Question

Suppose Lotus’ stock price is currently $50 and a dividend of $3 is expected in three months. A six-month European call option on the stock with an exercise price of $49 is selling for $6. A six-month European put option on the stock with an exercise price of $49 is selling for $3.50. The risk-free interest rate is $15% per year. a. Is there any arbitrage opportunity? b. If you answer yes to part a, please show your arbitrage strategy. Show the cash flows at t=0 and cash flows at t=6 months if the stock is $60 or $30.

Answer 1

The arbitrage opportunity may exist if the put call parity do not hold

As per put call parity equation

(Spot price - present value of dividends) + put option price = call option price + present value of strike price

Left hand side

= 50-3*exp(-0.15*3/12) + 3.5 (assuming risk free rate to be continuously compounded)

= 50-2.89 + 3.5 = 50.61

Right hand side

= 6+ 49*exp(-0.15*6/12) = 51.46

As the LHS is not equal to RHS , the put call parity do not hold

and the Arbitrage opportunity exists which can be encashed as under

i) Today, borrow $44.61 for six months and $2.89 for 3 months at the risk free rate of 15% p.a.

ii) Today, sell the call option for $6

iii)From the amount of ($44.61+$2.89+$6 = $53.5) so obtained, purchase one stock and the put option.

iv) After 3 months, get $3 as dividend and repay the loan of $2.89 fully (2.89*exp(0.15*3/12)=3)

v) After 6 months

If stock price < $49, sell the stock using your put option and get $49

repay the loan of $44.61*exp(0.15*6/12) = $48.08 and take the remaining amount of $0.92 as arbitrage profit

If stock price > $49, the call option will be exercised and the stock has to be sold to get $49

repay the loan of $44.61*exp(0.15*6/12) = $48.08 and take the remaining amount of $0.92 as arbitrage profit

If stock price = $49, sell the stock in the market to get $49

repay the loan of $44.61*exp(0.15*6/12) = $48.08 and take the remaining amount of $0.92 as arbitrage profit

Hence , in all situations, arbitrage profit of $0.92 can be made