The Statistical Mechanics of Financial Markets7.Derivative Pricing Beyond Black—Scholes.pdf
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1、7. Derivative Pricing Beyond BlackScholes In the two preceding chapters, we have observed that the price dynamics of real-world securities diff ers signifi cantly from geometric Brownian motion, most importantly by fat tails in the return distributions and by volatility correlations. The fundamental
2、 assumptions behind the BlackScholes theory of option pricing and hedging do not hold in real markets. More general methods which include these stylized facts are called for. 7.1 Important Questions This leads us to the following important questions concerning derivative pric- ing. Can the BlackScho
3、les theory of option pricing and hedging be worked out for non-Gaussian markets? Can we formulate a theory of option pricing which does not make any assumptions on the properties of the stochastic process followed by the underlying security, and for which BlackScholes obtains as a special limit? Are
4、 analytic expressions for option prices available when the underlying returns are taken from a stable L evy distribution? Are path-integral methods from physics useful in the elaboration of op- tion pricing schemes for non-Gaussian markets, and can we formulate a quantum theory of fi nancial markets
5、? How are American-style options priced? Can option prices and hedges be simulated numerically? 7.2 An Integral Framework for Derivative Pricing In Chap. 4, we determined exact prices for derivative securities. In particular, we derived the BlackScholes equation for (simple European) options. Our de
6、rivation relied on the construction of a risk-free portfolio, i.e., a perfect hedge of the option position was possible. The derivation was subject, however, to a few unrealistic assumptions: (i) security prices performing geometric Brownian motion, (ii) continuous 1987. Derivative Pricing Beyond Bl
7、ackScholes adjustment of the portfolio, (iii) no transaction fees. That (i) is unrealistic was demonstrated at length in Chap. 5. It is clear that transaction fees forbid a continuous adjustment of the portfolio. Also liquidity problems may prevent this. Both factors imply that a portfolio adjustmen
8、t at discrete time steps is more realistic. However, both with non-Gaussian statistics, and with discrete- time portfolio adjustment, a complete elimination of risk is no longer possible. A generalization of the BlackScholes framework, using an integral rep- resentation of global wealth balances, wa
9、s formulated by Bouchaud and Sor- nette 17, 164. To explain the basic idea, we take the perspective of a fi nancial institution writing a derivative security. In order to hedge its risk, it uses the underlying security, say a stock, and a certain amount of cash. In other words, it constitutes a port
10、folio made up of the short position in the derivative, the long position in the stock, and some cash. The stock and cash positions are adjusted according to a strategy which we wish to optimize. The optimal strategy, of course, should minimize the risk of the bank (it cant eliminate it completely).
11、However, in a non-Gaussian world, this strategy will depend on the quantity used by the bank to measure risk, and in contrast to the Black Scholes framework, where the risk is eliminated instantaneously, here one can minimize the global risk, incurred over the entire time interval to maturity. While
12、 the BlackScholes theory was diff erential, this method is integral. To formalize this idea, we establish the wealth balance of the bank over the time interval t = 0,.,T up to the maturity time T of the derivative. The unit of time is a discrete subinterval of length t = tn+1 tn. The asset has a pri
13、ce Snat time tn, it is held in a (strategy dependent) quantity (Sn,tn) nand has a return . The amount of cash is Bn, and its return is the risk-free interest rate r. At t = tn, the wealth of the bank then is Wn= (Sn,tn)Sn+ Bn.(7.1) How does it evolve from n n + 1? The updated cash position is Bn+1=
14、Bnert Sn+1(n+1 n) .(7.2) The fi rst term accounts for the interest, and the second term is due to the portfolio adjustment n n+1, due to stock price changes Sn Sn+1. The diff erence in wealth between tnand tn+1is then Wn+1 Wn= n(Sn+1 Sn) + Bn ?ert 1?.(7.3) Bncan be eliminated from this equation by u
15、sing (7.1), the resulting equation can be iterated, and the wealth of the bank after n time steps can be expressed in terms of the stock position alone: Wn= W0ernt+ n1 ? k=0 ker(nk1)t ?S k+1 Skert ? .(7.4) The term in parentheses is the stock price change discounted over one time step, and its prefa
16、ctor in the sum is the cost of the portfolio adjustment. 7.3 Application to Forward Contracts199 7.3 Application to Forward Contracts As a simple application, we consider a forward contract. In a forward, the underlying asset of price SNis delivered at maturity T = Nt for the forward price F, to be
17、fi xed at the moment of writing the contract. As we have seen in Sect. 4.3.1, there are no intrinsic costs associated with entering a forward contract because the contract is binding for both parties. The value of the banks portfolio at any time before maturity therefore is n= Wnat tn 2, and for an
18、odd integer, one can derive closed expressions for the hedging functions ? k above. Figure 7.1 shows the price C of a European call option at seven days from maturity, in units of the standard deviation, as a function of the price of the under- lying, using the optimal hedge derived from the formali
19、sm of this chapter (crosses). It also shows the residual risk which cannot be hedged away, as the -1 0 1 2 3 4 5 6 7 8 -4-2024 C/sigma, hedged S(0)-X/sigma Fig. 7.1. Price of a European call option sevend days from maturity, determined from the optimal hedging strategy discussed in this chapter, for
20、 IID random vari- ables drawn from a Student-t distribution (crosses) together with residual risk (dashed error bars). For comparison, the price and residual risk of the same call is shown when the return process is Gaussian in discrete time (solid error bars). Due to discreteness of time, a fi nite
21、 residual risk remains even for a Gaussian return process, unlike in the continuous-time BlackScholes theory. Both the call price C and the initial diff erence between the price of the underlying and the strike price, S(0) X, are measured in units of the standard deviation of the daily returns. By c
22、ourtesy of K. Pinn. Reprinted with permission from Elsevier Science from K. Pinn: Physica A 276, 581 (2000). c ?2000 Elsevier Science 2047. Derivative Pricing Beyond BlackScholes dashed error bars. A Student-t distribution with = 3 has been assumed. For comparison, the solid error bars show the call
23、 price and residual risk of a Gaussian return process in discrete time. While for a continuous-time Gaussian return process, the risk can be hedged away completely by follow- ing the BlackScholes -hedging strategy (cf. Chap. 4), for a discrete-time process, a residual risk always remains 165. The fi
24、 gure nicely demonstrates both the eff ects of the fat-tailed distribution, and of discrete trading time. What about real markets? Figure 7.2 compares the market price of an option on the BUND German government bond, traded at the London futures exchange, to the BlackScholes price. The inset shows t
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