17. Exotic Options

17.1. Packages

A package is a portfolio of standard options. The main difference between a package and an option strategy is that the package is sold as a whole product, whereas an option strategy involves trading different options at the same time.

We have studied many option strategies such as bull spreads, bear spreads, straddles, strangles, butterflies, and condors. Some of these strategies could be sold as a package.

One popular package is a range forward contract. When applied to foreign currencies, a range forward has the effect of ensuring that the exchange rate paid or received lies within a certain range.

A long range forward involves selling a put with strike \(K_{1}\) and buying a call with strike \(K_{2}\) (with \(K_{1} < K_{2}\)). This would be similar to a long forward position.

A short range forward involves buying a put with strike \(K_{1}\) and selling a call with strike \(K_{2}\). This would be similar to a short forward position. Normally the price of the put equals the price of the call so the contract has zero cost.

17.2. Variations of the Black & Scholes Framework

17.2.1. Gap Options

A gap option uses different strike prices to determine when and how much to pay. For example, a gap call pays \(S_{T} - K_{1}\) when \(S_{T} > K_{2}\), and zero otherwise. Therefore, \(K_{2}\) determines if the option is ITM at expiration whereas \(K_{1}\) determines how much the option pays if it is ITM. Therefore, a gap call is equivalent to a European call with strike if \(K_{1} = K_{2}.\)

The price of the gap call can then be obtained by computing its expected risk-neutral payoff and discounting it at the risk-free rate, that is,

\[\begin{align*} \text{Gap Call} & = \ev\left( (S_{T} - K_{1}) \1{S_{T} > K_{2}} \right) e^{-r T} \\ & = \ev\left( S_{T} \1{S_{T} > K_{2}} \right) e^{-r T} - K_{1} e^{-r T} \ev\left( \1{S_{T} > K_{2}} \right) \\ & = S e^{-q T} \cdf(d_{1}) - K_{1} e^{-r T} \cdf(d_{2}) \\ \end{align*}\]

where

\[\begin{align*} d_{1} & = \frac{\ln(S/K_{2}) + (r - q + 0.5 \sigma^{2}) T}{\sigma \sqrt{T}} \\ d_{2} & = d_{1} - \sigma \sqrt{T} \end{align*}\]

Example 17.1

Suppose you want to price an option that pays \(S_{T} - 180\) if \(S_{T} > 220\) in 10 months. The option is written on a non-dividend paying asset whose current price is $200 and that has a volatility of returns of 40%. The risk-free rate is 6% per year with continuous compounding.

The option in the example is a gap call option where \(K_{1} = 180\) and \(K_{2} = 220.\) This implies,

\[\begin{align*} d_{1} & = \frac{\ln(200/220) + (0.06 + 0.5 (0.40)^{2}) (10/12)}{0.40 \sqrt{10/12}} = 0.0585 \\ d_{2} & = d_{1} - 0.40 \sqrt{10/12} = -0.3067 \\ \end{align*}\]

The gap call price is then given by,

\[\begin{equation*} \text{Gap Call} = 200 \cdf(0.0585) - 180 e^{-0.06 \times 10/12} \cdf(-0.3067) = \$39.68 \end{equation*}\]

A gap put is similarly defined. It pays \(K_{1} - S_{T}\) when \(S_{T} < K_{2}\), and zero otherwise. Its price is given by

\[\begin{equation*} \text{Gap Put} = K_{1} e^{-r T} \cdf(-d_{2}) - S e^{-q T} \cdf(-d_{1}) \end{equation*}\]

Example 17.2

Using the data in example Example 17.1, the price of a gap put is

\[\begin{equation*} \text{Gap Put} = 180 e^{-0.06 \times 10/12} \cdf(0.3067) - 200 \cdf(-0.0585) = \$10.90 \end{equation*}\]

17.2.2. Binary Options

Binary options pay either a certain amount of cash or units of a risky asset if the underlying asset is above or below a certain strike price. For example, a cash-or-nothing call could pay \(Q\) if the stock price is above the strike price \(K\) or nothing otherwise. An asset-or-nothing call could pay 1 unit of the stock if the stock price is above the strike \(K\) or nothing otherwise.

