Reduction to parabolic equation with constant coefficients

∂V

∂t +1

2σ²S²∂²V

∂S² + rS∂V

∂S − rV = 0, (0, T ] × (0, ∞). (3.12) This P.D.E. is a parabolic equation with variable coefficients. Notice that this equation is invariant under S → λS. That is, it is homogeneous in S with degree 0. We therefore make the following change-of-variable:

x = logS E.

The fraction S/E makes x dimensionless. The domain S ∈ (0, ∞) becomes x ∈ (−∞, ∞) and

∂V Next, let us reverse the time by letting

τ = T − t.

Then the Black-Scholes equation becomes

∂V

∂τ = 1 2σ²∂²V

∂x² +

 r −1

2σ² ∂V

∂x − rV.

We can also make V dimensionless by setting v = V /E. Then v satisfies

∂v

∂τ = 1 2σ²∂²v

∂x² +

 r − 1

2σ² ∂v

∂x− rv. (3.13)

The initial and boundary conditions for v becomes

v(x, 0) = ¯Λ(x) = Λ(Ee^x)/E,

• call option: ¯Λ(x) = max{e^x− 1, 0}, v(−∞, τ ) = 0, v(x, τ ) → e^x− e^−rτ as x → ∞;

• put option: ¯Λ(x) = max{1 − e^x, 0}, v(−∞, τ ) = e^−rτ, v(x, τ ) → 0 as x → ∞.

Our goal is to solve v for 0 ≤ τ ≤ T . 3.4.2 Further reduction

Let us divide the v equation by σ²/2:

2 σ²

∂v

∂τ = ∂²v

∂x² + 2r

σ² − 1 ∂v

∂x −2r σ²v.

Let us make a change of variable and define the following new constants:

s = τ /(1 2σ²) a = 1 − r/(1

2σ²) b = r/(1

2σ²) Then the equation becomes

vs+ avx+ bv = vxx. (3.14)

The part, vs+ av_xis call the advection term. The term bv is called the source term, and the term vxxis called the diffusion term. The advection part:

v_τ+ av_x = (∂_τ+ a∂_x) v is a direction derivative along the curve (called characteristic curve)

dx dτ = a.

This suggests the following change-of-variable:

y = x − as s⁰ = s.

Then the direction derivative become

∂_s⁰ = ∂_s+ a∂_x

∂y = ∂x

Hence the equation is reduced to

vs⁰ + bv = vyy.

Since s⁰ = s, we will still use s below instead of s⁰. Therefore, the equation becomes vs+ bv = vyy.

Next, the equation vs+ bv suggests that v behaves like e^bsalong the characteristic curves. Thus, it is natural to make the following change-of-variable

v = e^−bsu.

Then

∂su = ∂s(e^bsv) = e^bs∂sv + be^bsv = e^bs(∂s+ b)v = e^bs∂²_yv = ∂²_yu.

Thus, the equation is reduced to

us= uyy, y ∈ (−∞, ∞), s > 0.

This is the standard heat equation on the real line. Its solution can be expressed as u(y, s) =

Z ∞

−∞

√1

4πse^{−(y−z)24s} f (z) dz, s > 0, where f is the initial data.

3.4.3 Black-Scholes formula

Let us return to the Black-Scholes equation (5.6). Thus, the solution v

v(x, τ ) = e^−rτ Z ∞

−∞

√ 1

2πσ²τe⁻

(x−z+(r− 12σ2)τ )2

2σ2τ Λ(z) dz¯ (3.15)

In terms of the original variables, with the change of variable S⁰ = Ee^z, we have the following Black-Scholes formula:

V (S, t) = e^{−r(T −t)} Z ∞

0

1

p2πσ²(T − t)S⁰ e⁻

(ln( S

S0)+(r− 12σ2)(T −t))2

2σ2(T −t) Λ(S⁰) dS⁰ (3.16)

We may express it as

This is the transition probability density of an asset price model with growth rate r and volatility σ. In other words, V is the present value (deducted by e^{−r(T −t)}) of the expectation of the payoff Λ under an asset price model with volatility σ and growth rate r. It is important to note that it depends on r, not on the growth rate of the underlying asset. We shall come back to this point later.

