or Options
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Options Pricing
Learning Outcome Statement The candidate should be able to: • Discuss the factor influencing option price • Describe put-call parity • Discuss the basic principles of the binomial option model • Define and discuss delta-hedging • Explain risk-neutral valuation • Calculate an option price using a one-step binomial model • Define the symbols and letters of inputs into the binomial model • Describe the basic principles of the Black-Scholes-Merton model • State the Black-Scholes-Merton formula for pricing a call option • Calculate an option price using Black-Scholes-Merton model • Identify and discuss the graphic representations of a put and a call • Define delta, gamma, vega, theta and rho • Define and discuss implied volatility • Define a volatility smile • Define intrinsic value and time value
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Notations • • • • • •
c: p: S0 : K: T: σ:
European call option price European put option price Stock price today Strike price Life of option Volatility of stock price
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• • • • •
C: P: ST: D: r:
3
American Call option price American Put option price Stock price at option maturity Present value of dividends during option’s life Risk-free rate for maturity T with cont comp
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Factors influencing option price
Parameter
European Call
European Put
American Call
American Put
Current price
Increase
Decrease
Increase
Decrease
Strike price
Decrease
Increase
Decrease
Increase
Volatility
Increase
Increase
Increase
Increase
Time to expiration
-
-
Increase
Increase
Risk-free rate
Increase
Decrease
Increase
Decrease
Positive Cash flow
Decrease
Increase
Decrease
Increase
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Example True or False • If the spot price increases the Value of put decreases • If the strike price increases the Value of put Increases • In American option the increase time to expiration will increases the Value of Put • As risk free rate increases the value of the option will increase • As dividend increases the value of the put will decrease • As volatility increases the value of the put will decrease
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Lower and Upper Bounds for Options : on Non-Dividend stocks
• A call option (either American or European) can never be worth more than the underlying asset • Since on exercising the call option one becomes long in the underlying asset, therefore, if the cal option is worth more than the underlying asset than one can arbitrage by buying the underlying asset & selling the call option.
• Similarly, a put (American or European) can never be worth more than the strike price • Since on exercising the put option one receives K, therefore if put is worth more than K then one could make a riskless profit by selling the put option and investing the sale proceeds at the riskfree interest rate. Since European put can be exercised only at the end of the period, a European upper bound can be refined by saying that it will never be more than Ke-rt , where t is time to maturity.
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Lower and Upper Bounds for Options: on Non-Dividend Stocks (Cont…) • Let “c” and “C” represent a European & American call respectively on an underlying and “p” and “P” represent a European & American put respectively on an underlying • c > S - Ke-rt
Equation 1
– Equation 1 defines the lower bound of an European call option which is S - Ke-rt. Lower bound of the European call option is its intrinsic value
• p > Ke-rt - S
Equation 2
– Equation 2 defines the lower bound of a European put option which is Ke -rt - S. Lower bound of the European put option is its intrinsic value
• p+S
= c + Ke-rt
Equation 3
– Equation 3 explains the put call parity of an European put and call. – Assumption in the above equations is r > 0.
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Lower & Upper bounds for options on dividend paying stocks In case of yielding asset (say dividend paying stock): • c > S - D - Ke-rt
Equation 4
– Equation 4 defines lower bound of an European call option paying dividends which is S - Ke-rt – D
• p > Ke-rt - S + D
Equation 5
– Equation 5 defines lower bound of a European put option paying dividends which is Ke -rt - S + D
• p+S
= c + D + Ke-rt
Equation 6
– Equation 6 explains the put call parity of a European put and call wherein the underlying pays dividends. – Assumption in the above equations is r > 0.
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Example
1. Consider a 1 year European call option with a Strike price of $27.50 that is currently valued at $4.1 on a $25 stock. The 1 year risk free rate is 6%. Which of the following is the closest to the value of the corresponding put option? A. 0 B. 3.12 C. 5 D. 6.6 Answer C; p + S0 = c + D + Xe -rt 2. According to Put Call parity for European options, purchasing a put option on ABC stock will be equivalent to A. Buying a call, buying ABC stock and buying a Zero Coupon bond. B. Buying a call, selling ABC stock and buying a Zero Coupon bond. C. Selling a call, selling ABC stock and buying a Zero Coupon bond. D. Buying a call, selling ABC stock and selling a Zero Coupon bond Answer B
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Example (Cont…) 3. Early exercise of an option is more likely for: I. European calls options on stocks paying large dividends. II. American call options on stocks paying small dividends. III. American call options close to maturity. IV. American put options on stocks paying large dividends. A. I and IV B. II and IV C. III only D. III and IV. Answer C 4. If the current USD/AUD rate is 0.6650 (1 AUD=0.6650USD) & the risk-free rates for the USD & AUD are 1.0% & 4.5% respectively, what is the lower bound of a 5-month European put option on the AUD with a strike price of 0.6880? Answer: The lower bound for a European option is given by the formula: Xe-rT - Se-rfT, where X is the strike price, r is the risk-free rate of the USD, rf is the risk-free rate of the AUD,T is the time to maturity and S is the spot rate of the AUD/USD. Thus, the lower bound = 0.6880 x [exp-(0.01x 5/12)] – 0.6650 x [exp-(0.045x 5/12]] = 0.6880 x (0.9958) –0.6650 x (0.9814) = 0.6851 – 0.6526 = 0.0325
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A Simple Binomial Model • A stock price is currently $50 • In three months it will be either $55 or $48
Stock Price = $55 Stock price = $50
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Stock Price = $48
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A Call Option • A 3-month call option on the stock has a strike price of 51.
