Option Pricing: Unit 2 - Self Check
An investor believes that there will be a big jump in a stock price, but is uncertain as to the direction. Identify four different strategies the investor can follow
- Strangle - Straddle - Strip - Strap - Reverse calendar spread - Reverse butterfly spread
A call option with a strike price of $50 costs $2. A put option with a strike price of $45 costs $3. Explain how a strangle can be created from these two options. What is the pattern of profits from the strangle?
A strangle is created by buying both options. The pattern of profits is as follows: Stock Price, ST Profit ST <45 (45 − ST)−5 45 < ST < 50 −5 ST > 50 (ST −50)−5
The price of a non-dividend paying stock is $19 and the price of a three-month European call option on the stock with a strike price of $20 is $1. The risk-free rate is 4% per year. What is the price of a three-month European put option with a strike price of $20?
In this case, c = 1, T = 0.25, S0= 19, K = 20, and r = 0.04. From put-call parity p = c + Ke -rT - S0 or p= 1+20e -0.004×0.25 −19 =1.80 so that the European put price is $1.80. https://www.chegg.com/homework-help/questions-and-answers/3-price-non-dividend-paying-stock-19-price-three-month-european-call-option-stock-strike-p-q32355699?trackid=8e3e98de0288&strackid=b35132a2ae46 Good response to this question
A stock price is $29. An investor buys one call option contract on the stock with a strike price of $30 and sells a call option contract on the stock with a strike price of $32.50. The market prices of the options are $2.75 and $1.50, respectively. The options have the same maturity date. What kind of strategy is this?
A Bull Spread
"A box spread comprises four options. Two can be combined to create a long forward position and two can be combined to create a short forward position." Explain this statement
A box spread is a bull spread created using calls and a bear spread created using puts. With the notation in the text it consists of a) a long call with strike K1 b) a short call with strike K2 c) a long put with strike K2 d) a short put with strike K1 a) and d) give a long forward contract with delivery price K1 b) and c) give a short forward contract with delivery price K2 . The two forward contracts taken together give the payoff of K2 - K1
When is it appropriate for an investor to purchase a butterfly spread?
A butterfly spread involves a position in options with three different strike prices (K1,K2 and K3 ). A butterfly spread should be purchased when the investor considers that the price of the underlying stock is likely to stay close to the central strike price, K2 .
A call with a strike price of $60 costs $6. A put with the same strike price and expiration date costs $4. Construct a table that shows the profit from a straddle. For what range of stock prices would the straddle lead to a loss?
A straddle is created by buying both the call and the put. This strategy costs $10. The profit/loss is shown in the following table: Stock Price Payoff Profit ST > 60 ST −60 ST − 70 ST ≤ 60 60 −ST 50 − ST This shows that the straddle will lead to a loss if the final stock price is between $50 and $70.
Calculate the price of a three-month European put option on a non-dividend-paying stock with a strike price of $50 when the current stock price is $50, the risk-free interest rate is 10% per annum, and the volatility is 30% per annum.
In this case S0=50, K=50, r=0.1, σ =0.3, T=0.25, and D1 = 0.2417 D2 = 0.0917 The European put price is 50N(-0.0917)e -0.1×0.25−50N(−0.2417) =50×0.4634e -0.1×0.25−50×0.4045 =2.37 or $2.37.
What difference does it make to your calculations in Problem 29 if a dividend of $1.50 is expected in two months?
In this case we must subtract the present value of the dividend from the stock price before using Black-Scholes-Merton. Hence the appropriate value of S0 is S0 = 50−1.50e -0.1667×0.1=48.52 As before K=50, r=0.1, σ=0.3, and T=0.25. In this case solution D1 = 0.0414 D2 = -0.1086 The European put price is 50N(0.1086)e -0.1×0.25−48.52N(−0.0414) =50×0.5432e -0.1×0.25−48.52×0.4835 =3.03 or $3.03. On Excel Sheet
Give an intuitive explanation of why the early exercise of an American put becomes more attractive as the risk-free rate increases and volatility decreases.