3.4.4 Special cases

1. European call option. The rescaled payoff function for a European call option is Λ(z) = max{e¯ ^z− 1, 0}. This can be integrated. Finally, we get the exact solution for the European call option

c (S, t) = SN (d1) − Ee^{−r(T −t)}N (d₂), (3.20)

The calculation is done as below. Let us use the following notations for abbreviation.

a = r − σ²/2, D = σ²τ, d1

I =

• The formula is a modification of the previous lower bound formula. The function N is the error function. N (+∞) = 1. The two parameters d1− d₂ = σ√

T − t, the standard deviation of the asset price model.

Exercise. Show that

(a) N (0) = 1/2, N (x) → 1 as x → ∞;

(b) N (x) + N (−x) = 1 for any x ∈ R Exercise. Prove the formula (3.20).

2. European put option. Recall the put-call parity

c + Ee^{−r(T −t)}= p + S.

We can obtain the price for p from c:

p(S, t) = Ee^{−r(T −t)}N (−d₂) − SN (−d1). (3.24) Exercise. Prove (3.24).

3. Forward contract Recall that a forward contract is an agreement between two parties to buy or sell an asset at certain time in the future for certain price. Its value V also satisfies the B-S equation. The payoff function for such a forward contract is

Λ(S) = S − E.

In terms of the rescaled variable, ¯Λ(z) := Λ(Ee^z)/E = e^z−1. Let x = ln(S/E), τ = T −t) and u(x, τ ) = e^rτV (Ee^x, t)/E. Then u satisfies the advect heat equation. Its solution with initial data ¯Λ(x). Its solution is given by

u(x, τ ) = 1

√ 2πσ²τ

Z ∞

−∞

e⁻

(x−z−(r−σ2/2)τ )2

2σ2τ (e^z− 1) dz

= e^x+rτ− 1.

Hence,

V (S, t) = S − Ee^{−r(T −t)}. (3.25)

This means that the current value of a forward contract is nothing but the difference of S and the discounted E. Notice that this value is independent of the volatility σ of the underlying asset.

Exercise. Show that the payoff function of a portfolio c − p is S − E. From this and the Black-Scholes formula (3.16), show the formula of the put-call parity.

4. Cash-or-nothing. A contact with cash-or-nothing is just like a bet. If S_T > E, then the reward is B. Otherwise, you get nothing. The payoff function is

Λ(S) =

 B if S > E 0 otherwise.

Using the Black-Scholes formula (3.16), we obtain the value of a cash-or-nothing contract to be

V (S, t) = Be^{−r(T −t)}N (d₂). (3.26) Exercise. Verify this formula.

5. Supershare. Supershare is a binary option whose payoff function is defined to be Λ(S) =

 B if E1< S < E2

0 otherwise.

One can show that the value for this binary option is

V (S, t) = Be^{−r(T −t)}(N (d2(E1)) − N (d2(E2))) , where d₂(E) is given by (3.23).

Exercise. Verify this formula.

6. Deterministic case (σ = 0). In this case, the Black-Scholes equation is reduced to V_t+ rSV_S− rV = 0.

Or in τ, x and u variables:

u_τ − ru_x = 0 with initial data

u(x, 0) = Λ(Ee^x)/E, Thus, its solution is given by

u(x, τ ) = u(x + rτ, 0) = Λ(Ee^x+rτ)/E = Λ(Se^rτ)/E.

Or

V (S, t) = e^{−r(T −t)}Λ(Se^{r(T −t)}).

This means that when the process is deterministic, the value of the option is the payoff func-tion evaluated at the future price of S at T (that is Se^{r(T −t)}), and then discounted by the factor e^{−r(T −t)}.

3.5 Risk Neutrality

Notice that the growth rate µ of the underlying asset does not appear in the Black-Scholes equation.