Stock Price = $55 Option Price =$4 Stock price = $50 Option Price=?
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Stock Price = $48 Option Price =$0
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Setting Up a Riskless Portfolio • Consider the Portfolio: – Long Δ shares – Short 1 call option
• Portfolio is riskless when 55Δ – 4 = 48Δ or • Δ = 4/7=0.5714
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55 – 4
48
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Valuing the Portfolio (Risk-Free Rate is 12%) • The riskless portfolio is: – Long 0.57 shares – Short 1 call option
• The value of the portfolio in 3 months is : 55*0.57 – 4 = 27.43 • The value of the portfolio today is : 27.43.5e
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–0.12*0.25 =
14
26.62
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Valuing the Option • The portfolio that is – Long 0.57 shares , short 1 option is worth 26.20
• The value of the shares is 28.57 (= 0.57*50 ) • The value of the option is therefore 1.95 (= 28.57 - 26.620 )
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Generalization • Consider a stock whose price is S0 and an option whose current price is ƒ • The option life is T and the option can go upto S0u or down to S0d during its lifetime
S0u fu
S0 f
S0d fd • The option price ƒ is given by ƒ = [ pƒu + (1 – p)ƒd ]e–rT – where p is the probability of an upward movement in the stock price from S0 to S0u and is given by
e rT d p u d
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Original Example Revisited S0u = 55 ƒu = 4 S0 ƒ S0d = 48 ƒd = 0 • Since p is the probability that gives a return on the stock equal to the risk-free rate. We can find it from 50e0.12 *0.25 = 55p + 48(1 – p ), which gives p = 0.503 • Alternatively, we can use the formula e rT d e 0.120.25 0.96 p 0.503 ud 1.1 0.96
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Example 1. Question: Assume that a binomial interest-rate tree indicates a 6-month period spot rate of 2.5%, and the price of the bond if rates decline is 98.45, & if rates increase is 96. The risk-neutral probabilities respectively associated with a decline and increase in rates if the market price of the bond is 97 correspond to: A. B. C. D.
0.1/0.9. 0.9/0.1. 0.2/0.8. 0.8/0.2.
Answer B, [p*98.45 + (1-p)*96] / [1+ (0.025/2)] = 97 2. Question: A binomial interest-rate tree indicates a 1-year spot rate of 4%, & the price of the bond if rates decline is 95.25 and 93.75 if rates increase. The risk neutral probability of an interest rate increase is 0.55. You hold a call option on the bond that expires in one year & has an exercise price of 93.00. The option value is closest to: A. B. C. D.
1.17. 0.97. 1.44. 1.37
Answer D, The call has a payoff of 95.25- 93 = 2.25 if rate declines; 93.75 – 93 = 0.75 if rates increase; the discounted value of payoff is 0.55*0.75 + 0.45*2.25 = 1.37
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Black Scholes - Assumptions 1. The price of underlying asset follows a lognormal distribution. 2. Market is frictionless i.e. There are no transaction costs or taxes. Underlying asset is fully divisible and short selling is allowed with full usage of proceeds permitted. 3. There are no cash flow (dividends or otherwise) from underlying asset during the tenor of the option. 4. Risk free rate is constant, known and same for all maturities. 5. There are no risk less arbitrage opportunities. 6. Volatility of the underlying asset is known and constant. 7. Trading of underlying asset is continuous. 8. Option can be exercised only at expiry of the option. i.e. options are European.
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Black Scholes Formula • The Black-Scholes formula for pricing the European call and put option for non yielding underlying asset are: – Price of European Call
c = S0N(d1) – Ke-rt*N(d2) – Price of European Put
p = Ke-rt*N(-d2) – S0N(-d1) – Where: • • • • •
S0 = Current stock price, K = Strike price, r = Continuously compounded risk free rate. d1 = [ln (S0 / K) + (r + σ2 / 2) t] / σ √t d2 = d1 - σ √t N(x) = cumulative probability distribution function for a standardized normal distribution (Area under normal curve)
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Black Scholes Few pointers on the Black – Scholes formula: 1. N(d1) is the delta of the option and therefore S0N(d1) represents the current price of delta. 2. N(d2) is the probability that the option will expire in the money and will be exercised in a risk-neutral world. (Risk neutrality means that the investor overlooks the risks while making investments). Therefore KN(d2) represents the strike price times the probability that the strike price will be paid. 3. Since as explained earlier early exercise of American call option (whose underlying is a non yielding asset) is not optimal therefore value of an European call option will be equal to American call option everything else remaining same. Hence equation 1 also holds true for American call option. 4. The same however could not be said about the American put option as early exercise in certain situations may be optimal.
