( ) since this is the benefit of buying the asset at the strike price rather

Transcription

1 Review of some financial models for MAT 483 Parity and Other Option Relationships The basic parity relationship for European options with the same strike price and the same time to expiration is: C( KT ) P( KT ) PV 0T ( F 0T K ) e rt ( F 0T K ) where C( KT )- the price of European call with strike price K and time to expiration T; P KT - the price of European put with strike price K and time to expiration T; F 0T - the forward price for the underlying asset; K the strike price; T the time to expiration of the options; PV 0T - the present value over the life of the options. The intuition is that buying a call and selling a put with the strike equal to the forward price ( F 0T K ) creates a synthetic forward contract and hence must have a zero price. If we create a synthetic long forward position at a price lower than the forward we have to since this is the benefit of buying the asset at the strike price rather pay PV 0T F 0T K than the forward price. Put-call parity implies that C( KT ) P( KT ) S PV ( K ) so that PV ( K ) S C( KT ) + P( KT ). The creation of a synthetic risk-free bond by buying the stock writing a call and buying a put is called a conversion. Its reverse is called a reverse conversion. Generalized parity and exchange options In general options can be designed to exchange any asset for any other asset and relative put and call premiums are determined by prices of prepaid forwards on the underlying and strike assets. Assume we are exchanging an asset A with price S t for a strike asset B P with price Q t. Let F tt S denote the time t price of a prepaid forward on the underlying ( Q) denote the time t price of a prepaid forward P asset paying S T at time T and let F tt on the asset B paying Q T at time T. Then the parity equation at time t is: P C S t Q t T t ( S) F tt ( Q). P P( S t Q t T t) F tt The Black-Scholes Formula Let S t be a random variable describing the value of an asset (a stock or some other asset) whose price follows a process we will be describing at time t with S 0 S. Let f t be the risk-neutral PDF of S t. The value of a European call with exercise price K and exercise time T is

2 + C e rt ( x K ) f T ( x)dx. K Assume that S t is lognormally distributed so that the continuously compounded rate of S return on the underlying over t years has the normal distribution with mean tµ and variance tσ. Based on the properties of the lognormal distribution E S T S and f T ( x) e T µ + σ xσ πt e x S T µ σ Tσ. Assuming risk-neutral valuation S E( e rt S T ) e rt Se T µ + or e rt e T µ + σ. This means that µ r σ. Combining the above we get + x S T r σ x K C e rt xσ πt e Tσ dx K x y S T r σ K S T r σ x K y F dy x dx x y σ Integration by substitution Se y +T r + K e rt e Se y +T r y Ke + y dy e rt y dy F π F π T +rt S eyσ Tσ + y rt e + dy Ke rt y π π dy e ( y ) + e + S dy Ke rt y π π dy F z y y F z F S dz dy y z We define Integration by substitution in the first integral F F σ e + z + π dz e Ke rt F F F y π dy.

3 K S d ( F ) T r F + σ + S K + T r σ + σ T S K + T r + σ S K + T r d F σ d. This gives C S e + z π dz e + Ke rt y F F ( ) Ke rt N F S N F ( ) Ke rt N F π dy S N F SN ( d ) Ke rt N d. This is the Black-Scholes formula. Note also that the coefficient of Ke rt N ( d ) e + y F π dy Pr( Y > F) Pr( S T > K ) so that N d money. On the other hand the coefficient of S is is ( ) can be interpreted as the risk-neutral probability of the call expiring in the N ( d ) + e z π dz > F e + z π dz N ( d ). F The Black-Scholes formula is a limiting case (as number of steps in binomial tree goes to infinity) of the binomial formula for the price of European options. In general the Black- Scholes formula gives the price of a European call option on a stock that pays dividends at the continuous rate δ as: C( SKσrT δ ) Se δt N ( d ) Ke rt N ( d ) where Se δt Ke rt + σ T S K + ( r δ )T + σ T d Se δt Ke rt σ T d d. In the above S is the current price of the stock K is the strike price of the option σ is the volatility of the stock (standard deviation of the continuously compounded rate of return on the stock) r is the continuously compounded risk-free rate T is the time to

