(atm) Option (time) value by discounted risk-neutral expected value

Transcription

1 (atm) Option (time) value by discounted risk-neutral expected value Model-based option Optional - risk-adjusted inputs P-risk neutral S-future C-Call value value S*Q-true underlying (not Current Spot (S0) probability spot Max(S-X,0) =P*C probability option!) value Exercise Price (X) % % Domestic Rate (R) 6.00% 24% % Yield [Foreign Rate] (Y) 6.00% 37% % Standard deviation (VOL) 10.00% 26% % Maturity (T) 1.0 7% % Probability Sum 100% 0 100% Expected value N/A Expected value with yield " Discount at rate Riskless R = 6.0% R=6.0% ER = 10% Expected return Discounted value - PV ~100.0 ~3.74 ~100.0 Discrete Probability Density 40% 30% 20% 10% 0% Binomial - 3

2 12) Four Periods in T maturity (set N input to 4, H should be 0.25, and y = 6%) [PgDn] for easy spot tree generation. S0*U S0*U^2 S0*U^3 S0*U^2*D S0*U^4 S0*U^3*D S0 S0*U*D S0*U^2*D^2 S0*D S0*D^2 S0*U*D^2 S0*D^3 S0*U*D^3 S0*D^4 Binomial - 26

3 16) Valuation by the Risk-Neutral Density (Direct valuation of terminal payouts) # of node paths Each period, prob. up, P, prob down, 1-P, (A) (B) (C) (D=A*C) (E=B*D) Outcomes Final Prob. of Prob. Of path # * Risk-neutral & count # Value a path a path path prob. Expectation U P^4 5.65% 5.65% U3D P^3*(1-P) 5.94% 23.75% U2D P^2*(1-P)^2 6.24% 37.45% UD P*(1-P)^3 6.56% 26.25% D (1-P)^4 6.90% 6.90% Sum % Discounted *EXP(-R*T) [PgDn] for martingale discussion (minding p's and q's) Binomial - 30

4 17) Minding p's and q's - Martingales and Measure-a 1 Period Risk Neutral True Measure Expected Risk neutral-r 6.00% Risk premium (RP) 4.00% volatility Vol 10.00% Vol 10.00% er=rp+r 10.00% prob. up p 48.75% q 58.80% =(exp((r-y)*h)-d)/(u-d) =(exp((er-y)*h)-d)/(u-d) discounted spot value =(p*su+(1-p)*sd)*exp(-(r-y)*h) =(q*su+(1-q)*sd)*exp(-(er-y)*h) 4 Periods p-prob. p*paths* q-prob # paths* q*paths* Spot-Paths of each spot value of Each q-prob. Spot Value % % 11.95% % % 33.50% % % 35.21% % % 16.45% % % 2.88% Expected Total=100% "Discount" EXP(-(R-Y)T) EXP(-(er+R-Y)T) [PgDn] for appropriate option value measure.. Binomial - 31

5 20) Generalizing the model to even more periods is a straight-forward extension. Using the "tree structures" defined in the "The derivative with multiperiods" spreadsheet sections (right triangle with the hypotenuse on upper left, and right angle on lower right) makes copying by macros relatively easy. As in the two period model example, American options are treated by augmenting valuation at each node with an early exercise value check. Direct European option valuation with maturity probabilities can be handled by calculating the probability of each path and using the binomial formula to determine how many paths go to a particular outcome. [Pgup] for analysis Binomial - 34

6 BINOML.XLS - James N. Bodurtha, Jr. Copyright ALL RIGHTS RESERVED, version 4.1 Spreadsheet works through example calculations of the Cox-Ross-Rubinstein (1979) binomial approach to options valuation. The constant dividend yield adjustment follows Bodurtha-Courtadon (1987). Column J is set up for filling in answers and printing. Users of this program must be aware that it is subject to error. Under no circumstances will any party involved in its development or dissemination be liable for any special, consequential or incidental damages arising from its use. STRIKE [Home] TO BEGIN. Binomial - 35

7 Algebraically, this equation can be written as A Binomial Model and It's Convergence Cox-Ross-Rubinstein (1979) or Cox-Rubinstein (1985) Risk neutral Each potential probability of C = * cash flow at * e the potential maturity cashflow C = n n! j= 0 j!(n j)! p j 1 p n j Max 0, u j d n j r t S X e ( ) 0 In this equation, p = e ( r d y) h d u d n! = j!( n j)! p j ( 1 p) n j j n j Max 0,u d S X 0, the risk-neutral probably of an up move the number of paths to a particular outcome at maturity, and j indexes number of up moves = the risk neutral probability of each of the paths leading to the same outcome at maturity (j ups) = the call value associated with the outcome Binomial - 36

