OM-W4

Options Trading Strategies

Objectives

• A strategy is defined as a set of options positions aimed at achieving a specific risk/return profile.

• Focus on strategies involving only European options on the same underlying asset and with the same expiration date.

• Assume the availability of a zero-coupon bond and an underlying forward of the same maturity.

• Objectives of studying these strategies include:

  • Risk Profile Analysis: For a given strategy (a list of derivative positions), determine its risk profile by evaluating the payoff of the strategy at expiry under various market conditions (different levels of underlying security prices).

  • Strategy Design: Given a targeted risk profile for a particular maturity, design a strategy using bonds, forwards, and options to meet this profile.

  • Familiarization: Gain knowledge about the most commonly used simple option strategies, including but not limited to straddles, strangles, butterfly spreads, risk reversals, and bull/bear spreads.

Put-Call Conversions

• Analysis of various combinations of calls/puts and forwards at the same strike price (K) and maturity (T) to understand the corresponding payoffs.

• Tasks:

  1. Long a call and short a forward: Compare payoff to long a put.

  2. Short a call and long a forward: Compare payoff to short a put.

  3. Long a put and long a forward: Compare payoff to long a call.

  4. Short a put and short a forward: Compare payoff to short a call.

  5. Long a call and short a put: Compare payoff to long a forward.

  6. Short a call and long a put: Compare payoff to short a forward.

Payoff Comparisons

• Example Payoff at K = 100:

  • Various payoffs with the corresponding spot prices at expiry (ST):

  • Output ranges demonstrate the correlation of payoffs with different strategies involving combinations of calls, puts, and forwards.

• Interpreted graphs illustrate the relationship and comparisons of the different strategies mentioned above.

Linkage Between Put, Call, and Forward

• The conversions highlight the following payoff parity conditions:

  • Payoff Relationships:

  • Payoff from a call − Payoff from a forward = Payoff from a put

  • Payoff from a put + Payoff from a forward = Payoff from a call

  • Payoff from a call − Payoff from a put = Payoff from a forward

  • When payoffs are equivalent, their present values also equate:

  • Put-Call Parity: $c<em>t - p</em>t = e^{-r(T-t)} (F_{t,T} - K)$

  • At a fixed strike (K) and maturity (T), knowing the prices of any two contracts out of the three ($c<em>t$, $p</em>t$, $F_{t,T}$) makes the third redundant.

Creating Forward Payoffs Using Spot and Bond

• Understand how to replicate a forward payoff without forwards by utilizing spot prices and bonds.

• Determine the payoff function of a zero-coupon bond and its role in constructing forward payoffs.

Popular Payoff Structures

• Bull Spread:

  • A bull spread can be generated using various combinations:

  • Two Calls: Long call at $K<em>1 = 90$, short call at $K</em>2 = 110$, along with a short bond with $10$ par.

  • Two Puts: Long a put at $K<em>1 = 90$, short a put at $K</em>2 = 110$, alongside a long bond with $10$ par.

  • A Call, a Put, and a Forward: Long put at $K<em>1 = 90$, short call at $K</em>2 = 110$, long forward at $K = 100$.

• Understanding who might desire such a payoff structure helps in identifying target markets and investor preferences.

Replicating Payoffs

• General Procedure using Calls, Forwards, and Bonds:

  • Progress through the payoff graph from the left (starting at $S_T = 0$) and move to each kink point sequentially. For each kink:

  1. If the payoff is $x$ dollars, long a zero-coupon bond with $x$ par value (short if $x$ is negative).

  2. Analyze slopes at these points to determine needs for calls/forwards.

  3. Continue this process until no slope changes remain.

• For Puts, Forwards, and Bonds:

  • The procedure is similar but begins from the right of the payoff graph. Different determine actions are prescribed based on positive or negative slope developments.

Examples of Various Payoff Strategies

• Bear Spread:

  • How to replicate the payoff and the minimum options required for replication. Understanding who tans to utilize a bear spread.

• Straddle:

  • Minimum options required to replicate the straddle strategy, with an exercise to gain familiarity with replication.

• Strangle:

  • Explanation on how many (at minimum) options are needed, insights into who might want this strategy, and the replication process.

• Risk Reversal:

  • Similar exploration into the replication of risk reversal structures and identification of market interests.

• Butterfly Spread:

  • Replication knowledge discussed alongside target market analysis.

Butterflies and Probabilities

• A constructed butterfly with center strike at $100$, and side strikes at $99$ and $101$ pays out $1$ if the stock price is $100$ at expiry.

• Insights into how the price of a butterfly represents risk-adjusted probability that the stock will fall within designated ranges.

• Various center strikes can be analyzed, leading to probabilities reflecting future price levels.

• Reference theoretical work by Breeden and Litzenberger (1978) and subsequent practical applications of their concepts.

Smoothing Out Kinks in Payoffs

• Examine the need for a continuum of options to replicate more complex payoffs represented by non-linear structures.

• Determine requirements based on deviations from standard payoff shapes and identify market interests.

• Discuss variance of stock prices and the relation to variance swap contracts actively traded on major stock indexes.

Formula for Replicating Payoffs

• Formula provided:
  $f(S<em>T) = f(F</em>t) ext{ (bonds)} + f'(F<em>t)(S</em>T - F<em>t) ext{ (forwards)} + \int</em>0^\infty f''(K)(K - S<em>T) dK+ \int</em>\infty f''(K)(S_T - K) dK ext{ (OTM options)}$

• This formula demonstrates the capability to replicate any terminal payoff structure using bonds, forwards, and European options.

• Notes for context include that exotic options may deal with complexities concerning path dependence and correlations.

• Memorization of the formula is not necessary; reference citation for proofs included (Carr and Madan, Quantitative Finance, 2001).

VIX—CBOE’s Volatility Index

• Definition and purpose of VIX, aimed at capturing expected annualized volatility of S&P 500 Index return over the upcoming 30 days.

• Creation method: weighted average price of 30-day S&P 500 Index options across all available strikes, weighted by a factor of $\frac{1}{K^2}$.

• Historical references for further technical details provided: Carr and Liuren Wu, "A Tale of Two Indices," Journal of Derivatives, 2006, 13(3), 13-29.