The early exercise of an American put is attractive when the interest earned on the strike price is greater than the insurance element lost. When interest rates increase, the value of the interest earned on the strike price increases making early exercise more attractive. When volatility decreases, the insurance element is less valuable. Again this makes early exercise more attractive.
A trader sells a strangle by selling a call option with a strike price of $50 for $3 and selling a put option with a strike price of $40 for $4. For what range of prices of the underlying asset does the trader make a profit?
The trader makes a profit if the total payoff is less than $7. This happens when the price of the asset is between $33 and $57.
Explain two ways in which a bear spread can be created.
A bear spread can be created using two call options with the same maturity and different strike prices. The investor shorts the call option with the lower strike price and buys the call option with the higher strike price. A bear spread can also be created using two put options with the same maturity and different strike prices. In this case, the investor shorts the put option with the lower strike price and buys the put option with the higher strike price.
Suppose that put options on a stock with strike prices $30 and $35 cost $4 and $7, respectively. How can the options be used to create (a) a bull spread and (b) a bear spread? Construct a table that shows the profit and payoff for both spreads.
A bull spread is created by buying the $30 put and selling the $35 put. This strategy gives rise to an initial cash inflow of $3. The outcome is as follows: Stock Price Payoff Profit ST ≥ 35 0 3 30 ≤ ST < 35 ST −35 ST − 32 ST < 30 -5 -2 A bear spread is created by selling the $30 put and buying the $35 put. This strategy costs $3 initially. The outcome is as follows: Stock Price Payoff Profit ST ≥ 35 0 -3 30 ≤ ST < 35 35 − ST 32−ST ST < 30 5 2
What is meant by a protective put? What position in call options is equivalent to a protective put?
A protective put consists of a long position in a put option combined with a long position in the underlying shares. It is equivalent to a long position in a call option plus a certain amount of cash. This follows from put-call parity: p+S0 = c+Ke -rT +D
Explain how an aggressive bear spread can be created using put options.
An aggressive bull spread using call options is discussed in the text. Both of the options used have relatively high strike prices. Similarly, an aggressive bear spread can be created using put options. Both of the options should be out of the money (that is, they should have relatively low strike prices). The spread then costs very little to set up because both of the puts are worth close to zero. In most circumstances the spread will provide zero payoff. However, there is a small chance that the stock price will fall fast so that on expiration both options will be in the money. The spread then provides a payoff equal to the difference between the two strike prices, K2−K1.
Call options on a stock are available with strike prices of $15, $17.50, and $20 and expiration dates in three months. Their prices are $4, $2, and, $0.50 respectively. Explain how the options can be used to create a butterfly spread. Construct a table showing how profit varies with stock price for the butterfly spread.
An investor can create a butterfly spread by buying call options with strike prices of $15 and $20 and selling two call options with strike prices of $17 ½. The initial investment is 4 + ½ - 2×2 = $½. The following table shows the variation of profit with the final stock price: Stock Price, ST Profit ST < 15 -½ 15 < ST < 17½ (ST −15) −½ 17½ < ST <20 (20 - ST) -½ ST > 20 -½
A stock price is currently $40. It is known that at the end of three months it will be either $45 or $35. The risk-free rate of interest with quarterly compounding is 8% per annum. Calculate the value of a three-month European put option on the stock with an exercise price of $40. Verify that no-arbitrage arguments and risk-neutral valuation arguments give the same answers.