4 expiration; δ is the dividend yield on the stock; and N(x) is the cumulative distribution function of the standard normal distribution. Put option The Black-Scholes formula for European put option on a stock that pays dividends at the continuous rate d is: P( SKσrT δ ) Ke rt N ( d ) Se δt N ( d ) where as for call options Se δt Ke rt + σ T S K + ( r δ )T + σ T d Se δt Ke rt σ T d d. We can get the put option price from the put-call parity: P C + Ke rt Se δt using this property of the standard normal distribution: N ( x) N ( x). Assumptions of the Black-Scholes Formula Assumptions about the distribution of the stock: Continuously compounded returns on the stock are normally distributed and independent over time (no jumps). The volatility of continuously compounded returns is known and constant. Future dividends are known (a dollar amount or fixed dividend rate). Assumptions about economic environment: The risk-free rate is known and constant. There are no transaction costs or taxes. It is possible to sell short without any cost and to borrow at the risk-free rate. Option Greeks Option Greeks are formulas that express the change in the option price when one input to the formula changes taking all the other inputs as fixed. In other words a Greek is a partial derivative of the option price with respect to one particular input. These formulas are used in investment practice to assess risk exposure. We will talk about the Greeks for a purchased option (i.e. for a long position in an option). The Greek for a written option is opposite in sign to that for the same purchased option. Delta Δ If S is the price of the underlying and C is the price of the derivative then delta is defined as Δ C. Delta is also the value obtained in the process of replication of a S derivative with underlying and cash resulting in: Derivative price Δ S + B. Delta is positive for calls and negative for puts. In-the-money option is more sensitive to the stock price (larger delta) than an out-of-the-money option. As time to expiration increases

5 delta is less for high stock prices and greater at low stock prices. If Black-Scholes formula holds call price equals Se δt N ( d ) Ke rt N ( d ) and the call delta is e δt N ( d ). and N ( d ) are This might seem to be merely the first approximation because N d also functions of S. But let us consider this in some more detail. Let us write ϕ for the PDF of the standard normal distribution. Note that d S d S S Se δt Ke rt Ke rt Se e δt δt Ke rt S. We have C S ( S Se δt N ( d ) Ke rt N ( d )) S S e δt N d + Se δt N ( d ) S Ke rt N ( d ) S e δt N ( d ) + Se δt ϕ ( d ) d S Ke rt ϕ ( d ) d S e δt N d + Se δt ϕ ( d ) Ke rt ϕ ( d ) + Se δt e d Ke rt e d e δt N d S π Now consider d d d d This implies that d Se δt and therefore ( Se δt ) e d e S π and d + d d + d ( ) e δt N d ( d d ) S δt Se + e d e S π ( d ( d )) d d Se δt σ Ke rt T Se δt Ke rt + σ T σ T Ke rt rt ( Ke ) d d 0 + σ T Se δt Ke rt rt ( Ke ) d σ T ( Se δt ) ( Ke rt )..

6 C S e δt N ( d ) not approximately but exactly. Also based on put-call parity P S C Se δt + Ke rt e δt N ( d ) e δt e δt ( N ( d ) ) e δt N ( d ). S Note that since P Ke rt N ( d ) Se δt N ( d ) this is the same value we would get by taking the partial derivative with respect to S and ignoring the partial derivatives of N ( d ) and N ( d ). Gamma G Defined as Γ Δ S C S. Gamma is always positive for purchased call or put. When call-put parity holds C S P S ( C P) ( S PV ( K )) 0 S S gamma is the same for put and call with the same strike and time to expiration. Gamma is close to zero for deep in-the-money and deep out-of-the-money options. Vega Defined as C. Increase of volatility increases the price of call or put so Vega is σ positive. Vega is greater for at-the-money options. It is greater for options with moderate time to expiration than with short time to expiration. Under the assumptions of the putcall parity Vega is the same for put and call with the same strike and time to expiration. Theta Defined as θ C and it measures the rate of change in option price with respect to T decrease in the time to maturity. Options generally become less valuable as the time to expiration decreases. Time decay is greatest for the at-the-money short-term options. Time decay can be positive for deep in-the-money calls on stocks with high dividends and deep in-the-money puts. Rho Defined as ρ C. Positive for call negative for put (greater interest means smaller r present value of strike price). Psi

