Lecture 16: Delta Hedging

We are now going to look at the construction of binomial trees asa first technique for pricing options in an approximative way.

These techniques were first proposed in:

J.C. Cox, S.A. Ross, M. Rubinstein, Option Pricing: A SimplifiedApproach, Journal of Financial Economics 7 (1979), 229–264.

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Example

Suppose we want to value a European call option giving the rightto buy a stock for the strike price K = AC 21 (the price in thecontract) in 3 months (expiry date T ). The current stock price S0

is AC 20.

We make a very simplifying assumption which in reality is notsatisfied:in 3 months the stock price will be either AC 22 or AC 18. We do notknow the probability for the occurrence of the stock prices AC 22 orAC 18.

However, at expiry date, we know the value of the option:if the stock price goes up (to AC 22), the payoff ST − K is AC 1;if the stock price goes down (to AC 18), then the option isworthless, so the value is 0.

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Example

Suppose we want to value a European call option giving the rightto buy a stock for the strike price K = AC 21 (the price in thecontract) in 3 months (expiry date T ). The current stock price S0

is AC 20.

We make a very simplifying assumption which in reality is notsatisfied:in 3 months the stock price will be either AC 22 or AC 18. We do notknow the probability for the occurrence of the stock prices AC 22 orAC 18.

However, at expiry date, we know the value of the option:if the stock price goes up (to AC 22), the payoff ST − K is AC 1;if the stock price goes down (to AC 18), then the option isworthless, so the value is 0.

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Example

Suppose we want to value a European call option giving the rightto buy a stock for the strike price K = AC 21 (the price in thecontract) in 3 months (expiry date T ). The current stock price S0

is AC 20.

We make a very simplifying assumption which in reality is notsatisfied:in 3 months the stock price will be either AC 22 or AC 18. We do notknow the probability for the occurrence of the stock prices AC 22 orAC 18.

However, at expiry date, we know the value of the option:if the stock price goes up (to AC 22), the payoff ST − K is AC 1;if the stock price goes down (to AC 18), then the option isworthless, so the value is 0.

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A portfolio with one stock and ∆ sharesLook at a portfolio which is generated at time 0 by

buying ∆ many shares of a stock (long position) andselling one call option of the stock (short position).

If f is the price of the option the portfolio at time 0 has value

20∆ − f .

After three months the value of this portfolio can be computed:if the stock prices moves from AC 20 to AC 22 then the value of theshares is AC 22∆ and the value of the option is AC 1, so the totalvalue of the portfolio is

22∆ − 1

(in this case our contract partner has the right to buy from us onestock at the strike price AC 21, so we have to give him AC 1);If the stock prices moves from AC 20 to AC 18 then the value of theshares is AC 18∆ and the value of the option is zero, so the totalvalue of the portfolio 18∆.

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Strategy: delta hedging

The idea is now to choose ∆ such that the value of the portfolio inboth cases (stock price up or down) is the same: so we require

22∆ − 1 = 18∆, so ∆ = 0.25.

Control: in 3 monthsif ST = AC 22, then the portfolio is worth 22 · 0.25 − 1 = AC 4.50.if ST = AC 18, then the portfolio is worth 18 · 0.25 = AC 4.50.

DefinitionThe delta of an option is the number ∆ of shares we should holdfor one option (short position) in order to create a riskless hedge:so after maturity time T the value of the portfolio containing ∆share and selling one call option is for both cases ST > K andST ≤ K the same.The construction of a riskless hedge is called delta hedging.

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Strategy: delta hedging

The idea is now to choose ∆ such that the value of the portfolio inboth cases (stock price up or down) is the same: so we require

22∆ − 1 = 18∆, so ∆ = 0.25.

Control: in 3 monthsif ST = AC 22, then the portfolio is worth 22 · 0.25 − 1 = AC 4.50.if ST = AC 18, then the portfolio is worth 18 · 0.25 = AC 4.50.

