Investments - Exam - Options

What is an option?

a contract that gives you the right to trade an underlying asset at a fixed price in the future, but you can walk away if it is a bad deal

• Derivative: Value comes from an underlying asset like a stock

• Right, not obligation: The buyer (long) can choose to exercise or not

• Seller (short) has the obligation to fulfill the trade if the buyer exercises

• Used for hedging, income, or speculation depending on the strategy

Difference between call and put

Call: right to buy an underlying asset at strike price K at/by maturity

Put: right to sell an underlying asset at K at/by maturity

What is the Strike Price K

price in the contract at which you can buy or sell the asset

Maturity T

Last date (European) or deadline (American) when exercise can happen

Stock Price S_T

the price of the underyling stock at maturity

Difference between European and American options

European: exercise allowed only at date T

American: exercise allowed at any time up to and including T

American options are at least as valuable as European options (same underlying, K, T) and most often more valuable for this extra flexibility

What are long and short positions in options

Long: you buy the option and hold the right

Short: you sell the option and take the obligation

Long call/Long put: pay the premium now, can choose to exercise later if it is good

Short call/short put: receive the premium now, must deliver if the buyer exercises

Option premium

the price you pay today to obtain the option

For calls: premium is c_t (European) or C (American)

For puts: premium is p_t (European) or P (American)

Price of any option is strictly greater than zero before maturity (no free rights if abitrage)

Difference between payoff and profit

Payoff: cash you receive at maturity from exercising (never negative)

Profit: is payoff minus what you paid initially (can be negative)

What is the payoff and approximate profit of a long call at maturity?

A long call wins when the stock price ends above the strike and otherwise loses only the premium

Payoff: max(0, S_T - K)

Profit: max(0, S_T − K) − c

Example: K = 50, premium c = 4. If S_T = 70: payoff 20, profit 16. If S_T = 40: payoff 0, profit −4

What is the payoff and approximate profit of a short call at maturity?

A short call gains the premium if the stock stays below the strike and loses more and more if the stock rises

What is the payoff and approximate profit of a long put at maturity?

A long put wins when the stock price ends below the strike and otherwise loses only the premium

• Payoff: max(0, K − S_T)

• Approx profit: max(0, K − S_T) − p

Example: K = 50, premium p = 3. If S_T = 30: payoff 20, profit 17. If S_T = 60: payoff 0, profit −3.

What is the intrinsic value of an option?

what the option would be worth if you exercised it right now

• For a call at time t: max(0, S_t − K)

• For a put at time t: max(0, K − S_t)

• Defined for t ≤ T, based on current spot price S_t, not S_T

What does it mean for a call to be in the money, at the money, or out of the money?

For calls, you compare the current stock price to the strike price

• In the money (ITM): S_t > K so intrinsic value S_t − K > 0.

• At the money (ATM): S_t = K.

• Out of the money (OTM): S_t < K so intrinsic value 0

What does it mean for a put to be in the money, at the money, or out of the money?

For puts, you also compare the current stock price to the strike, but the inequalities flip

• In the money (ITM): S_t < K so intrinsic value K − S_t > 0

• At the money (ATM): S_t = K

• Out of the money (OTM): S_t > K so intrinsic value 0

What is the time value of a call or put at time t≤ T?

Time value is the extra part of the premium that comes from having time left for good movements in the stock price

• For a call: Time value = c_t − max(0, S_t − K)

• For a put: Time value = p_t − max(0, K − S_t)

• Time value is zero at maturity and usually positive before

How does stock price affect call and put values?

Calls move with the stock, puts move against it

• S ↑ ⇒ call price ↑ (stock price rises, call more valuable)

• S ↑ ⇒ put price ↓ (stock price falls, put more valuable)

• Formula: ∂c/∂S > 0, ∂p/∂S < 0

How does volatility σ affect call and put values?

More volatility → both calls and puts more valuable

• Big moves ↑ chance of high call payoff

• Big moves ↑ chance of high put payoff

How does the risk-free rate r affect call and put prices (no dividends)?

Higher r helps calls, hurts puts

• r ↑ ⇒ call price ↑ (you pay K later, cheaper in today’s money)

• r ↑ ⇒ put price ↓ (you receive K later, less valuable today)

How do dividends affect call and put prices?

Dividends hurt calls, help puts

• Stock price drops after dividend is paid

• Good for put (price goes down), bad for call

Example: Bigger dividend announced → call cheaper, put pricier

What are simple upper bounds for European call and put (no dividends)?