7 Defined as Ψ C. Negative for call positive for put (greater dividend yield means δ smaller present value of stock to be received). Greek Measures for Portfolios For a portfolio containing n options with a single underlying stock with ω i quantity of the i-th option: Δ portfolio n i ω i Δ i. Same for the other Greeks. Option Elasticity Computes the percentage change in the option price relative to the percentage change in the stock price (the percentage change in the option price for a % change in stock price). Let e be a change in stock price. Then ed is the change in option price. Ω % change in the value of C % change in the value of S εδ C ε SΔ C. S For a call W (call is replicated by a levered investment in the stock) for a put W 0 (involves shorting the stock). Volatility of the option σ option σ stock Ω. This makes sense since elasticity is a measure of leverage. Let a denote the expected rate of return on the stock g the expected rate of return on the option. Since the call price C Δ S + B γ Δ S C α + Δ S C r Ωα + ( Ω)r. Thus: γ r ( α r)ω. The risk premium on the option equals the risk premium on the stock times option elasticity. risk premium Sharpe ratio for an asset volatility Sharpe ratio for stock α r σ Sharpe ratio for call Ω α r α r Ωσ σ. For a portfolio containing n options with a single underlying stock with ω i fraction invested in the i-th option: Ω portfolio ω i Ω i. n i

8 Lognormal distribution Recall that a random variable W is lognormal if its natural logarithm is normal i.e. X W N ( µσ ). In this case also W e X and E ( W ) e µ+ σ. Note that e µ+ E e X σ > e E( X) e µ. This is a special case of the Jensen s Inequality which says the following: Let h be a twice differentiable real-valued function of a real variable and let X be a random variable such that d h E X h E X E h( X) E h( X) dx h x 0 at every x at which f x X > 0. Then ( ). Under the same assumptions if d ( ). dx h x 0 then If the stock price S t is lognormal for S t S 0 e X where X the continuously compounded return from 0 to t is normal. If returns over non-overlapping time intervals of the same length are IID the mean and variance of the continuously compounded returns are proportional to time. If we assume that S t ~ N α δ σ tσ t S 0 then S t S 0 e α δ σ t + ( σ t )Z where Z is standard normal. If current stock price is S0 the probability that a call option with exercise price K expiring at time t will expire in the money is Pr ( S t > K ) N ( ˆd ) where ˆd contains α the true expected return on the stock in place of r the risk-free rate continuously comnpounded. Brownian Motion and Ito s Lemma Brownian Motion is a stochastic process that is a random walk occurring in continuous time with movements that are continuous rather than discrete. Its definition is as follows: Brownian Motion is a stochastic process Z(t) with the following characteristics: Z 0 0. Z ( t) is normally distributed with mean 0 and variance s. Z ( t) is independent of Z ( t) Z t s Z t + s Z t + s where s > 0 s > 0. In other words increments over non-overlapping time intervals are independent. Z(t) is continuous as a function of t. These properties imply that Z(t) is a martingale a stochastic process for which E Z ( t + s) Z ( t). Z t The process Z(t) is also called a diffusion process.

