Exotic Options

Options, Futures and Derivative Securities

Lorenzo Naranjo

Spring 2026

Packages

Packages

  • A package is a portfolio of standard options sold to a client as a single contract.
  • Spreads, straddles and collars become packages when a dealer sells them as one product, usually designed to cost nothing up front.

Range Forward Contracts

  • A forward fixes the price; a range forward lets it float within a band [K_{1}, K_{2}]. Common in currencies.
  • A long range forward, for a buyer of the asset: long a call with strike K_{2}, short a put with strike K_{1}. The net price paid is S_{T} - \max(S_{T} - K_{2}, 0) + \max(K_{1} - S_{T}, 0) = \min(\max(S_{T}, K_{1}), K_{2})
    • Protection above K_{2}, paid for by giving up the savings below K_{1}.
  • A short range forward, for a seller: long a put with strike K_{1}, short a call with strike K_{2}.
  • The strikes are chosen so that the put pays for the call: zero cost. A narrow band around the forward price is a plain forward.

Range Forward Payoff

Net price paid against the price of the asset at maturity. A dotted 45-degree line shows the market price S T, a dashed horizontal line at F the forward. The solid range forward line is flat at K1 below K1, follows the 45-degree line between K1 and K2, and is flat at K2 above K2.

Net price paid at maturity by a buyer under a long range forward with strikes K_{1} < F < K_{2}, compared with no hedge and with a forward at F.

Valuing a Range Forward

  • A long range forward is worth C(K_{2}) - P(K_{1}), both priced with Black-Scholes.
    • Positive: the client pays the dealer up front. Negative: the dealer pays the client.
  • Zero cost: fix one strike and solve for the other so that C(K_{2}) = P(K_{1}), by trial and error.
  • Since C(F) = P(F), a zero-cost band always contains the forward price: K_{1} < F < K_{2}.

Example 1

  • An asset at $100 pays a dividend yield of 2%, with volatility 25%. The risk-free rate is 5%, so the one-year forward price is F = 100 e^{0.03} = \$103.05.
  • A one-year range forward with K_{1} = \$90 and K_{2} = \$115: \begin{aligned} C(115) & = 100 e^{-0.02} \operatorname{Φ}(-0.3140) - 115 e^{-0.05} \operatorname{Φ}(-0.5640) = \$5.60 \\ P(90) & = 90 e^{-0.05} \operatorname{Φ}(-0.4164) - 100 e^{-0.02} \operatorname{Φ}(-0.6664) = \$4.23 \end{aligned}
  • The client pays 5.603 - 4.227 = \$1.38 up front.
  • Keeping K_{1} = \$90, the range forward costs nothing when K_{2} = \$120.68.

Variations of the Black-Scholes Framework

Binary Options

  • A call pays Q if S_{T} > K; the put pays Q if S_{T} < K.
  • An call pays one share if S_{T} > K; the put pays one share if S_{T} < K.
Type Call Put
Asset-or-nothing S e^{-\delta T} \operatorname{Φ}(d_{1}) S e^{-\delta T} \operatorname{Φ}(-d_{1})
Cash-or-nothing Q e^{-r T} \operatorname{Φ}(d_{2}) Q e^{-r T} \operatorname{Φ}(-d_{2})
  • A European call is a long asset-or-nothing call and a short position in K cash-or-nothing calls paying $1: C = \underbrace{S e^{-\delta T} \operatorname{Φ}(d_{1})}_{\text{asset-or-nothing}} - \underbrace{K e^{-r T} \operatorname{Φ}(d_{2})}_{\text{cash-or-nothing}}

Binary Payoffs

Two payoff panels against the stock price at maturity. Left, the cash-or-nothing call pays zero below K and jumps to a flat Q above K. Right, the asset-or-nothing call pays zero below K and jumps to K at the strike, then rises one for one with S T.

Payoffs at maturity of a cash-or-nothing call paying Q and an asset-or-nothing call paying one share, both with strike K.

Example 2

  • A stock pays a dividend yield of 3%, has a volatility of 35%, and trades at $150. The risk-free rate is 8%.
  • For binaries with strike $150 and maturity 6 months, \begin{aligned} d_{1} & = \frac{(0.08 - 0.03 + 0.5 \times 0.35^{2}) \times 0.5}{0.35 \sqrt{0.5}} = 0.2248 \\ d_{2} & = 0.2248 - 0.35 \sqrt{0.5} = -0.0227 \end{aligned}
  • An asset-or-nothing call costs 150 e^{-0.03 \times 0.5} \operatorname{Φ}(0.2248) = \$87.02, and the put \$60.74.
  • A cash-or-nothing call paying $100 costs 100 e^{-0.08 \times 0.5} \operatorname{Φ}(-0.0227) = \$47.17, and the put \$48.91.