The option may be valued as if all random walks involved are risk neutral. This means that the drift term (growth rate) µ in the asset pricing model can be replaced by r. The option is then valued by calculating the present value of its expected return at expiry. Recall the lognormal probability density function with growth rate r, volatility σ is

P(S⁰, T, S, t) := 1

p2πσ²(T − t)S⁰ exp −[ln(^S_S⁰) − (r −¹₂σ²)(T − t)]² 2σ²(T − t)

!

. (3.27)

This is the transition probability density of an asset price model in a risk-neutral world:

dS

S = rdt + σdz. (3.28)

The expected return at time T in this risk-neutral world is Z

P(S⁰, T, S, t)Λ(S⁰)dS⁰.

At time t, this value should be discounted by e^{−r(T −T )}: V (S, t) = e^{−r(T −t)}

Z

P(S⁰, T, S, t)Λ(S⁰)dS⁰.

We may reinvestigate the function N and the parameters di in the Black-Scholes formula. After some calculation, we find

N (d₂) = Z ∞

E

P(S⁰, T, S, t)dS⁰. (3.29)

This is the probability of the event { ˜S ≥ E}, where ˜S obeys the risk-neutral pricing model:

d ˜S

S˜ = rdt + σdz.

Similarly, one can show that

N (d₁) = R∞

E P(S⁰, T, S, t)S⁰dS⁰

Se^{r(T −t)} . (3.30)

is the expectation of ˜S at T when ˜S = 1 at t and under the condition that ˜S ≥ E at T . Exercise. Check formulae (3.29) and (3.30).

3.6 Hedging

Hedging is the reduction of sensitivity of a portfolio to the movement of the underlying of asset by taking opposite position in different financial instruments. The Black-Scholes analysis is a dy-namical hedging strategy. The delta hedge is instantaneously risk free. It requires a continuous rebalancing of the portfolio and the ratio of the holdings in the asset and the derivative product. The delta (∆) for a whole portfolio is ∆ = ^∂Π_∂S. This is the sensitivity of Π against the change of S. By taking dΠ − ∆ · dS = 0, the sensitivity of the portfolio to the asset price change is instantaneously zero.

Besides the delta hedge, there are more sophisticated hedging strategies such as:

Gamma: Γ = ∂²Π

∂S², Theta: Θ = ∂Π

∂t, Vega: = ∂Π

∂σ, rho: ρ = ∂Π

∂r.

Hedging against any of these dependencies requires the use of another option as well as the asset itself. With a suitable balance of the underlying asset and other derivatives, hedgers can eliminate the short-term dependence of the portfolio on the movement in t, S, σ, r.

3.6.1 The delta hedging

Suppose a financial institution write 1, 000 European calls on 10,000 shares of a stock. How does this institution hedge the risk from the price fluctuation of the underlying stock? The Delta-hedge is such a strategy. It is a dynamic hedging strategy. The institution long ∆ shares of the underlying stock. The ∆ is chosen to be

∆ = ∂c

∂S.

If ∆ = 0.5, this means that it should long 0.5 × 10, 000 = 5, 000 shares. In this case, the total portfolio Π = c − ∆S is instantaneously neutral. That is dΠ = dc − ∆dS = 0 under a small change of dS. We call Π is delta neutral. However, this delta neutral holds only for a short period of time.

It should be adjusted periodically. This is called a dynamic hedging.

For the Delta-hedge for the European call and put options, we have the following propositions.

Proposition 1. For European call options, its ∆ hedge is given by

∆ = N (d₁).

Proof. By definition, ∆ = ∂c/∂S. Since c = SN (d1) − Ee^{−r(T −t)}N (d₂), we get

∂c

∂S = N (d₁) + S · N⁰(d₁) · d_1S− Ee^{−r(T −t)}N⁰(d₂)d_2S. Since

d1 = log(_E^S) + (r +^σ₂²)(T − t) σ√

T − t , d2= log(_E^S) + (r −^σ₂²)(T − t) σ√

T − t ,

we get

d1S = d2S = 1

Sσp(T − t), N⁰(di) = 1

√

2πe^−d2i/2. Hence,

∂c

∂S = N (d₁) +



SN⁰(d1) − Ee^{−r(T −t)}N⁰(d2)



/(Sσ√ T − t)

≡ N (d₁) + I/(Sσ√ T − t).

