Chapter 6: The black Scholes pricing formula Flashcards
Lemma 15.
process followed by f, value f derivative
Assume that a stock price S follows the Geometric Brownian motion
dS = µS dt + σS dB
where µ and σ are constants.
Let f = f(S, t) be the value at time t of any derivative contingent on the value of S at some t = T .
Assume f(s, t) is twice differentiable with respect to s and differentiable with respect to t. The process
followed by f is
df = [(∂f/∂S) µS + (∂f /∂t) + (1/2)(∂^2f/∂S^2)σ^2S^2] dt +
(∂f/∂S) σS dB.
(by Apply Ito’s Lemma with a(S, t) = µS and b(S, t) = σS.)
DISCRETE VERSION OF
process followed by f, value f derivative
∆f = [(∂f/∂S) µS + (∂f /∂t) + (1/2)(∂^2f/∂S^2)σ^2S^2] ∆t +
(∂f/∂S) σS∆B.
where ∆B is a normally distributed random variable with zero mean and variance ∆t
THM 16
portfolio of variable quantity of shares ∂f/∂S of shares and −1 derivatives; no arbitrage
∂f/∂t + rS ∂f/∂S + (1/2)σ^2S^2 ∂^2f/∂S^2 = rf
boundary conditions
To obtain prices from the Black-Scholes PDE we impose boundary conditions,
e.g., for a European
call option with strike X and expiring at time T the boundary condition is f(S, T ) =
max{S − X, 0} for all S.
For a European put option with strike X and expiring at time T the
boundary condition is f(S, T ) = max{X − S, 0} for all S.
unique sol
a unique solution by
imposing a further boundary condition at S = 0. with one already
if S_t = 0 at any time 0 ≤ t ≤ T , S_t = 0 for all t ≥ T and hence f(0, t) = e^{−r(T −t)} f(0, T ) which provides us with a boundary condition. (BANKRUPT)
E.g., European call options, f(0, t) = 0. (For numerical purposes we may impose boundary
conditions at S = ∞).
EXAMPLE: f(S, t) = e^{rt}S^{1−{2r/σ^2}}
is a solution of the Black-Scholes PDE (checked)
Consider a derivative on certain stock with single payoff at time T > 0 amounting to S_T ^{1−{2r/σ^2})
where r is the constant interest rate. Assume that 1 −{2r/σ^2} > 0. Find the value of the derivative
at time 0 ≤ t ≤ T .
Consider a portfolio consisting of e^rT derivatives and write v(S, t) for the price of this portfolio;
we have v(ST , T ) = e^{rT} S_T ^{1−2r/σ^2) = f(S_T , T ).
Now,
• Both v(S, t) and f(S, t) are solutions of the Black-Scholes PDE, and
• v and f have the same values at the boundary: v(0, t) = f(0, t) = 0, and v(S, T ) = f(S, T ).
But there is only one solution of the Black-Scholes PDE satisfying a given boundary condition and
this forces v(S, t) = f(S, t) = e^{rt}S^{1−{2r/σ^2}}
So the price of one derivative is e^{rT}e^{rt}S^{1−{2r/σ^2}}
= e^{-r(T-t)}S^{1−{2r/σ^2}}
.
Theorem 17 (The Black-Scholes pricing formulas).
Consider a European option at time t on stock with spot price S, with strike price X and expiring at time T .
Let σ be the annual volatility of the stock, and r the T -year interest rate. Define
d_1 = [log(S/X) + (r + (σ^2/2))(T-t)]/σ(√(T − t))
d_2= [log(S/X) + (r- (σ^2/2))(T-t)]/σ(√(T − t))
= d_1 − σ√(T − t)
Then the price of the call option at time t is
c =
SΦ(d_1)− Xe^{−r(T −t)}Φ(d_2)
and the price of the put option is
p =
Xe^{−r(T −t)} Φ(−d_2) − SΦ(−d_1)
where Φ is the standard normal distribution function
risk-aversion
risk-aversion of
investors does not affect the value of derivatives given as solutions of the Black-Scholes PDE. Since
this formula holds regardless of amount of risk-aversion and we might as well assume that investors
are risk neutral.
