22.2 Discrete Time Hedging Strategies and Dynamic
22.2.4 Local Quadratic Risk-Adjusted-Minimizing Hedging
In this section, we introduce another type of quadratic risk-minimizing hedging under consideration of risk-adjusted. Define the one-step-ahead risk-adjusted hedg- ing cost as
ıt.St/ı.Stet/;
where t logŒEt1f.St=Bt/=.St1=Bt1/g is the risk premium. Figure 22.4 illustrates the concepts of risk-adjusted with k risky securities and discounted values. Instead of the risk-minimizing criterion (22.6) mentioned in Sect.22.2.3, we consider to construct a trading strategy which minimizes the one-step-ahead quadratic discounted risk-adjusted hedging costs, that is,
min
h0t1;h1t1Et1fQıt.St/g2; (22.8) whereıQt.St/Dıt.St/=Bt D QFt.h0t1Ch1t1SQt/;andFQtDFt.Stet/=Btand SQtDStet=Bt denote the discounted adjusted values ofFt andSt, respectively.
Statistically speaking, the criterion (22.8) is equivalent to find a best linear approximation of FQt with the shortest L2-distance under the physical measure P. Hence, this best linear approximation will pass through the point .Et1.SQt/; Et1.FQt//. By the definition of t, we have Et1.SQt/ D QSt1: Therefore, the amountEt1.FQt/is treated as the discounted hedging capital for a given discounted stock priceSQt1at timet1and thus
Et1.FQt/D QFt1;
under the dynamic probability measure. The theoretical proof of this equality could be found inElliott and Madan(1998) andHuang and Guo(2009c).
Based on the optimal criterion (22.8), the closed-form expression ofh0t1 and h1t1can be obtained by solving
@
@h0t1Et1fQıt.St/g2D0 and @
@h1t1Et1fQıt.St/g2D0:
We call this trading strategy by local quadratic risk-adjusted-minimizing hedging, abbreviated by LQRA-hedging, and the corresponding dynamic programming is described as follows.
[Dynamic programming of LQRA-hedging]
8ˆ ˆˆ ˆˆ ˆ<
ˆˆ ˆˆ ˆˆ :
.hO0T;hO1T/D.VQT; 0/
hO1t1DCovt1.FQt;SQt/ Vart1.SQt/
hO0t1DEt1.FQt/ Oh1t1E.SQt/DEt1f Oh0tC.hO1t Oh1t1/SQtg
: (22.9)
In the following, we give an example to illustrate the LQRA-hedging in a trinomial model.
Example 5. Consider the same one-period trinomial model as in Example2. First, we compute the risk premium
Dlog n
E
er St
St1
oDerlog.up1Cdp2Cjp3/Dlog.1:0005/:
The discounted risk-adjusted stock prices at time 1 are SQ1.!1/DS1.!1/er D 22000
201 > K D100;
SQ1.!2/DS1.!2/er D 1800 201 < K;
and
SQ1.!3/DS1.!3/er D 16000 201 < K:
Hence, the corresponding discounted risk-adjusted option values are VQ1.!1/ D
1900
201, andVQ1.!2/D QV1.!3/D0. By the dynamic programming (22.9), the holding units of riskless bonds and the security are given by.hO00;hO10/D .38:06; 0:4326/. Thus the initial hedging capital is
F0D Oh00C Oh10S0D5:199;
which also lies in the interval of no-arbitrage prices,.5;203/. In other words, the criterion of quadratic risk-adjusted-minimizing not only provides a hedging strategy, but also a no-arbitrage price of the European call option in this incomplete market.
If the market model is discrete time and continuous state type, such as the GARCH model (Bollerslev 1986),Elliott and Madan(1998) showed that
Et1.FQt/DEt1Q .FQt/D QFt1;
where the measureQis the martingale measure derived by the extended Girsanov change of measure. In particular, Huang and Guo (2009c) showed that if the innovation is assumed to be Gaussian distributed, then the GARCH martingale measure derived by the extended Girsanov principle is identical to that obtained by Duan(1995). Moreover, the formula of the optimal.hO0t1;hO1t1/obtained in (22.9) can be expressed as
8ˆ ˆˆ
<
ˆˆ ˆ:
hO0t1 D FQt1Et1Q .SQt2/ QSt1Et1Q .FQtSQt/ VarQt1.SQt/
hO1t1 D CovQt1.FQt;SQt/ VarQt1.SQt/
(22.10)
under the risk-neutral measureQ, fort D1; ; T, where CovQt1 and VarQt1are the conditional covariance and variance givenFt1computed under the risk-neutral measureQ, respectively.