At the end of three months the value of the option is either $5 (if the stock price is $35) or $0 (if the stock price is $45). Consider a portfolio consisting of: -Δ: shares +1: option (Note: The delta, Δ, of a put option is negative. We have constructed the portfolio so that it is +1 option and -Δ shares rather than -1 option and +Δ shares so that the initial investment is positive.) The value of the portfolio is either -35Δ+5 or -45Δ. If: -35Δ+5 = -45Δ i.e., Δ = -0.5 the value of the portfolio is certain to be 22.5. For this value of Δ the portfolio is therefore riskless. The current value of the portfolio is -40Δ + ƒ where ƒ is the value of the option. Since the portfolio must earn the risk-free rate of interest (40×0.5+ƒ) ×1.02 = 22.5 Hence ƒ=2.06 i.e., the value of the option is $2.06. This can also be calculated using risk-neutral valuation. Suppose that p is the probability of an upward stock price movement in a risk-neutral world. We must have 45p + 35(1−p) = 40×1.02 i.e., 10p =5.8 or p=0.58 The expected value of the option in a risk-neutral world is: 0×0.58+5×0.42=2.10 This has a present value of 2.10 =2.06 1.02 On Excel Sheet
What is the difference between a strangle and a straddle?
Both a straddle and a strangle are created by combining a long position in a call with a long position in a put. In a straddle the two have the same strike price and expiration date. In a strangle they have different strike prices and the same expiration date.
A stock price is currently $50. It is known that at the end of six months it will be either $45 or $55. The risk-free interest rate is 10% per annum with continuous compounding. What is the value of a six-month European put option with a strike price of $50?
Consider a portfolio consisting of −1: Put option +Δ: Shares If the stock price rises to $55, this is worth 55Δ. If the stock price falls to $45, the portfolio is worth 45Δ−5. These are the same when 45Δ−5 = 55Δ or Δ = -0.50. The value of the portfolio in six months is −27.5 for both stock prices. Its value today must be the present value of −27.5, or −27.5e -0.1×0.5 = −26.16. This means that −ƒ + 50Δ = −26.16 where ƒ is the put price. Because Δ = −0.50, the put price is $1.16. As an alternative approach we can calculate the probability, p , of an up movement in a risk-neutral world. This must satisfy: 55p + 45(1 −p) = 50e 0.1×0.5 so that 10p = 50e 0.1×0.5 - 45 or p = 0.7564. The value of the option is then its expected payoff discounted at the risk-free rate: [0×0.7564+5×0.2436]e -0.1×0.5 =1.16 or $1.16. This agrees with the previous calculation.
For the situation considered in Problem 23, what is the value of a one-year European put option with a strike price of $100? Verify that the European call and European put prices satisfy put-call parity.
Figure 2 shows how we can value the put option using the same tree as in Problem 23. The value of the option is $1.92. The option value can also be calculated directly from equation (13.10): e -2×0.008×0.5[0.70412×0+2×0.7041×0.2959×1+0.29592×19]=1.92 or $1.92. The stock price plus the put price is 100 + 1.92 = $101.92. The present value of the strike price plus the call price is 100e -0.08×1 +9.61 = $101.92. These are the same, verifying that put-call parity holds.
A European call option and put option on a stock both have a strike price of $20 and an expiration date in three months. Both sell for $3. The risk-free interest rate is 10% per annum, the current stock price is $19, and a $1 dividend is expected in one month. Identify the arbitrage opportunity open to a trader.
If the call is worth $3, put-call parity shows that the put should be worth 3+20e -0.10×3/12 + e -0.1×1/12 −19 = 4.50 This is greater than $3. The put is therefore undervalued relative to the call. The correct arbitrage strategy is to buy the put, buy the stock, and short the call. This costs $19. If the stock price in three months is greater than $20, the call is exercised. If it is less than $20, the put is exercised. In either case the arbitrageur sells the stock for $20 and collects the $1 dividend in one month. The present value of the gain to the arbitrageur is -3−19+3+20e -0.10×3/12 + e -0.1×1/12 = 1.50
A call option on a non-dividend-paying stock has a market price of $2.50. The stock price is $15, the exercise price is $13, the time to maturity is three months, and the risk-free interest rate is 5% per annum. What is the implied volatility?
In the case c = 2.5, S0 = 15, K = 13, T = 0.25, r = 0.05. The implied volatility must be calculated using an iterative procedure. A volatility of 0.2 (or 20% per annum) gives c = 2.20, A volatility of 0.3 gives c =2.32. A volatility of 0.4 gives c = 2.507. A volatility of 0.39 gives c = 2.487. By interpolation the implied volatility is about 0.396 or 39.6% per annum.