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Black Scholes – Put Call Parity Recalling put call parity : p + S = c + Ke-rt . Checking the put call parity using the formula above: To prove: Ke-rt N(-d2) – S0N(-d1) + S0 = S0N(d1) – Ke-rt N(d2) + Ke-rt Since N(-x) = 1 – N(x) - Replacing this in above equation we get Ke-rt (1 – N(d2)) – S0(1-N(d1)) + S0 = S0N(d1) – Ke-rt N(d2) + Ke-rt Ke-rt - Ke-rt N(d2) – S0 + S0 N(d1) + S0 = S0N(d1) – Ke-rt N(d2)+ Ke-rt S0N(d1) – Ke-rt N(d2) + Ke-rt = S0N(d1) – Ke-rt N(d2) + Ke-rt So, put call parity holds good for Black Scholes formula
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Black Scholes – Few Parameters Discussed • S0 (Current Stock Price) – The higher the stock price the more valuable the call option. – In case S0 becomes very large, both d1 and d2 will become very large and hence both N(d1) and N(d2) are pretty close to 1. S0, the value of the call option becomes S0 - Ke-rt which is nothing but the valuation of a forward contract. – As S0 increases in value, N(d1) increases or delta of the option increases. As delta increases, the contribution of the intrinsic value in the total value of the option increases and the time value contribution decrease. In case the stock price becomes very large then time value becomes small enough to be ignored and the value of the option becomes equal to the intrinsic value. In that case value of the option will resemble vale of the forward contract.
• σ (Volatility) – If volatility increases, price of the option increases. However, if volatility falls to zero, both d1 and d2 will become very large and hence both N(d1) and N(d2) are pretty close to 1. The value of the option therefore will be its intrinsic value.
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Black Scholes – Dividend Paying Stocks • Assume that the underlying asset is a dividend paying stock and that the timing and amount of dividend can be accurately predicted. In that case it can be safely assumed that the stock price today S0 includes the value of dividend (which is certain) discounted to time of valuing the option at the risk free rate.
• So the option pricing formula as given above has to be modified by replacing S0 wherever it is appearing with S0 – D or S0e-qt where D is the discounted value (discounted at risk free rate “r”) o the dividend and “q” is continuous compounding rate.
c S0 e qT N(d1 ) Ke rT N(d2 )
p Ke rT N( d2 ) S0 e qT N( d1 ) ln(S0 /K) (r q σ 2 /2)T where d1 σ T ln(S0 /K) (r q σ 2 /2)T d2 σ T
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Example
1. The current price of a stock is $25. A put option with a $20 strike price that expires in six months is available. N(-d1) = 0.0263 and N(-d2) = 0.0349. If the underlying stock exhibits an annual standard deviation of 25%, and the current continuously compounded risk-free rate's 4.5%, the Black-ScholesMerton value of the put is closest to: A. B. C. D.
$5.00. $3.00. $1.00. $0.03.
Answer D; p = Ke-rt N(-d2) – S0N(-d1) 2. The current price of a stock is $25. A put option is available with a $20 strike price that expires in 6 months. If the underlying stock exhibits an annual standard deviation of 25%, the current risk free rate is 4.5%, N (-d1) = 0.0263 and N (-d2) = .0349, the Black - Scholes- Merton value of the put is closest to : A. B. C. D.
$5 $3 $1 $0.03
Answer D; Put: {Xe-rt*[1- N (d2)]} – {S0* [1- N (d1)]}
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Delta • Delta is the slope of the call option pricing function at the current stock price. The call options delta range from 0 to 1, while put option delta range from -1 to 0. Delta ∆, = c/s – c = change in the call option price – s = change in the stock price
• The delta of a forward position is equal to 1 (Forwards have a Zero Gamma & thus can’t be used to create a Gamma neutral portfolio), and the delta of futures position is erT on a stock, e(r-q) T on a dividend paying stock. Delta hedging involves maintaining a delta neutral portfolio Option Price Slope=∆
B
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Stock Price A
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Example
To create a delta-neutral portfolio, an investor who has written 5,000 call options that have deltas equal to 0.5 will be: A. Long 2,500 shares in the underlying. B. Short 2,500 shares in the underlying. C. Long 2,500 shares in the underlying and short 2,500 more options. D. Short 2,500 shares in the underlying and be short 2,500 more options. Answer A; To hedge the short position on Call Options, the investor should go long 0.5*5000 = 2,500 shares to create a delta-neutral portfolio.
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Gamma
• Represents the expected change in the delta of an option. It measure the curvature of the option price function not captured by delta and thus can be used to minimize the hedging error associated with a linear relationship (delta). When gamma is large, delta will be changing rapidly.
• Gamma neutral positions have to be created using instruments that are not linearly related to the underlying instruments, such as options. • Gamma is greatest for options that are close to the money.
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Gamma Addresses Delta Hedging Errors Caused By Curvature
Call price
C'‘ C' C
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Stock price S
S'
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Example An existing option short position is delta neutral, but has a -5,000 gamma exposure. An option is available that has a gamma of 2 and a delta of 0.7. What actions should be taken to create a gamma neutral position that will remain delta neutral? A. Go long 2,500 options and sell 1,750 shares of the underlying stock. B. Go short 2,500 options and buy 1,750 shares of the underlying stock. C. Go long 10,000 options and sell 1,750 shares of the underlying stock. D. Go long 10,000 options and buy 1,750 shares of the underlying stock.
Answer A; Since the current position is short gamma, to create a gamma-neutral portfolio (5000/2) = 2,500 long option. However this will change the position from delta-neutral portfolio to 2,500*.7 = 1,750 long delta. So sell 1,750 shares to be gamma and delta neutrality.