Interestingly, a European call option can be seen as a long asset-or-nothing call and a short cash-or-nothing call. Indeed, a European call pays \(S - K\) if \(S > K\) at maturity, which can be seen as receiving \(S\) and paying \(K\) whenever \(S > K,\) implying that

\[\begin{equation*} C = \underbrace{S e^{-q T} \cdf(d_{1})}_{\text{asset-or-nothing call}} - \underbrace{K e^{-r T} \cdf(d_{2})}_{\text{cash-or-nothing call}} \end{equation*}\]

Similarly, a European put can be seen as a long cash-or-nothing put and a short asset-or-nothing put:

\[\begin{equation*} P = \underbrace{K e^{-r T} \cdf(-d_{2})}_{\text{cash-or-nothing put}} - \underbrace{S e^{-q T} \cdf(-d_{1})}_{\text{asset-or-nothing put}} \end{equation*}\]

The following table shows the prices of asset-or-nothing options that pay 1 share of the stock, and cash-or-nothing options that pay \(Q\), whenever the option is ITM at maturity.

Type	Call	Put
Asset-or-nothing	\(S e^{-q T} \cdf(d_{1})\)	\(S e^{-q T} \cdf(-d_{1})\)
Cash-or-nothing	\(Q e^{-r T} \cdf(d_{2})\)	\(Q e^{-r T} \cdf(-d_{2})\)

Example 17.3

Consider a stock that pays a dividend yield of 3% and that has a volatility of returns of 35%. The stock price is $150 and the risk-free rate is 8%.

First, let’s price an asset-or-nothing call that pays 1 share of the stock if the stock price in 6 months is above $150. We can then compute

\[\begin{equation*} d_{1} = \frac{\ln(150/150) + (0.08 - 0.03 + 0.5 \times 0.35^{2}) \times 0.5}{0.35 \sqrt{0.5}} = 0.2248 \end{equation*}\]

The costs of the asset-or-nothing call is \(150 e^{-0.03 \times 0.5} \cdf(0.2248) = \$87.02.\) The price of an otherwise equivalent asset-or-nothing put is \(150 e^{-0.03 \times 0.5} \cdf(-0.2248) = \$60.74.\)

Example 17.4

Using the data in Example 17.3, consider a cash-or-nothing call that pays $100 if the stock price in 6 months is above $150. We can compute

\[\begin{equation*} d_{2} = d_{1} - 0.35 \sqrt{0.5} = -0.0227 \end{equation*}\]

The cost of the cash-or-nothing call is then \(100 e^{-0.08 \times 0.5} \cdf(-0.0227) = \$47.17.\) The cost of an otherwise equivalent asset-or-nothing put is \(100 e^{-0.08 \times 0.5} \cdf(0.0227) = \$48.91.\)

17.2.3. Forward Start Options

A forward start option is a European call or put that starts at a future time \(\tau\) and expires at time \(T > \tau\). They are typically implicit in employee stock option plans, and are often structured so that strike price equals asset price at time \(\tau\), that is, \(K = S_{\tau}\).

Therefore, the value of a call or put option at time \(\tau\) is:

\[\begin{align*} C_{\tau} & = S_{\tau} e^{-q (T - \tau)} \cdf(d_{1}) - S_{\tau} e^{-r (T - \tau)} \cdf(d_{2}) \\ P_{\tau} & = S_{\tau} e^{-r (T - \tau)} \cdf(-d_{2}) - S_{\tau} e^{-q (T - \tau)} \cdf(-d_{1}) \end{align*}\]

where \(d_{1} = \dfrac{(r - q + 0.5 \sigma^{2}) (T - \tau)}{\sigma \sqrt{T - \tau}}\) and \(d_{2} = d_{1} - \sigma \sqrt{T - \tau}\).

The price of the option today is then \(V = \ev(V_{\tau}) e^{-r \tau}\) where the expectation is taken of course with respect the risk-neutral measure.

