Tag Archives: stocks

Black-Scholes Pricing Model: Is the Hedging Argument Correct? Preface Many are familiar with the works of Myron Scholes and Fisher Black in the late 1970s. Their contributions revolutionized the way we price options. Out of the many sections of their proof, the most interesting one in my opinion is the hedging argument given midway through the paper. The reason for which I find it so intriguing is due to the fact that it is the one section that is criticized the most. This leads us to a now popularized question, is such argument valid? Flashback Time As we pull out the proof written more than four decades ago, we notice that the traditional Black-Scholes hedging argument strictly assumes that: markets are frictionless, there is no arbitrage, there is a constant interest rate denoted as “r”, no dividends are paid out and that the stock price process respects the Geometric Brownian Motion. Let’s not forget that GBM (Geometric Brownian Motion) is a continuous time-stochastic process that models stock prices in the Black Scholes Model and other similar works. Click to enlarge If we denote the following as a European Call Price process: We must then further assume the following: As being in conjunction with some “C^2,1″ function C (S, t). When applying Itô’s Lemma we observe the following: Click to enlarge Let us also not forget that Itô’s Lemma is an identity to find the differential of a time-dependent function of a stochastic process. Now let’s consider a portfolio with the goal of long one call and short the following shares. (The goal is therefore what the portfolio consequently consists of.) If we short shares , then we must presume the following: By extracting the textbook argument we can observe the following steps: STEP 1. (**Highlighted**) STEP 2. Click to enlarge STEP 3. Click to enlarge Recalling that arbitrage is not part of the environment of the model we may equate coefficients on ” dt” yields the Black Scholes PDE. (We don’t need to go as far as to solve PDE) Click to enlarge This brings to mind a fascinating question, was the previously highlighted step (Step 1) correct? Gains Process Solution? Let’s remember that the tradition argument that if: Then: Although this seems appropriate, if we integrate by parts, we are then required to obtain the second equation as: Click to enlarge In order to maintain the strategy, two terms must be added representing additional investment. Both terms are differentials processes which have unbounded variation. We therefore cannot claim that ” dHt” is riskless. Since the math to find PDE takes too long I referred to Peter Carr’s solution to this problem in his 1999 paper discussing the proposed question. Carr had found that by doing the math right, PDE could not be found. Carr as well as many other critics, use this position in order to claims that perhaps the Black Scholes Model is wrong. The popular belief is that if the result is right, but the derivation is wrong, then the argument cannot stand. Many have proposed however, that it is possible to derive PDE by a more complicated means and achieve “tenure” therefore making the argument safe from derivations. Although this seems like a solution we must not forget that there are some that believe the contrary. Others have argued that the derivation is indeed correct since the number of shares held is referred to being “instantaneously constant”. (Sort of like instantaneous speed or velocity) In the eyes of mathematicians this two sided argument is difficult since the total variation of the number of shares held by any finite time interval must be in fact infinite. From Carr’s response, it can actually be observed that the number of shares is changing so fast that the ordinary rules of calculus do not apply. (Crazy Right?) So Is the Derivation Right..? No, well kinda… I believe that the derivation is in fact wrong since it is only correct up to some discrepancy or “typo”. If we assume the hedge portfolio value at time ” t” represented by: Then the gain “gHt” in the hedge portfolio is observed as: Click to enlarge It can be thought that “gHt” is riskless and therefore should grow at the risk-free rate in order to cancel out arbitrage. (Risk Free Rate is usually US Treasury Bill Yield) In Carr’s examples, equating coefficients on “dt” does in fact yield the Black Scholes PDE. To make the theoretical world “error free” for derivatives, the above argument replaces the need of computing a total derivative with the financial operation of determining a gain. The portfolio in question of an option and a stock is not self-financing. This is like the positions with regard to riskless assets. Essentially by showing that the gains between two non-self-financing strategies are always equal under no arbitrage, the derivative value of the security can be determined. So why say no? I believe that the solution does bring validity but also brings forward inconsistency. This inefficient manner of providing the solution takes away from the integrity of the model. I do not disagree with the hedging argument, I simply criticise the need for extra material to prove a factor that should be safeguarded in pricing models such as BSM. Disclosure: I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it. I have no business relationship with any company whose stock is mentioned in this article.