Question

Form a long butterfly spread using the three call options in the table below CI X- $90 C2 X= $100 C3 X= $110 T- 180 daysT- 180 daysT 180 davs 6.0600 0.4365 0.0187 11.4208 27.6602 18.5394 16.3300 0.7860 0.0138 Price DELTA GAMMA THETA VEGA RHO 10.3000 0.6151 0.0181 12.2607 26.8416 25.2515 11.2054 20.4619 30.7085 What does it cost to establish the butterfly spread? explain how each can be interpreted achieved by doing so? C3 remain the same. Does this create an arbitrage opportunity? Explain. a) b) Calculate cach of the Greek measures for this butterfly spread position and c) How would you make this option portfolio delta neutral? What would be d) Suppose that tomorrow the price of C1 falls to $12.18 while the prices of C2 and

Answer 1

Before answering the question, let's understand what a butterfly spread is. A butterfly spread is achieved by:

1. Buying a call option with a particular strike price
2. Selling two call options on the same stock with higher strike price
3. Buying another call option on the same stock with even higher strike price

Let's apply this to create the butterfly spread using the three call options given in the question. We can create a butterfly spread as follows:

1. Buy 1 number of C1
2. Sell 2 numbers of C2
3. Buy 1 number of C3

Part (a)

Cost to establish butterfly spread = Cost to buy 1 number of C1 - Proceeds from sale of 2 numbers of C2 + Cost to buy 1 number of C3 = 16.33 - 2 x 10.30 + 6.06 = 1.79

Part (b)

Delta of the butterfly spread = 1 x Delta of C1 - 2 x Delta of C2 + 1 x Delta of C3 = 0.7860 - 2 x 0.6151 + 0.4365 = -0.0077

Interpretation: Delta is a measure of sensitivity of the value of the derivative instrument for a unit change in the value of the underlying asset. Thus the value of the butterfly spread will decrease by $ 0.0077 if the price of the underlying stock changes by $ 1.

Gamma of the butterfly spread = 1 x Gamma of C1 - 2 x Gamma of C2 + 1 x Gamma of C3 = 0.0138 - 2 x 0.0181 + 0.0187 = -0.0037

Interpretation: Gamma is a measure of rate of change the delta of the derivative instrument for a unit change in the value of the underlying asset. Thus the delta of the butterfly spread will decrease by $ 0.0037 if the price of the underlying stock increases by $ 1.

Theta of the butterfly spread = 1 x Theta of C1 - 2 x Theta of C2 + 1 x Theta of C3 = -11.2054 - 2 x (-12.2607) + (-11.4208) = 1.8952

Interpretation: Theta is an indicator of rate of change of the price of the derivative as we move closer to maturity. Thus the price of the butterfly spread will change by 1.8952 as we move closer to the expiry or maturity date.

Vega of the butterfly spread = 1 x Vega of C1 - 2 x Vega of C2 + 1 x Vega of C3 = 20.4619 - 2 x 26.8416 + 27.6602 = -5.5611

Interpretation: Vega is an indicator of rate of change of the price of the derivative with respect to change in volatility of the underlying stock. Thus the price of the butterfly spread will decrease by 5.5611 if volatility of the underlying stock increases by 1%.

Rho of the butterfly spread = 1 x Rho of C1 - 2 x Rho of C2 + 1 x Rho of C3 = 30.7085 - 2 x 25.2515 + 18.5394 = -1.2551

Interpretation: Rho is an indicator of rate of change of the price of the derivative with respect to change in the risk free rate. Thus the price of the butterfly spread will decrease by 1.2551 if risk free rate increases by 1%.

Part (c)

Delta of the underlying stock is 1.

Delta of the butterfly spread = -0.0077

If we buy 0.0077 number of stock and combine it with the butterfly spread, the delta of the resultant portfolio will be 1 x 0.0077 - 0.0077 = 0

We can buy 0.0077 number of underlying stock for every 1 butterfly spread we have, to make it delta neutral.

By making a portfolio delta neutral, we are protected against any change in the the value of the portfolio even if the price of the underlying stock changes in a small range. So, the value of our portfolio will remain constant, and unaffected by the small changes in the price of the underlying stock.

Part (d)

Yes there is an arbitrage strategy.

Let's say that price of the call option C1 is $ 12.18 while prices of C2 and C3 remain same.

A trader creates a butterfly spread today:

His cost for the butterfly spread will be = = Cost to buy 1 number of C1 - Proceeds from sale of 2 numbers of C2 + Cost to buy 1 number of C3 = 12.18 - 2 x 10.30 + 6.06 = -2.36

The cost is negative, that means the trader gets a positive cash flow or cash inflow at t=0 when he enters into this strategy. The butterfly spread always results into a zero or positive payoff on expiry. Thus, the trader has created a situation where he makes money today by creating a butterfly spread today, and makes no money or more money when options mature. Thus he creates a situation of positive payoff without any initial investment. This is an arbitrage.