公司财务精品教学（徐江旻）Session14 DynamicHedging.pdfVIP

Dynamic Hedging Jiangmin Xu Hedging • It is possible to hedge risk by holding off-setting positions – e.g. long and short positions in assets that have positively correlated returns. • Hedging can also be achieved by active adjustment of the portfolio in response to price changes. – this is called dynamic hedging • How to formulate procedures for dynamic hedging. • Limits of dynamic hedging – what can go wrong. © Jiangmin Xu Recall: Portfolio Insurance © Jiangmin Xu Portfolio insurance and the 1987 crash © Jiangmin Xu Portfolio insurance and the 1987 crash • 1980s, specialized (hedge) fund managers (led by some well known academics) put into practice the principles behind the Black-Scholes theory to create portfolio insurance • Portfolio insurance: attempts to replicate a put option by trading actively in the market. © Jiangmin Xu Price of a Put Option • Option value is positive even when the current stock price S is above the strike price X • Option value increases as S falls • Also the rate at which the option is increasing in value is itself increasing as S falls • This is option Δ © Jiangmin Xu Options: Replicating Portfolios • What is the price of a 1-period Call option with strike = 90? Risk-free rate is r = 5%. f • Replicating Portfolio: buy Δ = (C – C )/(S – S ) = 0.5 u d u d shares • Borrow, T-bill position B = - (S Δ – C )/(1 + r ) 0 d d f • Call price: C = Δ S + B = 28.57 • For puts: Δ is negative and B is positive © Jiangmin Xu More Generally (the Black-Scholes Formula) • Current Stock Price = S, Sto