{"id":14670,"date":"2021-05-03T15:55:14","date_gmt":"2021-05-03T15:55:14","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=14670"},"modified":"2022-03-15T17:29:14","modified_gmt":"2022-03-15T17:29:14","slug":"arbitrage-opportunities-involving-options","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/arbitrage-opportunities-involving-options\/","title":{"rendered":"Arbitrage Opportunities Involving Options"},"content":{"rendered":"\r\n

Call Option<\/h2>\r\n

A hedging portfolio can be created by going long \\(\\phi\\) units of the underlying asset and going short the call option such that the portfolio has an initial value of:<\/p>\r\n

$$\\text{V}_{0}=\\phi\\text{S}_{0}-\\text{C}_{0}$$<\/p>\r\n

Where:<\/p>\r\n

\\(S_{0}\\)= The current stock price<\/p>\r\n

\\(c_{0}=\\) Current call value<\/p>\r\n

After a one-time-period, this portfolio will be worth:<\/p>\r\n

\\(V_{1}=\\phi\\text{S}_{0}u-\\text{c}_{u}\\), if the asset price jumps up<\/p>\r\n

or<\/p>\r\n

\\(\\text{v}_{1}=\\phi\\text{S}_{0}\\text{d}-\\text{c}_{d}\\) if the asset price jumps down<\/p>\r\n

If we equate the values of the up and down portfolios, the number of units of the underlying asset can be obtained as:<\/p>\r\n

$$\\phi=\\frac{c_{u}-c_{d}}{S_{0}\\text{u}-S_{0}d}$$<\/p>\r\n

\\(\\phi\\) is referred to as the hedge ratio.\u00a0<\/em>It is the ratio that makes the trader indifferent to the movement of the underlying asset price. An arbitrageur creates a hedged portfolio to eliminate price risk, thus satisfying Rule 2: \u201cDo not take any price risk.\u201d<\/strong><\/em><\/p>\r\n

Suppose that at time step 0, a trader borrows the present value of:<\/p>\r\n

$$-\\phi\\text{S}_{0}\\text{d}+\\text{c}_{\\text{d}}$$<\/p>\r\n

Assuming no-arbitrage, we have<\/p>\r\n

$$\\text{c}_{0}-\\phi\\text{S}_{0}=\\text{PV}(-\\phi\\text{S}_{0}\\text{d}+\\text{c}_{d})$$<\/p>\r\n

Since \\(-\\phi\\text{S}_{0}\\text{d}+\\text{c}_{\\text{d}}=-\\phi\\text{S}_{0}\\text{u}+\\text{c}_{u}\\)<\/p>\r\n

This can also be expressed as:<\/p>\r\n

$$c_{0}-\\phi\\text{S}_{0}=\\text{PV}(-\\phi\\text{S}_{0}\\text{u}+\\text{c}_{\\text{u}})$$<\/p>\r\n

The no-arbitrage single-period<\/strong><\/em> valuation approach leads to the following single-step call option valuation equation for call options:<\/p>\r\n

$$\\text{c}_{0}-\\phi\\text{S}_{0}=\\text{PV}(-\\phi\\text{S}_{0}\\text{d}+\\text{c}_{d})$$<\/p>\r\n

$$c_{0}-\\phi\\text{S}_{0}=\\text{PV}(-\\phi\\text{S}_{0}\\text{u}+\\text{c}_{\\text{u}})$$<\/p>\r\n

Thus, a call option is similar to owning \u00a0units of the underlying asset and borrowing \\(\\text{PV}(-\\phi\\text{S}_{0}\\text{d}+\\text{c}_{\\text{d}})\\).\u00a0This makes the transaction completely arbitrage-free, hence satisfying Rule 1: \u201cDo not use own money.\u201d<\/strong><\/em> Moreover, we can view a call option as a leveraged position in the underlying asset.<\/p>\r\n

We can use the idea that a hedged portfolio returns the risk-free rate to determine the initial value of a call or a put option.<\/p>\r\n

Example: Value of a Call Option<\/h3>\r\n

Consider a one-period binomial model of a non-dividend-paying stock whose current price is $20. Suppose that:<\/p>\r\n

    \r\n\t
  • Over the single period under consideration, the stock price can either jump up to $25 or down to $16<\/li>\r\n\t
  • The continuously compounded risk-free rate of return is 4% per period<\/li>\r\n<\/ul>\r\n

    The current value of a one-period European call option that has an exercise price of $20 is closest to<\/em>:<\/p>\r\n

    Solution<\/h4>\r\n

    The binomial tree in respect of the stock price is as follows:<\/p>\r\n\r\n\r\n\r\n

    \"ExampleSimilarly, consider a corresponding binomial tree with respect to the payoff provided by the call option at time 1, I.e., \u00a0<\/em>the profit paid at exercise:<\/p>\r\n\r\n\r\n\r\n

