6.3. Option Portfolios and Exotic Options

In addition to the pricing and properties of vanilla European and American options, you may be expected to be familiar with the construction and payoff of basic option-based trading strategies—covered call, protective put, bull/bear spread, butterfly spread, straddle, etc. Furthermore, if you are applying for a derivatives-related position, you 159 should also have a good understanding of pricing and hedging of some of the common exotic derivatives—binary option, barrier option, Asian option, chooser option, etc.

Bull spread

What are the price boundaries for a bull call spread?

Solution: A bull call spread is a portfolio with two options: long a call $c_1$ with strike $K_1$ and short a call $c_2$ with strike $K_2$ ($K_1<K_2$). The cash flow of a bull spread is summarized in table 6.3.

Since $K_1<K_2$, the initial cash flow is negative. Considering that the final payoff is bounded by $K_2-K_1$, the price of the spread, $c_1-c_2$, is bounded by $e^{-rT}(K_2-K_1)$. Besides, the payoff is also bounded by $\dfrac{K_2-K_1}{K_2}S_T$, so the price is also bounded by $\dfrac{K_2-K_1}{K_2}S$.

Straddle

Explain what a straddle is and when you want to purchase a straddle.

Solution: A straddle includes long positions in both a call option and a put option with the same strike price $K$ and maturity date $T$ on the same stock. The payoff of a long straddle is $|S_T-K|$. So a straddle may be used to bet on large stock price moves. In practice, a straddle is also used as a trading strategy for making bets on volatility. If an investor believes that the realized (future) volatility should be much higher than the implied volatility of call and put options, he or she will purchase a straddle. For example, 160 the value of an at-the-money call or put is almost a linear function of volatility. If the investor purchases an at-the-money straddle, both the call and the put options have the price $c\approx p\approx0.4\sigma_iS\sqrt\tau$, where $\sigma_i$ is the implied volatility. If the realized volatility $\sigma_r>\sigma_i$, both options are undervalued. When the market prices converge to the prices with the realized volatility, both the call and the put will become more valuable.

Although initially a straddle with an at-the-money call and an at-the-money put ($K=S$) has a delta close to 0, as the stock price moves away from the strike price, the delta is no longer close to 0 and the investor is exposed to stock price movements. So a straddle is not a pure bet on stock volatility. For a pure bet on volatility, it is better to use volatility swaps or variance swaps.¹¹ For example, a variance swap pays $N\times(\sigma_r^2-K_{\text{var}})$, where $N$ is the notional value, $\sigma_r^2$ is the realized variance and $K_{\text{var}}$ is the strike for the variance.

Binary options

What is the price of a binary (cash-or-nothing digital) European call option on a non-dividend paying stock if the stock price follows a geometric Brownian motion? How would you hedge a cash-or-nothing call option and what’s the limitation of your hedging strategy?

Solution: A cash-or-nothing call option with strike price $K$ pays $1 if the asset price is above the strike price at the maturity date, otherwise it pays nothing. The price of the option is $c_B=e^{-r\tau}N(d_2)$ if the underlying asset is a non-dividend paying stock. As we have discussed in the derivation of the Black-Scholes formula, $N(d_2)$ is the probability that a vanilla call option finishes in the money under the risk-neutral measure. So its discounted value is $e^{-r\tau}N(d_2)$.

Theoretically, a cash-or-nothing call option can be hedged using the standard delta hedging strategy. Since $\Delta=\dfrac{\partial c_B}{\partial S}=e^{-r\tau}N'(d_2)\dfrac1{S\sigma\sqrt\tau}$, a long position in a cash-or-nothing call option can be hedged by shorting $e^{-r\tau}N'(d_2)\dfrac1{S\sigma\sqrt\tau}$ shares (and a risk-free money market position). Such a hedge works well when the difference between $S$ and $K$ is large and $\tau$ is not close to 0. But when the option is approaching maturity $T$ ($\tau\to0$) 161 and the stock price $S$ is close to $K,\ \Delta$ is extremely volatile¹² and small changes in the stock price cause very large changes in $\Delta$. In these cases, it is practically impossible to hedge a cash-or-nothing call option by delta hedging.