Gap Options

  • A gap call pays S_{T} - \textcolor{blue}{K_{1}} when S_{T} > \textcolor{red}{K_{2}}, and zero otherwise.
  • It delivers one share and takes K_{1} in cash, so it is a long asset-or-nothing call and a short position in K_{1} cash-or-nothing calls, both triggered at K_{2}: \begin{aligned} \text{Gap Call} & = S e^{-\delta T} \operatorname{Φ}(d_{1}) - \textcolor{blue}{K_{1}} e^{-r T} \operatorname{Φ}(d_{2}) \\ \text{Gap Put} & = \textcolor{blue}{K_{1}} e^{-r T} \operatorname{Φ}(-d_{2}) - S e^{-\delta T} \operatorname{Φ}(-d_{1}) \end{aligned} where d_{1} = \frac{\ln(S/\textcolor{red}{K_{2}}) + (r - \delta + 0.5 \sigma^{2}) T}{\sigma \sqrt{T}}, \quad d_{2} = d_{1} - \sigma \sqrt{T}
  • K_{2} enters d_{1} and d_{2} because it decides when the binaries pay; K_{1} only scales the cash leg.

Example 3

  • An option pays S_{T} - 180 if S_{T} > 220 in 10 months, on a non-dividend paying asset worth $200 with volatility 40%. The risk-free rate is 6%.
  • It is a gap call with K_{1} = 180 and K_{2} = 220: \begin{aligned} d_{1} & = \frac{\ln(200/220) + (0.06 + 0.5 (0.40)^{2}) (10/12)}{0.40 \sqrt{10/12}} = 0.0585 \\ d_{2} & = 0.0585 - 0.40 \sqrt{10/12} = -0.3067 \end{aligned}
  • Hence, \begin{aligned} \text{Gap Call} & = 200 \operatorname{Φ}(0.0585) - 180 e^{-0.06 \times 10/12} \operatorname{Φ}(-0.3067) = \$39.68 \\ \text{Gap Put} & = 180 e^{-0.06 \times 10/12} \operatorname{Φ}(0.3067) - 200 \operatorname{Φ}(-0.0585) = \$10.90 \end{aligned}
  • If K_{1} > K_{2} the payoff can be negative, and so can the price: here K_{1} = \$289.90 makes the gap call free.

Gap Payoffs

Two payoff panels with a dashed standard call struck at K2. Left, with K1 below K2, the gap call jumps up at K2 and runs parallel above the call. Right, with K1 above K2, it jumps down at K2 to a negative payoff, crosses zero at K1, and runs parallel below the call; the negative triangle between K2 and K1 is shaded.

A gap call pays S_{T} - K_{1} when S_{T} > K_{2}, compared with a standard call with strike K_{2}. The shaded area marks negative payoffs when K_{1} > K_{2}.

Forward Start Options

  • The option starts at a future time \tau and expires at T > \tau, typically struck at the money when it starts, K = S_{\tau}.
    • Implicit in employee stock option plans. We do not know the strike today!
  • Since \ln(S_{\tau}/K) = 0, d_{1} and d_{2} are known today, and at \tau C_{\tau} = S_{\tau} \left(e^{-\delta (T - \tau)} \operatorname{Φ}(d_{1}) - e^{-r (T - \tau)} \operatorname{Φ}(d_{2})\right) where d_{1} = \dfrac{(r - \delta + 0.5 \sigma^{2}) (T - \tau)}{\sigma \sqrt{T - \tau}} and d_{2} = d_{1} - \sigma \sqrt{T - \tau}.
  • Today, C = \operatorname{E}^{*}(C_{\tau}) e^{-r \tau} with \operatorname{E}^{*}(S_{\tau}) e^{-r \tau} = S e^{-\delta \tau}, so C = e^{-\delta \tau} \times \text{(value of a similar option starting today)}

Example 4

  • S = \$100, r = 10\%, \delta = 4\%, \sigma = 30\%. An at-the-money forward start call starts in 6 months and expires in 12 months. d_{1} = \frac{(0.10 - 0.04 + 0.5 \times 0.09) \times 0.5}{0.30 \sqrt{0.5}} = 0.2475, \quad d_{2} = 0.0354 \begin{aligned} C & = \left(100 e^{-0.04 \times 0.5} \operatorname{Φ}(0.2475) - 100 e^{-0.10 \times 0.5} \operatorname{Φ}(0.0354)\right) e^{-0.04 \times 0.5} \\ & = 9.69 \times 0.9802 = \$9.50 \end{aligned}
  • A cliquet is a series of forward start options: for example, 20 at-the-money three-month options covering five years.