Noting that the futures price of a contract expiring at time \(\tau\) is given by \(f = \ev(S_{\tau}) = S e^{(r - q) \tau}\), we have that \(\ev(S_{\tau}) e^{-r \tau} = S e^{-q \tau}\). Therefore:

\[\begin{align*} C & = \left(S e^{-q (T - \tau)} \cdf(d_{1}) - S e^{-r (T - \tau)} \cdf(d_{2})\right) e^{-q \tau} \\ P & = \left(S e^{-r (T - \tau)} \cdf(-d_{2}) - S e^{-q (T - \tau)} \cdf(-d_{1})\right) e^{-q \tau} \end{align*}\]

where \(d_{1} = \dfrac{(r - q + 0.5 \sigma^{2}) (T - \tau)}{\sigma \sqrt{T - \tau}}\) and \(d_{2} = d_{1} - \sigma \sqrt{T - \tau}\).

We can then see that the value of a forward start option is \(e^{-q \tau}\) times the value of similar option starting today.

Forward start options can be packed together in what is known as a cliquet option, which consists in a series of call or put options with rules determining how the strike price is determined.

For example, a cliquet might consist of 20 at-the-money three-month options. The total life would then be five years. When one option expires a new similar at-the-money is coming into existence. As you can see, this would be a portfolio of 20 forward starting options that we just saw how to value.

17.2.4. Chooser Options

A chooser option allows the buyer to choose at an intermediate time before expiration whether to get a European call or put option. If the chooser option starts at time 0 and matures at \(T,\) then at time \(\tau\) \((0 < \tau < T)\) the buyer chooses whether it is a put or call with strike \(K\) and expiring at \(T.\)

Therefore, the value of the chooser option at \(\tau\) is \(\max(C_{\tau}, P_{\tau})\) since the buyer will choose to get whichever option is more valuable. From put-call parity we get that

\[\begin{equation*} P_{\tau} = C_{\tau} + K e^{-r (T - \tau)} - S_{\tau} e^{-q (T - \tau)} \end{equation*}\]

which implies that:

\[\begin{equation*} \max(C_{\tau}, P_{\tau}) = C_{\tau} + e^{-q (T - \tau)} \max\left(K e^{-(r - q)(T - \tau)} - S_{\tau}, 0\right) \end{equation*}\]

This is the payoff of a call with strike \(K\) and expiring at \(T\) plus \(e^{-q (T - \tau)}\) puts with strike \(\widetilde{K} = K e^{-(r - q)(T - \tau)}\) and expiring at time \(\tau\).

17.2.5. Compound Options

We can also write options on other options, which are also known as compound options. We can then have four possible combinations:

• Call on call

• Put on call

• Call on put

• Put on put

These options can be valued analytically (we will not cover this in class, though). Intuitively, the price of such options is quite low compared with the underlying option.

17.3. Path-Dependent Options

The payoffs of some options might depend not only on the final value of the stock but on the full history of the stock price until maturity. These type of options are commonly called path-dependent options.

17.3.1. Lookback Options

The payoff of a lookback option depends on the maximum \((S_{\textit{max}})\) or minimum \((S_{\textit{min}})\) value of the stock until maturity. There are two type of lookback options, namely floating and fixed lookbacks.

A floating lookback call pays \(S_{T} - S_{\textit{min}}\) at time \(T\). The idea is that the strike of the call is now replaced by the minimum value that the stock can have until maturity. Since by definition \(S_{\textit{min}} \leq S_{T},\) we always have that the payoff of the floating lookback call is non-negative. Similarly, the payoff of a floating lookback put is defined as \(S_{\textit{max}} - S_{T}\) and by construction is also non-negative.

A fixed lookback call is like a regular call whose final payoff depends on the maximum value of the stock during the lifetime of the option, that is, it pays \(\max(S_{\textit{max}} - K, 0)\) at maturity. A fixed lookback put pays \(\max(K - S_{\textit{min}}, 0)\) at maturity.

The table below summarizes the payoffs of the different looback options.

Type	Call	Put
Floating lookback	\(S_{T} - S_{\textit{min}}\)	\(S_{\textit{max}} - S_{T}\)
Fixed lookback	\(\max(S_{\textit{max}} - K, 0)\)	\(\max(K - S_{\textit{min}}, 0)\)

Even though is possible to derive analytic formulas for all types, we can easily price any lookback option by using a binomial tree as the next example shows.

