1.1.1 Digital Options

A European digital or binary option pays off $1 if the underlying asset price is above the strike K at maturity T, and 0 otherwise:

In its American version, which is more uncommon, the option pays off $1 as soon as the strike level is hit.

The Black-Scholes price formula for a digital option is simply:

where F is the forward price of S for maturity T, r is the continuous interest rate, and σ is the volatility parameter. When there is an implied volatility smile this formula is inaccurate and a corrective term must be added (see Section 2-1.3).

Digital options are not easy to dynamically hedge because their delta can become very large near maturity. Exotic traders tend to overhedge them with a tight call spread whose range may be determined according to several possible empirical rules, such as:

• Daily volatility rule: Set the range to match a typical stock price move over one day. For example, if the annual volatility of the underlying stock is 32% annually; that is, 32%/√252 ≈ 2% daily, a digital option struck at $100 would be overhedged with $98–$100 call spreads.
• Normalized liquidity rule: Set the range so that the quantity of call spreads is in line with the market liquidity of call spreads with 5% range. The quantity of call spreads is N/R where N is the quantity of digitals and R is the call spread range. If the tradable quantity of call spreads with range 5% is V, the normalized tradable quantity of call spreads with range R would be V × R / 0.05. Solving for R gives . In practice V is either provided by the option trader or estimated using the daily trading volume of the stock.

1.1.2 Asian Options

In an Asian call or put, the final underlying asset price is replaced by an average:

where for a set of pre-agreed fixing dates t₁ < t₂ < < t_n ≤ T. For example, a one-year at-the-money Asian call on the S&P 500 index with quarterly fixings pays off , where S₀ is the current spot price and S_0.25,…, S₁ are the future spot prices observed every three months.

On occasion, the strike may also be replaced by an average, typically over a short initial observation period.

Fixed-strike Asian options are always cheaper than their European counterparts, because A_T is less volatile than S_T.

There is no closed-form Black-Scholes formula for arithmetic Asian options. However, for geometric Asian options where , the Black-Scholes formulas may be used with adjusted volatility and dividend yield , as shown in Problem 1.3.3.

A common numerical approximation for the price of arithmetic Asian options is obtained by fitting a lognormal distribution to the actual risk-neutral moments of A_T.

1.1.3 Barrier Options

In a barrier call or put, the underlying asset price must hit, or never hit, a certain barrier level H before maturity:

• For a knock-in option, the underlying must hit the barrier, or else the option pays nothing.
• For a knock-out option, the underlying must never hit the barrier, or else the option pays nothing.

Barrier options are always cheaper than their European counterparts, because their payoff is subject to an additional constraint. On occasion, a fixed cash “rebate” is paid out if the barrier condition is not met.

Similar to digital options, barrier options are not easy to dynamically hedge: their delta can become very large near the barrier level. Exotic traders tend to overhedge them by shifting the barrier a little in their valuation model.

Continuously monitored barrier options have closed-form Black-Scholes formulas, which can be found, for instance, in Hull (2012). The preferred pricing approach is the local volatility model (see Chapter 4).

In practice the barrier is often monitored on a set of pre-agreed fixing dates t₁ < t₂ < < t_n ≤ T. Monte Carlo simulations are then commonly used for valuation.

Broadie, Glasserman, and Kou (1997) derived a nice result to switch between continuous and discrete barrier monitoring by shifting the barrier level H by a factor where β ≈ 0.5826, σ is the underlying volatility, and Δt is the time between two fixing dates.

1.1.4 Lookback Options

A lookback call or put is an option on the maximum or minimum price reached by the underlying asset until maturity:

Lookback options are always more expensive than their European counterparts: about twice as much when the strike is nearly at the money, as shown in Problem 1.3.5.

Continuously monitored lookback options have closed-form Black-Scholes formulas, which can be found, for instance, in Hull (2012). The preferred pricing approach is the local volatility model (see Chapter 4).

In practice the maximum or minimum is often monitored on a set of pre-agreed fixing dates t₁ < t₂ < < t_n ≤ T. Monte Carlo simulations are then commonly used for valuation.

1.1.5 Forward Start Options

In a forward start option the strike is determined as a percentage k of the spot price on a future start date t₀ > 0:

At t = t₀ a forward start option becomes a regular option. Note that the forward start feature is not specific to vanilla options and can be added to any exotic option that has a strike.

Forward start options have closed-form Black-Scholes formulas. The preferred pricing approach is to use a stochastic volatility model (see Chapter 4).

1.1.6 Cliquet Options

A cliquet or ratchet option consists of a series of consecutive forward start options, for example:

where 5% is the local cap amount. In other words, this particular cliquet option pays off the greater of zero and the sum of monthly returns, each capped at 5%.

Cliquet options can be very difficult to value and especially hedge.

1.2.1 Spread Options

The payoff of a spread option is based on the difference in gross return between two underlying assets:

where k is the residual strike level (in %). For example, a spread option on Apple Inc. vs Google Inc. with 5% strike pays off the outperformance of Apple over Google in excess of 5%: if Apple's return is 13% and Google's is 4%, the option pays off 13% − 4% − 5% = 4%.