Chooser Options

  • At time \tau (0 < \tau < T) the buyer chooses whether the option is a call or a put with strike K and maturity T.
  • As with forward start options, we first value the contract at \tau: it is worth \max(C_{\tau}, P_{\tau}).
  • From put-call parity: \max(C_{\tau}, P_{\tau}) = C_{\tau} + e^{-\delta (T - \tau)} \max\left(K e^{-(r - \delta)(T - \tau)} - S_{\tau}, 0\right)
  • A call with strike K and maturity T plus e^{-\delta (T - \tau)} puts with strike \widetilde{K} = K e^{-(r - \delta)(T - \tau)} and maturity \tau.

Example 5

  • S = K = \$100, r = 8\%, \delta = 0, \sigma = 30\%, T = 1 year, choice at \tau = 0.5 years.
  • Call with K = 100 and T = 1: \begin{aligned} d_{1} & = 0.4167, \quad d_{2} = 0.1167 \\ C & = 100 \operatorname{Φ}(0.4167) - 100 e^{-0.08} \operatorname{Φ}(0.1167) = \$15.71 \end{aligned}
  • One put with \widetilde{K} = 100 e^{-0.08 \times 0.5} = \$96.08 and maturity 0.5 years: \begin{aligned} d_{1} & = 0.4831, \quad d_{2} = 0.2710 \\ P & = 96.08 e^{-0.04} \operatorname{Φ}(-0.2710) - 100 \operatorname{Φ}(-0.4831) = \$4.85 \end{aligned}
  • The chooser costs 15.71 + 4.85 = \$20.56.

Path-Dependent Options

Path-Dependent Options

  • Some payoffs depend on the whole history of the stock price, not only on S_{T}.
  • The closed-form prices above work because each payoff depends only on S_{T}, whose risk-neutral distribution is lognormal.
  • A binomial tree handles path dependence directly, as long as we follow each path separately.
    • The stock tree still recombines, but the path information does not: two paths that end at the same price can have different maxima or averages.
  • A tree with n steps has n + 1 final prices but 2^{n} paths, over a million for n = 20.
    • In practice: record at each node what the payoff needs (running maximum, barrier hit), or simulate risk-neutral paths and average the discounted payoffs.

One Path, Its Extremes and Its Average

A simulated stock price path over one year starting at 100, falling to a minimum of 81.75 after mid-year and rising to a maximum of 121.13 near month eleven, ending at 117.75. The running maximum steps up to 121.13, the running minimum steps down to 81.75, and the running average stays near 100.

One simulated year of daily stock prices starting at S_{0} = 100 with \sigma = 0.30, with the running maximum, minimum, and average.

Lookback Options

  • A floating lookback call pays S_{T} - S_{\textit{min}}: buy at the lowest price the stock reaches.
  • A floating lookback put pays S_{\textit{max}} - S_{T}: sell at the highest price the stock reaches.
  • A fixed lookback call pays \max(S_{\textit{max}} - K, 0).
  • A fixed lookback put pays \max(K - S_{\textit{min}}, 0).
  • S_{\textit{max}} and S_{\textit{min}} are taken over the whole life of the option, including S_{0}.
  • Analytic formulas exist for all four types.

Asian Options

  • The payoff depends on the average stock price \bar{S}_{T} from time 0 until T; on a tree, the arithmetic mean of the prices along the path, including t = 0.
  • Average price: call \max(\bar{S}_{T} - K, 0), put \max(K - \bar{S}_{T}, 0).
  • Average strike: call \max(S_{T} - \bar{S}_{T}, 0), put \max(\bar{S}_{T} - S_{T}, 0).
  • Averaging smooths the path, so an average price option is usually cheaper than the European one: popular with firms that buy or sell regularly over a period.
  • No exact analytic formula for the arithmetic average (options on the geometric average have one); approximations assume the average is lognormal.

Example 6

  • A non-dividend paying stock trades for $100 and every 3 months can rise or fall by 5%. The risk-free rate is 6%.
  • We price a floating lookback put paying S_{\textit{max}} - S_{T} and an average price call paying \max(\bar{S}_{T} - 100, 0), both in 6 months.
Two-period non-recombining stock tree over dates 0, 3/12 and 6/12, starting at 100.00 and moving to 105.00 or 95.00, with up probability q. Final prices 110.25, 99.75, 99.75 and 90.25. Next to each, the path maximum is 110.25, 105.00, 100.00 and 100.00, the lookback payoff is 0.00, 5.25, 0.25 and 9.75, the path average is 105.08, 101.58, 98.25 and 95.08, and the Asian call payoff is 5.08, 1.58, 0.00 and 0.00.