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Theta • Theta (Θ) of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to the passage of time • The theta of a call or put is usually negative. This means that, if time passes with the price of the underlying asset and its volatility remaining the same, the value of the option declines • Theta is also termed as “time decay” of an option
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Vega & Rho • Vega (٧) is the rate of change of the value of a derivatives portfolio with respect to volatility – Vega tends to be greatest for options that are close to the money – For Example, a Vega of .8 indicates that for a 1% increase in volatility, the option price will increase by $.08
• Rho: measures the sensitivity of the option price to changes in the Risk free rate. “In the money” calls and Puts are most sensitive to changes in the rates than “Out of the Money”. Rho is much more important risk factor for fixed income derivatives than equity options.
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Questions Which of the following statement is true regarding options? A. B. C. D.
Theta tends to be large and positive for at the money options. Gamma is greatest for in the money options with long times remaining to expiration. Vega is greatest for at the money options with long times remaining to expiration. Delta of deep in the money put options tends towards +1.
Answer: C
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Questions True or False A. Theta affects the value of a call and a put in similar way B. Theta is more pronounced when the option is “in the money” C. Theta usually decreases in absolute terms as expiration approaches D. It is possible for a European put option that is “in the money” to have a positive theta value E. Rho for fixed income is small F. Call option delta range from 0 to 1 G. A Vega of .1 suggests that for 1% increase in volatility, Option price will increase by 0.10 H. Theta is the most negative for out of money options I. Options are most sensitive to changes to volatility, when they are “At the money”
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Answers A. B. C. D. E. F. G. H. I.
True False False True False True True False True
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Implied Volatilities • Put-call parity p +S0e-qT = c +X e–rT holds regardless of the assumptions made about the stock price distribution
• It follows that the call option pricing error caused by using the wrong distribution is the same as the put option pricing error when both have the same strike price and maturity. i.e. pmkt-pbs=cmkt-c • Where “mkt” subscript means actual market prices and “bs” means as suggested by Black Scholes formula • The implied volatility calculated from a European call option should be the same as that calculated from a European put option when both have the same strike price and maturity. The same is approximately true of American options • A volatility smile shows the variation of the implied volatility with the strike price • The volatility smile should be the same whether calculated from call options or put options
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The Volatility Smile for Foreign Currency Options
• The volatility smile used by traders to price foreign currency options has the general form shown in figure below. The volatility is relatively low for at-the-money options. It becomes progressively higher as an option moves either in the money or out of the money.
Implied Volatility
• The smile used by traders for foreign currency options implies that they consider that the lognormal distribution understates the probability of extreme movements in exchange rates. Exchange rates not log-normally distributed because two of the conditions for an asset price to have a lognormal distribution i.e. – The volatility of the asset is constant – The price of the asset changes smoothly with no jumps
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Strike Price 37
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Possible Causes of Volatility Smile • Asset price exhibiting jumps rather than continuous change • Volatility for asset price being stochastic (One reason for a stochastic volatility in the case of equities is the relationship between volatility and leverage)
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Intrinsic Value And Time Value • The price of an option is the sum of its intrinsic value and its time value • Intrinsic Value – The in-the-money portion of the option – For a call option, this is the greater of zero and the difference between the current stock price and the strike price – For a put option, this is the greater of zero and the difference between the strike price and the current strike price
• Time value – Also known as extrinsic value, or instrumental value, it is the portion of the option premium tha is attributed to the amount of time remaining until the expiration of the option – Time Value = Option Value - Intrinsic Value
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Questions 1. As the time to expiration of a European call and an American put option increases A. The price of both the options remain the same B. European call price will remain the same, American put price will increase C. The price of both Increase D. American put price will increase, European call price may increase, decrease or remain the same
2. A 6-month European call of strike price $50 is trading for $3. What will be the price of a 6month European put of strike price $50 if the underlying stock price is $48, term structure is flat and the risk free rate is 10%. A. $2.56 B. $2.76 C. $2.36 D. $3.56
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Questions 3. In Black Scholes Model, the stock prices and returns on stocks are A. Both are normally distributed B. Both are log-normally distributed C. Normally and log-normally distributed D. Log-normally and normally distributed
4. A non dividend paying stock with volatility of 20% per annum is currently trading at $50. A European call option on the stock with a strike price of $49 has a time to maturity of 5 months. The risk free rate is 6%. The price of the option is A. $3.78 B. $4.28 C. $3.44 D. $3.14
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Questions 5. Which of the following is false regarding the Greeks? A. A position in the underlying has a zero vega B. Theta is greatest for at the money options close to expiry C. Vega is greatest for at the money options D. Delta for put options is negative
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Answers 1. As the time to expiration of a European call and an American put option increases D. American put price will increase, European call price may increase, decrease or remain the same 2. p = c + Ke-rt– S0 = 3 +50e-0.1x0.5 – 48 = 2.56 A. $2.56
3. Stock prices are log-normally and returns are normally distributed D. Log-normally and normally distributed 4. $3.48 – d1 = 0.415, d2 = 0.286 – c = 50 x N(0.415) – 48e-0.06x5/12N(0.286) = $3.78