DefinitionThe delta of an option is the number ∆ of shares we should holdfor one option (short position) in order to create a riskless hedge:so after maturity time T the value of the portfolio containing ∆share and selling one call option is for both cases ST > K andST ≤ K the same.The construction of a riskless hedge is called delta hedging.

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So choosing ∆ = 0.25 leads to a portfolio where there is nouncertainty about the value of the portfolio in 3 months, namelyAC 4.50. Since the portfolio has no risk we can compute its value attime 0 by discounting the price with the risk-free rate (no arbitrageopportunities). This means that we can find the value of theportfolio at time 0 by discounting the value in 3 months:

If we suppose that r = 12%, then the portfolio at time 0 is worth

4.50e−0.12·0.25 = AC 4.367.

Let f be the price of the call option today. Then the portfolio attime 0 has worth

20 · 0.25 − f = 4.367,

and sof = 20 · 0.25 − 4.367 = 0.633.

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So choosing ∆ = 0.25 leads to a portfolio where there is nouncertainty about the value of the portfolio in 3 months, namelyAC 4.50. Since the portfolio has no risk we can compute its value attime 0 by discounting the price with the risk-free rate (no arbitrageopportunities). This means that we can find the value of theportfolio at time 0 by discounting the value in 3 months:

If we suppose that r = 12%, then the portfolio at time 0 is worth

4.50e−0.12·0.25 = AC 4.367.

Let f be the price of the call option today. Then the portfolio attime 0 has worth

20 · 0.25 − f = 4.367,

and sof = 20 · 0.25 − 4.367 = 0.633.

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So choosing ∆ = 0.25 leads to a portfolio where there is nouncertainty about the value of the portfolio in 3 months, namelyAC 4.50. Since the portfolio has no risk we can compute its value attime 0 by discounting the price with the risk-free rate (no arbitrageopportunities). This means that we can find the value of theportfolio at time 0 by discounting the value in 3 months:

If we suppose that r = 12%, then the portfolio at time 0 is worth

4.50e−0.12·0.25 = AC 4.367.

Let f be the price of the call option today. Then the portfolio attime 0 has worth

20 · 0.25 − f = 4.367,

and sof = 20 · 0.25 − 4.367 = 0.633.

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The general case

We want to value the price f of an European option (put or call).The current stock price is S0.

We assume that after maturity time T the stock price ST will beeither

S0 · u (u > 1) or S0 · d (d < 1),

where u stands for up, and d for down.After maturity time T we compute the value (payoff) of the optiondepending on the stock price ST and the strike price K : there areonly two possibilities and we denote the value of the option by fu ifthe stock has gone up and by fd if it has gone down.

In our example:if the stock price goes up (to AC 22), the value fu is AC 1;if the stock price goes down (to AC 18), then the value fd = 0.

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The general case

We want to value the price f of an European option (put or call).The current stock price is S0.

We assume that after maturity time T the stock price ST will beeither

S0 · u (u > 1) or S0 · d (d < 1),

where u stands for up, and d for down.After maturity time T we compute the value (payoff) of the optiondepending on the stock price ST and the strike price K : there areonly two possibilities and we denote the value of the option by fu ifthe stock has gone up and by fd if it has gone down.

In our example:if the stock price goes up (to AC 22), the value fu is AC 1;if the stock price goes down (to AC 18), then the value fd = 0.

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The general case

We want to value the price f of an European option (put or call).The current stock price is S0.

We assume that after maturity time T the stock price ST will beeither

S0 · u (u > 1) or S0 · d (d < 1),

where u stands for up, and d for down.After maturity time T we compute the value (payoff) of the optiondepending on the stock price ST and the strike price K : there areonly two possibilities and we denote the value of the option by fu ifthe stock has gone up and by fd if it has gone down.

In our example:if the stock price goes up (to AC 22), the value fu is AC 1;if the stock price goes down (to AC 18), then the value fd = 0.

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The delta of a stockLook at a portfolio at time 0 generated by

buying ∆ shares of a stock (long position) at stock price S0

selling one option of the stock (short position).

If f is the price of the option at time 0 then the portfolio at time 0has value

∆S0 − f .