Example 17.5

Consider a non-dividend paying stock that trades for $100. Every 3-months, the stock price can increase or decrease by 5%. The risk-free rate is 6% per year with continuous compounding. We will compute the price of a floating lookback put that pays \(S_{\textit{max}} - S_{T}\) in 6 months.

For lookback options is better to draw the full tree for the stock in order to see all four possible histories for the stock price. We can then compute the maximum stock price of each history and then the final payoff of the option, as the tree below shows.

The risk-neutral probability of an up-move is

\[\begin{equation*} p = \frac{e^{-0.06 \times 3/12} - 0.95}{1.05 - 0.95} = 0.6511 \end{equation*}\]

The tree for the lookbak put is presented below.

We can then compute:

\[\begin{align*} L_{u} & = (0 p + 5.25 (1-p)) e^{-0.06 \times 3/12} = 1.80 \\ L_{d} & = (0.25 p + 9.75 (1-p)) e^{-0.06 \times 3/12} = 3.51 \\ L & = (1.80 p + 3.51 (1 - p)) e^{-0.06 \times 3/12} = 2.36 \\ \end{align*}\]

Alternatively, we could obtain the price of the lookback put directly from the final payoffs:

\[\begin{equation*} L = (0 p^{2} + 5.25 p (1-p) + 0.25 (1-p) p + 9.75 (1-p)^{2}) e^{-0.06 \times 6/12} = 2.36 \end{equation*}\]

17.3.2. Asian Options

The payoff of an Asian option is related to the average stock price \(\bar{S}\) from time 0 until \(T\):

\[\begin{equation*} \bar{S}_{T} = \frac{1}{T} \int_{0}^{T} S_{t} dt \end{equation*}\]

As with lookback options, there are two types of Asian options:

• An average price call pays \(\max(\bar{S}_{T} - K, 0)\) whereas an average price put pays \(\max(K - \bar{S}_{T}, 0)\) at maturity.

• An average strike call pays \(\max(S_{T} - \bar{S}_{T}, 0)\) whereas an average strike put pays \(\max(\bar{S}_{T} - S_{T}, 0)\)

There is no exact analytic valuation for Asian options, but they can be approximately valued by assuming that the average stock price is lognormally distributed. Alternatively, it is straightforward to price them using a binomial tree.

Example 17.6

Consider a non-dividend paying stock that trades for $100. Every 3-months, the stock price can increase or decrease by 5%. The risk-free rate is 6% per year with continuous compounding. We will compute the price of an average price call that pays \(\bar{S}_{T} - K\) in 6 months, where \(K = 100.\)

As we did for lookback options, it is better to draw the full tree for the stock in order to see all four possible histories for the stock price. We can then compute the average stock price of each history and then the final payoff of the option, as the tree below shows.

The risk-neutral probability of an up-move is

\[\begin{equation*} p = \frac{e^{-0.06 \times 3/12} - 0.95}{1.05 - 0.95} = 0.6511 \end{equation*}\]

The tree for the Asian call is presented below.

We can then compute:

\[\begin{align*} A_{u} & = (5.08 p + 1.58 (1-p)) e^{-0.06 \times 3/12} = 3.80 \\ A_{d} & = (0 p + 0 (1-p)) e^{-0.06 \times 3/12} = 0 \\ A & = (3.80 p + 0 (1 - p)) e^{-0.06 \times 3/12} = 2.44 \\ \end{align*}\]

Alternatively, we could obtain the price of the lookback put directly from the final payoffs:

\[\begin{equation*} L = (5.08 p^{2} + 1.58 p (1-p) + 0 (1-p) p + 0 (1-p)^{2}) e^{-0.06 \times 6/12} = 2.44 \end{equation*}\]

17.3.3. Barrier Options

Barrier options are either call or put options that get activated or deactivated depending on whether the stock hits a barrier from above or below.

In options come into existence only if stock price hits the barrier before option maturity. Out options die if stock price hits the barrier before option maturity.