5. Theta is lowest (large and negative) for at the money options close to expiry B. Theta is greatest for at the money options close to expiry is false
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End of Option Pricing
Options Pricing
Learning Outcome Statement The candidate should be able to: • Discuss the factor influencing option price • Describe put-call parity • Discuss the basic principles of the binomial option model • Define and discuss delta-hedging • Explain risk-neutral valuation • Calculate an option price using a one-step binomial model • Define the symbols and letters of inputs into the binomial model • Describe the basic principles of the Black-Scholes-Merton model • State the Black-Scholes-Merton formula for pricing a call option • Calculate an option price using Black-Scholes-Merton model • Identify and discuss the graphic representations of a put and a call • Define delta, gamma, vega, theta and rho • Define and discuss implied volatility • Define a volatility smile • Define intrinsic value and time value
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Notations • • • • • •
c: p: S0 : K: T: σ:
European call option price European put option price Stock price today Strike price Life of option Volatility of stock price
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• • • • •
C: P: ST: D: r:
3
American Call option price American Put option price Stock price at option maturity Present value of dividends during option’s life Risk-free rate for maturity T with cont comp
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Factors influencing option price
Parameter
European Call
European Put
American Call
American Put
Current price
Increase
Decrease
Increase
Decrease
Strike price
Decrease
Increase
Decrease
Increase
Volatility
Increase
Increase
Increase
Increase
Time to expiration
-
-
Increase
Increase
Risk-free rate
Increase
Decrease
Increase
Decrease
Positive Cash flow
Decrease
Increase
Decrease
Increase
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Example True or False • If the spot price increases the Value of put decreases • If the strike price increases the Value of put Increases • In American option the increase time to expiration will increases the Value of Put • As risk free rate increases the value of the option will increase • As dividend increases the value of the put will decrease • As volatility increases the value of the put will decrease
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Lower and Upper Bounds for Options : on Non-Dividend stocks
• A call option (either American or European) can never be worth more than the underlying asset • Since on exercising the call option one becomes long in the underlying asset, therefore, if the cal option is worth more than the underlying asset than one can arbitrage by buying the underlying asset & selling the call option.
• Similarly, a put (American or European) can never be worth more than the strike price • Since on exercising the put option one receives K, therefore if put is worth more than K then one could make a riskless profit by selling the put option and investing the sale proceeds at the riskfree interest rate. Since European put can be exercised only at the end of the period, a European upper bound can be refined by saying that it will never be more than Ke-rt , where t is time to maturity.
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Lower and Upper Bounds for Options: on Non-Dividend Stocks (Cont…) • Let “c” and “C” represent a European & American call respectively on an underlying and “p” and “P” represent a European & American put respectively on an underlying • c > S - Ke-rt
Equation 1
– Equation 1 defines the lower bound of an European call option which is S - Ke-rt. Lower bound of the European call option is its intrinsic value
• p > Ke-rt - S
Equation 2
– Equation 2 defines the lower bound of a European put option which is Ke -rt - S. Lower bound of the European put option is its intrinsic value
• p+S
= c + Ke-rt
Equation 3
– Equation 3 explains the put call parity of an European put and call. – Assumption in the above equations is r > 0.
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Lower & Upper bounds for options on dividend paying stocks In case of yielding asset (say dividend paying stock): • c > S - D - Ke-rt
Equation 4
– Equation 4 defines lower bound of an European call option paying dividends which is S - Ke-rt – D
• p > Ke-rt - S + D
Equation 5
– Equation 5 defines lower bound of a European put option paying dividends which is Ke -rt - S + D
• p+S
= c + D + Ke-rt
Equation 6
– Equation 6 explains the put call parity of a European put and call wherein the underlying pays dividends. – Assumption in the above equations is r > 0.
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Example
1. Consider a 1 year European call option with a Strike price of $27.50 that is currently valued at $4.1 on a $25 stock. The 1 year risk free rate is 6%. Which of the following is the closest to the value of the corresponding put option? A. 0 B. 3.12 C. 5 D. 6.6 Answer C; p + S0 = c + D + Xe -rt 2. According to Put Call parity for European options, purchasing a put option on ABC stock will be equivalent to A. Buying a call, buying ABC stock and buying a Zero Coupon bond. B. Buying a call, selling ABC stock and buying a Zero Coupon bond. C. Selling a call, selling ABC stock and buying a Zero Coupon bond. D. Buying a call, selling ABC stock and selling a Zero Coupon bond Answer B
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Example (Cont…) 3. Early exercise of an option is more likely for: I. European calls options on stocks paying large dividends. II. American call options on stocks paying small dividends. III. American call options close to maturity. IV. American put options on stocks paying large dividends. A. I and IV B. II and IV C. III only D. III and IV. Answer C 4. If the current USD/AUD rate is 0.6650 (1 AUD=0.6650USD) & the risk-free rates for the USD & AUD are 1.0% & 4.5% respectively, what is the lower bound of a 5-month European put option on the AUD with a strike price of 0.6880? Answer: The lower bound for a European option is given by the formula: Xe-rT - Se-rfT, where X is the strike price, r is the risk-free rate of the USD, rf is the risk-free rate of the AUD,T is the time to maturity and S is the spot rate of the AUD/USD. Thus, the lower bound = 0.6880 x [exp-(0.01x 5/12)] – 0.6650 x [exp-(0.045x 5/12]] = 0.6880 x (0.9958) –0.6650 x (0.9814) = 0.6851 – 0.6526 = 0.0325
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A Simple Binomial Model • A stock price is currently $50 • In three months it will be either $55 or $48
Stock Price = $55 Stock price = $50
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Stock Price = $48
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A Call Option • A 3-month call option on the stock has a strike price of 51.
