Leverage in the financial markets is one of the oldest techniques to increase one’s gains in an investment. It has also has lead to colossal losses and defaults. Leverage within an investment exists when an investor is exposed to a higher capital base than his or her original capital inlay. The margin mechanism of buying futures, as explained in Chapter 1, is a typical example of leverage. One posts margin of 5%–15% of the futures contract value but is exposed to 100% of the gains or losses of the notional amount of the futures contract. Exchanges will reduce the risk of this leverage in futures contracts by remargining daily using margin calls. Derivatives securities are another way to increase leverage. The call and put options described in Chapter 1 are standard ways to go long or short an underlying asset using leverage. A call option costing $5 and expiring $10 in the money creates a 200% return on investment. If this call expires out of the money, the loss is 100%. The stylized empirical facts of fat tails, term structure of moments, autocorrelation of squared returns, volatility clustering, and the leverage effect (not to be confused with the more generic leverage being discussed here) are brought to the forefront when dealing with leveraged financial products. For instance, say an investor has bought an unlevered stock position worth $100. If this stock drops by 20% in one day (a fat-tailed event), the investor has lost 20% of their investment, which is a substantial but not huge loss. If on the other hand, the investor had spent $100 on at-the-money call options, the stock drop would have had a much larger effect on the marked-to-market value of these levered products (the exact loss depending on time to expiration, volatility, etc.). If these options were close to expiry, the loss could have approached 100%. Yet in the Black-Scholes world alluded to in Chapter 4, none of these stylized facts were taken into account when pricing options. The optimal hedging Monte Carlo (OHMC) methodology presented here is a framework to allow for realistic option pricing without the largely inaccurate assumptions needed in the standard risk-neutral pricing framework. Philosophically, the OHMC method has the following scenario in mind when pricing a derivative security. Suppose someone undertakes to trade derivatives on his own. • What’s the first thing this trader will need? Risk capital. • Who will be the investor (the provider of risk capital)? • What will be the required return on this risk capital? www.it-ebooks.info Chapter 5 ■ Optimal Hedging Monte Carlo Methods 196 One of the main goals of the OHMC method is to quantify the question of risk capital using a fully simulated P&L distribution of a hedging strategy for a derivative security. This chapter deals with the question of what exactly is a hedging strategy. Dynamic Hedging and Replication Dynamic hedging is the periodic hedging of a nonlinear financial instrument (i.e., an option) with linear instruments like spot instruments and futures or with other nonlinear instruments. The amount of the linear hedge is generically called the delta. The deltas of the nonlinear position are calculated and executed using either linear or other nonlinear instruments. The original nonlinear position along with its delta hedge yields an instantaneous zero delta risk position. For instance, a short call option is delta hedged with a long position in the underlying asset. A short put option is delta hedged with a short position in the underlying. The direction of these hedges can be assessed by the direction of the payoff the hedger is trying to simulate or replicate with the chosen linear hedge. A sold call position is a levered long position in the underlying for the holder and therefore the seller must also go long in order to replicate its payoff. A sold put position exposes the seller to a levered short position in the underlying and therefore, he must also go short in order to replicate the investment he has sold. However, as the underlying asset value changes, the delta of the nonlinear position changes, indicating its implied leverage, while that of a linear hedge stays the same. Therefore, the deltas no longer offset, and the linear hedge has to be adjusted (increased or decreased) to restore the offsetting delta hedge. This process of continually adjusting a linear position to maintain a delta hedge is one aspect of dynamic hedging. Dynamic hedging also refers to the hedging of the changing volatility exposure of an option (called vega hedging) or the changing of the delta position itself (gamma hedging). These types of hedges may involve other derivatives (see [Bouchaud, 2003]). Figure 5-1 indicates that geometrically, dynamic replication is attempting to replicate the curved option value with tangent lines that represent the deltas.