"Frontmatter". In: Analysis of Financial Time Series

238 CONTINUOUS-TIME MODELS(a) Call options(b) Put optionsValue of a call0 10 20 30 40Value of a put0 20 40 60 800 20 40 60 80 100 120Current stock price0 20 40 60 80 100 120Current stock priceFigure 6.3. Marginal effects of the current stock price on the price of an option with K = 80,T − t = 0.25, σ = 0.3, and r = 0.06: (a) call option, and (b) put option.1. The current stock price P t : c t is positively related to ln(P t ). In particular, c t →0asP t → 0andc t →∞as P t →∞. Figure 6.3(a) illustrates the effects withK = 80, r = 6% per annum, T − t = 0.25 years, and σ = 30% per annum.2. The strike price K : c t is negatively related to ln(K ). In particular, c t → P t asK → 0andc t → 0asK →∞.3. Time to expiration: c t is related to T − t in a complicated manner, but we canobtain the limiting results by writing h + and h − ash + = ln(P t/K )σ √ T − t + (r + σ 2 /2) √ T − t,σh − = ln(P t/K )σ √ T − t + (r − σ 2 /2) √ T − t.σIf P t < K , then c t → 0as(T − t) → 0. If P t > K , then c t → P t − K as(T − t) → 0andc t → P t as (T − t) →∞. Figure 6.4(a) shows the marginaleffects of T −t on c t for three different current stock prices. The fixed variablesare K = 80, r = 6%, and σ = 30%. The solid, dotted, and dashed lines of theplot are for P t = 70, 80, and 90, respectively.