Example 6 (cont’d)

  • The risk-neutral probability of an up-move is q = \frac{e^{0.06 \times 3/12} - 0.95}{1.05 - 0.95} = 0.6511
  • Working backward, the lookback put L and the Asian call A are worth \begin{aligned} L_{u} & = (0 q + 5.25 (1 - q)) e^{-0.06 \times 3/12} = 1.80 \\ L_{d} & = (0.25 q + 9.75 (1 - q)) e^{-0.06 \times 3/12} = 3.51 \\ L & = (1.80 q + 3.51 (1 - q)) e^{-0.06 \times 3/12} = 2.36 \\[4pt] A_{u} & = (5.08 q + 1.58 (1 - q)) e^{-0.06 \times 3/12} = 3.80 \\ A_{d} & = 0 \\ A & = (3.80 q + 0 (1 - q)) e^{-0.06 \times 3/12} = 2.44 \end{aligned}

Barrier Options

  • Calls or puts that are activated or deactivated when the stock hits a barrier.
    • In options come into existence only if the stock hits the barrier before maturity; out options die if it does.
    • Up options require the stock to hit the barrier from below, down options from above.
  • Eight combinations: up/down, in/out, call/put.
  • An in and an out option with the same direction, strike, barrier, and maturity add up to the plain European option.
  • A binomial tree tracks, along each path, whether the barrier has been hit; analytic formulas also exist.

In-Out Parity

Option value against the barrier H from 60 to 100. A dashed line at 14.23 is the European call. The down-and-out call starts near 14.23 for low barriers and falls to zero at 100; the down-and-in call mirrors it, rising from zero to 14.23. The two add up to the dashed line at every barrier.

Black-Scholes prices of a one-year down-and-in and down-and-out call against the barrier H, with S = K = 100, r = 5\% and \sigma = 30\%. They add up to the European call, dashed.

Example 7

  • A down-and-in call with barrier $90, strike $110, and maturity 6 months, on a stock at $100.
    • The stock falls to $85 and finishes at $120: the call pays 120 - 110 = \$10.
    • The stock never goes below $90 and finishes at $140: the call pays nothing, since it was never activated.
    • The stock falls to $85 and finishes at $105: the call is activated but out-of-the-money, and pays nothing.

Barrier Options in Practice

  • Monitoring. The contract states how often the barrier is checked: continuously, or at specified times such as daily closes.
    • A barrier checked less often is hit less often: discrete monitoring raises knock-out prices and lowers knock-in prices.
    • A tree checks the barrier only at its nodes, so it needs many steps to price a continuously monitored barrier.
  • Hedging. Near the barrier of an up-and-out call, a small rise in the stock can wipe out a large payoff.
    • The value falls steeply and delta is large and negative.
    • Cash-or-nothing options have the same problem near the strike close to maturity.
    • A delta hedger must trade large amounts of stock for small price moves.

Hedging Near the Barrier

Option value against the stock price from 60 to 130 with a vertical line at the barrier. With one year left the value stays below 2. With three months it peaks near 5 at 110 and falls to zero at 130. With one week it tracks the payoff up to about 18 at 120, then drops steeply to zero at the barrier.

Black-Scholes prices of an up-and-out call with K = 100 and H = 130 for three maturities, with r = 5\% and \sigma = 30\%. The dotted line is the call payoff if never knocked out.

Other Exotic Options

Compound and Exchange Options

  • Compound options are options on options: call on call, put on call, call on put, put on put.
    • The equity of a leveraged firm is a call on its assets, so an option on the equity is a compound option.
    • Valued analytically in the Black-Scholes framework; cheap relative to the underlying option.
  • Exchange options give the right to exchange one asset U for another V: payoff \max(V_{T} - U_{T}, 0).
    • In units of U, the payoff is U_{T} \max(V_{T}/U_{T} - 1, 0): a call on V/U with strike 1.
    • The interest rate drops out, since the holder pays with U, not cash.
    • The volatility of V/U depends on the correlation: assets that move together make a cheap exchange option.

Basket, Shout, and Non-Standard American Options

  • Basket options are options on a portfolio of assets, valued by matching the first two moments of the basket to a lognormal.
  • Shout options let the buyer shout once, at \tau, locking in the intrinsic value: a call pays \max(S_{T} - K, S_{\tau} - K) = \max(S_{T} - S_{\tau}, 0) + S_{\tau} - K
    • Similar to a lookback, but cheaper.
  • Non-standard American options: exercisable only on specific dates (Bermudans), after a lock-out period, or with a strike that changes over time (warrants, convertibles).