as computed above would be 

appropriate. 

Dilution from Warrant Exercise

Warrants differ from standard equity options in that the shares that a warrant holder receives, if exercised, 

are issued by the company and are not currently outstanding. As such, the exercise of the warrants 

triggers an issuance of shares by the company, and a potential dilution of existing shareholders’ values. 

The Treasury makes no adjustment to the Black-Scholes value for this dilution effect, due to the fact that 

this dilution effect is rationally anticipated by market participants and already included in the current 

stock price input into the modified Black-Scholes formula. 

Justification 

There are two issues that arise due to warrant exercise resulting in the issuance of new shares. These are 

called sequential exercise and strategic exercise. Sequential exercise occurs when a large trader or 

monopolist owns most of the shares and they can create more value by exercising sequentially rather than 

as a block (see Constantinides [1984], Emanuel [1983], Spatt and Sterbenz [1988], and Linder and 

Trautmann [2009]). Strategic exercise occurs when a large trader or a group of small traders (acting 

independently via a Nash equilibrium) can create more value by exercising the shares only if the market 

price exceeds a value greater than the strike price (see Cox and Rubinstein [1985], p. 396, Galai and 

Schneller [1978], and Crouchy and Galai1[1994]). This is due to a potential transfer of wealth from 

shareholders to the liability holders due to dilution and an inflow of cash into the firm. 

Since the Treasury is mandated to determine a fair market price, sequential exercise is not a relevant 

consideration because it only applies to a monopolist. Strategic exercise is a potential consideration, but 

to obtain a realistic representation of both the dilution and cash inflow effects, one must explicitly model 

 

6 There is no economic rationale for why the bank's high funding costs (higher borrowing rate) should be 

used as a justification for obtaining a lower selling price. The logic underlying using the bank’s selling 

price is that the bank wants to buy back the warrants from the Treasury while simultaneously creating a 

short position in the warrants to finance the purchase (using the proceeds from the short position). This 

argument effectively retains the economic position of the outstanding warrants on the bank's balance 

sheet. 
the liability structure of the bank. Given the complexity of a large financial institutions balance sheet 

(including off and on balance sheet items), this is an impossible task.7

An alternative approach, consistent with both theory and empirical evidence (see Schulz and Trautmann, 

[1994] is to assume that the market rationally anticipates the dilution and cash flow effects, and that these 

are embedded in the current stock price input into the modified Black-Scholes formula. Note also that the 

adjustment for stochastic volatility is consistent with this implementation. This is the approach that the 

Treasury adopts.

Warrant Contract Terms

The terms of the Treasury’s warrants are specified in the Form of the Warrant documentation available on 

www.financialstability.gov. Certain of these terms can affect the warrant’s value in a way not captured 

by an unadjusted Black-Scholes model. The Treasury considers each of these effects and includes them, 

when possible, in the valuation of the warrants. 

Dividend protection. The warrant document (see 

http://www.financialstability.gov/docs/CPP/warrant.pdf) specifies protective adjustments to the terms of 

the warrants in the case that dividends in excess of certain levels are paid. This dividend protection 

would never decrease and would sometimes increase the value of the warrant. The exact effect on the 

value of the warrant depends on many factors including dividends at the time of funding, current 

dividends, and expected future dividend activity. 

Business combinations. The warrant document also specifies certain adjustments to be made under 

certain business combinations. The effect of the terms is that some out-of-the-money warrants could 

become worthless (i.e. lose all their time value) if the underlying equity is purchased for cash by another 

company. Business combinations could also change the volatility of the underlying equity of a warrant. 

Conclusion 

As documented above, it is my belief that the Treasury’s modeling methodology for valuing the warrants 

is consistent with industry best practice and the highest academic standards. The methodology uses the 

industry standard model for pricing options, the Black-Scholes model, in a modified form to account for 

the size of the warrant position, stochastic interest rates, stochastic volatility, as well as numerous market 

imperfections. 

Furthermore, as previously detailed, the Treasury’s financial model is only one component of a robust 

valuation procedure. For warrant positions that are evaluated, the Treasury also collects market prices 

(where available) or indications from market participants and valuations from outside 

consultants/financial agents. All valuation information is considered in the determination of an 

appropriate fair market value for the warrants of a specific institution. 

The valuation process results in a warrant valuation that is fair to both the participating banks and the U.S. 

taxpayers. 

 

7

 Note that the existing academic literature only considers too simple and unrealistic capital structures. 
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