Binary Options. Hamish Raw

to expiry and one attempts to roughly approximate the time decay of a bet over the forthcoming single day, which theta would provide the most accurate estimate?

      a) the current 8-day theta

      b) the 7-day theta, or

      c) a theta somewhere in between the seventh and eighth day?

      4. For each bet find the time decay over the requested number of days using the associated theta.

      5. The following table provides the prices of three bets at five separate points in time (t) where t–1 has one day less to expiry than t. Find an approximation to theta for each bet at time t.

      2.11 Answers

      1. Thetas are negative when out-of-the-money, positive when in the money and zero when at-the-money. Therefore,

      a) Positive

      b) Negative

      c) Zero

      d) Positive

      2. Since the $49 strike downbet and the $51 strike downbet are equidistant around the underlying they will have roughly the same absolute theta, i.e. 5.1. Therefore the $51 strike downbet will have a theta of roughly +5.1 since it is in the money and will therefore increase in value as time passes..

      3. Both a) and b) provide average thetas at the single points of 8 days and 7 days to expiry exactly. The former will underestimate the one day time decay while the latter will overestimate it; therefore c) will provide the most accurate forecast.

      4. Using the formula:

      Time Decay = 100 x (No. of Days x Theta ) / 365

      Bet 1: 100 ¥ ((1 ¥ 5) / 365) = 1.37 points

      Bet 2: 100 ¥ ((2 ¥ 5) / 365) = 2.74 points

      Bet 3: 100 ¥ ((5 ¥ 10) / 365) = 13.70 points

      5. Ignore t–2 and t+2, as the most accurate theta is obtained from making the increment in time as small as possible, i.e. as dt→0. Therefore subtracting the price at t+1 from the price at t–1 gives the difference in price over two days, and then multiplying by 365 and dividing by 100 provides the correct theta:

      Theta = [( Pt–1 – Pt+1 ) / ((t + 1) – (t – 1))]*365/100

      Bet 1 = [( 50.280 – 50.331 ) / (( t + 1 ) – ( t – 1))] ¥ 365 / 100 = –0.0931

      Bet 2 = [( 11.709 – 15.792 ) / (( t + 1 ) – ( t – 1))] ¥ 365 / 100 = –7.4515

      Bet 3 = [( 88.291 – 84.208 ) / (( t + 1 ) – ( t – 1))] ¥ 365 / 100 = 7.4515