8 Define one additional parameter, a, such that: j n j An Alternative Binominal Model Max 0,u d S X = 0 for j < a and Max 0,u d S X = u d S X for j a 0 j n j j n j The parameter, a, represents the minimum number of upward movements in the spot price necessary for the call option to finish in the money after n movements in the spot price. Substituting in the above equation, we obtain: yt C(X, t) = S0e Ψ (a; n, p' ) Xe Ψ (a;n, p) 0 where Ψ ( a; n, p ) = n j= a n! j!( n j)! p j ( 1 p) n j (the cumulative binominal distribution) ( r y) h ( r y) h p' = ue p, and ( 1 p') = de ( 1 p) (Note, the prime over p does not indicate a derivative.) In turn, the binomial put pricing model is the following: P(X,t) = Xe [1 Ψ (a; n, p)] S e [1 Ψ (a;n, p')] 0 yt Binomial - 37

9 Convergence to Black-Scholes u = the multiplicative factor by which the spot price will increase if it appreciates (u-1 σ h is the percentage change); u is set equal to e d = the multiplicative factor by which the spot price will decrease if it depreciates (d-1 σ h is the percentage change); d is set equal to e ~q = e ( µ + σ 2 2) u d h d binomial up probability, with limiting value q µ = σ µ = the instantaneous drift of the Brownian motion, followed by the continuously compounded rate of change in the spot price σ = the volatility of the Brownian motion ~ µ = the mean one period rate of change in the currency price implied by the binomial model ~ σ = the standard deviation of the one period rate of change in the currency price implied by the binomial model. h Binomial - 38

10 With the previous definitions, True Distribution Convergence ~ u µ n = q ln + ln d n µ t d [ ( )] 2 u [ ] ~ 2 σ ( 1 ) ln σ 2 2 n = q q n = µ h t σ t, as n d [ ] 1 Ψ ( j; n, q) Pr ob S * < Z = N S* is the future spot currency price S is the spot price today, j is the smallest integer such that ln(z/s) = j ln(u/d) + n ln d. [ ln ( Z µ t / S) ] σ t Log-normal Probability Density 40% 35% 30% 25% 20% 15% 10% 5% 0% Binomial - 39

11 as n, ~ 2 σ p n = p ( 1 p ) 1 n u d Risk-Neutral Distribution Convergence ~ u µ p n = p ln + ln d n 1 2 r y σ t d 2 2 n σ 2 [ 1 ( / ) + ( σ / 2) ] Ψ ( a; n, p) N n S X r y t σ t yt C( S, t) = Se N( x + t ) Xe N( x) 2 t, p is the risk-neutral probability, as above. σ,, and N(.) is the cumulative normal distribution x = S X r y t σ t 2 ln( / ) + σ /2, yt C( S, t) = e N( x + σ t ) > 0, Cash = Xe N( x) Puts -- P S t Xe rt yt (, ) = [ 1 N ( x) ] Se [ 1 N ( x + σ t ) ] yt [ ] P( S, t) = e N( x + t ) < σ 1 0, Cash = Xe [ N(x) 1] Binomial - 40

12 Binomial (puts - substitute P u for C u, etc.) Delta Δ = Spot Option Risk Management Parameters Cu Cd ( u d) S e uc dc Cash B = ( u d) Gamma Γ Vega = = C Call Theta Θ = Put Theta Θ = Rho ρ = o e yh d u rh Δu S u Δ S d d ( σ ) C ( σ ) u u d d u d Black-Scholes: See previous page See previous page yt ( ) e N x+σ t Sσ t yt σ σ Sσ te N ( x σ t) Cud C 2 t / n P P 2t / n o + yse, N is the normal density +, N is normal density e ( +σ t ) rxe N(x) σxn (x) N x ud o yt ( σ ) ( ) ( ) C r C r r r u u d d u d yt [ ] + yse N x + t 1 rxe yt e 2 [ N(x) 1] σxn' (x) 2 = = Calls Xte N(x), Puts Xte N( x) t t Binomial - 41