Financial Derivatives Section 2 (of part II) Futures Options and Greeks

Michail Anthropelos [email protected] http://web.xrh.unipi.gr/faculty/anthropelos/ University of Piraeus

Spring 2016

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Outline

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Futures Options Payoff Issues on pricing

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The Greek Letters The Delta The rest letters

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The Nature of Futures Options Definition A futures option is the right to enter into a futures contract at a specific delivery price at a specific date. Call futures option gives the right to enter into a long position. Put futures option gives the right to enter into a short position.

Payoff When a call futures is exercised, a long position in a futures is acquired plus a cash amount which is equal to the latest settlement futures price minus the strike price of the option. When a put futures is exercised, a short position in a futures is acquired plus a cash amount which is equal to the strike price of the option minus the latest futures settlement price. X At the majority of the cases, the futures options are American and their expiration is usually on (or few days before) the maturity of the underlying futures contract. M. Anthropelos (Un. of Piraeus)

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An Example Suppose that an investor has one Dec 16 futures call option contract on gold with strike price $1,250 per ounce. One futures contract is on 100 ounces. The futures price of gold for delivery in Dec 16 is currently F=$1,282 and the closest settlement price was $1,277. If the option is exercised, the investor receives $100 × (1, 277 − 1, 250) = $2, 700 and a long position in a futures contract to buy 100 ounces of gold in Dec 16. If he wants to close his position on this futures, he also gets: $100 × (1, 282 − 1, 277) = $500 which reflects the change in the futures price since the last settlement. The total payoff from exercising the option today is $3,200, that is Call option payoff = (Contract Size) × (F − K ), where F is the futures price at the time of the exercise and K is the strike price of the option. The price of this option is around $7, 950. M. Anthropelos (Un. of Piraeus)

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An Example cont’d Suppose that an investor has one Dec 16 futures put option contract on gold of $1,300 per ounce. The futures price of gold for delivery in Dec 15 is currently F=$1,282 and the closest settlement price was $1,277. If the option is exercised, the investor receives $100 × (1, 300 − 1, 277) = $2, 300 and a short position in a futures contract to sell 100 ounces of gold in Dec 16.If he wants to closed his position on this futures he pays: $100 × (1, 282 − 1, 277) = $500 which reflects the change in the futures price since the last settlement. The total payoff from exercising the option today is $1,800, that is Put option payoff = (Contract Size) × (K − F ), where F is the futures price at the time of the exercise and K is the option strike price. The price of this option is around $5, 180. M. Anthropelos (Un. of Piraeus)

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Facts about Futures Options The futures options are referred to by the delivery month of the underlying futures contract - not by the expiration month of the option. Usually the maturity of the option is the same as the maturity of the futures contract or some days before. They have became really popular and the most popular contract in exchange markets are on consumption commodities, gold, bonds and stock indices.

Reasons for popularity 1

The main reason of the popularity is that in some circumstances it is more liquid and less costly to trade futures contracts than the underlying asset.

2

For some underlying assets the spot price is not immediately known, whereas the futures price is readily available.

3

The exercise of a futures option leads normally to cash settlement.

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In some cases the transaction costs on futures options are less than the transaction costs on spot options.

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European Futures Options Payoff The payoff of the European call futures options is max{FT − K , 0} where FT is the futures price at the option’s maturity. In case where the underlying future contract has the same maturity as the option, then the European call futures option has exactly the same payoff as the European call (spot) option.

Put-call parity Similarly as in the case of the European call spot options c(t) + Ke −r (T −t) = p(t) + F (t, τ )e −r (T −t) where τ is the maturity of the futures contract. The proof of this parity relation is left as an exercise. M. Anthropelos (Un. of Piraeus)

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The Differential Equation We have seen that if the underlying asset follows a GBM model, then the futures payoff follows a GBM too but with different expected rate of return dF (t, τ ) = (µ − r ) F (t, τ )dt + σF (t, τ )dz(t) This can be seen as a stock that pays dividend yield equal to r . In other words, under the risk-neutral probability the dynamics of the futures price is given as dF (t, τ ) = σF (t, τ )dz(t) By following the same arguments as the ones in the proof of the BSM pde, we get the following pde for the derivative products written on the futures contract ∂f 1 ∂2f 2 2 + σ F = rF . ∂t 2 ∂F 2

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The Black-Scholes Price for Futures Options The BS pricing for the futures options follows exactly the same recipe as in the case of spot options. The only differences are: a dividend is considered to be equal to r the spot price is equal to F (t, τ ). Hence, the call/put futures option prices are given as: c(t) = e −r (T −t) (F (t, τ )N(d1 ) − KN(d2 )) and p(t) = e −r (T −t) (KN(−d2 ) − F (t, τ )N(−d1 )) where d1 = and d2 =

ln(F (t, τ )/K ) + σ 2 /2(T − t) √ σ T −t

√ ln(F (t, τ )/K ) − σ 2 /2(T − t) √ = d1 − σ T − t. σ T −t

X Is the price of American call futures option equal to the price of European call futures option on an asset that pays no dividend? M. Anthropelos (Un. of Piraeus)

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An Example Let the maturity of the option to be equal to 4 months. The current futures price with maturity in 5 months is equal to $20 and the option’s exercise price is also $20. Let the annual interest rate to be equal to 9% and the volatility of the futures price (and the spot price) is 25%. Hence, we have: T = 4/12, τ = 5/12, F (0, τ ) = $20, K = $20, r = 9% and σ = 25%. We calculate d1 = 0.072 and d2 = −0.072. Thus, N(−d1 ) = 0.47 and N(−d2 ) = 0.52. Therefore the put futures option price is 4

p(0) = e −0.09× 12 (20 × 0.52 − 20 × 0.47) = $1.12.

