European Call Option Dynamic and Static Hedging Strategies

Consider an European call option with strike equal to 10, T = 1, r = 0.05 and σ = 0.2. Using the following time-series:

1. For every time moment t = 0, 1/360, 2/360:

• Calculate the Black-Scholes price.
• Calculate the corresponding delta.
• Calculate the price of the corresponding replicating portfolio and its composition, if we consider a dynamic hedging.

2. Calculate a static hedging. What is the final value of the portfolio? Compare with the dynamic hedging results.

Solution

• At time 0, time to maturity is 1, and then
  • The Black-Scholes price is given by S0N(d1)−Ke−rT N(d2) = 10N(0.35)−10e−0.05N(0.15) = 1.0450
  • The Delta is given by N(d1) = N(0.35) = 0.6368
  • At time t = 0, the value of the replicating portfolio coincides with the Black-Scholes price 1.0450. This portfolio consists of 0.6368 assets and a risk-free investment of 1.0450−0.6368×10 = −5.323.
• At time 1/360, we apply the Black-Scholes formula with initial price S1/360 = 10.08 and time to maturity T equal to 1 − 1/360, and then:
  • The Black-Scholes price is given by S1/360N(d1) − Ke−rT N(d2) = 10.08N(0.3894) − 10e−0.05(1−1/360)N(0.1896) = 1.0948
  • The Delta is given by N(d1) = N(0.3894) = 0.6515
  • The new value of the replicating portfolio depends on the interest rate and the new value of S. That is, it is −5.323e0.05/360 + 0.6368 × 10.08 = 1.0952. Now we rebalance it: the number of assets will be the Delta: 0.6515 and the risk-free investment will be 1.0952 − 0.6515 × 10.08 = −5.472.
• At time 2/360, time to maturity is 1 − 2/360, and then:
  • The Black-Scholes price is given by S2/360N(d1) − Ke−rT N(d2) = 9.98N(0.3389) − 10e−0.05(1−2/360)N(0.1395) = 1.0288 (2)
  • The Delta is given by N(d1) = N(0.3389) = 0.6327
  • The new value of the replicating portfolio is −5.472e0.05/360 + 0.6515 × 9.98 = 1.0292. Now we rebalance it: the number of assets will be 0.6327 and the risk-free investment will be 1.0292 − 0.6327 × 9.98 = −5.285
• Under a static hedging, we do not rebalance our portfolio. At time 0, this portfolio consists of 0.6368 assets and a risk-free investment of 1.0450 − 0.6368 × 10 = −5.323. At time t = 2/360 the value of this portfolio will be 0.6368 × 9.98 − 5.323e0.05×2/360 = 1.0308
• The value of the dynamic hedging portfolio is closer to the Black-Scholes price of the option.