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29 Mar 1998

In this issue, Johan G.B. Beumée, a partner in Riskcare Limited, and Paul Wilmott, professor of mathematics from Imperial College London, present some approximations of the warrant pricing method introduced in a previous Learning Curve (DW 1/12).


One of the differences between the classical Black-Scholes expression for a share call options and the warrant price as presented in equations 1, 2 and 3. is that the volatility relates to the value of the company rather than the volatility of the share price which was pointed out by John Hull. This effect may be considerable as the examples will demonstrate. The other important difference is that equation 1 is actually non-linear.

To approximate the contribution of the non-linear dividend term we can rewrite equation 1 to obtain compensation for the dividend (an approximation again) such that the effective dividend is related to the dividend yield (q) as;

This is yet another circular argument that can be circumvented by first calculating the solution with the dividend yield q and then recalculating equation 3 using the adjusted dividend. The assumption that the ratio between spot warrant price and spot share price remains constant may actually be more or less true for a period following issuance but the assumption is certainly unreasonable towards expiry as the warrant will either migrate into-the-money or move out-of-the-money.

From equation 1 we can find a rough-and-ready relationship between the volatility of the share price (which can be measured) and the volatility of the asset value. Taking the differentials of both sides of equation 1 it is found that the corresponding stochastic parts yield;

where gamma is the portfolio hedge ratio. To resolve the circularity problem again we first calculate the warrant price, see equation 3, using the measured share price volatility and then recalculate equation 3 using the estimate for the total company value volatility. It can be shown that C>=W(t)/S(t) so the total company value volatility is always larger than the share price volatility. Notice the volatility and dividend adjustments increase the premium of the warrant while the dilution decreases this effect.

As an example consider a company that issues 100 European style warrants while currently there are 1000 shares outstanding. The warrant has a 10 year expiry, we assume a 9% flat interest rate curve (LIBOR ACT/360 & SWAP semi-annual ACT/365 rates at 9%) 4% dividend, S = 100 per share and a 35% share price volatility. To determine the price of the warrant correctly we first assume the total company value volatility is also 35% and the dividend yield equals 4%. In the first iteration the price for the warrant is calculated using Black-Scholes which subsequently can be recalculated using the adjustments.

These calculations demonstrate that the potential dilution of an extra 100 shares reduces the Black-Scholes price by about 50 cents of its original value. The first column in Table 1 shows the Black Scholes price ignoring any changes in volatility or dividend resulting from dilution. The effective volatility is approximately 35.65% and this adjustment is almost independent of the strike. The effective dividend rate has to be adjusted to 3.90%(K=190) to 3.87% (K=130). The last column provides the warrant prices employing the effective volatility and dividend in the second and third column.

This 50 cent correction is independent of strike. Imagine that a company is in demand of a sizeable quantity of capital but instead of issuing shares and diluting the present shareholders immediately the company issues a large quantity of warrants. For example, if the number of warrants issued equals 500 on a 1000 outstanding shares the dilution effect is much more substantial. The Black Scholes approximation for the warrant premium can in principle be found again using equation 1.

The effect of a 50% share dilution at warrant expiry leads to a 2.00 (strike=190) ­ 2.40 (strike=130) drop in warrant price.

Just as in the European case the delta decreases under a smaller dividend which decreases the dilution effect. The effective dividend to be used in this particular case is approximately 3.5% and the total value volatility is approximately 3% higher than the volatility in share price. This dilution seems too large to occur in practice, however, there are some examples where a company raised substantial capital in precisely this fashion preferring to issue a large quantity of warrants rather than issuing shares outright.


If American-style exercise is allowed we demonstrated above that once the optimal boundary W(t)>=S(t)-k is reached it is optimal to exercise at least some of the warrants. To side-step the issue of the optimal exercise policy for the moment let us assume that all warrants will be exercised immediately by the warrant holders. In this case we can obtain a premium for the American style warrant by solving equation 1 with the boundary condition W(t)>=(A(t)-Nsk)/(Ns+Nw) and final condition equation 2. So, for example, let the final expiry be 10 years with a 9% interest rate, 4% dividend, S = 100 per share and a 35% share price volatility. The warrant can be exercised at any time before year 10 and the corresponding warrant premiums are presented in Table 2 as a function of the strike. The corresponding option prices were obtained using a standard tree algorithm and the warrant premiums were obtained with the aid of the delta and gamma approximation, see previous Learning Curve.

Since the delta is larger than in the European case the effect of dilution is somewhat less pronounced than in the European case. Using the adjusted dividend and volatility we can find a second order approximation to the premium in the same fashion as for the European style warrants.

For American style warrants the fashion of exercise may be of influence on the premium. Our analysis suggests that at the optimal exercise boundary the premium of the warrant, the share price and the asset value for the company are all continuous. An argument can be presented to suggest that immediate exercise of all warrants is the optimal policy, however, the reasoning is rather heuristic. The time that the warrant reaches the boundary is a Markov time implying that the underlying stochastic process resets itself from that point onward. Now, if the value of the remaining warrants can be represented by the adjusted differential equation from this time onward and if the initial condition for these warrants can be given by the total asset value of the company plus the new cash then the exercise boundary remains unaltered. The remaining warrants are still at the boundary and should therefore also be exercised.

Though in theory this approach to calculate an exchange warrant is correct the required hedge necessitates a liquid market of corporate bonds which is often not a reasonable assumption. A great many warrants are associated with relatively illiquid corporate bond markets. In that case alternative interest rate hedges can be considered (treasury instruments, for instance), however, this will result in a certain amount of basis risk. This lack of liquidity in hedging instruments suggests the warrant premium is actually higher than the premium implied by the exchange warrant model. The imperfection of the market, the longevity of the warrant, the potential dilution at expiry and the non-linearity of the equation of motion insure that pricing a warrant remains something of an art.

29 Mar 1998