Finance Technicals
Returns, risk, portfolios, bonds and options, as the parts of the course cover them.
01Call and put at expiry
A call and a put on the same stock both have the strike 105 and expire on the same day. The call costs 3 dollars and the put costs 2.50 dollars. At expiry the stock trades at 112. What is the profit of someone who bought one call, and of someone who bought one put?
Hint
At expiry a call pays the amount by which the stock is above the strike, and a put pays the amount by which it is below. Then subtract the price.
Answer
The call makes a profit of 4 dollars, and the put loses its whole price of 2.50 dollars.
Solution
The call pays max(112 - 105, 0) = 7 dollars at expiry. After its price of 3 dollars the profit is 4 dollars.
The put pays max(105 - 112, 0) = 0 dollars, so the buyer loses the 2.50 dollars paid for it.
A call buyer can lose at most the price but gains with every dollar above the strike, which makes the payoff a hockey stick.
02Parity out of line
A stock trades at 50 dollars and pays no dividend. A European call and a European put on it both have the strike 50 and expire in one year. The call costs 6.00 dollars and the put costs 3.50 dollars. A safe bond that pays 1 dollar in one year costs 0.96 dollars today. Is there an arbitrage, and how large is it per option pair if you ignore trading costs?
Hint
A call minus a put with the same strike pays the stock minus the strike at expiry, whatever happens. Compare its price with the price of that payoff today.
Answer
Yes, there is one. Parity says the call minus the put should cost 2.00 dollars, and it costs 2.50 dollars. Sell the call, buy the put, buy the stock and borrow 48 dollars, which locks in 0.50 dollars today.
Solution
At expiry, a long call plus a short put with strike K pays S_T - K in every state, the same as owning the stock and owing K.
Owning the stock and owing 50 dollars in a year is worth 50 - 50 × 0.96 = 2.00 dollars today. So C - P should be 2.00 dollars.
The market charges 6.00 - 3.50 = 2.50 dollars. The call is rich relative to the put, so you sell the expensive side.
Sell the call (+6.00), buy the put (-3.50), buy the stock (-50.00) and borrow 48.00 against the 50 due in a year. The cash today is +0.50 dollars.
At expiry the stock plus the put minus the call is worth exactly 50 dollars in every state, which repays the loan. The 0.50 dollars stay with you without any risk.
03Long vega with no gamma
Two options on the same stock are available. Option X has delta 0.50, gamma 0.040 and vega 0.10 per option. Option Y has delta 0.45, gamma 0.020 and vega 0.25 per option. Build a position from X, Y and shares of the stock that has gamma 0, delta 0 and a vega of +30. What do you hold, and what risk is left?
Hint
Shares have delta 1 and no gamma or vega. So first solve for X and Y with the gamma and vega conditions, and fix the delta with shares at the end.
Answer
Buy 150 of Y, sell 75 of X and sell 30 shares. The position gains when implied volatility rises, while small stock moves leave it flat for the moment. The Greeks drift as the stock and time move, so it needs regular rebalancing.
Solution
Let x and y be the numbers of X and Y. Gamma 0 means 0.040x + 0.020y = 0, so x = -0.5y.
Vega +30 means 0.10x + 0.25y = 30. With x = -0.5y this becomes 0.20y = 30, so y = 150 and x = -75.
The options then have delta 0.50 × (-75) + 0.45 × 150 = -37.5 + 67.5 = 30. Selling 30 shares brings the delta to 0.
Gamma and delta are 0 only at the current price and time. The theta of the two options, vega that differs by expiry and a skew that moves are the risks that remain.
A good follow-up answer says when you would rebalance and what the position costs to hold each day.
Full collection This category has 12 more questions with worked solutions (4 easy, 4 medium, 4 hard). They open with Foundations or Pro. See pricing, or sign in to your account.
Compare your own answer with the worked solution.