After expiry date T we can compute the value of the portfolio:

if stock goes up the portfolio is worth S0 · u · ∆ − fu,

if stock goes down the portfolio is worth S0 · d · ∆ − fd ,

We choose ∆ such that the value of the portfolio at time T inboth cases (stock price up or down) is the same:

S0 · u · ∆ − fu = S0 · d · ∆ − fd ,

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The delta of a stock

From S0 · u · ∆ − fu = S0 · d · ∆ − fd , we find

∆ =fu − fd

S0(u − d).

TheoremAssume that the stock price S0 after time T goes either up toS0 · u with u > 1 or down to S0 · d with d < 1. Let fu and fd bethe payoffs at maturity time T in the case of up or downmovement. Then the delta of one option is

∆ =fu − fd

S0(u − d)=

difference of payoffs at time T

difference of prices of stock at time T.

The delta of a call option is positive, whereas the delta of a putoption is negative.

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Using the choice

∆ =fu − fd

S0(u − d)

the value of portfolio at time T is in both cases (stock up ordown) the same, namely:

S0 · u · ∆ − fu = S0 · u · fu − fdS0(u − d)

− fu

which is equal to

u · fu − fdu − d

− fu =u · (fu − fd) − (u − d) · fu

u − d=

d · fu − u · fdu − d

.

Thus the value of the portfolio at time T is

PT :=d · fu − u · fd

u − d.

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The portfolio is riskless. Since no arbitrage opportunities exist theportfolio at time 0 is worth

P0 = e−rTPT = e−rT dfu − ufdu − d

.

The portfolio at time 0 is also worth

P0 = S0∆ − f =fu − fdu − d

− f .

Comparing these two values, we find

f =fu − fdu − d

− e−rT dfu − ufdu − d

=e−rT

u − d

(erT (fu − fd) − (dfu − ufd)

)=

e−rT

u − d

(fu(erT − d) + fd(u − erT )

).

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The portfolio is riskless. Since no arbitrage opportunities exist theportfolio at time 0 is worth

P0 = e−rTPT = e−rT dfu − ufdu − d

.

The portfolio at time 0 is also worth

P0 = S0∆ − f =fu − fdu − d

− f .

Comparing these two values, we find

f =fu − fdu − d

− e−rT dfu − ufdu − d

=e−rT

u − d

(erT (fu − fd) − (dfu − ufd)

)=

e−rT

u − d

(fu(erT − d) + fd(u − erT )

).

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Put now

p =erT − d

u − d.

Then

f = e−rT

(pfu +

u − erT

u − dfd

).

Since

1 − p =u − d

u − d− erT − d

u − d=

u − erT

u − d

we obtain the final formula for the price f of an option:

f = e−rT[pfu + (1 − p)fd

].

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Summary

TheoremAssume that the stock price S0 after time T goes either up toS0 · u with u > 1 or down to S0 · d with d < 1. Let f be the priceof an option (either call or put) at time 0 with payoff fu and fdrespectively at time T . If r is the riskless interest rate then

f = e−rT[pfu + (1 − p)fd

]where p =

erT − d

u − d.

In our previous example we had u = 1.1, d = 0.9, fu = 1, fd = 0,r = 0.12 and T = 0.25, so

p =e0.12·0.25 − 0.9

1.1 − 0.9= 0.6523,

f = e−0.12·0.25(0.6523 · 1 + 0.3477 · 0) = 0.633.

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Summary

TheoremAssume that the stock price S0 after time T goes either up toS0 · u with u > 1 or down to S0 · d with d < 1. Let f be the priceof an option (either call or put) at time 0 with payoff fu and fdrespectively at time T . If r is the riskless interest rate then

f = e−rT[pfu + (1 − p)fd

]where p =

erT − d

u − d.

In our previous example we had u = 1.1, d = 0.9, fu = 1, fd = 0,r = 0.12 and T = 0.25, so

p =e0.12·0.25 − 0.9

1.1 − 0.9= 0.6523,

f = e−0.12·0.25(0.6523 · 1 + 0.3477 · 0) = 0.633.

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