Stock Price = $55 Option Price =$4 Stock price = $50 Option Price=?
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Stock Price = $48 Option Price =$0
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Setting Up a Riskless Portfolio • Consider the Portfolio: – Long Δ shares – Short 1 call option
• Portfolio is riskless when 55Δ – 4 = 48Δ or • Δ = 4/7=0.5714
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55 – 4
48
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Valuing the Portfolio (Risk-Free Rate is 12%) • The riskless portfolio is: – Long 0.57 shares – Short 1 call option
• The value of the portfolio in 3 months is : 55*0.57 – 4 = 27.43 • The value of the portfolio today is : 27.43.5e
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–0.12*0.25 =
14
26.62
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Valuing the Option • The portfolio that is – Long 0.57 shares , short 1 option is worth 26.20
• The value of the shares is 28.57 (= 0.57*50 ) • The value of the option is therefore 1.95 (= 28.57 - 26.620 )
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Generalization • Consider a stock whose price is S0 and an option whose current price is ƒ • The option life is T and the option can go upto S0u or down to S0d during its lifetime
S0u fu
S0 f
S0d fd • The option price ƒ is given by ƒ = [ pƒu + (1 – p)ƒd ]e–rT – where p is the probability of an upward movement in the stock price from S0 to S0u and is given by
e rT d p u d
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Original Example Revisited S0u = 55 ƒu = 4 S0 ƒ S0d = 48 ƒd = 0 • Since p is the probability that gives a return on the stock equal to the risk-free rate. We can find it from 50e0.12 *0.25 = 55p + 48(1 – p ), which gives p = 0.503 • Alternatively, we can use the formula e rT d e 0.120.25 0.96 p 0.503 ud 1.1 0.96
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Example 1. Question: Assume that a binomial interest-rate tree indicates a 6-month period spot rate of 2.5%, and the price of the bond if rates decline is 98.45, & if rates increase is 96. The risk-neutral probabilities respectively associated with a decline and increase in rates if the market price of the bond is 97 correspond to: A. B. C. D.
0.1/0.9. 0.9/0.1. 0.2/0.8. 0.8/0.2.
Answer B, [p*98.45 + (1-p)*96] / [1+ (0.025/2)] = 97 2. Question: A binomial interest-rate tree indicates a 1-year spot rate of 4%, & the price of the bond if rates decline is 95.25 and 93.75 if rates increase. The risk neutral probability of an interest rate increase is 0.55. You hold a call option on the bond that expires in one year & has an exercise price of 93.00. The option value is closest to: A. B. C. D.
1.17. 0.97. 1.44. 1.37
Answer D, The call has a payoff of 95.25- 93 = 2.25 if rate declines; 93.75 – 93 = 0.75 if rates increase; the discounted value of payoff is 0.55*0.75 + 0.45*2.25 = 1.37
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Black Scholes - Assumptions 1. The price of underlying asset follows a lognormal distribution. 2. Market is frictionless i.e. There are no transaction costs or taxes. Underlying asset is fully divisible and short selling is allowed with full usage of proceeds permitted. 3. There are no cash flow (dividends or otherwise) from underlying asset during the tenor of the option. 4. Risk free rate is constant, known and same for all maturities. 5. There are no risk less arbitrage opportunities. 6. Volatility of the underlying asset is known and constant. 7. Trading of underlying asset is continuous. 8. Option can be exercised only at expiry of the option. i.e. options are European.
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Black Scholes Formula • The Black-Scholes formula for pricing the European call and put option for non yielding underlying asset are: – Price of European Call
c = S0N(d1) – Ke-rt*N(d2) – Price of European Put
p = Ke-rt*N(-d2) – S0N(-d1) – Where: • • • • •
S0 = Current stock price, K = Strike price, r = Continuously compounded risk free rate. d1 = [ln (S0 / K) + (r + σ2 / 2) t] / σ √t d2 = d1 - σ √t N(x) = cumulative probability distribution function for a standardized normal distribution (Area under normal curve)
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Black Scholes Few pointers on the Black – Scholes formula: 1. N(d1) is the delta of the option and therefore S0N(d1) represents the current price of delta. 2. N(d2) is the probability that the option will expire in the money and will be exercised in a risk-neutral world. (Risk neutrality means that the investor overlooks the risks while making investments). Therefore KN(d2) represents the strike price times the probability that the strike price will be paid. 3. Since as explained earlier early exercise of American call option (whose underlying is a non yielding asset) is not optimal therefore value of an European call option will be equal to American call option everything else remaining same. Hence equation 1 also holds true for American call option. 4. The same however could not be said about the American put option as early exercise in certain situations may be optimal.