9 Arithmetic Brownian Motion We generalize Brownian Motion to allow an arbitrary variance and a nonzero mean: dx t αdt + σdz ( t) We say that α is an instantaneous mean per unit of time and σ is an instantaneous variance per unit of time. The term αdt introduces a nonrandom drift to the process. Since arithmetic Brownian Motion is a basically a scaled Brownian Motion we see that X ( T ) X ( 0) ~ N ( αt σ T ). In this model X may assume negative values so this is not a good model for stock prices. Ito Process This is a process where both the drift ˆα and volatility ˆσ are functions of the stock (or a capital asset in general) price X(t): dx ( t) ˆα ( X ( t) )dt + ˆσ ( X ( t) )dz ( t) Geometric Brownian Motion This is an Ito process where both the drift ˆα and volatility ˆσ are linear functions of stock price X(t); more precisely multiples: ˆα X t ( ) α X ( t) and ˆσ ( X ( t) ) σ X ( t). We can represent geometric Brownian motion as: dx ( t) α X ( t)dt + σ X ( t)dz ( t) or in this more popular form: dx ( t) αdt + σdz t X ( t). Thus the percentage change in the asset value is normally distributed with instantaneous mean α and instantaneous variance σ. This is the most popular model of stock returns. For the change over time period of length h we have: X ( t + h) X ( t) α X ( t)h + σ X ( t)y ( t) h. Thus over short period of time the character of Brownian motion is determined almost entirely by the random component. Over time interval longer than a year the mean becomes more important than standard deviation. Lognormality A variable that follows geometric Brownian motion is lognormally distributed. More precisely if X(t) satisfies dx t αdt + σdz ( t X t ) then the distribution of its logarithm is

10 ( X ( t) ) ~ N ( ( X ( 0) ) + ( α 0.5σ )tσ t) or equivalently the solution of the above differential equation is ( X ( t) X ( 0)e α 0.5σ )t+σ tz where Z ~ N ( 0). Note that if U ~ N ( µσ ) then E e U expected value of X(t) ( E X ( t) e α 0.5σ )te e σ ty ( e α 0.5σ )te 0.5σ t X ( 0)e αt. X 0 X 0 e µ + σ we can compute the Thus we can see that α is the expected continuously compounded rate of return on X. The Sharpe Ratio Recall that the Sharpe ratio for an asset is its risk premium α r per unit of volatility σ : Sharpe Ratio α r σ. We will show that two perfectly correlated assets must have the same Sharpe ratio or otherwise there is an arbitrage opportunity. Consider the following processes for nondividend-paying stocks which have the same source of randomness. ds ( t) α S ( t)dt + σ S ( t)dz ( t) ds ( t) α S ( t)dt + σ S ( t)dz ( t). Assume that Sharpe ratio for asset is greater than that for asset. We then buy σ S shares of asset and short shares of asset. These two positions will have σ S generally different costs so we invest (or borrow) the cost difference by σ σ buying (or borrowing) the risk-free bond which has the instantaneous rate of return rdt. The return on the two assets and the risk-free bond is ds ds + σ S σ S σ σ rdt α r σ α r σ dt. This demonstrates that if Sharpe ratio of asset is greater than that for asset we can construct a zero investment portfolio with a positive risk-free return. Ito s Lemma Suppose that a stock with an expected instantaneous return of ˆα dividend yield of ˆδ and instantaneous volatility ˆσ follows an Ito process: ds( t) ( ˆα ( S( t)t) ˆδ ( S( t)t) )dt + ˆσ ( S( t)t)dz ( t). If C( S( t)t) is a twice-differentiable function of S( t) then the change in C dc S( t)t is

