# Vanilla Options

3. Vanilla Options

In this chapter, we won’t recall basics about vanilla options but will discuss a couple of concepts useful for the understanding of more elaborated exotic derivatives.

3.1. Put-Call Parity and Synthetic Options

Put–call parity specifies a relationship between the prices of call and put options with the identical strike price K and expiry T. To derive the put–call parity relationship, we must assume that the call and put options involved are European options. Perhaps the most important feature of put–call parity is that it must be satisfied at all times, in a model independent manner. A violation of this leads to arbitrage opportunities.

Call (K,T) +KerT  = Put (K,T) + S0eqT

• Portfolio A: Synthetic. Purchase one call on an underlying asset S, struck at K and expiring at T. Sell a put on the same underlying, with the same strike price and maturity.

• Portfolio B. Purchase a forward contract that gives the obligation to buy S at a price K at maturity date T.

In all states, Portfolio A has the same payoff at maturity as Portfolio B. For European options, early exercise is not possible. If the values of these two portfolios are the same at the expiry of the options, then the present values of these portfolios must also be the same, otherwise, an investor can arbitrage and make a risk-free profit by purchasing the less expensive portfolio, selling the more expensive one and holding the long-short position to maturity. Accordingly, we have the price equality:

Call (K, T) – Put (K, T) = Forward (K, T)

Forward (K,T) = S0 e−qT − Ke−rT

3.2. Black-Scholes model

Black and Scholes developed a closed-form pricing formula for European options. The market assumptions behind their model are quite strong and contained constant volatility, constant IR, log-normality distributed stock prices (= log returns of S are normally distributed) and constant dividend yield.

3.2.1. Risk-neutral Pricing

Risk neutrality is the middle point between being risk seeking and risk averse. In finance, when pricing an asset, a common technique is to figure out the probability of a future CF, then discount that CF at the RFR. This is called the expected value, using real-world probabilities. In the theory of risk-neutral pricing, the real-world probabilities assigned to future CFs are irrelevant, and we must obtain what are known as risk-neutral probabilities.

The fundamental assumption behind risk-neutral valuation is to use a replicating portfolio of assets with known prices to remove any risk. The amounts of assets needed to hedge determine the risk-neutral probabilities. Under the aforementioned assumptions, the Black– Scholes theory considers options to be redundant in the sense that one can replicate the payoff of a European option on a stock using the stock itself and risk-free bonds. As such, the key feature of the Black–Scholes framework is that it is preference-free: since options can be replicated, their theoretical values do not depend upon investors’ risk preferences. Therefore, an option can be valued as though the return on the underlying is riskless.

The risk-neutral assumption behind the Black–Scholes model constitutes a great advantage in a trading environment.

3.3. Pricing European Options

European Call:     C = Seqt N(d1) − Kert N(d2)

European Put:      P = Kert N(−d2) − Se−qt N(−d1)

The first thing to note is that the expected rate of the return of the underlying S does not enter into this equation. In fact the relevant parameter is the risk-free rate of interest.

Plotting the Call price wrt the spot price shows an ascending convex curve.

Ascending curve --> long call --> positive delta --> long forward

Convex curve --> long call -->positive gamma

The call price is always bigger than the intrinsic value of the option. The difference is the time value, which measures the uncertainty of the option ending ITM. It is always positive for a call and reaches its maximum ATM. As time to maturity decreases, the time value decreases to be equal to zero at expiry date. A call option usually loses two-thirds of its time value during the last third of its life.

Plotting the Put price wrt the spot price shows a descending convex curve.

Descending curve à long put à negative delta à short forward

Convex curve à long put à positive gamma

The time value is not always positive for a put.

3.4. The Cost of Hedging

It is very important to remember that the price of an option should reflect the cost of hedging it.

When a trader sells an option, he charges a premium for the risks he is bearing, which depends on many parameters (IR, vol, div, repo, correl, etc.). These parameters can be used in the Black–Scholes formula to obtain the prices of vanilla options, but for more complicated payoffs we may not have closed formulas that directly reflect the cost of hedging. As we move to more complicated options we must keep in mind that the cost of an option should reflect the cost of hedging the risks it entails.

Financial products are not necessarily hedged to replication as this is typically not possible, but they are hedged to risk exposures that are tolerable. There are many risks to be understood, managed and priced into the ask and bid prices of structured products. Of course, the fewer the set of risks that are acceptable, the wider is the bid–ask spread. Hence a proposed product coming from a tailored customer request must be analysed for the risks associated with its issue. One may then determine strategies to control the risk exposure and the likely costs of doing so. The latter can then be built into the price.

3.5. American Options

American options can be exercised at any time during their life. Since investors have the freedom to exercise their American options at any point during the life of the contract, they are more valuable than European options.

3.5.1. American Calls

D/K > r * (T - t)

If this formula is verified at any time t, this means that it is optimal to exercise the American call. Intuitively, if one exercises the American call, he pays a specific amount of money to buy the underlying shares. On the one hand, he doesn’t receive interest on this cash amount; and, on the other, he would receive future dividends for holding the stocks. In other words, if the dividend yield is higher than the interest rate until maturity, it is optimal to exercise the American call. For stocks not paying dividends, it is never optimal to exercise the American call.

3.5.1. American Puts

Ultimately, it can be optimal for the holder of an American put option to choose to exercise if the interest rate that would be received on a cash deposit equal to K is higher than the dividend payments until maturity. For non-dividend-paying stocks, an American put should always be exercised when it is sufficiently deep ITM.

3.6. Asian Options

An Asian option is a derivative with a payoff at maturity date T based on the average performance of the underlying recorded at different dates against the initial date during the product life.

Since the payoff of Asian options is based on the average of the underlying asset prices during the term of the product, the uncertainty concerning the fluctuations of the underlying price at maturity decreases. Therefore, the risk exposure to the spot price and volatility is lower for an Asian option compared to a regular European option. Also, the higher the number of observations, the lower the price of the option.

Asian in: uniform averaging periods during the life of the option.

Asian out: average is computed during a specific period near the maturity date.

In the general case, averaging-in style options are less risky than averaging-out Asian options. Indeed, the uncertainty about future spot prices is lower when the average is computed periodically since inception date.

Moreover, there are many ways to compute the average of the stock returns. Indeed, Saverage can be a geometric or arithmetic average. In the case of a geometric average, it is possible to find analytical formulas for pricing Asian call and put options. This is due to the fact that the geometric average of log-normal variables is log-normally distributed in the risk-neutral world.

However, in most of the cases, the average of the underlying asset prices is arithmetic, and there are no closed formulas for pricing arithmetic average options since the arithmetic average of log-normal variables is not log-normal. In general, one will simply obtain the implied volatility of the option implied from vanilla derivatives, and perform a Monte Carlo simulation of a log-normal process using this volatility, where the paths are simulated to reflect the period over which the averaging is taking place.

3.7. Struturing Process: A way of thinking

In an environment where volatility is high, the value of a call option will also be high in comparison with a lower volatility environment, and the prices offered on such a structure will not be appealing. In cases such as these, there are more suitably fitted structures one can use to obtain equity exposure, but where the seller of the instrument is able to buy volatility. As we go along, observing the volatility position of each product is key in understanding the environment for which it is best suited, and in doing so one can know which structures would offer the most appealing deals to investors. The process described serves as a good example of combining an investor’s requirements with price constraints, and that the only real limit is the structurer’s imagination.