Question

A market-maker sells a straddle on a stock. You are given: (i) The stock's price follows the Black-Scholes framework. (ii) S(O)= 45. (iii) The continuously compounded risk-free rate is 0.10. (iv) The stock pays no dividends. (v) The annual volatility of the stock is 0.2. (vi) The straddle consists of European options and expires in one year. The market-maker delta-hedges the sale by buying shares of the underlying stock. Calculate the amount of money the market-maker spends on the stock.

Answer 1

Solution: Straddle will be computed to be sum of an at-the-money put and an at-the- money call 0.10+0.5(0.22) d =- -= 0.6 0.2 N(d1)= N(0.6)=0.72575 By put-call parity, we have: Aput = Acall-e-61 We have e-ot =1. Delta for straddle: 2Acall -1=2(0.72575)-1=0.4515 Thus market-maker buys 0.4515 shares of stock and will be spending 20.318 = 0.4515 *45)