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Black Scholes – Put Call Parity Recalling put call parity : p + S = c + Ke-rt . Checking the put call parity using the formula above: To prove: Ke-rt N(-d2) – S0N(-d1) + S0 = S0N(d1) – Ke-rt N(d2) + Ke-rt Since N(-x) = 1 – N(x) - Replacing this in above equation we get Ke-rt (1 – N(d2)) – S0(1-N(d1)) + S0 = S0N(d1) – Ke-rt N(d2) + Ke-rt Ke-rt - Ke-rt N(d2) – S0 + S0 N(d1) + S0 = S0N(d1) – Ke-rt N(d2)+ Ke-rt S0N(d1) – Ke-rt N(d2) + Ke-rt = S0N(d1) – Ke-rt N(d2) + Ke-rt So, put call parity holds good for Black Scholes formula
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Black Scholes – Few Parameters Discussed • S0 (Current Stock Price) – The higher the stock price the more valuable the call option. – In case S0 becomes very large, both d1 and d2 will become very large and hence both N(d1) and N(d2) are pretty close to 1. S0, the value of the call option becomes S0 - Ke-rt which is nothing but the valuation of a forward contract. – As S0 increases in value, N(d1) increases or delta of the option increases. As delta increases, the contribution of the intrinsic value in the total value of the option increases and the time value contribution decrease. In case the stock price becomes very large then time value becomes small enough to be ignored and the value of the option becomes equal to the intrinsic value. In that case value of the option will resemble vale of the forward contract.
• σ (Volatility) – If volatility increases, price of the option increases. However, if volatility falls to zero, both d1 and d2 will become very large and hence both N(d1) and N(d2) are pretty close to 1. The value of the option therefore will be its intrinsic value.
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Black Scholes – Dividend Paying Stocks • Assume that the underlying asset is a dividend paying stock and that the timing and amount of dividend can be accurately predicted. In that case it can be safely assumed that the stock price today S0 includes the value of dividend (which is certain) discounted to time of valuing the option at the risk free rate.
• So the option pricing formula as given above has to be modified by replacing S0 wherever it is appearing with S0 – D or S0e-qt where D is the discounted value (discounted at risk free rate “r”) o the dividend and “q” is continuous compounding rate.
c S0 e qT N(d1 ) Ke rT N(d2 )
p Ke rT N( d2 ) S0 e qT N( d1 ) ln(S0 /K) (r q σ 2 /2)T where d1 σ T ln(S0 /K) (r q σ 2 /2)T d2 σ T
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Example
1. The current price of a stock is $25. A put option with a $20 strike price that expires in six months is available. N(-d1) = 0.0263 and N(-d2) = 0.0349. If the underlying stock exhibits an annual standard deviation of 25%, and the current continuously compounded risk-free rate's 4.5%, the Black-ScholesMerton value of the put is closest to: A. B. C. D.
$5.00. $3.00. $1.00. $0.03.
Answer D; p = Ke-rt N(-d2) – S0N(-d1) 2. The current price of a stock is $25. A put option is available with a $20 strike price that expires in 6 months. If the underlying stock exhibits an annual standard deviation of 25%, the current risk free rate is 4.5%, N (-d1) = 0.0263 and N (-d2) = .0349, the Black - Scholes- Merton value of the put is closest to : A. B. C. D.
$5 $3 $1 $0.03
Answer D; Put: {Xe-rt*[1- N (d2)]} – {S0* [1- N (d1)]}
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Delta • Delta is the slope of the call option pricing function at the current stock price. The call options delta range from 0 to 1, while put option delta range from -1 to 0. Delta ∆, = c/s – c = change in the call option price – s = change in the stock price
• The delta of a forward position is equal to 1 (Forwards have a Zero Gamma & thus can’t be used to create a Gamma neutral portfolio), and the delta of futures position is erT on a stock, e(r-q) T on a dividend paying stock. Delta hedging involves maintaining a delta neutral portfolio Option Price Slope=∆
B
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Stock Price A
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Example
To create a delta-neutral portfolio, an investor who has written 5,000 call options that have deltas equal to 0.5 will be: A. Long 2,500 shares in the underlying. B. Short 2,500 shares in the underlying. C. Long 2,500 shares in the underlying and short 2,500 more options. D. Short 2,500 shares in the underlying and be short 2,500 more options. Answer A; To hedge the short position on Call Options, the investor should go long 0.5*5000 = 2,500 shares to create a delta-neutral portfolio.
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Gamma
• Represents the expected change in the delta of an option. It measure the curvature of the option price function not captured by delta and thus can be used to minimize the hedging error associated with a linear relationship (delta). When gamma is large, delta will be changing rapidly.
• Gamma neutral positions have to be created using instruments that are not linearly related to the underlying instruments, such as options. • Gamma is greatest for options that are close to the money.
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Gamma Addresses Delta Hedging Errors Caused By Curvature
Call price
C'‘ C' C
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Stock price S
S'
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Example An existing option short position is delta neutral, but has a -5,000 gamma exposure. An option is available that has a gamma of 2 and a delta of 0.7. What actions should be taken to create a gamma neutral position that will remain delta neutral? A. Go long 2,500 options and sell 1,750 shares of the underlying stock. B. Go short 2,500 options and buy 1,750 shares of the underlying stock. C. Go long 10,000 options and sell 1,750 shares of the underlying stock. D. Go long 10,000 options and buy 1,750 shares of the underlying stock.
Answer A; Since the current position is short gamma, to create a gamma-neutral portfolio (5000/2) = 2,500 long option. However this will change the position from delta-neutral portfolio to 2,500*.7 = 1,750 long delta. So sell 1,750 shares to be gamma and delta neutrality.