We claim that I = 0. Or equivalently, S E

N⁰(d₁)

N⁰(d₂) = e^−rτ This follows from the computation below.

S E

N⁰(d1)

N⁰(d2) = e^x· e^{−(d21−d22)/2}.

From (3.22), (3.23),

d²₁− d²₂ = 1 σ²τ



(x + rτ +σ

2τ )²− (x + rτ − σ 2τ )²

= 2(x + rτ ) Hence,

S E

N⁰(d1)

N⁰(d2) = e^x· e^−x−rτ = e^−rτ.

Proposition 2. For European put options, its ∆ hedge is given by

∆ = N (−d₁).

Proof. From the put-call parity,

∆ = ∂p

∂S = ∂c

∂S − 1 = N (d₁) − 1 = −N (−d₁),

3.7 Time-Dependent r, σ and µ for Black-Sholes equation

We can extend Black-Schole analysis to the case when r, σ, µ are functions of t, but still determin-istic. We use the change-of-variables:

S = Ee^x, V = Ev, τ = T − t.

The Black-Scholes equation is converted to vτ = σ²(τ )

2 vxx+ (r(τ ) −σ²(τ )

2 )vx− r(τ )v (3.31)

We look for a new time variable ˆτ such that

dˆτ = σ²(τ )dτ For instance, we can choose

ˆ τ =

Z τ 0

σ²(τ ) dτ.

Then the equation becomes

v_τˆ = 1

2v_xx+ a(ˆτ )v_x− b(ˆτ )v. (3.32) To handle the term a(ˆτ ), we consider the following characteristic equation:

dx

dˆτ = −a(ˆτ )

This can be integrated and yields

x = − Z τˆ

0

a(τ⁰)dτ⁰+ y,

where y is an integration constant. The lines with constant y’s are called characteristic curves.

Along the characteristic curves, the derivative ∂τˆ+ a(ˆτ )∂_xis just the derivative in time with fixed y. In fact, we introduce the following change-of-variable:

x The equation (3.32)is transformed to

v_τˆ1 = 1

2v_yy− b(ˆτ₁)v.

Let B(ˆτ₁) =Rτˆ1

0 b(τ⁰)dτ⁰and u = e^B(ˆτ1)v, then u_τˆ1 = ¹₂u_yy. And we can solve this heat equation explicitly.

3.8 Trading strategy involving options

An investor can design his/her payoff by using options and underlying stocks. The options whose payoff are max{ST − E, 0} or max{E − ST, 0} are called vanilla option. They are the simplest payoff functions. In this section, we shall discuss how to design more general payoff functions to fit an investor’s anticipation.

3.8.1 Strategies involving a single option and stock There are four ways:

a. Π = S − c (writing a covered call option). When an investor short a call, we say he write a nake call. The investor has cash inflow at the beginning. However, short selling is highly risky. If the stock rises S_T > E, then the investor needs to buy the share on S_T and sell to the call holder on E. The investor loses ST− E. This loss can be unlimited because ST may rise very high. To limit this risk, the investor can long a share Stat time t. That is, his portfolio is Π = −c + S. At expiry, if S_T > E, then he can use this share to sell to the call owner on E, instead of S_T. His risk is limited. We say that this share covers the call, and say the writer writes a covered call option. The payoff of Π is Λ = ST − max(S_T − E, 0) = min{S_T, E}.

b. Π = c − S (reverse of a covered call). Suppose an investor anticipates the stock price will decrease. So he shorts a share and receive money Stat time t. If ST does decrease, he earns money. If unfortunately, the stock price increases, then he loses ST − S_tat time T . This risk is unlimited. Thus, short selling has unlimited risk. However, he can buy a call to limit this risk. That is, his portfolio Π = −S + c. Since c < S, he still receive some money from Π at beginning. At expiry, if the price goes beyond E, then he can exercise this call to cover the shorted share. The payoff now is Λ = − min{S, E} ≥ −E. Thus, his loss is at most E.