BS PDE is
(∂f /∂t) + rS(∂f /∂S) + (1/2)((∂^2f /∂S^2) σ^2S^2 = rf
Notice that µ does not appear here! The quantity µ − r is the excess return that investors demand
when investing in an asset whose volatility measure is σ.
risk neutral investors
Risk-neutral investors only care about the expected return of their investments, and they do not care about uncertainty regarding these returns. Hence all investments in a risk neutral world must have the same expected return r equal to the risk-free interest rate.
Definition (The principle of risk-neutral valuation)
Let f be the price of a derivative which
pays H(S_T ) for some function H at a future time T . In our risk-neutral world the stock price has expected return r, the risk-free T -year interest rate. In a risk-neutral world the current value of the
derivative f(S, 0) is the present value of the expected value of the derivative payoff at time T , i.e.,
f(S, 0) = e^{−rT} E~ ( H (S_T ))
where E denotes expected values in our risk-neutral world.
digital or binary options
Consider a derivative on a stock which at expiration time T pays £1 if ST ≤ a, for some positive number a, and zero otherwise
Let the
volatility of the stock price be σ and assume all interest rates are constant and equal to r.
Apply a
risk neutral valuation argument to show that, for any 0 ≤ t ≤ T , the value of this derivative equals
e^{−rT}Φ [[log(a/S) - (r- (σ^2/2))(T-t)]/σ(√(T − t))]
where Φ is the cumulative distribution function of the standard normal distribution.
digital or binary options
Apply a
risk neutral valuation argument to show that, for any 0 ≤ t ≤ T , the value of this derivative equals
e^{−rT}Φ [[log(a/S) - (r- (σ^2/2))(T-t)]/σ(√(T − t))]
where Φ is the cumulative distribution function of the standard normal distribution.
We are assuming S follows the process
dS = µSdt + σSd
for constants µ and σ, and so at time 0 ≤ t ≤ T , log S_T is normally distributed with mean log S +(µ − (σ^2/2))(T − t) and standard deviation σ√(T − t).
In a risk neutral world we set µ = r and now log S_T is normally distributed with mean log S + (r −(σ^2/2))(T − t) and standard deviation σ√(T − t).
The event S_T ≤ a is equivalent to the event log S_T ≤ log a and so the probability in this risk
neutral world of the event S_T ≤ a is Φ [(log (a)−(log S+(r− (σ^2/2)(T −t))] / [σ√ (T −t)])
= Φ ([log (a/S)−(r− σ^2/2)(T −t))]/[σ√(T −t)])
In our risk neutral world the value of the derivative is the present value of the expected value of its payoff,
ie
e^(-r(T-t) * Φ ([log (a/S)−(r− σ^2/2)(T −t))]/[σ√(T −t)])
Lemma 18
∫_[X, ∞ ] (y-X)φ(y) .dy
Se^{rT}Φ(d_1) - XΦ(d_2)
where
d_1 = [log (a/S)+(r+ σ^2/2)T]/[σ√(T )])
d_2 = [log (a/S)+(r− σ^2/2)T]/[σ√(T )])
=d_1 - σ√(T )
and thus
c = SΦ (d_1) − e^{−rT} XΦ (d_2)
in our risk neutral world (Back to European call/put options)
assumption:
*logS_T is normally distributed with mean log S + (µ −σ^2/2)T and
variance σ^2 *T
*in our risk neutral valuation argument we set µ = r
- Risk-neutral investors also
expect the current value of the derivative f(S, 0) to be the expected value of the present value of
the derivative payoff at time T , i.e. f(S,0) = e^{-rT} E~ (H(S_T)) - case where H(y) = max{y − X, 0} with X being the strike price of the call option.
We have c = e^{-rT} E~ (max{ST − X, 0})
*
Let φ be the density function of the lognormal random variable ST in our risk neutral world
c = e^{-rT}*∫_[X, ∞ ] (y-X)φ(y) .dy
lemma 18 and put option
a similar argument shows that the price of a European put option with strike price X is
p = Xe^{−rT}Φ(−d_2) − SΦ(−d_1
