European Option

Study guide: Exam FAM

A European option gives its holder the right, but not the obligation, to buy (a call) or sell (a put) an underlying asset at a fixed strike price $K$ on one date only, the expiry date $T$ . More generally, a simple European-style derivative pays an amount $g(S_T)$ that depends only on the asset's price at the single date $T$ , so calls, puts and cash-or-nothing payoffs are all of this kind. Its price is the risk-neutral expected present value of the payoff.

$\text{Call payoff} = (S_T - K)^+$

$\text{Put payoff} = (K - S_T)^+$

$\max\!\left(0,\; S_0e^{-\delta T} - Ke^{-rT}\right) \leq C \leq S_0e^{-\delta T}$

$\max\!\left(0,\; Ke^{-rT} - S_0e^{-\delta T}\right) \leq P \leq Ke^{-rT}$

Symbols: Following McDonald, $S_T$ is the asset's price at expiry, $K$ is the strike, $r$ is the continuously compounded risk-free rate and $\delta$ is the continuous dividend yield, and $(x)^+ = \max(x, 0)$ . FAM's objectives use a non-dividend-paying asset, so $\delta = 0$ there. $C$ and $P$ are the prices of the European call and put, and the last two blocks are their no-arbitrage bounds.
European and American: An American option can be exercised at any time up to $T$ , so it is worth at least as much as the European option with the same terms. For a call on a non-dividend-paying stock the two are worth the same, because exercising that call early is never optimal.
Moneyness: A call is in the money when $S > K$ , at the money when $S = K$ and out of the money when $S < K$ , and a put is the reverse. The intrinsic value is what exercising now would pay, $(S - K)^+$ for a call or $(K - S)^+$ for a put, and the time value is the option's price less its intrinsic value.
Pricing: The price comes from the binomial model or the Black-Scholes formula, and a call and a put with the same strike and expiry are tied together by put-call parity. The bounds hold in every arbitrage-free model, so a quote outside them can be exploited with the asset and a bond.

A Deep-in-the-Money Put Worth Less Than Its Intrinsic Value A non-dividend-paying stock has fallen to $S_0 = 60$ . A 2-year European put on it has strike $K = 100$ . With $r = 0.05$ and $\sigma = 0.20$ , find the put's Black-Scholes price, its intrinsic and time values, and its lower bound.
Answer
 $\begin{align*} d_1 &= \frac{\ln(60/100) + (0.05 + 0.02)(2)}{0.20\sqrt{2}} \\ &= \frac{-0.51083 + 0.14}{0.28284} \\ &= -1.31107 \end{align*}$ 
So $d_2 = -1.31107 - 0.28284 = -1.59391$ , $N(-d_2) = 0.94452$ , $N(-d_1) = 0.90508$ and $Ke^{-rT} = 100e^{-0.10} = 90.4837$ .
 $\begin{align*} P &= 90.4837(0.94452) - 60(0.90508) \\ &= 85.46 - 54.30 \\ &= 31.16 \end{align*}$ 
The intrinsic value is $100 - 60 = 40$ , so the time value is $31.16 - 40 = -8.84$ . The lower bound is $90.48 - 60 = 30.48$ , and the price is above it. The holder cannot collect the strike until $T$ , so the strike is worth only its present value today. An American put on the same terms would be worth at least 40, because it could be exercised now.

Arbitrage from a Call Quoted Below Its Lower Bound A non-dividend-paying stock trades at $50$ , and $r = 0.06$ . A 1-year European call with strike $45$ is quoted at $7.00$ . Show that the quote allows an arbitrage, and find the riskless profit per share.
Answer
 $\begin{align*} \text{Lower bound} &= S_0 - Ke^{-rT} \\ &= 50 - 45e^{-0.06} \\ &= 50 - 42.3794 \\ &= 7.6206 \end{align*}$ 
The call is cheap, so buy the call, short one share and lend $42.3794$ for one year.
 $\begin{align*} \text{Cash today} &= -7.00 + 50 - 42.3794 \\ &= 0.6206 \end{align*}$ 
At $T$ the loan returns $45$ . If $S_T \geq 45$ , exercising the call buys back the share for $45$ , and the position closes at zero. If $S_T < 45$ , the share is bought back for $S_T$ and $45 - S_T$ is left over. The profit of $0.62$ per share today is riskless, and the position can only add to it at expiry.

How it appears on the exam

FAM asks candidates to "identify the cash flows and characteristics of puts and calls", such as the payoff and profit at expiry for a given $S_T$ , whether an option is in or out of the money, and how a European option differs from an American one.

Pricing a simple European-style derivative: The next FAM objectives price "a simple European-style derivative on a single non-dividend paying asset" with a binomial tree or the Black-Scholes formula, then ask for its delta hedge or for the other option through put-call parity.

Bounds: A question can also give a quote outside the no-arbitrage bounds and ask which trades lock in a riskless profit, as in the second example.

Exam traps

Moneyness of a put: A put is in the money when $S < K$ . Reading a put's moneyness with the call's rule gives the wrong intrinsic value and the wrong sign for its delta.

Negative time value is not an arbitrage: A deep-in-the-money European put can sell below $K - S_0$ , because its holder must wait until $T$ . The bound that binds is $Ke^{-rT} - S_0$ , so a put priced between the two is fair.

No early exercise: A European option is valued from its payoff at $T$ alone. Adding an early-exercise check at the interior nodes of a tree values the American option instead, which is worth more for a put.

Dividends in the bounds: With a dividend yield, the stock enters the bounds as $S_0e^{-\delta T}$ . Using $S_0$ overstates the call's lower bound and can make a fair price look like an arbitrage.