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Theta • Theta (Θ) of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to the passage of time • The theta of a call or put is usually negative. This means that, if time passes with the price of the underlying asset and its volatility remaining the same, the value of the option declines • Theta is also termed as “time decay” of an option
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Vega & Rho • Vega (٧) is the rate of change of the value of a derivatives portfolio with respect to volatility – Vega tends to be greatest for options that are close to the money – For Example, a Vega of .8 indicates that for a 1% increase in volatility, the option price will increase by $.08
• Rho: measures the sensitivity of the option price to changes in the Risk free rate. “In the money” calls and Puts are most sensitive to changes in the rates than “Out of the Money”. Rho is much more important risk factor for fixed income derivatives than equity options.
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Questions Which of the following statement is true regarding options? A. B. C. D.
Theta tends to be large and positive for at the money options. Gamma is greatest for in the money options with long times remaining to expiration. Vega is greatest for at the money options with long times remaining to expiration. Delta of deep in the money put options tends towards +1.
Answer: C
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Questions True or False A. Theta affects the value of a call and a put in similar way B. Theta is more pronounced when the option is “in the money” C. Theta usually decreases in absolute terms as expiration approaches D. It is possible for a European put option that is “in the money” to have a positive theta value E. Rho for fixed income is small F. Call option delta range from 0 to 1 G. A Vega of .1 suggests that for 1% increase in volatility, Option price will increase by 0.10 H. Theta is the most negative for out of money options I. Options are most sensitive to changes to volatility, when they are “At the money”
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Answers A. B. C. D. E. F. G. H. I.
True False False True False True True False True
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Implied Volatilities • Put-call parity p +S0e-qT = c +X e–rT holds regardless of the assumptions made about the stock price distribution
• It follows that the call option pricing error caused by using the wrong distribution is the same as the put option pricing error when both have the same strike price and maturity. i.e. pmkt-pbs=cmkt-c • Where “mkt” subscript means actual market prices and “bs” means as suggested by Black Scholes formula • The implied volatility calculated from a European call option should be the same as that calculated from a European put option when both have the same strike price and maturity. The same is approximately true of American options • A volatility smile shows the variation of the implied volatility with the strike price • The volatility smile should be the same whether calculated from call options or put options
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The Volatility Smile for Foreign Currency Options
• The volatility smile used by traders to price foreign currency options has the general form shown in figure below. The volatility is relatively low for at-the-money options. It becomes progressively higher as an option moves either in the money or out of the money.
Implied Volatility
• The smile used by traders for foreign currency options implies that they consider that the lognormal distribution understates the probability of extreme movements in exchange rates. Exchange rates not log-normally distributed because two of the conditions for an asset price to have a lognormal distribution i.e. – The volatility of the asset is constant – The price of the asset changes smoothly with no jumps
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Strike Price 37
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Possible Causes of Volatility Smile • Asset price exhibiting jumps rather than continuous change • Volatility for asset price being stochastic (One reason for a stochastic volatility in the case of equities is the relationship between volatility and leverage)
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Intrinsic Value And Time Value • The price of an option is the sum of its intrinsic value and its time value • Intrinsic Value – The in-the-money portion of the option – For a call option, this is the greater of zero and the difference between the current stock price and the strike price – For a put option, this is the greater of zero and the difference between the strike price and the current strike price
• Time value – Also known as extrinsic value, or instrumental value, it is the portion of the option premium tha is attributed to the amount of time remaining until the expiration of the option – Time Value = Option Value - Intrinsic Value
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Questions 1. As the time to expiration of a European call and an American put option increases A. The price of both the options remain the same B. European call price will remain the same, American put price will increase C. The price of both Increase D. American put price will increase, European call price may increase, decrease or remain the same
2. A 6-month European call of strike price $50 is trading for $3. What will be the price of a 6month European put of strike price $50 if the underlying stock price is $48, term structure is flat and the risk free rate is 10%. A. $2.56 B. $2.76 C. $2.36 D. $3.56
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Questions 3. In Black Scholes Model, the stock prices and returns on stocks are A. Both are normally distributed B. Both are log-normally distributed C. Normally and log-normally distributed D. Log-normally and normally distributed
4. A non dividend paying stock with volatility of 20% per annum is currently trading at $50. A European call option on the stock with a strike price of $49 has a time to maturity of 5 months. The risk free rate is 6%. The price of the option is A. $3.78 B. $4.28 C. $3.44 D. $3.14
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Questions 5. Which of the following is false regarding the Greeks? A. A position in the underlying has a zero vega B. Theta is greatest for at the money options close to expiry C. Vega is greatest for at the money options D. Delta for put options is negative
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Answers 1. As the time to expiration of a European call and an American put option increases D. American put price will increase, European call price may increase, decrease or remain the same 2. p = c + Ke-rt– S0 = 3 +50e-0.1x0.5 – 48 = 2.56 A. $2.56
3. Stock prices are log-normally and returns are normally distributed D. Log-normally and normally distributed 4. $3.48 – d1 = 0.415, d2 = 0.286 – c = 50 x N(0.415) – 48e-0.06x5/12N(0.286) = $3.78
5. Theta is lowest (large and negative) for at the money options close to expiry B. Theta is greatest for at the money options close to expiry is false
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End of Option Pricing