c. Π = p + S (protective put). When an investor anticipates a stock will increase, he set up a portfolio Π = S. This costs him St. If the prices ST goes up, he earns ST − S_t. However, if the price goes down, he sells it and loses St− S_T. If he wants to limit his loss, he can set up a portfolio Π = S + p. This costs him more at beginning. But at expiry, if the price goes down, he can sell the share on E instead of a lower price ST. Thus, he limits his loss. In this portfolio, the payoff is Λ = S + max{E − S, 0} = max{S, E}. The downside is limited (by E) and the upside is unlimited (S_T can be arbitrary large).

d. Π = −p − S (reverse of a protective put). In this portfolio, the investor anticipates the stock price will decrease. He shorts a share St and also a put. He receives money from shorting these. At expiry, the payoff is − max{S, E}. This means that he pays E if S_T ≤ E. However, if S_T is large, then his risk is unlimited.

Below are the payoff functions for the above four cases.

3.8.2 Bull spreads

In this strategy, an investor anticipates the stock price will increase. However, he would like to give up some of his right if the price goes beyond certain price, say E2. Indeed, he does not anticipate the stock price will increase beyond E2. Therefore he does not want to own a right beyond E2. Such a portfolio can be designed as

Π = CE1− C_E2, E1 < E2,

where CEi is a European call option with exercise price Ei and CE1, CE2 have the same expiry.

The payoff

Λ = max{ST − E₁, 0} − max{ST − E₂, 0}

=







0 if S_T < E1

S_T − E₁ if E1 < S_T < E₂ E2− E₁ if ST > E2

Since E1 < E2, we have CE1 > CE2. A bull spread, when created from CE1− C_E2, requires an initial investment. We can describe the strategy by saying that the investor has a call option with a strike price E1 and has chosen to give up some upside potential by selling a call option with strike price E2> E1. In return, the investor gets E2− E₁ if the price goes up beyond E2.

E S Λ

(a)

E S

Λ

(b)

E S

Λ

(c)

E S

Λ

(d)

Example: CE1 = 3, CE2 = 1 and E1 = 30, E2= 35. The cost of the strategy is 2. The payoff is







0 if ST ≤ 30

ST − 30 if 30 < ST < 35

5 if S_T ≥ 35

The bull spread can also be created by using put options Π = PE1− P_E2, E1 < E2. 3.8.3 Bear spreads

An investor entering into a bull spread is hoping that the stock price will increase. By contrast, an investor entering into a bear spread is expecting the stock price will go down. The bear spread is

Π = CE2− C_E1, E1 < E2. There is cash flow entered (CE2− C_E1). The payoff is

Λ = − max{S_T − E₁, 0} + max{S_T − E₂, 0}

=







0 if ST < E₁

−S_T + E1 if E1 < ST < E2

−E₂+ E1 if ST > E2

E1 E

2 S

Λ

E2−E 1

3.8.4 Butterfly spread

If an investor anticipate the stock price will stay in certain region, say, E1 < ST < E3, he or she can have a butterfly spread such that the payoff function is positive in that region and he or she gives up the return outside that region.

1. Butterfly spread using calls: Define the portfolio:

Π = C_E1 − 2C_E2 + C_E3, with E1< E2< E3.

where E3 = E2 + (E2 − E₁). Its payoff function is a piecewise linear function and is determined by Λ(E₁) = Λ(E₃) = 0, Λ(E₂) = E₂− E₁. Below is the graph of its payoff function.

E₁ S−E1

E2−E 1

E₂ E₃

E2−S

S Λ

Example: Suppose a certain stock is currently worth 61. A investor who feels that it is unlikely that there will be significant price move in the next 6 month. Suppose the market of 6 month calls are

E C

55 10 60 7 65 5

55

5

60 65 S

Λ

The investor creates a butterfly spread by

Π = CE1 − 2C_E2 + CE3.

The cost is 10 + 5 − 2 × 7 = 1. The payoff is the figure below (Figure (1)).