Up options require that the stock hits the barrier from below, whereas down options require that the stock hit the barrier from above.

Therefore, there are eight possible combinations.

Example 17.7

Consider an in-and-down call with barrier $90, strike price $110, and expiring in 6 months. The stock price is currently $100.

• Case 1: During the next six months, the stock goes down to $85 and but then finishes at $120. The in-and-out call pays \(120 - 110 = \$10.\)

• Case 2: During the next six months, the stock never goes below $90 and finishes at $140. The in-and-out call pays nothing since even though it is in-the-money it never gets activated.

• Case 3: During the next six months, the stock goes down to $85 and and finishes at $105. Even though the in-and-out call gets activated, it is out-of-the-money at maturity and hence pays nothing.

17.4. Other Exotic Options

17.4.1. Exchange Options

An exchange option is an option to exchange one asset for another. For example, an option to exchange one unit of \(U\) for one unit of \(V\).The payoff is then \(\max(V_{T} - U_{T}, 0)\).

In order to price such an option we need to know how the two assets are correlated. It is hard to value such an option using a binomial tree since we need to take into account the correlation of the two assets. It is possible, however, to find an analytical formula using the Black & Scholes framework.

17.4.2. Basket Options

A basket option is an option to buy or sell a portfolio of assets. This can be valued by calculating the first two moments of the value of the basket at option maturity and then assuming it is lognormal.

17.4.3. Non-Standard American Options

In practice, many American type options have restrictions about when they can be exercised. Some American options are exercisable only on specific dates (Bermudans), whereas other options can be exercised early only after a certain lock out period has elapsed. For some

Some American options have a strike price that changes over the life of the contract, such as warrants or convertibles.

17.5. Practice Problems

Problems 1 to 6 are based on an asset with the following characteristics:

Variable	Value
\(S_{0}\)	100
\(r\)	10%
\(q\)	4%
\(\sigma\)	30%

Exercise 17.1

Compute the price of a long range forward with maturity 9 months, and strikes \(K_{1} = \$90\) and \(K_{2} = \$120\).

Solution to Exercise 17.1

The long range forward is a long call with strike \(K_{2}\) and a short put with strike \(K_{1}\). For the call, we have:

\[\begin{align*} d_{1} & = \frac{\ln(100/120) + (0.10 - 0.04 + \frac{1}{2} 0.30^{2})(9/12)}{0.30 \sqrt{9/12}} = -0.3986 \\ \cdf(d_{1}) & = 0.3451 \\ d_{2} & = -0.3986 - 0.30 \sqrt{9/12} = -0.6585 \\ \cdf(d_{2}) & = 0.2551 \\ C & = 100 e^{-0.04 (9/12)} \times 0.3451 - 120 e^{-0.10 (9/12)} \times 0.2551 = \$5.09 \end{align*}\]

For the put we have:

\[\begin{align*} d_{1} & = \frac{\ln(100/90) + (0.10 - 0.04 + \frac{1}{2} 0.30^{2}) (9/12)}{0.30 \sqrt{9/12}} = 0.7086 \\ \cdf(-d_{1}) & = 0.2393 \\ d_{2} & = 0.7607 - 0.30 \sqrt{9/12} = 0.4488 \\ \cdf(-d_{2}) & = 0.3268 \\ P & = 90 e^{-0.10 (9/12)} \times 0.3268 - 100 e^{-0.04 (9/12)} \times 0.2393 = \$4.06 \end{align*}\]

Therefore, the long range forward costs \(5.09 - 4.06 = \$1.02\).

Exercise 17.2

Compute the price of a gap put option with maturity 9 months, that pays \(90 - S_{T}\) whenever \(S_{T} < 110\).

