FIN Investments Option Pricing. Options: Definitions. Arbitrage Restrictions on Call Prices. Arbitrage Restrictions on Call Prices

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1 FIN Investments Option Pricing imple arbitrage relations s to call options Black-choles model Put-Call Parity Implied Volatility Options: Definitions A call option gives the buyer the right, but not the obligation, to purchase a specific asset for a prespecified price on a specific future date American vs. European Options exercisable before maturity? 1) C > on Call Prices Consider the portfolio formed by buying the call option. Later Today * < * > -C * - > 2) C < on Call Prices Consider the following portfolio: Buy the stock and sell the call. Later Today * < * > C - * * - (* - )= on Call Prices 3) C > - PV() 1) C > on Call Prices C Consider the following portfolio: Buy the call, sell the stock, and lend PV(). Later Today * < * > - C - PV() - * (* - ) - * + > = 2) C < 3) C > - PV() C= PV() C=-PV() Prof. chwert 1-6 pring 1997

2 Early Exercise of American Options uppose you own a call option and you want to close out your position. You can exercise and receive - Or you can sell your option for its current market price C You choose the alternative that yields the greatest profit (e.g., exercise if C < - and sell if C > - ) on American Call Prices uppose C < - between ex-dividend days. Then buy 1 call, short stock, lend close out position just before ex-dividend day Today At Ex-dividend Day * < * > -C+- > -*+(1+r)> (*-)-*+(1+r) = r > on American Call Prices C > - except at expiration or just prior to an ex-dividend day It is never optimal to exercise an American Call Option except at expiration or possibly just before an ex-dividend day. A Call Option is "worth more alive than dead" 1) P > 2) P < 3) P > PV() - 4) P > - on Put Prices (American Put Only) P P= PV() P=PV()- PV() s Diagrams for Contingent Claims hows relation between $ and tock Price for claims with an exercise price, = $1 Options: Diagrams ignores cost of buying the contingent claims/options ignores transaction costs useful for seeing relations among different contracts = max [ (* - ), ) ] = call at maturity t=t *= stock at maturity = exercise price * Prof. chwert 7-12 pring 1997

3 s to Buying & elling Call Options [Exercise Price, =$1] s to Buying & elling Put Options [Exercise Price, =$1] tock Price Buy Call tock Price Buy Call ell (write) Call tock Price Buy Put tock Price Buy Put ell (write) Put s to Buying & elling traddles [Put & Call Combinations, =$1] s to Buying & hort-selling tock [Purchase Price, =$1] tock Price Buy traddle tock Price Buy traddle ell (write) traddle tock Price Buy tock tock Price Buy tock hort-sell tock s to hort-selling tock & Buying a Call vs Buying a Put [=$1] s to Buying tock & elling a Call vs elling a Put [=$1] tock Price tock Price tock Price tock Price hort-sell tock & Buy Call Buy Put Buy tock & ell Call ell Put Prof. chwert pring 1997

4 s Add Up: Useful for Pricing imple Contingent Claims Put-call parity is nothing more than the observation that buying a put is equivalent to short-selling the stock & buying a call invest the net proceeds in a riskfree bond You can combine basic options with stocks & riskfree bonds to create any payoff structure you like presumably the market will price it "fairly" i.e., you will be correctly compensated for the risk you choose to bear The Intuition Behind the Black/choles Model It is possible to create a portfolio of stocks and bonds that has the exact same payoff as a call option over a very short period of time ince the stock and bond portfolio and the call option have the exact same payoffs, they must have the same price or there would be pure arbitrage opportunities Thus, we can value options by identifying this replicating portfolio of stocks and bonds. The stock and bond prices are directly observable The Black/choles Model A imple Example Assume T-1= $ T = $1 $25 r = 1.25 A call option is available with = $. What is the value of this call option? The Black/choles Model A imple Example Consider the following portfolio: T T-1 T = 25 T = 1 Write 3 calls 3C -1 Buy 2 shares -1 2 Borrow $ Total No arbitrage implies 3C = or C = $2 The Black/choles Model A imple Example We were able to value the call option in this case because we were able to find a stock and bond portfolio (buy 2/3 of a share and borrow $13.33) that had the exact same payoff as the call option over this one period. If two portfolios have identical payoffs, then the no arbitrage condition implies that they must have the same price. Lets try to generalize this reasoning. Black/choles Model uppose the risk-free interest rate is 5%. What is the price of a call option with an exercise price of 1? T-1 T 15 5 = 95 9 Create a portfolio of shares of stock and B dollars of bonds where and B are chosen so that the stock and bond portfolio has the exact same payoffs as the call option. Prof. chwert pring 1997