2. Butterfly spread using puts.

P_E1 + P_E3 − 2P_E2, E₁< E₃, E₂= E₁+ E₃

2 .

ϕE2 =





 linear

∆E if S = E2

0 if S < E₂− ∆E, or S > E₂− ∆E

Remark 1. Suppose European options were available for every possible strike price E, then any payoff function could be created theoretically:

Λ(S) =X

Λ_iϕ(S − E_i) where E_i = i∆E, Λ_iis constant and

ϕ(S − E_i) =







1 if S − Ei = 0 0 if |S − E_i| ≥ ∆E

linear for |S − Ei| ≤ ∆E.

Then Λ(Ei) = Λi and Λ is linear on every interval (Ei, Ei+1) and Λ is continuous. As ∆E → 0, we can approximate any payoff function by using butterfly spreads.

Remark 2. One can also use cash-or-nothing to create any payoff function:

Λ(S) =X

Λiψ(S − Ei), where

ψ(S) := H(S) − H(S − ∆E) =

 1 if 0 ≤ S < ∆E 0 otherwise.

and H is called the Heaviside function. The value for such a portfolio is V = e^{−r(T −t)}

Z

P(S⁰, T, S, t)Λ(S⁰)dS⁰,

= e^{−r(T −t)}ΣΛiP(E_i≤ S ≤ E_i+1).

3.9 Derivation of heat equation and its exact solution

3.9.1 Derivation of the heat equation on R The heat equation on R reads

 ut= uxx, x ∈ R, t > 0, u(x, 0) = f (x).

This equation models heat propagation on the line. Here, u(·, t) represents the temperature distribu-tion at time t and f the initial temperature distribudistribu-tion. The derivadistribu-tion of this equadistribu-tion is based on a physical law – the conservation of energy, and the Fourier law of heat flux. Given a temperature distribution u(x, t), the Fourier law describes that the temperature flows from high temperature to low temperature at a rate q called heat flux. It is the energy passing through a point (per unit area) per unit time. The Fourier law states that

q = −κux.

Here, the constant κ is called thermal conductivity. It is a positive constant. It differs for different material. For instant, κ of cooper is much larger than that of woods. According to thermal dynamics, the energy density e is related to temperature by e = cvu. The constant cv is called specific heat (at constant volume). The conservation of energy states that the change of energy in an interval (a, b) per unit time is the same as the heat flux flows into (a, b) from boundaries. Mathematically, it is

∂_t Z b

a

e dx = q(a) − q(b) = − Z b

a

q_xdx.

This gives

Z b a

c_v∂_tu dx = Z b

a

(κu_x)_xdx.

This holds for any interval (a, b). Thus, the integrands much be the same. This gives

∂_tu = Du_xx. where D = κ/cv > 0 is the heat conductivity.

3.9.2 Exact solution of heat equation on R.

If we rescale time by t⁰ = Dt, then the heat equation becomes

∂_t⁰u = u_xx.

Thus, without loss of generality, we consider the heat equation with D = 1. We shall show that its solution is given by

u(x, t) = Z ∞

−∞

G(x − y, t)f (y) dy, G(x, t) := 1

√4πte^−x2/(4t).

The function G(·, t) is the Gaussian distribution. It corresponds to the temperature distribution at time t when the initial temperature distribution is a hot spot at x = 0. The derivation of this formulation is decomposed into the following steps.

1. Superposition principle: If u and v are solutions, so is au + bv, where a and b are constants.

This is called superposition principle for linear partial differential equations.

Now, we imagine f can be approximated by piecewise step function: let h be a small mesh size, xj = jh and f can be approximated by f_hdefined by

f ≈ fh, fh(x) :=

∞

X

j=−∞

f (xj)δh(x − xj)h.

Here,

δ_h(x) := 1

hχ_h(x), χ_h(x) :=

 1 for − h/2 < x < h/2 0 otherwise.

Suppose each solution corresponding to f (xj)δh(x) is f (xj)Gh(x − xj). Because the equa-tion is linear, then the general soluequa-tion is just the linear superposiequa-tion of these soluequa-tions:

u(x, t) ≈X

j

G_h(x − x_j, t)f (x_j) · h ≈ Z ∞

−∞

G(x − y, t)f (y) dy.