11 dc( S( t)t) C S ds + C S ( ds ) + C t dt When working with Ito processes and Ito Lemma we could use the following multiplication rules : dt dz 0 0 dt. dt dz In the above Z(t) is the standard Brownian motion (i.e. not arithmetic or geometric Brownian motion). The reasoning behind these multiplication rules is that the multiplications resulting in powers of dt greater than vanish in the limit. In the case of geometric Brownian motion: ds( t) S( t) ( α δ )dt + σdz ( t ) Ito lemma becomes: dc( St) ( α δ )SC S + σ S C SS + C t dt + σ SC S dz since ( ds) σ S ( dz) σ S dt. The Black-Scholes Equation The process of valuation of securities in a continuous time model uses differential equations. Consider the basis equation relating the values of a stock at time t to the value of the same stock at time t + h: S ( t)( + r h ) D( t + h)h + S ( t + h). It can be rewritten as S( t)r D( t + h)h h + S ( t + h ) S( t). Return on stock Cash payment Change in stock price Dividing by h and letting h à 0 results in a differential equation: ds ( t) + D( t) rs ( t) dt with r being the continuously compounded rate of return as always. We obtain the differential equation that describes the evolution of the stock price over time to generate an appropriate rate of return. For dividend-paying stocks If the value of the stock at time T is S that is the terminal boundary condition: S(T ) S and the solution is

12 De S t T t ds r s t + Se. For bonds Let S(t) represent the price of a zero-coupon bond that pays $ at T. We have ds t + D( t ) rs ( t). dt 0 because no income is paid The general solution to this differential equation is S t Ae. With the terminal boundary condition S(T) $ the particular solution for the bond value is S t e Now consider the problem of creating a riskless hedge for an option position through trading in shares and bonds. Assume that the stock price follows geometric Brownian motion: ds S ( α δ )dt + σdz where α is the expected return in the stock σ is the stock s volatility and δ is the continuous dividend yield. If we invest W in riskless bonds the change in value of the bond is dw rwdt. Let I denote the total investment in the portfolio N stocks and W in the risk-free bonds so that the total investment is zero with V standing for the value of the derivative security I V (S t) + NS + W 0. Applying Itô s Lemma. di dv + N(dS + δsdt) + dw V t dt + V S ds + σ S V SS dt + N(dS + δsdt) + rwdt. Since the option delta is V S set N V S this will make W V S S V. Also di 0 because with zero investment the return should also be zero. Substituting all these and dividing by dt we obtain: V t + σ S V SS + ( r δ )SV S rv 0. This is the Black-Scholes stochastic partial differential equation (PDE) for any contingent claim assuming Underlying asset follows constant volatility geometric Brownian motion. Underlying asset pays a continuous proportional dividend at the rate δ. The contingent claim itself pays no dividend. The interest rate is fixed with equal borrowing and lending rates. There are no transaction costs. The call pricing formula Se δ (T t ) N(d ) Ke r(t t ) N(d ) satisfies the Black-Scholes PDE using the appropriate boundary condition. Similarly it can be shown that the put pricing formula and the formulas for asset-or-nothing options cash-or-nothing options and gap options satisfy the Black-Scholes PDE using the

13 appropriate boundary condition in each case. Black-Scholes PDE can be generalized to the case when one uses equilibrium expected return on the underlying asset instead of the risk-free return. Risk-neutral pricing The Black-Scholes partial differential equation does not contain the expected stock return α but only the risk-free rate and this implies that the equation must be consistent with any world in which no arbitrage is possible. The actual expected change in the option price is E dv dt dt E ( dv ) V t + σ S V SS + ( α δ )SV S. Let E* denote the risk-neutral expected value. Under the risk-neutral valuation the expected change in the stock price is E *( ds) ( r δ )dt. The drift of the option price can therefore be written as dt E *( dv ) V t + σ S V SS + ( r δ )SV S. If we compare this with the Black-Scholes equation V t + σ S V SS + ( r δ )SV S rv 0 we conclude that the equivalent form of the Black-Scholes equation is: dv dt E* rv indicating that the option appreciates on average at the risk-free rate. The solution of the equation is equivalent to computing the expected payoff of the derivative security and discounting it at the risk-free rate. For a simple European call option on a stock that pays dividends at a constant continuous rate δ that call s price is S T e K ( K ) f * s ds where s is a specific value of S(T). The integral form of the Black-Scholes equation is also called the Feynman-Kac equation. Option pricing Another approach to option pricing is to impose the condition that the actual expected return on the option must equal the equilibrium expected return. To avoid arbitrage two assets with returns generated by the same dz must have the same Sharpe ratio i.e. α r σ Ω Ω α option r σ option where σ option Ω σ and the option s elasticity is

