(i) Define the following Greeks algebraically: (a) delta (b) vega (c) theta (d) gamma Consider a call option with price C at time t (in years) written on an underlying non-dividend paying asset with price St at time t and volatility 0. Using Taylor's expansion, it can be shown that the change in value of the option is approximately given by: delta x dSt +0.5 x gamma x (dS+)2 + theta x dt + vega x do At time t= 0, the underlying asset price is $23 and the volatility is 20% per annum. The option is priced at $6.17 and has the following properties: de = 1. delta = 0.822 2. vega = 0.104 3. theta = -0.855 4. gamma - 0.033 At time t = 1, the security price has fallen to $20 and its volatility is now 15% per annum. (ii) Estimate the value of the call option at time t = 1. The delta of a call option is always positive, whilst the delta of a put option is always negative. (iii) Justify this result. The vega of both call and put options is always positive. (iv) Justify this result