We can also approximate a digital option using a bull spread with two calls. If call options are available for all strike prices and there are no transaction costs, we can long $1/2\varepsilon$ call options with strike price $K-\varepsilon$ and short $1/2\varepsilon$ call options with strike price $K+\varepsilon$. The payoff of the bull spread is the same as the digital call option if $S_T\le K-\varepsilon$ (both have payoff 0) or $S_T\ge K+\varepsilon$ (both have payoff $1). When $K-\varepsilon<S_T<K+\varepsilon$, their payoffs are different. Nevertheless, if we set $\varepsilon\to0$, such a strategy will exactly replicate the digital call. So it provides another way of hedging a digital call option. This hedging strategy suffers its own drawback. In practice, not all strike prices are traded in the market. Even if all strike prices were traded in the market, the number of options needed for hedging, $1/2\varepsilon$, will be large in order to keep $\varepsilon$ small.

Exchange options

How would you price an exchange call option that pays $\max(S_{T,1}-S_{T,2},0)$ at maturity. Assume that $S_1$ and $S_2$ are non-dividend paying stocks and both follow geometric Brownian motions with correlation $\rho$.

Solution: The solution to this problem uses change of numeraire. Numeraire means a unit of measurement. When we express the price of an asset, we usually use the local currency as the numeraire. But for modeling purposes, it is often easier to use a different asset as the numeraire. The only requirement for a numeraire is that it must always be positive.

The payoff of the exchange option depends on both $S_{T,1}$ (price of $S_1$ at maturity date $T$) and $S_{T,2}$ (price of $S_2$ at $T$), so it appears that we need two geometric Brownian motions:

Yet if we use $S_1$ as the numeraire, we can convert the problem to just one geometric Brownian motion. The final payoff is $\max(S_{T,2}-S_{T,1},0)=S_{T,1}\max\left(\dfrac{S_{T,2}}{S_{T,1}}-1,0\right)$. When 162 $S_1$ and $S_2$ are geometrical Browian motions, $f=\dfrac{S_2}{S_1}$ is a geometric Brownian motion as well. One intuitive explanation is that both $\ln S_1$ and $\ln S_2$ follow normal distributions, so $\ln f=\ln S_2-\ln S_1$ follows a normal distribution as well and $f$ follows a lognormal distribution. More rigorously, we can apply the Ito’s lemma to $f=\dfrac{S_2}{S_1}$:

To make $f=\dfrac{S_2}{S_1}$ a martingale, set $\mu_2-\mu_1+\sigma_1^2-\rho\sigma_1\sigma_2=0$ and we have $\widetilde E\left[\dfrac{S_{T,2}}{S_{T,1}}\right]=\dfrac{S_2}{S_1}$, and $\dfrac{S_{t,2}}{S_{t,1}}$ is a martingale under the new measure. The value of the exchange option using $S_1$ as the numeraire is $C_s=\widetilde E\left[\max\left(\dfrac{S_{T,2}}{S_{T,1}}-1,0\right)\right]$, which is just the value of a call option with underlying asset price $S=\dfrac{S_2}{S_1}$, strike price $K=1$, interest rate $r=0$, and volatility $\sigma_s=\sqrt{\sigma_1^2-2\rho\sigma_1\sigma_2+\sigma_2^2}$. So its value is $C_s=\dfrac{S_2}{S_1}N(d_1)-N(d_2)$, where $d_1=\dfrac{\ln(S_2/S_1)+0.5\sigma_s^2\tau}{\sigma_s\sqrt\tau}$ and $d_2=d_1-\sigma\sqrt\tau$. The payoff of the exchange option expressed in local currency is $S_1C_s=S_2N(d_1)-S_1N(d_2)$.