Both (22.9) and (22.10) provide recursive formulae for building the LQRA- hedging backward from the expiration date. In practical implementation, practition- ers may want to keep the holding units of the hedging portfolio constant for`units of time due to the impact of the transaction costs. If we denote the discounted hedging capital with hedging period`at timet by
FQt;`D Oh0t;`C Oh1t;`SQt; (22.11) wherehO0t;`andhO1t;`are the holding units of riskless bonds and the underlying asset, respectively, and are determined instead by the following optimal criterion
min
h0t;`;h1t;`EtQfQıtC`.StC`/g2: (22.12) Note that the optimal holding units .hO0t;`;hO1t;`/ are functions of the hedging period`. By similar argument as (22.10),hO0t;`andhO1t;`can be represented as
8ˆ
<
ˆ:
hO1t;`DCovQt .FQtC`;SQtC`/=VarQt .SQtC`/ hO0t;`DEtQ.FQtC`/ O1t;`SQtC`
: (22.13)
Equation (22.13) is really handy in building LQRA-hedging. For example, suppose that a practitioner writes a European call option with strike priceKand expiration dateT. She wants to set up a`-period hedging portfolio consisting of the underlying stock and the riskless bonds at timetwith the hedging capitalFt to hedge her short position, and the hedging portfolio remains constant till timetC`,0 < `T t. Huang and Guo(2009c) proved that the hedging capital of the `-period LQRA- hedging is independent of the hedging period` and is equal to the no-arbitrage price derived by the extended Girsanov principle. A dynamic programming of the
`-period LQRA-hedging for practical implementation is also proposed byHuang and Guo(2009c). The algorithm is summarized as follows.
[Dynamic programming of`-period LQRA-hedging]
1. For a given stock priceStat timet, generatenstock pricesfStC`;jgnjD1, at timetC` conditional onStfrom the risk-neutral model.
2. For eachStC`;j, derive the corresponding European call option prices,FtC`.StC`;j/, by either the dynamic semiparametric approach (DSA) (Huang and Guo 2009a,b) or empirical martingale simulation (EMS) method (Duan and Simonato 1998) for tC` < T. IftC`DT, thenFT.ST;j/D.ST;jK/C.
3. RegressFQtC`.StC`;j/onSQtC`;j,j D1; ; n. Then.hO0t;`;hO1t;`/are the corresponding regression coefficients.
Since the above trading strategy employs the simple linear regression to deter- mine the hedging positions, it is easy to be implemented and computed in a personal computer. The main computational burden might comes from Step-2 of the above algorithm, where we have to compute the European call option values by the DSA or EMS method. Herein, we give a brief introduction of this method. The DSA is proposed by Huang and Guo (2009a) for solving the multi-step conditional expectation problems where the multi-step conditional density doesn’t have closed- form representation. It is an iterative procedure which uses nonparametric regression to approximate derivative values and parametric asset models to derive the one- step conditional expectations. The convergence order of the DSA is derived under continuity assumption on the transition densities of the underlying asset models.
For illustration, suppose we want to compute the multi-step conditional expectation E0.FT/. We transform the problem into E0ŒE1Œ ŒET1.FT/ . and then compute the one-step backward conditional expectation. DenoteFt D Et.FtC1/, t D 0; ; T 1. In general,Ft is a nonlinear function of the underlying asset fort D 1; ; T 1, and the conditional expectationEt1.Ft/ does not have closed-form representation, which makes the multi-step conditional expectation complexity. Huang and Guo (2009a) adopted piecewise regression function to approximate the derivative value functionFt at each discrete time pointt, denoted byFOt, and then compute the conditional expectation ofFOt, that is,FQt D Et.FOtC1/ and treatedFQt as an approximation ofFt. The procedure keeps iterating till the initial time to obtain the derivative price. A flow chart of the DSA is given in Fig.22.5.
Stop Assume F~T
is given. Start from. i T
If i 1 0
If i 1 0, set i i 1 Fit the regression function of F~i
: Fˆi
Compute the conditional expectation : Ei1(Fˆi)
: ~ 1 Fi
Define the approximate option value
Fig. 22.5 Flow chart of the DSA