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A Complete Model of Warrant pricing that Maximizes Utility. Chen Rong. The stock price motion. The stock price motion. A common stock without dividends; Current price: X t The price of the stock n periods later: X t+n, a finite-variance multiplicative probability distribution
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A Complete Model of Warrant pricing that Maximizes Utility Chen Rong
The stock price motion • A common stock without dividends; Current price: Xt • The price of the stock n periods later: Xt+n, • a finite-variance multiplicative probability distribution • Where the price ratios Xt+n/X=Z=Z1Z2…Zn are assumed to be products of uniformly and independently distributed distributions of the form prob{Z1≤Z}=P(Z;1), and where for all integral n and m, the chapman-Kolmogorov relation Is satisfied.
The stock price motion(2) • All these mean that the geometric Brownian motion, which at lease asymptotically approach the familiar log-normal. • Another hypothesis: risk averter with concave utility and diminishing marginal utility. So, • Where αis the mean expected rate of return on the stock per unit time. (a constant independent of n.)
A special case • A special case: n=1，λ>1 • This simple geometric binomial random walk leads asymptotically to the log-normal distribution. • (7.2)becomes
The 1965 Model of Samuelson • Some assumptions • American warrants • A arbitrarily postulated gain as exogenously given for stocks (α) and warrants (β) • No-arbitrage pricing
No-arbitrage pricing • Rational pricing (arbitrage value ) • like binomial trees pricing for American options • One-step binomial tree • Then, for any length of life,
The perpetual warrant • The perpetual warrant • Where C is the critical level that the warrant will be exercised at once or not. • This critical level will be definite if β>α.
β=α • When β=α, the case is particularly simple. • When β=α, the conversion will never be profitable. • The value of the warrants of any duration can be evaluated by mere quadrature:
The main shortcomings of the 1965 model: • arbitrarily postulated gain • Not consider the utility of investors
New considerations • Expected utility maximization • Strictly concave utility function • Portfolio analysis • The expected utility is maximized when w=w*,where w is the fraction of wealth in the stock:
Further understanding • Figure 7.1---7.3 • If 0<w*<1,U’(0)>0; U’(1)<0 • The above relationship might well be called the Fundamental Equations of Optimizing Portfolio theory. • If borrowing or short-selling is permitted, we can get the inequalities by Kuhn-Tucker methods.
The contined example • P=1/2, Bernoulli logarithmic utility
How to determine warrants holding and price under these new considerations: a basis analysis
Determining warrant holdings and returns • We can deduce how many warrants we would hold and what warrants must yield by a maximizer of expected utility. • Some assumptions: • Safe asset; er-1 • Concave total utility • Every person has a constant elasticity of marginal utility at every level of wealth and that the value of this constant is the same for all individuals to free us from problems of aggregation. • The specified outstanding warrants to be voluntarily held. • To induce people to hold a larger amounts of warrants, their relative yields will have to be sweetened. • w1+w2+w3=1
To solve these equations, we can set T=1 for simpleness. • can be got as the solution to a simultaneous-equation supply and demand process that auctions off the exogenously given supplies of common stock and warrants at the prices that will just get them held voluntarily. (we will see it later)
can be got from arbitrage-conversion consideration. • So, (7.16 ) becomes an implicit equation enabling us to solve for the unknown function recursively in terms of the assumed known function .
The “incipient” case • We can enormously simplify the implicit equation in (7.16) by using w3*=0. • When w3*=0, the dependence of U’ on becomes zero, and (7.16) becomes a simple linear relationship.(P227) • When w3*=0, the warrant price is to be determined at the critical level just necessary to induce an incipient amount of them to be voluntarily held. And it is also the critical level at which hedging transactions, involving buying the common and selling a bit of the warrant short, just become desirable
How to get the wj* • T=1，fixed warrants supply of • The probability distribution of common-stock price changes as exogenously given, with P(Z;1). • Current stock price x • The amount of the safe asset is and the return is exp(r)
Then, (7.15),(7.16)
Above three equations are independent equations for the three unknown w1*, w2* and F1(x) • Hence we do have a determinate system. • When , we have the simpler theory.
Utility-maximizing warrant pricing: the important “incipient” case • For T=1, the “incipient” case become
Now, put (7.20),(7,21)and (7.9)(F0(X)=max(0,X-1)) together, we can get linear recursion relationships to solve our problem completely, provided we can be sure that the warrants will not be converted.
Will it be converted? • We can find that with dQ, the effective probability mean of every asset will be the same, which seems the same as the α＝βcase in the 1965 model. • But in fact αP＝Βp is not the same as αQ＝ΒQ • However, if without dividends, the warrants will not be converted before it matures. We can prove it later.
So, if we are assured of nonconversion, the value of a perpetual warrant can be determined. • is a solution; • is another solution but only for c=1 can we satisfy the two-side arbitrage conditions X≥F(X)≥X-1 • In fact cXm are solutions but only m=1 is relevant in view of our boundary condition.
Explicit solutions • Besides the step-by-step solution, we can also use quadrature or direct integration over the original F0(X) function to get the explicit formulas. • But there are some by-no–means obvious complications in our new theory which we need care about. • P222 (7.12)and P231(7.24-7.26)
But these relations are not valid. They would be valid only if we locked ourselves in at the beginning to a choice of portfolio that is frozen. There is no w in dP but there is in dQ. • So we need new iterated integrals .
Now we get the new generalized util-prob function Qt(Z), which can replace the Q(Z) to complete the iterated integrals. • At the same time, we found that the “subjective yields”αQ and βQ calculated for the new Qt(Z) do all equal r per unit time. See P234 (7.31).
Warrants never to be converted • The explicit assumption: no dividend. • We can show that in the present model the warrants are never converted. (i.e.Fn(X)>F0(X)). • Theorem 7.1 • If And we are in the case where the warrants need never be converted prior to expiration.
Corollary of this theorem: • Longer life of a warrant can at most enhance its value.i.e.Fn+1(X)≥Fn(X)
If Q(Z;1)>0 for all Z>0 and Q(Z;1)<1 for all Z<∞,we can write strong inequalities • But for some case as in the example, Q(Z;1)=0 for Z<λ-1<1,weak inequalities. • The crucial test is: if for a given X, one can in T steps end up above the conversion price of 1, then for r>0,FT(X)>F0(X) and Fn+T(X)>Fn(X) .
now, our new theory has arrived at such a conclusion: a perpetual warrant should sell for as much as the common stock itself. • How to explain it? • Common stock may pay dividends and the result will not right. • Some argue that the incipient case may be the reason but they are wrong • One dollar exercise price can be deemed of negligible percentage importance relative to the future value of the common. (and this one dollar is not paid now but in the future). • If people believe this, it will be a self-fulfilling belief; if most people doubt this ,the person who believe in it will average a greater gain by buying warrants.