Explain the no-arbitrage and risk-neutral valuation approaches to valuing a European option using a one-step binomial tree.
In the no-arbitrage approach, we set up a riskless portfolio consisting of a position in the option and a position in the stock. By setting the return on the portfolio equal to the risk-free interest rate, we are able to value the option. When we use risk-neutral valuation, we first choose probabilities for the branches of the tree so that the expected return on the stock equals the risk-free interest rate. We then value the option by calculating its expected payoff and discounting this expected payoff at the risk-free interest rate.
Calls were traded on exchanges before puts. During the period of time when calls were traded but puts were not traded, how would you create a European put option on a non-dividend-paying stock synthetically?
Put-call parity can be used to create a put from a call. A put plus the stock equals a call plus the present value of the strike price when both the call and the put have the same strike price and maturity date. A put can be created by buying the call, shorting the stock, and keeping an amount of cash that when invested at the risk-free rate will grow to be sufficient to exercise the call. If the stock price is above the strike price, the call is exercised and the short position is closed out for no net payoff. If the stock price is below the strike price, the call is not exercised and the short position is closed out for a gain equal to the put payoff.
What is the price of a European call option on a non-dividend-paying stock when the stock price is $52, the strike price is $50, the risk-free interest rate is 12% per annum, the volatility is 30% per annum, and the time to maturity is three months?
So = 52 K - 50 r = 0.12 rho = 30 T = 0.25 d1 = [ln(52/50) + (0.12+0.3^2/2)*0.25]/(0.3 * sqrt(0.25)) = 0.5365 d2 = d1 - 0.3*sqrt(0.25) = 0.3865 price of European call = 52 * N(0.5365) - 50 * e^(-0.23*0.25) * N(0.3865) = 52 * 0.7042 - 50 * e^-0.03 * 0.6504 = 5.06 On Excel Sheet
A stock index is currently 1,500. Its volatility is 18%. The risk-free rate is 4% per annum (continuously compounded) for all maturities and the dividend yield on the index is 2.5%. Calculate values for u, d, and p when a six-month time step is used. What is the value a 12-month American put option with a strike price of 1,480 given by a two-step binomial tree?
The option is exercised at the lower node at the six-month point. It is worth $78.41. On Excel Sheet
What is meant by the delta of a stock option?
The delta of a stock option measures the sensitivity of the option price to the price of the stock when small changes are considered. Specifically, it is the ratio of the change in the price of the stock option to the change in the price of the underlying stock.
A trader buys a call option with a strike price of $45 and a put option with a strike price of $40. Both options have the same maturity. The call costs $3 and the put costs $4. Draw a diagram showing the variation of the trader's profit with the asset price. What type of trading strategy is this?
The figure below shows the variation of the trader's position with the asset price. We can divide the alternative asset prices into three ranges: a) When the asset price is less than $40, the put option provides a payoff of 40− ST and the call option provides no payoff. The options cost $7 and so the total profit is 33− ST . b) When the asset price is between $40 and $45, neither option provides a payoff. There is a net loss of $7. c) When the asset price is greater than $45, the call option provides a payoff of ST − 45 and the put option provides no payoff. Taking into account the $7 cost of the options, the total profit is ST − 52. The trader makes a profit (ignoring the time value of money) if the stock price is less than $33 or greater than $52. This type of trading strategy is known as a strangle\ *Graph has Put looks like normal hockey stick Call is backwards hockey stick and the total is a V with a plateau around -7*
Explain the principle of risk-neutral valuation.
The price of an option or other derivative when expressed in terms of the price of the underlying stock is independent of risk preferences. Options therefore have the same value in a risk-neutral world as they do in the real world. We may therefore assume that the world is risk neutral for the purposes of valuing options. This simplifies the analysis. In a risk-neutral world all securities have an expected return equal to risk-free interest rate. Also, in a risk-neutral world, the appropriate discount rate to use for expected future cash flows is the risk-free interest rate.