Option cannot be worth more than its max possible payoff

• Call: c ≤ S_0

• Put: p ≤ K e^{−rT}

Example: Stock 50 ⇒ call cannot cost 60

What are lower bounds for European call and put (no dividends)?

No-arbitrage gives minimum prices

• Call: c ≥ max(S_0 − K e^{−rT}, 0)

• Put: p ≥ max(K e^{−rT} − S_0, 0)

What is put–call parity (no dividends)?

Call + bond = put + stock

• c + K e^{−rT} = p + S_0

• One option’s price gives the other

How does parity change with dividends?

Subtract present value of dividends on the stock side

• Let PV(D) = PV of dividends to T.

• c + K e^{−rT} = p + S_0 − PV(D)

What is a protective put?

Stock + long put = downside insurance

• Long stock + long put (same K, T)

• Minimum value at T is K

What is a long straddle and what are you betting on?

Call + put same K, T = bet on big move either way

• Long call (K) + long put (K)

• Profit if S_T is far from K (up or down)

Payoff: max(S_T−K,0) + max(K−S_T,0)

Example: K=100, S_T=60 or 150 → big payoff

What is a short straddle and what are you betting on?

Sell call + sell put = bet on little movement

• Short call (K) + short put (K)

• Keep premiums if S_T stays near K

Payoff = −straddle payoff + premiums

Example: If S_T ≈ K, both options expire worthless.

What is a long strangle?

Cheaper “wider” straddle

• Long put (low K₁) + long call (high K₂)

• Needs bigger move than straddle to profit

Payoff: max(K₁−S_T,0) + max(S_T−K₂,0)

Example: K₁=90, K₂=110, S_T must go below 90 or above 110

What is a bull call spread?

Limited-profit bullish bet

• Buy call K₁, sell call K₂ > K₁, same T

• Profit capped at (K₂ − K₁) minus net premium

Payoff: max(S_T−K₁,0) − max(S_T−K₂,0)

Example: K₁=325, K₂=350 → max payoff = 25

What is a (long) butterfly spread and what volatility view is it?

Bet that S_T ends near middle strike (low vol)

• Buy call K₁, sell 2 calls K₂, buy call K₃ (K₁<K₂<K₃)

• Highest payoff when S_T ≈ K₂

Example: K=250, 275, 300; profit max near 275

In a one-step binomial model, how do you find Δ and B for a call?

Match call payoffs in up and down states

• Up: Δ S_u + B e^{rΔt} = c_u

• Down: Δ S_d + B e^{rΔt} = c_d

• Solve these 2 equations for Δ, B

• Δ = (c_u − c_d)/(S_u − S_d)

Once Δ and B are known, what is the call price today in the binomial model?

It is just the cost of the replicating portfolio

c_0 = Δ S_0 + B

How do we price an American option in a binomial tree?

At each node: pick max(hold, exercise)

• Compute continuation value (like European)

• Compute intrinsic value at node

• Node value = max(continuation, intrinsic)

When is early exercise of an American call optimal?

• No dividends: never

• With dividends: maybe right before ex-div

• On non-dividend stocks: C = c

• With dividend: check if IV > continuation just before dividend

Why can an American put be worth more than a European put?

You can exercise early when very ITM

• You may want K now, not later.

• So P ≥ p, often strictly >

What is the Black–Scholes call formula (no dividends)?

Call = stock part − discounted strike part

• c_0 = S_0 N(d1) − K e^{−rT} N(d2)

• d2 = d1 − σ√T

How do we adapt Black–Scholes for a dividend yield q?

Discount S_0 by q

• Use S_0 e^{−qT} instead of S_0.

• Call: c_0 = S_0 e^{−qT} N(d1) − K e^{−rT} N(d2)

How do we get annual volatility σ from daily volatility?

Multiply by √(trading days)

If daily vol = σ_d and N days/year: σ = σ_d √N

σ_annual = √N · σ_daily

Example: σ_d=1%, N=240 ⇒ σ≈0.155

How use put–call parity to spot arbitrage (no dividends)?

Compare c + K e^{−rT} with p + S_0

• If LHS > RHS: sell LHS, buy RHS.

• If RHS > LHS: sell RHS, buy LHS

Can a European put be worth less than its intrinsic value today?

Yes, because you cannot exercise early

Lower bound: p ≥ K e^{−rT} − S_0, which may be < K − S_0

How does volatility affect a put (common trap)?

Higher volatility increases put value

Formula: ∂p/∂σ > 0

What is the key difference between payoff and profit?

Payoff ignores premium, profit subtracts it

Payoff ≥ 0; profit can be negative

Profit = Payoff − FV(premium)