Here, Gh(x−xj, t) is the solution of the heat equation corresponding to the initial data δh(x−

xj). Notice that we have used another important of heat equation, namely, the translation invariance: if u(x, t) is a solution of the heat equation with initial data u(x, 0), then u(x−a, t) is a solution of heat equation with initial data u(x − a, 0). This is because the equation is unchanged under the translation transformation x 7→ x − a. Now if G_h(x, t) is the solution with initial data δ_h(x), then G_h(x − x_j, t) is the solution corresponding to the initial data δh(x − xj).

2. Translational invariance. Because the equation is translationally invariant, we can solve the equation with initial data to be χh(x)/h. This initial function tends so-called δ-function as

h → 0. Thus, what we need is to solve the case when the initial function is a δ function. A δ-function is a generalized function having the property:

δ(x) = for any smooth function f .

3. Parabolic scaling. To solve the heat equation with an initial hot spot at x = 0, we use the parabolic invariance property of the heat equation. Namely, the equation and the initial condition are both invariant under the transformation

x 7→ λx, t 7→ λ²t, λ > 0.

This suggests that the solution is a function of ξ = x/√

t. So, let us look a solution of the form u(x, t) = U (x/√

t). Plug it into the equation, we obtain ut= U⁰(ξ)

4π for normalization. However, this is still not what we want. But we notice that if v(x, t) = U (x/√ Then G is a solution and satisfies

Z ∞

−∞

G(x, t) dx = 1.

4. Verification. Finally, one check

• u(x, t) :=R∞

−∞G(x − y, t)f (y) dy is a solution of the heat equation

• u(x, t) → f (x) as t → 0+ for any bounded continuous function f . To show the first one, we check that Gt= Gxx. This is left to you.

For the second one, the integral Z ∞ represents the average of f about x with weight G(y, t). We notice that G(y, t) is a Gaussian distribution with standard deviation√

t. As t → 0, this weight is more and more concentrated at 0. Therefore, this average becomes the average of f roughly in a√

t-neighborhood of x.

Its limit is f (x) for any bounded smooth function f . We can also give a more rigorous proof. We want to show

t→0lim|u(x, t) − f (x)| =

Here, we have used G(x, t) ≥ 0. To estimate this integral, we break it into two parts Z ∞

where δ > 0 is a small parameter to be chosen later. Since f is continuous, we have for any

 > 0, there exists a δ > 0 such that

For II, since we assume f is bounded, there exists a constant M > 0 such that |f (x)−f (y)| ≤ 2M . Thus,

II ≤ 2M Z

|x−y|≥δ

G(x − y, t) dy = 2M Z

|y|≥δ

G(y, t) dy.

This integral is the tail estimate of a Gaussian distribution:

Z

|y|≥δ

G(y, t) dy = 2 Z

y≥δ

√1

4πte^−y2/(4t)dy = Z

z≥δ/(2√ t)

√1

πe^−z2dz.

As t → 0+, this integral goes to 0. Therefore, we can choose a small t1 > 0 such that II ≤  for all 0 < t < t₁.

Combining I and II, we have for any  > 0, there exists t₁ > 0 such that for any 0 < t < t₁ we have I + II < 2. This shows I + II → 0 as t → 0.

Variations on Black-Scholes models

4.1 Options on dividend-paying assets

Dividends are payments to the shareholders out of the profits made by the company. We will con-sider two “deterministic” models for dividend. One has constant dividend yield. The other has discrete dividend payments.

4.1.1 Constant dividend yield

Suppose that in a short time dt, the underlying asset pays out a dividend D₀Sdt, where D₀ is a constant, called the dividend yield. This continuous dividend structure is a good model for index options and for short-dated currency options. In the latter case, D0 = r_f, the foreign interest rate.

As the dividend is paid, the return ^dS_S must fall by the amount of the dividend payment D0dt. It

As the dividend is paid, the return ^dS_S must fall by the amount of the dividend payment D0dt. It

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