14 % change in option price Ω % change in stock price εv S V εs SV S V S V V S. This produces the equilibrium condition that the option must obey: ( α r) α option r SV S V This means that the option must be priced so that its expected return is related to the expected return on the stock in a particular way. Under the Black-Scholes framework the price of an option V must satisfy Black-Scholes equation V t + σ S V SS + ( r δ )SV S rv 0 or else there is an arbitrage opportunity. Moreover the pricing formula must satisfy the appropriate boundary conditions which depend on the type of the option. Consider a European call option. Its Black-Scholes pricing formula is V S ( t)t N ( d ) Ke N ( d ) where d Se ( T t ) Se δ Ke + σ T t S K + r δ + σ T t d d. It satisfies the Black-Scholes equation and we will check for the boundary conditions: V ( S ( T )T ) max ( 0 S ( T ) K ). When t T e e. As t T we have: S > K S S K K > 0 + N ( d ) N ( d ) V S K. S < K S S K K < 0 N ( d ) N ( d ) 0 V 0. All or Nothing Options An asset-or-nothing call pays one share of stock if S T denote its value at time t by V 3 S( t)t > K and 0 otherwise. We will or just V 3. Note that V 3 ( S ( T )T ) S ( T ) if > K and 0 otherwise. There is also an asset-or-nothing put which pays one share < K and 0 otherwise. We will write V 4 ( S( t)t) or just V 4 for its value > K and 0 otherwise. We will denote ( S( t)t) or just V 5. Notice that V 5 ( S( T )T ) if S ( T ) > K S T of stock if S T at time t. A cash-or-nothing call pays $ if S T its value at time t by V 5 and 0 otherwise. There is also a cash-or-nothing put which pays $ if S T < K and 0

15 otherwise. We will write V 6 ( S( t)t) for its value at time t. These options: asset-or-nothing call asset-or-nothing put cash-or-nothing call and cashor-nothing put are also called binary options. The prices of them under Black-Scholes framework are: V 3 Se N ( d ) V 4 Se N ( d ) V 5 e N ( d ) V 6 e N ( d ) where d and d are calculated using the standard formulas in the Black-Scholes framework. Note also that the event S T { K} has zero probability so that V 3 + V 4 Se as well as V 5 + V 6 e. Note also that V 4 Se N ( d ) Se N ( d ) V 3 e V 6 e N d Se ( ) e N d V 5. Proof that V 3 V 4 V 5 and V 6 are solutions of the Black-Scholes equation Recall the Black-Scholes equation V t + σ S V S + r δ V S rv 0. S Consider V Se N ( d ) where ( Se) Ke + σ ( T t) d ( T t) σ S K + σ r δ + σ T t Note that d ( T t) t t σ S K + σ r δ + σ ( T t ) ( ) σ ( T t ) 3 S K + ( ) σ r δ + σ ( T t ) S T t K r δ + T t σ ( T t ) S T t K + r δ + σ ( T t ) r δ + T t σ ( T t ) d r δ + T t σ.