    \"ExampleWe can determine \\(c_{0}\\) by using the single-period call option valuation equation as follows:<\/p>\r\n

    $$\\text{c}_{0}-\\phi\\text{S}_{0}=\\text{PV}(-\\phi\\text{S}_{\\text{d}}+\\text{c}_{d})$$<\/p>\r\n

    $$\\begin{align*}\\phi&=\\frac{c_{u}-c_{d}}{S_{0}u-S_{0}d}\\\\&=\\frac{5-0}{25-16}\\\\&=0.56\\end{align*}$$<\/p>\r\n

    Thus,<\/p>\r\n

    $$c_{0}=0.56\\times20+e^{-0.04}[-0.56\\times16+0]=$2.59$$<\/p>\r\n

    This implies that buying a call option for $2.59 is equivalent to buying 0.56 units of the underlying stock for $11.20 and lending $8.61 such that the effective payment is $2.59<\/p>\r\n

    Put Options<\/h2>\r\n

    The no-arbitrage single period valuation equation for put options is expressed as:<\/p>\r\n

    $$\\text{p}=\\phi\\text{S}_{0}+\\text{PV}(-\\phi\\text{S}_{0}\\text{d}+\\text{p}_{\\text{d}})$$<\/p>\r\n

    Equivalently,<\/p>\r\n

    $$\\text{p}=\\phi\\text{S}_{0}+\\text{PV}(-\\phi\\text{S}_{0}\\text{u}+\\text{p}_{\\text{u}})$$<\/p>\r\n

    Where the hedge ratio, \\(\\phi\\) is given as:<\/p>\r\n

    $$\\phi=\\frac{p_{u}-p_{d}}{S_{0}u-S_{0}d}\\leq0$$<\/p>\r\n

    Note that the hedge ratio, in this case, will be negative as \\(p_{u}\\) is less than \\(p_{d}\\). Thus, the arbitrageur should short-sell the underlying and lend a portion of the proceeds to replicate a long-put position.<\/p>\r\n

    Therefore, a put option can be viewed as equivalent to shorting the underlying asset and lending \\(\\text{PV}(-\\phi\\text{S}_{\\text{u}}+\\text{p}_{\\text{u}})\\).<\/p>\r\n

    Example: Value of a Put Option<\/h3>\r\n

    Consider a one-period binomial model of a non-dividend-paying stock whose current price is $20. Suppose that:<\/p>\r\n

      \r\n\t
    • Over the single period under consideration, the stock price can either jump up to $25 or down to $16<\/li>\r\n\t
    • The continuously compounded risk-free rate of return is 4% per period<\/li>\r\n<\/ul>\r\n

      The current value of a one-period European put option that has an exercise price of $20 is closest to<\/em>:<\/p>\r\n

      Solution<\/h4>\r\n

      The payoff provided by the put option at time one is represented in the following binomial tree:<\/p>\r\n\r\n\r\n\r\n

      \"Example$$p=\\phi\\text{S}_{0}+\\text{PV}(-\\phi\\text{S}_{0}\\text{u}+\\text{p}_{u})$$<\/p>\r\n\r\n\r\n

      Where:<\/p>\r\n\r\n\r\n

      $$\\begin{align*}\\phi&=\\frac{p_{u}-p_{d}}{S_{0}u-S_{0}d}\\leq0\\\\&=\\frac{0-4}{25-16}\\\\&=-0.44\\end{align*}$$<\/p>\r\n\r\n\r\n

      $$p=-0.44\\times e^{-0.04}(-{-}0.44\\times25+0)=$1.77$$<\/p>\r\n\r\n\r\n

      Notice that buying a put option for $1.77 is equivalent to short selling 0.44 units of the underlying stock for $8.80 and lending $10.57.<\/p>\r\n\r\n\r\n

      Exploiting Arbitrage Opportunities<\/h2>\r\n\r\n\r\n

      Consider a one-period binomial model of a non-dividend-paying stock whose current price is $20. Suppose that:<\/p>\r\n\r\n\r\n

        \r\n\t
      • Over the single period under consideration, the stock price can either jump up to $25 or down to $16<\/li>\r\n\t
      • The continuously compounded risk-free rate of return is 4% per period<\/li>\r\n<\/ul>\r\n\r\n\r\n

        In the previous section, we determined the current value of this call option as $2.59 given a strike price of $20.<\/p>\r\n\r\n\r\n

        Now, assume that the call option has a market price of $4.50. Assuming that we trade 1,000 call options, we can illustrate how this opportunity can be exploited to earn an arbitrage profit.<\/p>\r\n\r\n\r\n

        Since the call option is overpriced, we will sell 1,000 call options and buy several shares of the underlying determined by the hedge ratio.<\/p>\r\n\r\n\r\n

        $$\\begin{align*}\\phi&=\\frac{c_{u}-c_{d}}{S_{0}u-S_{0}d}\\\\&=\\frac{$5-$0}{$25-$16}\\\\&=\\frac{5}{9} \\text{shares per option}\\end{align*}$$<\/p>\r\n\r\n\r\n