What is the result if the strike price of the put is higher than the strike price of the call in a strangle?
The result is shown in the figure below. The profit pattern from a long position in a call and a put when the put has a higher strike price than a call is much the same as when the call has a higher strike price than the put. Both the initial investment and the final payoff are much higher in the first case.
An investor believes that there will be a big jump in a stock price, but is uncertain as to the direction. Identify four different strategies the investor can follow & explain the differences
The strategies all provide positive profits when there are large stock price moves. A strangle is less expensive than a straddle, but requires a bigger move in the stock price in order to provide a positive profit. Strips and straps are more expensive than straddles but provide bigger profits in certain circumstances. A strip will provide a bigger profit when there is a large downward stock price move. A strap will provide a bigger profit when there is a large upward stock price move. In the case of strangles, straddles, strips and straps, the profit increases as the size of the stock price movement increases. By contrast in a reverse calendar spread and a reverse butterfly spread there is a maximum potential profit regardless of the size of the stock price movement
The volatility of a non-dividend-paying stock whose price is $78, is 30%. The risk-free rate is 3% per annum (continuously compounded) for all maturities. Calculate values for u, d, and p when a two-month time step is used. What is the value of a four-month European call option with a strike price of $80 given by a two-step binomial tree. Suppose a trader sells 1,000 options (10 contracts). What position in the stock is necessary to hedge the trader's position at the time of the trade?
The value of the option is $4.67. The initial delta is 9.58/(88.16 - 69.01) which is almost exactly 0.5 so that 500 shares should be purchased. On Excel Sheet
Section 11.1, of Hull's text, gives an example of a situation where the value of a European call option decreases with the time to maturity. Give an example of a situation where the same thing happens for a European put option.
There are some circumstances when it is optimal to exercise an European put option early. One such situation is when it is deep in the money with high interest rates. In this situation it is better to have a short-life European option than a long life European option. The strike price is almost certain to be received and the earlier this happens, the better.
The price of a European call that expires in six months and has a strike price of $30 is $2. The underlying stock price is $29, and a dividend of $0.50 is expected in two months and again in five months. Interest rates (all maturities) are 10%. What is the price of a European put option that expires in six months and has a strike price of $30?
Using the notation in the chapter, put-call parity [equation (11.10)] gives c + Ke -rT + D = p + S0 or p = c + Ke -rT + D − S0 In this case p = 2+30e -0.1×6.12 + (0.5e -0.1×2/12+ 0.5e -0.1x5/12) − 29 = 2.51 In other words the put price is $2.51. https://www.chegg.com/homework-help/questions-and-answers/57-price-european-call-expires-six-months-strike-price-30-2-underlying-stock-price-29-divi-q36333549?trackid=0aa132650289&strackid=f002baf02e0c
Explain why the arguments leading to put-call parity for European options cannot be used to give a similar result for American options.
When early exercise is not possible, we can argue that two portfolios that are worth the same at time T must be worth the same at earlier times. When early exercise is possible, the argument falls down. Suppose that P + S > C + Ke -rT. This situation does not lead to an arbitrage opportunity. If we buy the call, short the put, and short the stock, we cannot be sure of the result because we do not know when the put will be exercised.
A stock price is currently $100. Over each of the next two six-month periods it is expected to go up by 10% or down by 10%. The risk-free interest rate is 8% per annum with continuous compounding. What is the value of a one-year European call option with a strike price of $100?
n this case: u =1.10, d = 0.90, Δt =0.5, and r =0.08, so that p = e^(0.08*0.5) - .90 /(1.10 - .09) = 0.7041 The tree for stock price movements is shown in Figure 1. We can work back from the end of the tree to the beginning, as indicated in the diagram, to give the value of the option as $9.61. The option value can also be calculated directly from equation (13.10): [0.70412×21+2×0.7041×0.2959×0+0.29592×0]e -2×0.08×0.5 =9.61 or $9.61 On Excel Sheet