16 and d S S. Therefore δ V Se ( T t ) N d t t and ( ) δse N ( d ) + Se δse N ( d ) + Se ( ) V S Se δ ( T t) N d S V S e N ( d ) + Se e N ( d ) + e e Based on this N d e e d + e π Sσ d d d π d t d π d T t ( Se) e d π π d S d π d π S T t + ( e) σ π S e e d T t e d r δ + σ. π d S π S d

17 V t + σ S V S + r δ V S S rv δse N ( d ) + Se + σ S e + ( r δ )S e N ( d ) Se +Se d N d d π d T t Se π S e + e ( r) δ + r δ + 0 d π d r δ + π σ + d π S d + ( rse ) N d d π d r δ + T t σ + σ d T t + r δ 0. Now consider V 4 Se V 3. Because V 3 is a solution of the Black-Scholes Equation all we need to show is that Se is a solution of it. We have ( Se δ ( T t) ) ( Se δt e ) δt Se δt δe δt δ δse ( T t ) t t ( S Se) e Se δ ( T t) S and thus for V Se S e 0 V t + σ S V S + r δ V S S rv δse ( r δ )S e rse Now consider V e N d ( Se) d Note that Ke where σ T t T t 0 0 σ S K + σ r δ σ T t.

18 d t d r δ + T t d t d + r δ + σ + T t d T t + σ r δ + σ + d T t + σ r + δ σ + σ σ and d S d S S. Therefore V ( e N ( d )) t t and V S re N ( d ) + e ( ) r( T t) V e N d S S e T t σ σ σ σ re N ( d ) + e e d π S S d T t ( ) d r δ T t σ d π d t d π d T t ( e) e d π d π d S e e d e π d S S + e e Based on this d π d S σ T t e e d e d e d π S. r δ σ π S π S

19 V t + σ S V S + r δ V S S rv re N ( d ) + e + σ S e + ( r δ )S e e N d +e ( r r) + 0 d d π d T t e π d S σ T t d π S e d r δ π σ + d ( re ) N d π S + d π d r δ T t σ d T t σ + r δ 0. This proves that the formula for the value of the cash or nothing call is a solution of the Black-Scholes equation. Now consider V 6 e V 5. Because V 5 is a solution of the Black-Scholes Equation all we need to show is that e is a solution of it. We have ( e r( T t) ) ( e rt e ) rt e rt re rt r re ( T t ) t t 0 S e e r ( T t) 0 S and thus for V e V t + σ S V S + r δ V S S rv re 0. re Note that because derivative is a linear functional this implies that any linear combination of these two formulas is a solution of the Black-Scholes equation and one such linear combination is the Black-Scholes formula for the price a European call obtained as a linear combination of V 3 and V 5 while another one is the Black-Scholes formula for the price of a European put obtained as a linear combination of V 4 and V 6 : C Se r N d ( T t ) N d P Ke N d Ke Se 0 V 3 K V 5 K V 6 V 4. N d

20 In the process of proving the statements above we also obtained some values of Greeks for these binary options Delta of an asset-or-nothing call: V 3 S ( e) N ( d ) + e d π Delta of an asset-or-nothing put: e V 4 S ( S Se) V 3 Delta of a cash-or-nothing call: V 5 S ( e) e d π S Delta of a cash-or-nothing put: N d e d π V 6 S ( S e) V 5 V 5 S ( e) e d π S Gamma of an asset-or-nothing call: V 3 e δ ( T t) e d S π S ( e) e d π S d Gamma of an asset-or-nothing put: V 4 Se δ ( T t) V 3 V 3 S S S d e π S + e Gamma of a cash-or-nothing call: V 5 e r ( T t) e d S π d S σ T t Gamma of a cash-or-nothing put: V 6 e r ( T t) V 5 V 5 S S S e d π d S σ T t Theta of an asset-or-nothing call: e d e + e π S d e d e d π S π S