5 T-1 T 15 5 = 95 9 Black/choles Model B = B = 15 = 5; = (.3333) B = B = C = + B = 95(.3333) = 3.9 Black/choles Model T-2 T-1 T 12 2 b 11 a c: C= Black/choles Model b: B = B = 5 = B = 5 B = C = + B = 11 (1) = Black/choles Model a: B = B = 3.9 =.778 B = C = + B = 1 (.778) = 1.35 Black/choles Model T-2 T-1 T b: C= a: C= c: C= The Black/choles Model 9 Prof. chwert 25-3 pring 1997

6 The Black/choles Model The Black/choles Model The Black/choles Model The Black/choles Model f() The Black/choles Model Create a hedge portfolio ~ ~ ~ VH = Qs + C Qc ~ ~ ~ dvh = d Qs + dc Qc (1) o far, this looks like a standard calculus problem. The only problem is that C and are correlated random variables and the standard rules of calculus do not apply. Ito's Lemma If C = C(,t) where C and are random variables, then dc = (δc/δ) d + (δc/δt) dt + (1/2) (δ²c/δ²) σ² ² dt (2) Assumptions needed for Ito's lemma: tock prices are continuous tock prices have no memory Option price is function of current price but not a function of past price path Prof. chwert pring 1997

7 The Black/choles Model: A Riskfree Hedge ~ ~ dvh = d Q + [(δc/δ) d + (δc/δt) dt + (1/2) (d²c/d²) σ² ² dt] QC (3) Choose Q and QC so that d Q + (δc/δ) d QC = (4) In other words, VH is risk free as long as Q/QC = - (δc/δ) The Black/choles Model: A Riskfree Hedge ince VH is risk-free, its rate of return must be equal to the risk-free rate, i.e. dvh / VH = r dt. (5) If you make the necessary substitutions of (4) and (5) into (3), you are left with a partial differential equation (without any random variables). The solution to this equation [with the boundary condition = max (, - )] is the Black/choles Option Pricing Model The Black/choles Model ln (/) + (R + σ²/2)t C = N σ T ln (/) + (R - σ²/2)t - exp (-r T) N σ T The Black/choles Model Valuing an option with no uncertainty about exercising: C = PV () = PV [ max (, * - ) ] = PV (* - ) (if option is in the money) = - exp(-r T) The Black/choles Model & Boundary Conditions The Black/choles Model Out-of-the-Money: > C > Value with no uncertainty about exercise ( - PV()) Value with no uncertainty about exercise Black/choles Black/choles Value at maturity (* - ) Value at maturity PV() PV() Prof. chwert pring 1997

8 The Black/choles Model In-the-money: C > - PV() The Black/choles Model At-the-Money Options Are Valuable Value with no uncertainty about exercise Value with no uncertainty about exercise Black/choles Black/choles Value at maturity Value at maturity PV() PV() The Black/choles Riskfree Hedge Comparative tatics of the Black/choles Model C Black/choles PV() lope = (δc / δ) A hedge is risk-free if Q / QC = - (δc / δ) On Wall treet, this portfolio is referred to as a hedge C = C (,, T, σ², r, DIV) Comparative tatics of the Black/choles Model Comparative tatics of the Black/choles Model C = C (,, T, σ², r, DIV) Low σ² C = C (,, T, σ², r, DIV) Low σ² High σ² Prof. chwert pring 1997

9 Comparative tatics of the Black/choles Model Comparative tatics of the Black/choles Model C = C (,, T, σ², r, DIV) C = C (,, T, σ², r, DIV) Low σ² Low σ² High σ² High σ² Put-Call Parity With European options, there is a direct relation between put and call options: a put option gives the owner the right to sell stock to the issuer at an exercise price a put is equal to a call, minus the stock plus the discounted exercise price P = C - + exp(-rt) so the Black/choles Model can price puts by pricing the call Valuing American Put Options on Dividend Paying tocks It is not true, in general, that a put option is worth more alive than dead The optimal exercise strategy for American put options is more complicated than the optimal exercise strategy for American call options The most common time to exercise an American put option is just after an ex-dividend day, but this is not always the case There are likely to be larger differences between B/ prices and market prices for puts than for calls Estimating Volatility Using Options Prices -- Implied Volatility If you assume that the Black/choles model is correct you can observe all of the other variables necessary to calculate model prices then experiment with different values of volatility σ² until you find one that is consistent with the observed option price this is called the implied volatility Prof. chwert pring 1997