        Thus, we will purchase \\(1,000\\times\\frac{5}{9}=\\text{555.5555 shares}\\)<\/p>\r\n\r\n\r\n

        The net cost of a portfolio with 555.55 shares of the stock held long at $20 per share and 1,000 calls held short at $4.50 is:<\/p>\r\n

        $$\\text{Net cost of the portfolio}=(555.55\\times$20)-(1,000\\times$4.50)\\approx$6,611$$<\/p>\r\n

        Assume that we begin with $0<\/p>\r\n

        Then borrow $6,611 at 4%<\/p>\r\n

        At the end of the one-time period, we repay the loan of \\($6,611\\times1.04\\approx$6,875\\)<\/p>\r\n

        The portfolio value will be the same at maturity regardless of whether the stock price moves up to $25 or down to $16.<\/p>\r\n

        Value of the portfolio after stock price moves up:<\/p>\r\n

        $$V_{u}=(555.55\\times$25)-(1,000\\times$5)\\approx$8,889$$<\/p>\r\n

        Value of the portfolio after stock price moves down:<\/p>\r\n

        $$V_{d}=(555.55\\times$16)-(1,000\\times$0)\\approx$8,889$$<\/p>\r\n

        Arbitrage profit on this portfolio at the end of one year if the price moves up or down after repayment of loan \\(=$8,889-$6,875=$2,014\\)<\/p>\r\n

        The discounted value of the arbitrage profit is thus:<\/p>\r\n

        $$\\text{PV (Arbitrage profit)}=\\frac{$2,014}{1.04}=$1,936.54$$<\/p>\r\n

        Question<\/h2>\r\n

        Consider a non-dividend-paying stock with a current price of $50 and an exercise price of $50. The stock price can be modeled by assuming that it will either increase by 12% or decrease by 10% each year, independent of the price movement in other years. A trader constructs a portfolio consisting of 100 call options. If the call option is overpriced, the portfolio that leads to an arbitrage profit is most likely:<\/em><\/p>\r\n

          \r\n\t
        1. Buy 100 call options, short 54.55 shares<\/li>\r\n\t
        2. Buy 100 call options, short 45.45 shares<\/li>\r\n\t
        3. Sell 100 call options, buy 54.55 shares<\/li>\r\n<\/ol>\r\n

          Solution<\/h3>\r\n

          The correct answer is C:<\/strong>\u00a0<\/strong><\/p>\r\n

          u=1.12<\/p>\r\n

          d=0.90<\/p>\r\n

          We can represent the above information in the following binomial tree<\/p>\r\n

          \"ExampleThe call payoffs are as follows:<\/p>\r\n

          $$c_{u}=max($56-$50,0)=$6$$<\/p>\r\n

          $$c_{d}=max($45-50,0)=$0$$<\/p>\r\n

          Since the option is overpriced, the trader will sell 100 call options and purchase several shares determined by the hedge ratio:<\/p>\r\n

          $$\\begin{align*}\\phi&=\\frac{c_{u}-c_{d}}{S_{0}u-S_{0}d}\\\\&=\\frac{$6-$0}{$56-$45}\\\\&=0.5455 \\text{shares per option}\\end{align*}$$<\/p>\r\n

          $$\\text{Total number of shares to purchase}=100\\times0.5455=54.55$$<\/p>\r\n

          Buying 54.55 shares of stock will produce a riskless hedge. The payoff at expiry will return more than the risk-free rate on the hedge portfolio\u2019s net cost. Borrowing to finance the hedge portfolio and earning a higher rate than the borrowing rate produces riskless profits.<\/p>\r\n

          Reading 38: Valuation of Contingent Claims<\/em><\/p>\r\n

          LOS 38 (C) identify an arbitrage opportunity involving options and describe the related arbitrage<\/em><\/p>\r\n","protected":false},"excerpt":{"rendered":"

          Call Option A hedging portfolio can be created by going long \\(\\phi\\) units of the underlying asset and going short the call option such that the portfolio has an initial value of: $$\\text{V}_{0}=\\phi\\text{S}_{0}-\\text{C}_{0}$$ Where: \\(S_{0}\\)= The current stock price \\(c_{0}=\\)…<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[102,302],"tags":[320,321,216,304,323,322],"class_list":["post-14670","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-derivatives","tag-arbitrage-opportunities-involving-options","tag-call-option","tag-cfa-level-2","tag-derivatives","tag-exploiting-arbitrage-opportunities","tag-put-option","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"\nArbitrage Opportunities Involving Options - CFA, FRM, and Actuarial Exams Study Notes<\/title>\n<meta name=\"description\" content=\"\u03d5 is referred to as the hedge ratio. 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