Solution to Exercise 17.2

We have that:

\[\begin{align*} d_{1} & = \frac{\ln(100/110) + (0.10 - 0.04 + \frac{1}{2} 0.30^{2}) (9/12)}{0.30 \sqrt{9/12}} = -0.0637 \\ \cdf(-d_{1}) & = 0.5254 \\ d_{2} & = -0.0637 - 0.30 \sqrt{9/12} = -0.3235 \\ \cdf(-d_{2}) & = 0.6269 \end{align*}\]

Therefore, the gap put should cost

\[\begin{align*} 90 e^{-0.10 (9/12)} (0.6269) - 100 e^{-0.04 (9/12)} (0.5254) = \$1.35. \end{align*}\]

Exercise 17.3

Compute the price of a forward start put option that starts in 9 months, with final maturity in 15 months, and that is struck at-the-money when it starts.

Solution to Exercise 17.3

The difference between the final maturity and the starting date is 6 months, hence:

\[\begin{align*} d_{1} & = \frac{(0.10 - 0.04 + \frac{1}{2} 0.30^{2}) (6/12)}{0.30 \sqrt{6/12}} = 0.2475 \\ \cdf(-d_{1}) & = 0.4023 \\ d_{2} & = 0.2475 - 0.30 \sqrt{6/12} = 0.0354 \\ \cdf(-d_{2}) & = 0.4859 \end{align*}\]

Therefore, the forward start put should cost

\[\begin{align*} \left( 100 e^{-0.10 \times 6/12} \times 0.4859 - 100 e^{-0.04 \times 6/12} \times 0.4023 \right) e^{-0.04 \times 9/12} = \$6.59. \end{align*}\]

Exercise 17.4

Compute the price of a chooser option with maturity 15 months, strike \(K = \$110\), and where the buyer can choose whether is a call or a put in 6 months.

Solution to Exercise 17.4

The chooser option is equivalent to a call with strike 110 and maturity 15 months, plus \(e^{-0.04 (15 - 6)/12} = 0.9704\) puts with strike \(110 e^{-(0.10 - 0.04)(15 - 6)/12} = \$105.16\) and maturity 6 months. To compute the price of the call:

\[\begin{align*} d_{1} & = \frac{\ln(100/110) + (0.10 - 0.04 + \frac{1}{2} 0.30^{2}) (15/12)}{0.30 \sqrt{15/12}} = 0.1072 \\ \cdf(d_{1}) & = 0.5427 \\ d_{2} & = 0.1072 - 0.30 \sqrt{15/12} = -0.2283 \\ \cdf(d_{2}) & = 0.4097 \\ C & = 100 e^{-0.04 (15/12)} (0.5427) - 110 e^{-0.10 (15/12)} (0.4097) = \$11.85 \end{align*}\]

To compute the price of the put:

\[\begin{align*} d_{1} & = \frac{\ln(100/105.16) + (0.10 - 0.04 + \frac{1}{2} 0.30^{2}) (6/12)}{0.30 \sqrt{6/12}} = 0.0103 \\ \cdf(-d_{1}) & = 0.4959 \\ d_{2} & = 0.0103 - 0.30 \sqrt{6/12} = -0.2018 \\ \cdf(-d_{2}) & = 0.5800 \\ P & = 105.16 e^{-0.10 (6/12)} (0.5800) - 100 e^{-0.04 (6/12)} (0.4959) = \$9.41 \end{align*}\]

Therefore, the chooser option is worth \(11.85 + 0.9704 (9.41) = \$20.98\).

Exercise 17.5

Compute the price of a cash-or-nothing binary option with maturity 9 months that pays $50 whenever \(S_{T} > 100\).

Solution to Exercise 17.5

For the cash-or-nothing option we have:

\[\begin{align*} d_{1} & = \frac{\ln(100/100) + (0.10 - 0.04 + \frac{1}{2} 0.30^{2}) (9/12)}{0.30 \sqrt{9/12}} = 0.3031 \\ \cdf(d_{1}) & = 0.6191 \\ d_{2} & = 0.3031 - 0.30 \sqrt{9/12} = 0.0433 \\ \cdf(d_{2}) & = 0.5173 \end{align*}\]

Therefore, the cash-or-nothing costs

\[\begin{align*} 50 e^{-0.10 (9/12)} (0.5173) = \$23.99. \end{align*}\]

Exercise 17.6

Compute the price of an asset-or-nothing binary option with maturity 9 months that pays one unit of the stock whenever \(S_{T} > 100\).