The value of a spread option is very sensitive to the level of correlation between the two assets. Specifically the option value increases as correlation decreases: the lower the correlation, the wider the two assets are expected to spread apart.

In practice hedging spread options can be difficult because the spread is often nearly orthogonal to the basket .

When k = 0 a spread option is also known as an exchange option. A closed-form Black-Scholes formula is then available which can be found, for instance, in Hull (2012).

1.2.2 Basket Options

A basket call or put is an option on the gross return of a portfolio of n underlying assets:

where the weights w₁,…, w_n sum to 100% and the strike k is expressed as a percentage (e.g., 100% for at the money).

The value of a basket option is sensitive to the level of pairwise correlations between the assets. The lower the correlation, the less volatile the portfolio and the cheaper the basket option.

Basket options do not have closed-form Black-Scholes formulas. A common approximation technique is to fit a lognormal distribution to the actual moments of the basket and then use formulas for the single-asset case.

1.2.3 Worst-Of and Best-Of Options

A worst-of call or put is an option on the lowest gross return between n underlying assets:

where the strike k is expressed as a percentage (e.g., 100% for at the money). For example, a worst-of at-the-money call on Apple, Google, and Microsoft pays off the worst stock return between the three companies, if positive.

Similarly, a best-of call or put is an option on the highest gross return between n underlying assets.

Worst-of calls and best-of puts are always cheaper than any of their single-asset European counterparts, while best-of calls and worst-of puts are always more expensive.

1.2.4 Quanto Options

The payoff of a quanto option is paid out in a different currency from the underlying assets, at a guaranteed exchange rate. For example, a call on the S&P 500 index quanto euro pays off max(0, S_T − K) in euros instead of dollars, thereby guaranteeing an exchange rate of 1 euro per dollar.

The actual exchange rate between the asset currency and the quanto currency is in fact an implicit additional underlying asset. The value of quanto options is very sensitive to the correlation between the primary asset and the implicit exchange rate.

Quanto options are an example of hybrid exotic options involving different asset classes—here equity and foreign exchange.

In terms of pricing, the quanto feature is often approached using a technique called change of numeraire. In summary, this technique says that the risk-neutral dynamics of an asset quantoed in a different currency from its natural currency has the same volatility coefficient but an adjusted drift coefficient.

1.1 “Free” Option

Consider a European call option on an underlying asset S with strike K and maturity T where “you only pay the premium if you win,” that is, if S_T > K.

1. Draw the diagram of the net P&L of this “free” option at maturity. Is it really “free”?
2. Find a replicating portfolio for the “free” option using vanilla and exotic options.
3. Calculate the fair value of the “free” option premium using the Black-Scholes model with 20% volatility, S₀ = K = $100, one-year maturity, zero interest and dividend rates.

1.2 Autocallable

Consider an exotic option expiring in one, two, or three years on an underlying asset S with the following payoff mechanism:

• If after one year S₁ > S₀ the option pays off 1 + C and terminates;
• Else if after two years S₂ > S₀ the option pays off 1 + 2C and terminates;
• Else if after three years S₃ > 0.7 × S₀ the option pays off
• Otherwise, the option pays off S₃/S₀.

Assuming S₀ = $100, zero interest and dividend rates, and 25% volatility, estimate the level of C so that the option is worth 1 using Monte Carlo simulations.

1.3 Geometric Asian Option

Consider a geometric Asian option on an underlying S with payoff f(A_T) where . Assume that S follows a geometric Brownian motion with parameters (r − q, σ) under the risk-neutral measure.

1. Using the Ito-Doeblin theorem, show that
   
2. Using the Ito-Doeblin theorem, show that . What is the distribution of this quantity?
3. Show that A_T is lognormally distributed with parameters where and .

1.4 Change of Measure

In the context of Appendix 1.A, verify that using the expression for d/d.

1.5 At-the-Money Lookback Options

The Black-Scholes closed-form formula for an at-the-money lookback call is given as:

where and .

Using a first-order Taylor expansion of the cumulative normal distribution N(·) show that for reasonable rates and maturities we have the proxy:

which is twice as much as the European call proxy: .

1.6 Siegel's Paradox

This problem is about foreign exchange rates and goes beyond the scope of equity derivatives.

Consider two currencies, say dollars and euros, and suppose that their corresponding interest rates, r_$ and r_€, are constant. Let X be the euro–dollar exchange rate defined as the number of dollars per euro. The traditional risk-neutral process for X is thus:

where W is a standard Brownian motion.

1. Using the Ito-Doeblin theorem, show that the risk-neutral dynamics for the dollar-euro exchange rate, that is, the number 1/X of euros per dollar, is:
   
2. Symmetry suggests that the drift of 1/X should be r_€ − r_$ instead—this is Siegel's paradox. Use your knowledge of quantos (see Section 1.2.4) to resolve the paradox.