21 V 3 V t T t 3 δse N ( d ) Se Theta of an asset-or-nothing put: V 4 4 Se δ ( T t) V 3 T t t t V δse δse + δse N ( d ) + Se d π d T t + ( Se) e d π + V 3 t d π d T t ( Se) e d π δe N ( d ) + Se d π d T t ( Se) e d π Theta of a cash-or-nothing call: V 5 V t T t 5 re N ( d ) e and theta of a cash-or-nothing put: V 6 6 e T t t t V re + re N ( d ) + e re N ( d ) + e ( r( T t) V ) 5 re d π d T t e r δ + σ d π d T t + ( e) e d π + V 5 t d π d T t ( e) e d π e d r δ + σ r δ + σ σ T t r δ σ r δ σ σ r δ π σ. T t Interest Rate Models We model a zero-coupon bond by assuming that the price of a unit zero coupon bond P( tt ) follows an Ito process: dp( rtt ) P( rtt ) α ( rtt )dt + q ( rtt )dz. Here the coefficients cannot be constant. They have to be carefully modeled to reflect the bond boundary condition: has to be worth $ at maturity as well as the fact that the volatility of the bond should decrease as the bond approaches maturity. Equivalently we can assume that the short-term interest rate follows the Ito process:

22 where Z t dr a( r)dt + σ ( r)dz { } is a standard Brownian motion under the true probability measure. We call it the true stochastic process of the short rates. Now we can use Ito Lemma and get an expression for α ( rtt ) and q( rtt ) We have dp( rtt ) P r dr + a r P P ( dr) + P r t dt r + P r σ ( r ) + P t dt + P r σ r dz. + P( tt ) + W of one bond maturing at time T We analyze a portfolio I N P tt which we delta-hedge by buying N bonds maturing at T and we finance the difference by borrowing W. We use Ito Lemma and set di 0. This gives us two equivalent formulas. First we get that the Sharpe ratio for two bonds is equal (both bond prices are driven by the same random term dz they must have the same Sharpe ratio if they are fairly priced) α rtt r r q( rtt ) φ( rt) α rtt q( rtt ) Equivalently we obtain the partial differential equation that must be satisfied by any zero-coupon bond: ( σ ( r) P ) r + ( a( r) σ ( r )φ ( rt )) P r + P rp 0. t The risk-neutral process for the interest rate is obtained by subtracting risk premium from the drift: dr ( a( r) σ ( r)φ( rt) )dt + σ ( r)dz. Different specifications of a( r) σ ( r) and φ rt models. lead to different bond pricing The Rendelman-Bartter Model We assume here that the short-rate follows geometric Brownian motion: dr ardt + σrdz. Interest rates can never be negative in this model but they can be arbitrary high (while in reality they should exhibit mean reversion if rates are high we expect them on average to decrease). The Vasicek Model We assume here that the short-rate follows Brownian motion with mean reversion: dr a( b r)dt + σdz. Since the random term dz is multiplied only by σ the variability of rates is independent of the level of rates and the rates can become negative. In this model the Sharpe ratio is constant:

23 r q( rtt ) α rtt φ( rt) φ The price formula that solves the partial differential equation for zero-coupon bonds is: P( tt r( t) ) A( tt )e B( tt )r( t) where e r B tt ( +t T ) B σ e a T t 4a A tt B tt a r b + σφ a σ a where r is the yield to maturity of an infinitely lived bond a consol. Note that we can remember the formula e a T t B tt a in the Vasicek model P( tt r( t) ) A( tt )e B( tt as a T tδ a the present value of a continuous annuity-certain of rate payable for T t years and evaluated at force of interest a where a is the speed of mean reversion for the associated short-rate process. a T tδ a v ( T t) ( T t ) e δ ( T t ) e a. δ δ a The Cox-Ingersoll-Ross Model We assume here that the short-rate follows the following Ito process: dr a b r dt + σ rdz. Here the volatility of interest rates is proportional to r and thus the rates cannot become negative. They also exhibit mean reversion. In this model the Sharpe ratio is of the form: α rtt r q( rtt ) φ( rt) φ r σ. The price formula that solves the partial differential equation for zero-coupon bonds is similar to that of Vasicek model P tt r( t) e B( tt )r( t) A tt and B( tt ) are more complicated. Note that like in Vasicek but the formulas for A tt model both functions A and B depend on (T t) rather than on t and T separately.