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The Delta When an investor sells a derivative, he encounters the risk of paying its payoff. For example, short selling a call means receiving c now, but also facing the payment max{S(T ) − K , 0} at time T . One way to hedge this risk is to follow the corresponding Delta hedging, according to which the investor should buy/short ∆ units of the underlying asset, where ∆ is given by ∂c ∆= ∂S where c is the price of the derivative. Delta is in fact an approximation of the change of the derivative price when the price of the underlying asset slightly changes (this approximation is good as long as the change in the price of the underlying asset is small... or in fact very very small). Since the price of the derivative as well as the price of the underlying asset change, the value of Delta also changes. Hence, the so-called delta hedging should be frequently rebalanced. A continuous rebalancing delta hedging is the one imposed in the BSM model. The transaction costs is a very important part of the delta hedging. M. Anthropelos (Un. of Piraeus)

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Delta in BS Model Consider for example a European call option on a non-dividend-paying stock and assume the GBM for the stock price. By the BS formula we calculate (this is one of your HM’s) that ∆(call) = where d1 =

∂c = N(d1 ) ∂S

ln(S(t)/K )+(r +σ 2 /2)(T −t) √ . σ T −t

Since Delta is positive, if an investor takes the short position on the call, according to the delta hedging, he has to take long position on the underlying asset. For put option, Delta is negative.

Example Let S(0) = $49, K = $50, r = 5%, T = 0.38 (20 weeks), σ = 20%. Hence, we ln(49/50)+(0.05+(0.2)2 /2)0.38 √ calculate d1 = = 0.05 and therefore, 0.2 0.38 N(d1 ) = N(0.05) = 0.522. This means that if the price of the stock increases by $2, we approximate that the call option price will increase by $2×0.522=$1.044. M. Anthropelos (Un. of Piraeus)

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Weekly Delta Hedging Example Consider the time evolution of the Delta hedging of the above example (see page 356 of Hull’s book, 7th edition).

.. .. .. . . .

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Another Weekly Delta Hedging Example Consider the time evolution of the Delta hedging of the above example (see page 357 of Hull’s book, 7th edition).

.. .. .. . . .

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Delta of Portfolios Suppose that the investor has a portfolio of options and he wants to hedge the risk. One way to do so is to follow the delta hedging for the whole portfolio: ∂Π ∂S where Π is the value of the portfolio. If there are n options on the same asset and the portfolio consists of wi units of each option i, then the Delta of the portfolio is given as: ∆ = w1 ∆1 + w2 ∆2 + ... + wn ∆n

Example Consider the portfolio that consists of: → Long position on 100,000 call options on stock ABC, each of which has delta 0.53. → Short position on 200,000 call options on stock ABC each of which has delta 0.46. → Short position on 50,000 short options on stock ABC each of which has delta −0.508.

The delta of the portfolio is then 100, 000 × 0.53 − 200, 000 × 0.46 − 100, 000 × (−0.508) = −14, 900. M. Anthropelos (Un. of Piraeus)

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Theta Theta is the partial derivative of the option with respect to time to maturity (also called time decay). It approximates the rate of change of the value of the option with respect to the passage of time. Under the BS model, we can calculate that Θ(call) = where N 0 (x) =

∂c S(0)N 0 (d1 )σ √ =− − rKe −rT N(d2 ) ∂t 2 T

2

x √1 e − 2 2π

, it is the pdf of the Normal distribution (0,1).

Similarly, for the put option Θ(put) =

∂p S(0)N 0 (d1 )σ √ =− + rKe −rT N(−d2 ). ∂t 2 T

Theta is measured in time units (usually in days). As we have seen, Theta is negative for both call and put options.

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Gamma Gamma of a portfolio of options on a specific underlying asset is an approximation of the rate change of the portfolio delta with respect to the price of the underlying asset. In other words, it is the second partial derivative of the portfolio value with respect to underlying asset price. ∂2Π ∂∆ = . ∂S ∂S 2 Large values of Gamma mean that a small change in the underlying asset price results in a large change in the portfolio’s Delta. This means that delta hedging may still be risky. For BS model on European call option, we calculate that Γ=

Γ(call) =

N 0 (d1 ) ∂2c √ = ∂S 2 S(0)σ T

Note that Γ(call) = Γ(put) (it is left as an exercise). The BSM pde implies that 1 Θ(t) + rS(t)∆(t) + σ 2 S 2 (t)Γ(t) = r Π(t) 2 for every portfolio of derivatives written on the same asset. M. Anthropelos (Un. of Piraeus)

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Vega and Rho Vega is the derivative of the value of the option portfolio with respect to the volatility: ∂Π V= ∂σ In BS model √ ∂c = S(0) T N 0 (d1 ). V(call) = ∂σ Note that V(call) = V(put). Rho is the derivative of the value of the option portfolio with respect to the interest rate: ∂Π rho = ∂r In BS model rho(call) = KTe −rT N(d2 ).

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