Path Integral Warmup The Black Scholes Model

The reader may already be familiar with the Black-Scholes (BS) model. The goal here is partly to put old wine in new bottles and to exhibit the path integral formalism for those who are unfamiliar with it. We start with demonstrating the compatibility of path integrals with the standard "no-arbitrage" framework. At the end of the section, we re-derive the same results using a more compact and more straightforward approach in which "no-arbitrage" appears as a simple parameter specification. Appendix B contains more no-arbitrage details.

Therefore, we begin with stock options. Similar models are used for FX (foreign exchange) options, commodity options, and some other types of options.

the situation in perspective: "In particular, the contribution of axiomatic field theory to calculations has been less than any pre-assigned positive number, however small".

Nonetheless, I repeat that the application of path integrals applied to finance can be made as rigorous as you like.

10The Stochastic Equations are in the Path Integrals: For details, see the end of this chapter, Appendix B, and also the next chapter "Path Integrals and Options II"

Textbook Discussion in a Path Integral Framework

The usual discussion11 starts with assuming that Ns shares of stock with stochastic price S(t) per share at time t are contained in a portfolio along with Nc options with price per option C(S,t). The portfolio value V is

In order that there is "no arbitrage", the return of V has to be the same as holding risk-free securities12, so V is assumed to satisfy dV

Here r0 is the risk-free interest rate (presumed constant for simplicity)13. We define the volatility <r0, also held constant in time for the moment14. We assume

11 More No Arbitrage and Hedging: See Appendix B for a general approach discussing no-arbitrage and hedging.

12 No Arbitrage Warm-up and a Joke: The reader might argue that no arbitrage is nonsense. If one cannot do better than buying treasuries that produce a risk-free rate, why would people go to the trouble of buying options and dynamically hedging them with stock? Why should we assume that time-averaged stock returns are equal to the risk-free rate, when we all know that stock is riskier than debt, so the stockholder deserves a greater return than the bondholder (who because of corporate credit risk, already receives a coupon above the risk-free rate). Nonetheless, options are priced using no arbitrage. The answers to the questions are what you need to understand to become a quant or a trader.

Here is the no-arbitrage joke. The professor and the trader are walking along when they both spot a $10 bill on the sidewalk. The professor says, "This is impossible as demonstrated by no arbitrage; it must be a mirage". The trader picks up the $10.

13 The "Risk-Free Rate": This rate is assumed constant in this section. It is actually specified over a time period relevant for the option. The type of rate is not unique. It can for example be taken as a treasury rate, Libor, a cost-of-funds rate based on Libor plus a spread, Fed Funds, a stock rebate rate, etc. Libor is the Street standard. The appropriate Libor rate for a given option is obtained by interpolation from the Eurodollar futures and swaps markets. For FX options there are two interest rates - the "domestic" and the "foreign" rate that must be considered. See earlier chapters for details.

14 A Little Essay on Volatility: For those readers starting the book here, we give a practical synopsis of volatility. Relaxing the constant volatility assumption is one of the central complicating features of options pricing and hedging. The volatility is taken as different for different times ("volatility term structure"). It may also include stock-price effects to produce "skew", needed to match market prices of options with different strikes. The volatility is sometimes taken as obeying a stochastic equation, with a "volatility of volatility" describing fluctuations of the volatility itself. The volatility takes significance from the model in which it is defined, and models are not unique because no the lognormal stochastic equation dS / S = jusdt + cr0dz(t) with the Wiener

The Option Diffusion Equation

Setting the hedge ratio Ns / Nc = -dC / 3S cancels out the stochastic quantity dS/dt . This produces the diffusion equation for C(S,t) as model describes the statistical properties of the underlying variable except in approximation.

In practice, the option volatility is backed out from interpolating values of the volatility needed to obtain agreement with options trading in the market; this defines the "implied" volatility. Only a small fraction of possible options actually trade - and your option may not trade at all - so the implied volatility may be an interpolated or extrapolated quantity.

The implied volatility is usually compared with the volatility of the stock price observed in the past (the "historical" volatility). It is often said that the implied volatility is the market's estimate of future historical volatility, and this is the assumption made in the option pricing formalism. However there are all kinds of technical issues affecting option prices, and therefore affecting implied volatilities (option supply/demand being an example). Therefore, it is hard to know to what approximation this association is true. Another complication is that the value of the historical volatility depends on the size of the data window.

Traders naturally hedge options with stock, and therefore the relation of the implied volatility to the historical volatility forms an obsessive topic in determining whether trading makes or loses money. Sometimes the stock of the hedge is the same as the stock (or index) on which the option is written, but often for practical reasons it isn't.

15 Path Integrals and (dz(t))2 = dt: This is actually just a statement of the width of individual Wiener measures that begin the path integral approach. There is nothing mysterious about it at all. This equation is valid for the expectation value, not for some individual pick of a random number, of course.

16 Brownian Motion Limitations: The infinitesimal limit dt —> 0 with the expectation (dz(t)) = dt assumes that a Brownian-motion diffusion random-walk process of the underlying stochastic variable occurs down to the smallest time scales. In the real world this idealization of infinitesimal time scale price changes cannot occur (not even a computer can react in one picosecond, and people go to sleep sometimes). Following the standard literature on options models, we temporarily ignore this problem along with other issues of importance, such as discontinuous jumps, possible feedback non-linearities in the options price itself, effective phase transitions from disordered to coherent actions among investors, etc.

measure satisfying15' 16 (dz(t)) = dt. Next, we use the expansion for the full time derivative of the option value C , dt dt dS dt 2 dS2 dt dt dt dS dt 2 dS2 dt

17 Extension of the no-arbitrage derivation: See Appendix B.

at ox dx

Here * = 111(5) acts as a co-ordinate18, while pQ - rQ -^a^ functions as a drift. The stock-specific drift /J,s does not enter since dS/dt terms cancelled. Also, Ito's rule (or alternatively the need to obtain the same diffusion equation under change of variable) means the stochastic variable satisfies dx = dS/S -jtr^dt,

while — - S— and —=---- S2 —as ordinary variables.

dx dS dx2 dx dS2

Solution of the Option Diffusion Equation

The solution to this equation is classic. Let us take a moment to recall its derivation. Write the formal Taylor expansion for a time step At as

Here, 9, = r0 -M0dx -^o^lx* where =d/dx, d2^ = 32 / dx2 dt - dldt.

We continue using Fourier Transform (FT) methods XV1. We set

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