Solution to Exercise 17.6

For the asset-or-nothing option we have:

\[\begin{align*} d_{1} & = \frac{\ln(100/100) + (0.10 - 0.04 + \frac{1}{2} 0.30^{2}) (9/12)}{0.30 \sqrt{9/12}} = 0.3031 \\ \cdf(d_{1}) & = 0.6191 \end{align*}\]

Therefore, the asset-or-nothing is worth

\[\begin{align*} 100 e^{-0.04 (9/12)} (0.6191) = \$60.08. \end{align*}\]

Exercise 17.7

Consider a non-dividend paying stock that trades for $100. Every 3-months, the stock price can increase or decrease by 10%. The risk-free rate is 10% per year with continuous compounding. Compute the price of the following path-dependent options expiring in 6 months:

1. A floating lookback put that pays \(S_{\textit{max}} - S_{T}\) at maturity.

2. A fixed lookback call that pays \(\max(S_{\textit{max}} - 100, 0)\) at maturity.

3. An average price Asian call option that pays \(\max(\bar{S} - 100, 0)\) at maturity.

4. A floating lookback call that pays \(S_{T} - S_{\textit{min}}\) at maturity.

Solution to Exercise 17.7

To price these options we need to compute the maximum, minimum and average price per path. This means that, even though the tree is recombinant, we need to specify the payoffs for each possible history or path.

Path	\(T = 0\)	\(T = 0.25\)	\(T = 0.5\)
1	100	110	121
2	100	110	99
3	100	90	99
4	100	90	81

The risk-neutral probability of the stock going up each period is

\[\begin{equation*} p = \frac{e^{0.10 \times 3/12} - 0.9}{1.1 - 0.9} = 0.6266. \end{equation*}\]

The risk-neutral probability of the stock price going up twice is \(p^{2} = 0.3926,\) whereas the risk-neutral probability of the stock price going up and then down, or down and then up is \(p (1 - p) = 0.2340.\) Finally, the risk-neutral probability of the stock price going down twice is \((1 - p)^{2} = 0.1394.\)

1. The possible payoffs at maturity of the floating lookback put are:

   Path	\(S_{\textit{max}}\)	\(S_{\textit{max}} - S_{T}\)
   1	121	0
   2	110	11
   3	100	1
   4	100	19

Therefore, the price of the floating lookback put is

\[\begin{equation*} \left( 0 p^{2} + 11 p (1 - p) + 1 (1 - p) p + 19 (1 - p)^{2} \right) e^{-0.10 \times 6/12} = \$5.19. \end{equation*}\]

2. The possible payoffs at maturity of the fixed lookback call are:

   Path	\(S_{\textit{max}}\)	\(\max(S_{\textit{max}} - 100, 0)\)
   1	121	21
   2	110	10
   3	100	0
   4	100	0

Therefore, the price of the fixed lookback call is

\[\begin{equation*} \left( 21 p^{2} + 10 p (1 - p) + 0 (1 - p) p + 0 (1 - p)^{2} \right) e^{-0.10 \times 6/12} = \$10.07. \end{equation*}\]

3. The possible payoffs at maturity of the average price Asian call are:

   Path	\(\bar{S}\)	\(\max(S_{\textit{max}} - 100, 0)\)
   1	110.33	10.33
   2	103	3
   3	96.33	0
   4	90.33	0

Therefore, the price of the average price Asian call is

\[\begin{equation*} \left( 10.33 p^{2} + 3 p (1 - p) + 0 (1 - p) p + 0 (1 - p)^{2} \right) e^{-0.10 \times 6/12} = \$4.53. \end{equation*}\]

4. The possible payoffs at maturity of the floating lookback call are:

   Path	\(S_{\textit{min}}\)	\(S_{T} - S_{\textit{min}}\)
   1	100	21
   2	99	0
   3	90	9
   4	81	0

Therefore, the price of the floating lookback put is

\[\begin{equation*} \left( 21 p^{2} + 0 p (1 - p) + 9 (1 - p) p + 0 (1 - p)^{2} \right) e^{-0.10 \times 6/12} = \$9.85. \end{equation*}\]